diff --git a/milli/src/lib.rs b/milli/src/lib.rs
index 66dd33036..594405891 100644
--- a/milli/src/lib.rs
+++ b/milli/src/lib.rs
@@ -3,7 +3,53 @@
 
 #[cfg(test)]
 #[global_allocator]
-static ALLOC: mimalloc::MiMalloc = mimalloc::MiMalloc;
+pub static ALLOC: mimalloc::MiMalloc = mimalloc::MiMalloc;
+
+// #[cfg(test)]
+// pub mod allocator {
+//     use std::alloc::{GlobalAlloc, System};
+//     use std::sync::atomic::{self, AtomicI64};
+
+//     #[global_allocator]
+//     pub static ALLOC: CountingAlloc = CountingAlloc {
+//         max_resident: AtomicI64::new(0),
+//         resident: AtomicI64::new(0),
+//         allocated: AtomicI64::new(0),
+//     };
+
+//     pub struct CountingAlloc {
+//         pub max_resident: AtomicI64,
+//         pub resident: AtomicI64,
+//         pub allocated: AtomicI64,
+//     }
+//     unsafe impl GlobalAlloc for CountingAlloc {
+//         unsafe fn alloc(&self, layout: std::alloc::Layout) -> *mut u8 {
+//             self.allocated.fetch_add(layout.size() as i64, atomic::Ordering::SeqCst);
+//             let old_resident =
+//                 self.resident.fetch_add(layout.size() as i64, atomic::Ordering::SeqCst);
+
+//             let resident = old_resident + layout.size() as i64;
+//             self.max_resident.fetch_max(resident, atomic::Ordering::SeqCst);
+
+//             // if layout.size() > 1_000_000 {
+//             //     eprintln!(
+//             //         "allocating {} with new resident size: {resident}",
+//             //         layout.size() / 1_000_000
+//             //     );
+//             //     // let trace = std::backtrace::Backtrace::capture();
+//             //     // let t = trace.to_string();
+//             //     // eprintln!("{t}");
+//             // }
+
+//             System.alloc(layout)
+//         }
+
+//         unsafe fn dealloc(&self, ptr: *mut u8, layout: std::alloc::Layout) {
+//             self.resident.fetch_sub(layout.size() as i64, atomic::Ordering::Relaxed);
+//             System.dealloc(ptr, layout)
+//         }
+//     }
+// }
 
 #[macro_use]
 pub mod documents;
diff --git a/milli/src/search/new/graph_based_ranking_rule.rs b/milli/src/search/new/graph_based_ranking_rule.rs
index 264686b0a..e5a0fbad6 100644
--- a/milli/src/search/new/graph_based_ranking_rule.rs
+++ b/milli/src/search/new/graph_based_ranking_rule.rs
@@ -3,12 +3,11 @@ use roaring::RoaringBitmap;
 
 use super::db_cache::DatabaseCache;
 use super::logger::SearchLogger;
-use super::ranking_rule_graph::cheapest_paths::KCheapestPathsState;
 use super::ranking_rule_graph::edge_docids_cache::EdgeDocidsCache;
 use super::ranking_rule_graph::empty_paths_cache::EmptyPathsCache;
-use super::ranking_rule_graph::paths_map::PathsMap;
+
 use super::ranking_rule_graph::{RankingRuleGraph, RankingRuleGraphTrait};
-use super::{QueryGraph, RankingRule, RankingRuleOutput};
+use super::{BitmapOrAllRef, QueryGraph, RankingRule, RankingRuleOutput};
 
 use crate::{Index, Result};
 
@@ -24,9 +23,40 @@ impl<G: RankingRuleGraphTrait> GraphBasedRankingRule<G> {
 
 pub struct GraphBasedRankingRuleState<G: RankingRuleGraphTrait> {
     graph: RankingRuleGraph<G>,
-    cheapest_paths_state: Option<KCheapestPathsState>,
     edge_docids_cache: EdgeDocidsCache<G>,
     empty_paths_cache: EmptyPathsCache,
+    all_distances: Vec<Vec<u64>>,
+    cur_distance_idx: usize,
+}
+
+fn remove_empty_edges<'transaction, G: RankingRuleGraphTrait>(
+    graph: &mut RankingRuleGraph<G>,
+    edge_docids_cache: &mut EdgeDocidsCache<G>,
+    index: &Index,
+    txn: &'transaction RoTxn,
+    db_cache: &mut DatabaseCache<'transaction>,
+    universe: &RoaringBitmap,
+    empty_paths_cache: &mut EmptyPathsCache,
+) -> Result<()> {
+    for edge_index in 0..graph.all_edges.len() as u32 {
+        if graph.all_edges[edge_index as usize].is_none() {
+            continue;
+        }
+        let docids = edge_docids_cache
+            .get_edge_docids(index, txn, db_cache, edge_index, &*graph, universe)?;
+        match docids {
+            BitmapOrAllRef::Bitmap(bitmap) => {
+                if bitmap.is_disjoint(universe) {
+                    graph.remove_edge(edge_index);
+                    empty_paths_cache.forbid_edge(edge_index);
+                    edge_docids_cache.cache.remove(&edge_index);
+                    continue;
+                }
+            }
+            BitmapOrAllRef::All => continue,
+        }
+    }
+    Ok(())
 }
 
 impl<'transaction, G: RankingRuleGraphTrait> RankingRule<'transaction, QueryGraph>
@@ -41,18 +71,31 @@ impl<'transaction, G: RankingRuleGraphTrait> RankingRule<'transaction, QueryGrap
         txn: &'transaction RoTxn,
         db_cache: &mut DatabaseCache<'transaction>,
         _logger: &mut dyn SearchLogger<QueryGraph>,
-        _universe: &RoaringBitmap,
+        universe: &RoaringBitmap,
         query_graph: &QueryGraph,
     ) -> Result<()> {
         // TODO: update old state instead of starting from scratch
-        let graph = RankingRuleGraph::build(index, txn, db_cache, query_graph.clone())?;
+        let mut graph = RankingRuleGraph::build(index, txn, db_cache, query_graph.clone())?;
+        let mut edge_docids_cache = EdgeDocidsCache::default();
+        let mut empty_paths_cache = EmptyPathsCache::new(graph.all_edges.len());
+
+        remove_empty_edges(
+            &mut graph,
+            &mut edge_docids_cache,
+            index,
+            txn,
+            db_cache,
+            universe,
+            &mut empty_paths_cache,
+        )?;
+        let all_distances = graph.initialize_distances_cheapest();
 
-        let cheapest_paths_state = KCheapestPathsState::new(&graph);
         let state = GraphBasedRankingRuleState {
             graph,
-            cheapest_paths_state,
-            edge_docids_cache: <_>::default(),
-            empty_paths_cache: <_>::default(),
+            edge_docids_cache,
+            empty_paths_cache,
+            all_distances,
+            cur_distance_idx: 0,
         };
 
         self.state = Some(state);
@@ -70,34 +113,42 @@ impl<'transaction, G: RankingRuleGraphTrait> RankingRule<'transaction, QueryGrap
     ) -> Result<Option<RankingRuleOutput<QueryGraph>>> {
         assert!(universe.len() > 1);
         let mut state = self.state.take().unwrap();
-        if state.cheapest_paths_state.is_none() {
+        remove_empty_edges(
+            &mut state.graph,
+            &mut state.edge_docids_cache,
+            index,
+            txn,
+            db_cache,
+            universe,
+            &mut state.empty_paths_cache,
+        )?;
+
+        if state.cur_distance_idx
+            >= state.all_distances[state.graph.query_graph.root_node as usize].len()
+        {
+            self.state = None;
             return Ok(None);
         }
+        let cost =
+            state.all_distances[state.graph.query_graph.root_node as usize][state.cur_distance_idx];
+        state.cur_distance_idx += 1;
 
-        let mut paths = PathsMap::default();
+        let paths = state.graph.paths_of_cost(
+            state.graph.query_graph.root_node as usize,
+            cost,
+            &state.all_distances,
+            &state.empty_paths_cache,
+        );
 
-        while paths.is_empty() {
-            let Some(cheapest_paths_state) = state.cheapest_paths_state.take() else {
-                break;
-            };
-            if let Some(next_cheapest_paths_state) = cheapest_paths_state
-                .compute_paths_of_next_lowest_cost(
-                    &mut state.graph,
-                    &state.empty_paths_cache,
-                    &mut paths,
-                )
-            {
-                state.cheapest_paths_state = Some(next_cheapest_paths_state);
-            } else {
-                break;
-            }
-        }
-
-        if paths.is_empty() && state.cheapest_paths_state.is_none() {
-            return Ok(None);
-        }
-
-        G::log_state(&state.graph, &paths, &state.empty_paths_cache, logger);
+        G::log_state(
+            &state.graph,
+            &paths,
+            &state.empty_paths_cache,
+            universe,
+            &state.all_distances,
+            cost,
+            logger,
+        );
 
         let bucket = state.graph.resolve_paths(
             index,
diff --git a/milli/src/search/new/logger/detailed.rs b/milli/src/search/new/logger/detailed.rs
index a7a3f8793..d2ce627dc 100644
--- a/milli/src/search/new/logger/detailed.rs
+++ b/milli/src/search/new/logger/detailed.rs
@@ -1,6 +1,8 @@
+
 use rand::random;
 use roaring::RoaringBitmap;
 use std::fs::File;
+use std::time::Instant;
 use std::{io::Write, path::PathBuf};
 
 use crate::new::ranking_rule_graph::typo::TypoGraph;
@@ -9,7 +11,7 @@ use crate::new::query_term::{LocatedQueryTerm, QueryTerm, WordDerivations};
 use crate::new::ranking_rule_graph::empty_paths_cache::EmptyPathsCache;
 use crate::new::ranking_rule_graph::{Edge, EdgeDetails, RankingRuleGraphTrait};
 use crate::new::ranking_rule_graph::{
-    paths_map::PathsMap, proximity::ProximityGraph, RankingRuleGraph,
+    proximity::ProximityGraph, RankingRuleGraph,
 };
 
 use super::{RankingRule, SearchLogger};
@@ -19,14 +21,18 @@ pub enum SearchEvents {
         ranking_rule_idx: usize,
         query: QueryGraph,
         universe: RoaringBitmap,
+        time: Instant,
     },
     RankingRuleNextBucket {
         ranking_rule_idx: usize,
         universe: RoaringBitmap,
+        candidates: RoaringBitmap,
+        time: Instant,
     },
     RankingRuleEndIteration {
         ranking_rule_idx: usize,
         universe: RoaringBitmap,
+        time: Instant,
     },
     ExtendResults {
         new: Vec<u32>,
@@ -36,20 +42,27 @@ pub enum SearchEvents {
     },
     ProximityState {
         graph: RankingRuleGraph<ProximityGraph>,
-        paths: PathsMap<u64>,
+        paths: Vec<Vec<u32>>,
         empty_paths_cache: EmptyPathsCache,
+        universe: RoaringBitmap,
+        distances: Vec<Vec<u64>>,
+        cost: u64,
     },
     TypoState {
         graph: RankingRuleGraph<TypoGraph>,
-        paths: PathsMap<u64>,
+        paths: Vec<Vec<u32>>,
         empty_paths_cache: EmptyPathsCache,
+        universe: RoaringBitmap,
+        distances: Vec<Vec<u64>>,
+        cost: u64,
     },
-    RankingRuleSkipBucket { ranking_rule_idx: usize, candidates: RoaringBitmap },
+    RankingRuleSkipBucket { ranking_rule_idx: usize, candidates: RoaringBitmap, time: Instant, },
 }
 
 pub struct DetailedSearchLogger {
     folder_path: PathBuf,
     initial_query: Option<QueryGraph>,
+    initial_query_time: Option<Instant>,
     initial_universe: Option<RoaringBitmap>,
     ranking_rules_ids: Option<Vec<String>>,
     events: Vec<SearchEvents>,
@@ -58,17 +71,19 @@ impl DetailedSearchLogger {
     pub fn new(folder_path: &str) -> Self {
         Self {
             folder_path: PathBuf::new().join(folder_path),
-            initial_query: <_>::default(),
-            initial_universe: <_>::default(),
-            ranking_rules_ids: <_>::default(),
-            events: <_>::default(),
+            initial_query: None,
+            initial_query_time: None,
+            initial_universe: None,
+            ranking_rules_ids: None,
+            events: vec![],
         }
     }
 }
 
 impl SearchLogger<QueryGraph> for DetailedSearchLogger {
-    fn initial_query(&mut self, query: &QueryGraph) {
+    fn initial_query(&mut self, query: &QueryGraph, time: Instant) {
         self.initial_query = Some(query.clone());
+        self.initial_query_time = Some(time);
     }
 
     fn initial_universe(&mut self, universe: &RoaringBitmap) {
@@ -84,11 +99,13 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
         _ranking_rule: &dyn RankingRule<'transaction, QueryGraph>,
         query: &QueryGraph,
         universe: &RoaringBitmap,
+        time: Instant,
     ) {
         self.events.push(SearchEvents::RankingRuleStartIteration {
             ranking_rule_idx,
             query: query.clone(),
             universe: universe.clone(),
+            time,
         })
     }
 
@@ -97,10 +114,14 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
         ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, QueryGraph>,
         universe: &RoaringBitmap,
+        candidates: &RoaringBitmap,
+        time: Instant,
     ) {
         self.events.push(SearchEvents::RankingRuleNextBucket {
             ranking_rule_idx,
             universe: universe.clone(),
+            candidates: candidates.clone(),
+            time,
         })
     }
     fn skip_bucket_ranking_rule<'transaction>(
@@ -108,10 +129,12 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
         ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, QueryGraph>,
         candidates: &RoaringBitmap,
+        time: Instant,
     ) {
         self.events.push(SearchEvents::RankingRuleSkipBucket {
             ranking_rule_idx,
             candidates: candidates.clone(),
+            time
         })
     }
 
@@ -120,10 +143,12 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
         ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, QueryGraph>,
         universe: &RoaringBitmap,
+        time: Instant,
     ) {
         self.events.push(SearchEvents::RankingRuleEndIteration {
             ranking_rule_idx,
             universe: universe.clone(),
+            time
         })
     }
     fn add_to_results(&mut self, docids: &[u32]) {
@@ -134,18 +159,19 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
         self.events.push(SearchEvents::WordsState { query_graph: query_graph.clone() });
     }
 
-    fn log_proximity_state(&mut self, query_graph: &RankingRuleGraph<ProximityGraph>, paths_map: &PathsMap<u64>, empty_paths_cache: &EmptyPathsCache) {
-        self.events.push(SearchEvents::ProximityState { graph: query_graph.clone(), paths: paths_map.clone(), empty_paths_cache: empty_paths_cache.clone() })
+    fn log_proximity_state(&mut self, query_graph: &RankingRuleGraph<ProximityGraph>, paths_map: &[Vec<u32>], empty_paths_cache: &EmptyPathsCache, universe: &RoaringBitmap, distances: Vec<Vec<u64>>, cost: u64,) {
+        self.events.push(SearchEvents::ProximityState { graph: query_graph.clone(), paths: paths_map.to_vec(), empty_paths_cache: empty_paths_cache.clone(), universe: universe.clone(), distances, cost })
     }
     
-    fn log_typo_state(&mut self, query_graph: &RankingRuleGraph<TypoGraph>, paths_map: &PathsMap<u64>, empty_paths_cache: &EmptyPathsCache) {
-        self.events.push(SearchEvents::TypoState { graph: query_graph.clone(), paths: paths_map.clone(), empty_paths_cache: empty_paths_cache.clone() })
+    fn log_typo_state(&mut self, query_graph: &RankingRuleGraph<TypoGraph>, paths_map: &[Vec<u32>], empty_paths_cache: &EmptyPathsCache, universe: &RoaringBitmap, distances: Vec<Vec<u64>>,  cost: u64,) {
+        self.events.push(SearchEvents::TypoState { graph: query_graph.clone(), paths: paths_map.to_vec(), empty_paths_cache: empty_paths_cache.clone(), universe: universe.clone(), distances,  cost })
     }
 
 }
 
 impl DetailedSearchLogger {
     pub fn write_d2_description(&self) {
+        let mut prev_time = self.initial_query_time.unwrap();
         let mut timestamp = vec![];
         fn activated_id(timestamp: &[usize]) -> String {
             let mut s = String::new();
@@ -164,13 +190,16 @@ impl DetailedSearchLogger {
             writeln!(&mut file, "{idx}: {rr_id}").unwrap();
         }
         writeln!(&mut file, "results").unwrap();
+        // writeln!(&mut file, "time").unwrap();
         for event in self.events.iter() {
             match event {
-                SearchEvents::RankingRuleStartIteration { ranking_rule_idx, .. } => {
-
+                SearchEvents::RankingRuleStartIteration { ranking_rule_idx, time, .. } => {
+                    let elapsed = time.duration_since(prev_time);
+                    prev_time = *time;
                     let parent_activated_id = activated_id(&timestamp);
                     timestamp.push(0);
                     let self_activated_id = activated_id(&timestamp);
+                    // writeln!(&mut file, "time.{self_activated_id}: {:.2}", elapsed.as_micros() as f64 / 1000.0).unwrap();
                     if *ranking_rule_idx != 0 {
                         let parent_ranking_rule_idx = ranking_rule_idx - 1;
                         writeln!(
@@ -186,16 +215,22 @@ impl DetailedSearchLogger {
     }}
 }}").unwrap();
                 }
-                SearchEvents::RankingRuleNextBucket { ranking_rule_idx, .. } => {
+                SearchEvents::RankingRuleNextBucket { ranking_rule_idx, time, universe, candidates } => {
+                    let elapsed = time.duration_since(prev_time);
+                    prev_time = *time;
                     let old_activated_id = activated_id(&timestamp);
+                    // writeln!(&mut file, "time.{old_activated_id}: {:.2}", elapsed.as_micros() as f64 / 1000.0).unwrap();
                     *timestamp.last_mut().unwrap() += 1;
                     let next_activated_id = activated_id(&timestamp);
                     writeln!(&mut file, 
-                        "{ranking_rule_idx}.{old_activated_id} -> {ranking_rule_idx}.{next_activated_id} : next bucket",)
+                        "{ranking_rule_idx}.{old_activated_id} -> {ranking_rule_idx}.{next_activated_id} : next bucket {}/{}", candidates.len(), universe.len())
                         .unwrap();
                 }
-                SearchEvents::RankingRuleSkipBucket { ranking_rule_idx, candidates } => {
+                SearchEvents::RankingRuleSkipBucket { ranking_rule_idx, candidates, time } => {
+                    let elapsed = time.duration_since(prev_time);
+                    prev_time = *time;
                     let old_activated_id = activated_id(&timestamp);
+                    // writeln!(&mut file, "time.{old_activated_id}: {:.2}", elapsed.as_micros() as f64 / 1000.0).unwrap();
                     *timestamp.last_mut().unwrap() += 1;
                     let next_activated_id = activated_id(&timestamp);
                     let len = candidates.len();
@@ -203,8 +238,12 @@ impl DetailedSearchLogger {
                         "{ranking_rule_idx}.{old_activated_id} -> {ranking_rule_idx}.{next_activated_id} : skip bucket ({len})",)
                         .unwrap();
                 }
-                SearchEvents::RankingRuleEndIteration { ranking_rule_idx, .. } => {
+                SearchEvents::RankingRuleEndIteration { ranking_rule_idx, time, .. } => {
+                    let elapsed = time.duration_since(prev_time);
+                    prev_time = *time;
                     let cur_activated_id = activated_id(&timestamp);
+                    // writeln!(&mut file, "time.{cur_activated_id}: {:.2}", elapsed.as_micros() as f64 / 1000.0).unwrap();
+
                     timestamp.pop();
                     let parent_activated_id = activated_id(&timestamp);
                     let parent_ranking_rule = if *ranking_rule_idx == 0 {
@@ -254,43 +293,48 @@ results.{random} {{
     link: \"{id}.d2.svg\"
 }}").unwrap();
                 },
-                SearchEvents::ProximityState { graph, paths, empty_paths_cache } => {
+                SearchEvents::ProximityState { graph, paths, empty_paths_cache, universe, distances, cost } => {
                     let cur_ranking_rule = timestamp.len() - 1;
                     let cur_activated_id = activated_id(&timestamp);
                     let id = format!("{cur_ranking_rule}.{cur_activated_id}");
                     let new_file_path = self.folder_path.join(format!("{id}.d2"));
                     let mut new_file = std::fs::File::create(new_file_path).unwrap();
-                    Self::ranking_rule_graph_d2_description(graph, paths, empty_paths_cache, &mut new_file);
+                    Self::ranking_rule_graph_d2_description(graph, paths, empty_paths_cache, distances.clone(), &mut new_file);
                     writeln!(
                         &mut file,
                         "{id} {{
     link: \"{id}.d2.svg\"
-}}").unwrap();
+    tooltip: \"cost {cost}, universe len: {}\"
+}}", universe.len()).unwrap();
                 },
-                SearchEvents::TypoState { graph, paths, empty_paths_cache } => {
+                SearchEvents::TypoState { graph, paths, empty_paths_cache, universe, distances, cost } => {
                     let cur_ranking_rule = timestamp.len() - 1;
                     let cur_activated_id = activated_id(&timestamp);
                     let id = format!("{cur_ranking_rule}.{cur_activated_id}");
                     let new_file_path = self.folder_path.join(format!("{id}.d2"));
                     let mut new_file = std::fs::File::create(new_file_path).unwrap();
-                    Self::ranking_rule_graph_d2_description(graph, paths, empty_paths_cache, &mut new_file);
+                    Self::ranking_rule_graph_d2_description(graph, paths, empty_paths_cache, distances.clone(), &mut new_file);
                     writeln!(
                         &mut file,
                         "{id} {{
     link: \"{id}.d2.svg\"
-}}").unwrap();
+    tooltip: \"cost {cost}, universe len: {}\"
+}}", universe.len()).unwrap();
                 },
             }
         }
         writeln!(&mut file, "}}").unwrap();
     }
     
-    fn query_node_d2_desc(node_idx: usize, node: &QueryNode, file: &mut File) {
+    fn query_node_d2_desc(node_idx: usize, node: &QueryNode, distances: &[u64], file: &mut File) {
         match &node {
             QueryNode::Term(LocatedQueryTerm { value, .. }) => {
                 match value {
-                    QueryTerm::Phrase(_) => todo!(),
-                    QueryTerm::Word { derivations: WordDerivations { original, zero_typo, one_typo, two_typos, use_prefix_db } } => {
+                    QueryTerm::Phrase { phrase } => {
+                        let phrase_str = phrase.description();
+                        writeln!(file,"{node_idx} : \"{phrase_str}\"").unwrap();
+                    },
+                    QueryTerm::Word { derivations: WordDerivations { original, zero_typo, one_typo, two_typos, use_prefix_db, synonyms, split_words } } => {
                         writeln!(file,"{node_idx} : \"{original}\" {{
 shape: class").unwrap();
                         for w in zero_typo {
@@ -302,9 +346,19 @@ shape: class").unwrap();
                         for w in two_typos {
                             writeln!(file, "\"{w}\" : 2").unwrap();
                         }
+                        if let Some((left, right)) = split_words {
+                            writeln!(file, "\"{left} {right}\" : split_words").unwrap();
+                        }
+                        for synonym in synonyms {
+                            writeln!(file, "\"{}\" : synonym", synonym.description()).unwrap();
+                        }
                         if *use_prefix_db {
                             writeln!(file, "use prefix DB : true").unwrap();
                         }
+                        // for (i, d) in distances.iter().enumerate() {
+                        //     writeln!(file, "\"distances\" : {d}").unwrap();
+                        // }
+                        
                         writeln!(file, "}}").unwrap();
                     },
                 }
@@ -324,14 +378,14 @@ shape: class").unwrap();
             if matches!(query_graph.nodes[node], QueryNode::Deleted) {
                 continue;
             }
-            Self::query_node_d2_desc(node, &query_graph.nodes[node], file);
+            Self::query_node_d2_desc(node, &query_graph.nodes[node], &[], file);
             
             for edge in query_graph.edges[node].successors.iter() {
                 writeln!(file, "{node} -> {edge};\n").unwrap();
             }
         }        
     }
-    fn ranking_rule_graph_d2_description<R: RankingRuleGraphTrait>(graph: &RankingRuleGraph<R>, paths: &PathsMap<u64>, empty_paths_cache: &EmptyPathsCache, file: &mut File) {
+    fn ranking_rule_graph_d2_description<R: RankingRuleGraphTrait>(graph: &RankingRuleGraph<R>, paths: &[Vec<u32>], _empty_paths_cache: &EmptyPathsCache, distances: Vec<Vec<u64>>, file: &mut File) {
         writeln!(file,"direction: right").unwrap();
 
         writeln!(file, "Proximity Graph {{").unwrap();
@@ -339,7 +393,8 @@ shape: class").unwrap();
             if matches!(node, QueryNode::Deleted) {
                 continue;
             }
-            Self::query_node_d2_desc(node_idx, node, file);
+            let distances = &distances[node_idx];
+            Self::query_node_d2_desc(node_idx, node, distances.as_slice(), file);
         }
         for edge in graph.all_edges.iter().flatten() {
             let Edge { from_node, to_node, details, .. } = edge;
@@ -362,26 +417,39 @@ shape: class").unwrap();
         }
         writeln!(file, "}}").unwrap();
 
+        // writeln!(file, "Distances {{").unwrap();
+        // Self::paths_d2_description(graph, paths, file);
+        // writeln!(file, "}}").unwrap();
+
+        
         writeln!(file, "Shortest Paths {{").unwrap();
-        Self::paths_d2_description(graph, "", paths, file);
+        Self::paths_d2_description(graph, paths, file);
         writeln!(file, "}}").unwrap();
 
-        writeln!(file, "Empty Path Prefixes {{").unwrap();
-        Self::paths_d2_description(graph, "", &empty_paths_cache.empty_prefixes, file);
-        writeln!(file, "}}").unwrap();
+        // writeln!(file, "Empty Edge Couples {{").unwrap();            
+        // for (i, (e1, e2)) in empty_paths_cache.empty_couple_edges.iter().enumerate() {
+        //     writeln!(file, "{i} : \"\" {{").unwrap();
+        //     Self::edge_d2_description(graph, *e1, file);
+        //     Self::edge_d2_description(graph, *e2, file);
+        //     writeln!(file, "{e1} -- {e2}").unwrap();
+        //     writeln!(file, "}}").unwrap();
+        // }
+        // writeln!(file, "}}").unwrap();
 
-        writeln!(file, "Removed Edges {{").unwrap();
-        for edge_idx in empty_paths_cache.empty_edges.iter() {
-            writeln!(file, "{edge_idx}").unwrap();
-        }
-        writeln!(file, "}}").unwrap();
+        // writeln!(file, "Removed Edges {{").unwrap();
+        // for edge_idx in empty_paths_cache.empty_edges.iter() {
+        //     writeln!(file, "{edge_idx}").unwrap();
+        // }
+        // writeln!(file, "}}").unwrap();
     }
-    fn edge_d2_description<R: RankingRuleGraphTrait>(graph: &RankingRuleGraph<R>, paths_idx: &str, edge_idx: u32, file: &mut File) -> String {
+    fn edge_d2_description<R: RankingRuleGraphTrait>(graph: &RankingRuleGraph<R>, edge_idx: u32, file: &mut File) {
         let Edge { from_node, to_node, cost, .. } = graph.all_edges[edge_idx as usize].as_ref().unwrap() ;
         let from_node = &graph.query_graph.nodes[*from_node as usize];
         let from_node_desc = match from_node {
             QueryNode::Term(term) => match &term.value {
-                QueryTerm::Phrase(_) => todo!(),
+                QueryTerm::Phrase { phrase } => {
+                    phrase.description()
+                },
                 QueryTerm::Word { derivations } => derivations.original.clone(),
             },
             QueryNode::Deleted => panic!(),
@@ -391,27 +459,28 @@ shape: class").unwrap();
         let to_node = &graph.query_graph.nodes[*to_node as usize];
         let to_node_desc = match to_node {
             QueryNode::Term(term) => match &term.value {
-                QueryTerm::Phrase(_) => todo!(),
+                QueryTerm::Phrase { phrase } => phrase.description(),
                 QueryTerm::Word { derivations } => derivations.original.clone(),
             },
             QueryNode::Deleted => panic!(),
             QueryNode::Start => "START".to_owned(),
             QueryNode::End => "END".to_owned(),
         };
-        let edge_id = format!("{paths_idx}{edge_idx}");
-        writeln!(file, "{edge_id}: \"{from_node_desc}->{to_node_desc} [{cost}]\" {{
+        writeln!(file, "{edge_idx}: \"{from_node_desc}->{to_node_desc} [{cost}]\" {{
             shape: class
         }}").unwrap();
-        edge_id
     }
-    fn paths_d2_description<R: RankingRuleGraphTrait, T>(graph: &RankingRuleGraph<R>, paths_idx: &str, paths: &PathsMap<T>, file: &mut File) { 
-        for (edge_idx, rest) in paths.nodes.iter() {
-            let edge_id = Self::edge_d2_description(graph, paths_idx, *edge_idx, file);
-            for (dest_edge_idx, _) in rest.nodes.iter() {
-                let dest_edge_id = format!("{paths_idx}{edge_idx}{dest_edge_idx}");
-                writeln!(file, "{edge_id} -> {dest_edge_id}").unwrap();
+    fn paths_d2_description<R: RankingRuleGraphTrait>(graph: &RankingRuleGraph<R>, paths: &[Vec<u32>], file: &mut File) { 
+        for (path_idx, edge_indexes) in paths.iter().enumerate() {
+            writeln!(file, "{path_idx} {{").unwrap();
+            for edge_idx in edge_indexes.iter() {
+                Self::edge_d2_description(graph, *edge_idx, file);
             }
-            Self::paths_d2_description(graph, &format!("{paths_idx}{edge_idx}"), rest, file);
+            for couple_edges in edge_indexes.windows(2) {
+                let [src_edge_idx, dest_edge_idx] = couple_edges else { panic!() };
+                writeln!(file, "{src_edge_idx} -> {dest_edge_idx}").unwrap();
+            }
+            writeln!(file, "}}").unwrap();
         }
     }
 }
diff --git a/milli/src/search/new/logger/mod.rs b/milli/src/search/new/logger/mod.rs
index 4e119ae42..079bb892c 100644
--- a/milli/src/search/new/logger/mod.rs
+++ b/milli/src/search/new/logger/mod.rs
@@ -2,28 +2,31 @@
 pub mod detailed;
 
 use roaring::RoaringBitmap;
+use std::time::Instant;
 
 use super::{
     ranking_rule_graph::{
-        empty_paths_cache::EmptyPathsCache, paths_map::PathsMap, proximity::ProximityGraph,
-        typo::TypoGraph, RankingRuleGraph,
+        empty_paths_cache::EmptyPathsCache, proximity::ProximityGraph, typo::TypoGraph,
+        RankingRuleGraph,
     },
     RankingRule, RankingRuleQueryTrait,
 };
 
 pub struct DefaultSearchLogger;
 impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
-    fn initial_query(&mut self, _query: &Q) {}
+    fn initial_query(&mut self, _query: &Q, _time: Instant) {}
 
     fn initial_universe(&mut self, _universe: &RoaringBitmap) {}
 
     fn ranking_rules(&mut self, _rr: &[&mut dyn RankingRule<Q>]) {}
+
     fn start_iteration_ranking_rule<'transaction>(
         &mut self,
         _ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, Q>,
         _query: &Q,
         _universe: &RoaringBitmap,
+        _time: Instant,
     ) {
     }
 
@@ -32,6 +35,8 @@ impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
         _ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, Q>,
         _universe: &RoaringBitmap,
+        _candidates: &RoaringBitmap,
+        _time: Instant,
     ) {
     }
     fn skip_bucket_ranking_rule<'transaction>(
@@ -39,6 +44,7 @@ impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
         _ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, Q>,
         _candidates: &RoaringBitmap,
+        _time: Instant,
     ) {
     }
 
@@ -47,6 +53,7 @@ impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
         _ranking_rule_idx: usize,
         _ranking_rule: &dyn RankingRule<'transaction, Q>,
         _universe: &RoaringBitmap,
+        _time: Instant,
     ) {
     }
 
@@ -57,22 +64,28 @@ impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
     fn log_proximity_state(
         &mut self,
         _query_graph: &RankingRuleGraph<ProximityGraph>,
-        _paths_map: &PathsMap<u64>,
+        _paths_map: &[Vec<u32>],
         _empty_paths_cache: &EmptyPathsCache,
+        _universe: &RoaringBitmap,
+        _distances: Vec<Vec<u64>>,
+        _cost: u64,
     ) {
     }
 
     fn log_typo_state(
         &mut self,
-        query_graph: &RankingRuleGraph<TypoGraph>,
-        paths: &PathsMap<u64>,
-        empty_paths_cache: &EmptyPathsCache,
+        _query_graph: &RankingRuleGraph<TypoGraph>,
+        _paths: &[Vec<u32>],
+        _empty_paths_cache: &EmptyPathsCache,
+        _universe: &RoaringBitmap,
+        _distances: Vec<Vec<u64>>,
+        _cost: u64,
     ) {
     }
 }
 
 pub trait SearchLogger<Q: RankingRuleQueryTrait> {
-    fn initial_query(&mut self, query: &Q);
+    fn initial_query(&mut self, query: &Q, time: Instant);
     fn initial_universe(&mut self, universe: &RoaringBitmap);
 
     fn ranking_rules(&mut self, rr: &[&mut dyn RankingRule<Q>]);
@@ -83,24 +96,29 @@ pub trait SearchLogger<Q: RankingRuleQueryTrait> {
         ranking_rule: &dyn RankingRule<'transaction, Q>,
         query: &Q,
         universe: &RoaringBitmap,
+        time: Instant,
     );
     fn next_bucket_ranking_rule<'transaction>(
         &mut self,
         ranking_rule_idx: usize,
         ranking_rule: &dyn RankingRule<'transaction, Q>,
         universe: &RoaringBitmap,
+        candidates: &RoaringBitmap,
+        time: Instant,
     );
     fn skip_bucket_ranking_rule<'transaction>(
         &mut self,
         ranking_rule_idx: usize,
         ranking_rule: &dyn RankingRule<'transaction, Q>,
         candidates: &RoaringBitmap,
+        time: Instant,
     );
     fn end_iteration_ranking_rule<'transaction>(
         &mut self,
         ranking_rule_idx: usize,
         ranking_rule: &dyn RankingRule<'transaction, Q>,
         universe: &RoaringBitmap,
+        time: Instant,
     );
     fn add_to_results(&mut self, docids: &[u32]);
 
@@ -109,14 +127,20 @@ pub trait SearchLogger<Q: RankingRuleQueryTrait> {
     fn log_proximity_state(
         &mut self,
         query_graph: &RankingRuleGraph<ProximityGraph>,
-        paths: &PathsMap<u64>,
+        paths: &[Vec<u32>],
         empty_paths_cache: &EmptyPathsCache,
+        universe: &RoaringBitmap,
+        _distances: Vec<Vec<u64>>,
+        cost: u64,
     );
 
     fn log_typo_state(
         &mut self,
         query_graph: &RankingRuleGraph<TypoGraph>,
-        paths: &PathsMap<u64>,
+        paths: &[Vec<u32>],
         empty_paths_cache: &EmptyPathsCache,
+        universe: &RoaringBitmap,
+        _distances: Vec<Vec<u64>>,
+        cost: u64,
     );
 }
diff --git a/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs b/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
index e46f6ce66..2377f1c84 100644
--- a/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
+++ b/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
@@ -1,10 +1,8 @@
-use std::collections::{BTreeMap, HashSet};
-
-use roaring::RoaringBitmap;
+#![allow(clippy::too_many_arguments)]
 
 use super::empty_paths_cache::EmptyPathsCache;
-use super::paths_map::PathsMap;
-use super::{Edge, RankingRuleGraph, RankingRuleGraphTrait};
+use super::{RankingRuleGraph, RankingRuleGraphTrait};
+use std::collections::VecDeque;
 
 #[derive(Debug, Clone, PartialEq, Eq, Hash)]
 pub struct Path {
@@ -12,226 +10,119 @@ pub struct Path {
     pub cost: u64,
 }
 
-struct DijkstraState {
-    unvisited: RoaringBitmap, // should be a small bitset?
-    distances: Vec<u64>,      // or binary heap, or btreemap? (f64, usize)
-    edges: Vec<u32>,
-    edge_costs: Vec<u8>,
-    paths: Vec<Option<u32>>,
-}
-
-pub struct KCheapestPathsState {
-    cheapest_paths: PathsMap<u64>,
-    potential_cheapest_paths: BTreeMap<u64, PathsMap<u64>>,
-    pub kth_cheapest_path: Path,
-}
-
-impl KCheapestPathsState {
-    pub fn next_cost(&self) -> u64 {
-        self.kth_cheapest_path.cost
-    }
-
-    pub fn new<G: RankingRuleGraphTrait>(
-        graph: &RankingRuleGraph<G>,
-    ) -> Option<KCheapestPathsState> {
-        let Some(cheapest_path) = graph.cheapest_path_to_end(graph.query_graph.root_node) else {
-            return None
-        };
-        let cheapest_paths = PathsMap::from_paths(&[cheapest_path.clone()]);
-        let potential_cheapest_paths = BTreeMap::new();
-        Some(KCheapestPathsState {
-            cheapest_paths,
-            potential_cheapest_paths,
-            kth_cheapest_path: cheapest_path,
-        })
-    }
-
-    pub fn remove_empty_paths(mut self, empty_paths_cache: &EmptyPathsCache) -> Option<Self> {
-        self.cheapest_paths.remove_edges(&empty_paths_cache.empty_edges);
-        self.cheapest_paths.remove_prefixes(&empty_paths_cache.empty_prefixes);
-
-        let mut costs_to_delete = HashSet::new();
-        for (cost, potential_cheapest_paths) in self.potential_cheapest_paths.iter_mut() {
-            potential_cheapest_paths.remove_edges(&empty_paths_cache.empty_edges);
-            potential_cheapest_paths.remove_prefixes(&empty_paths_cache.empty_prefixes);
-            if potential_cheapest_paths.is_empty() {
-                costs_to_delete.insert(*cost);
-            }
-        }
-        for cost in costs_to_delete {
-            self.potential_cheapest_paths.remove(&cost);
-        }
-
-        if self.cheapest_paths.is_empty() {}
-
-        todo!()
-    }
-
-    pub fn compute_paths_of_next_lowest_cost<G: RankingRuleGraphTrait>(
-        mut self,
-        graph: &mut RankingRuleGraph<G>,
+impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
+    pub fn paths_of_cost(
+        &self,
+        from: usize,
+        cost: u64,
+        all_distances: &[Vec<u64>],
         empty_paths_cache: &EmptyPathsCache,
-        into_map: &mut PathsMap<u64>,
-    ) -> Option<Self> {
-        if !empty_paths_cache.path_is_empty(&self.kth_cheapest_path.edges) {
-            into_map.add_path(&self.kth_cheapest_path);
+    ) -> Vec<Vec<u32>> {
+        let mut paths = vec![];
+        self.paths_of_cost_rec(
+            from,
+            all_distances,
+            cost,
+            &mut vec![],
+            &mut paths,
+            &vec![false; self.all_edges.len()],
+            empty_paths_cache,
+        );
+        paths
+    }
+    pub fn paths_of_cost_rec(
+        &self,
+        from: usize,
+        all_distances: &[Vec<u64>],
+        cost: u64,
+        prev_edges: &mut Vec<u32>,
+        paths: &mut Vec<Vec<u32>>,
+        forbidden_edges: &[bool],
+        empty_paths_cache: &EmptyPathsCache,
+    ) {
+        let distances = &all_distances[from];
+        if !distances.contains(&cost) {
+            panic!();
         }
-        let cur_cost = self.kth_cheapest_path.cost;
-        while self.kth_cheapest_path.cost <= cur_cost {
-            if let Some(next_self) = self.compute_next_cheapest_paths(graph, empty_paths_cache) {
-                self = next_self;
-                if self.kth_cheapest_path.cost == cur_cost
-                    && !empty_paths_cache.path_is_empty(&self.kth_cheapest_path.edges)
+        let tos = &self.query_graph.edges[from].successors;
+        let mut valid_edges = vec![];
+        for to in tos {
+            self.visit_edges::<()>(from as u32, to, |edge_idx, edge| {
+                if cost >= edge.cost as u64
+                    && all_distances[to as usize].contains(&(cost - edge.cost as u64))
+                    && !forbidden_edges[edge_idx as usize]
                 {
-                    into_map.add_path(&self.kth_cheapest_path);
-                } else {
-                    break;
+                    valid_edges.push((edge_idx, edge.cost, to));
                 }
-            } else {
-                return None;
-            }
+                std::ops::ControlFlow::Continue(())
+            });
         }
-        Some(self)
-    }
 
-    fn compute_next_cheapest_paths<G: RankingRuleGraphTrait>(
-        mut self,
-        graph: &mut RankingRuleGraph<G>,
-        empty_paths_cache: &EmptyPathsCache,
-    ) -> Option<KCheapestPathsState> {
-        // for all nodes in the last cheapest path (called spur_node), except last one...
-        for (i, edge_idx) in self.kth_cheapest_path.edges[..self.kth_cheapest_path.edges.len() - 1]
-            .iter()
-            .enumerate()
-        {
-            let Some(edge) = graph.all_edges[*edge_idx as usize].as_ref() else { continue; };
-            let Edge { from_node: spur_node, .. } = edge;
-
-            let root_path = &self.kth_cheapest_path.edges[..i];
-            if empty_paths_cache.path_is_empty(root_path) {
+        for (edge_idx, edge_cost, to) in valid_edges {
+            prev_edges.push(edge_idx);
+            if empty_paths_cache.empty_prefixes.contains_prefix_of_path(prev_edges) {
                 continue;
             }
-
-            let root_cost = root_path.iter().fold(0, |sum, next| {
-                sum + graph.all_edges[*next as usize].as_ref().unwrap().cost as u64
-            });
-
-            let mut tmp_removed_edges = vec![];
-            // for all the paths already found that share a common prefix with the root path
-            // we delete the edge from the spur node to the next one
-            for edge_index_to_remove in self.cheapest_paths.edge_indices_after_prefix(root_path) {
-                let was_removed =
-                    graph.node_edges[*spur_node as usize].remove(edge_index_to_remove);
-                if was_removed {
-                    tmp_removed_edges.push(edge_index_to_remove);
-                }
+            let mut new_forbidden_edges = forbidden_edges.to_vec();
+            for edge_idx in empty_paths_cache.empty_couple_edges[edge_idx as usize].iter() {
+                new_forbidden_edges[*edge_idx as usize] = true;
+            }
+            for edge_idx in empty_paths_cache.empty_prefixes.final_edges_ater_prefix(prev_edges) {
+                new_forbidden_edges[edge_idx as usize] = true;
             }
 
-            // Compute the cheapest path from the spur node to the destination
-            // we will combine it with the root path to get a potential kth cheapest path
-            let spur_path = graph.cheapest_path_to_end(*spur_node);
-            // restore the temporarily removed edges
-            graph.node_edges[*spur_node as usize].extend(tmp_removed_edges);
-
-            let Some(spur_path) = spur_path else { continue; };
-            let total_cost = root_cost + spur_path.cost;
-            let total_path = Path {
-                edges: root_path.iter().chain(spur_path.edges.iter()).cloned().collect(),
-                cost: total_cost,
-            };
-            let entry = self.potential_cheapest_paths.entry(total_cost).or_default();
-            entry.add_path(&total_path);
+            if to == self.query_graph.end_node {
+                paths.push(prev_edges.clone());
+            } else {
+                self.paths_of_cost_rec(
+                    to as usize,
+                    all_distances,
+                    cost - edge_cost as u64,
+                    prev_edges,
+                    paths,
+                    &new_forbidden_edges,
+                    empty_paths_cache,
+                )
+            }
+            prev_edges.pop();
         }
-        while let Some(mut next_cheapest_paths_entry) = self.potential_cheapest_paths.first_entry()
+    }
+
+    pub fn initialize_distances_cheapest(&self) -> Vec<Vec<u64>> {
+        let mut distances_to_end: Vec<Vec<u64>> = vec![vec![]; self.query_graph.nodes.len()];
+        let mut enqueued = vec![false; self.query_graph.nodes.len()];
+
+        let mut node_stack = VecDeque::new();
+
+        distances_to_end[self.query_graph.end_node as usize] = vec![0];
+        for prev_node in
+            self.query_graph.edges[self.query_graph.end_node as usize].predecessors.iter()
         {
-            let cost = *next_cheapest_paths_entry.key();
-            let next_cheapest_paths = next_cheapest_paths_entry.get_mut();
+            node_stack.push_back(prev_node as usize);
+            enqueued[prev_node as usize] = true;
+        }
 
-            while let Some((next_cheapest_path, cost2)) = next_cheapest_paths.remove_first() {
-                assert_eq!(cost, cost2);
-                // NOTE: it is important not to discard the paths that are forbidden due to a
-                // forbidden prefix, because the cheapest path algorithm (Dijkstra) cannot take
-                // this property into account.
-                if next_cheapest_path
-                    .iter()
-                    .any(|edge_index| graph.all_edges[*edge_index as usize].is_none())
-                {
-                    continue;
-                } else {
-                    self.cheapest_paths.insert(next_cheapest_path.iter().copied(), cost);
-
-                    if next_cheapest_paths.is_empty() {
-                        next_cheapest_paths_entry.remove();
+        while let Some(cur_node) = node_stack.pop_front() {
+            let mut self_distances = vec![];
+            for succ_node in self.query_graph.edges[cur_node].successors.iter() {
+                let succ_distances = &distances_to_end[succ_node as usize];
+                let _ = self.visit_edges::<()>(cur_node as u32, succ_node, |_, edge| {
+                    for succ_distance in succ_distances {
+                        self_distances.push(edge.cost as u64 + succ_distance);
                     }
-                    self.kth_cheapest_path = Path { edges: next_cheapest_path, cost };
-
-                    return Some(self);
+                    std::ops::ControlFlow::Continue(())
+                });
+            }
+            self_distances.sort_unstable();
+            self_distances.dedup();
+            distances_to_end[cur_node] = self_distances;
+            for prev_node in self.query_graph.edges[cur_node].predecessors.iter() {
+                if !enqueued[prev_node as usize] {
+                    node_stack.push_back(prev_node as usize);
+                    enqueued[prev_node as usize] = true;
                 }
             }
-            let _ = next_cheapest_paths_entry.remove_entry();
         }
-        None
-    }
-}
-
-impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
-    fn cheapest_path_to_end(&self, from: u32) -> Option<Path> {
-        let mut dijkstra = DijkstraState {
-            unvisited: (0..self.query_graph.nodes.len() as u32).collect(),
-            distances: vec![u64::MAX; self.query_graph.nodes.len()],
-            edges: vec![u32::MAX; self.query_graph.nodes.len()],
-            edge_costs: vec![u8::MAX; self.query_graph.nodes.len()],
-            paths: vec![None; self.query_graph.nodes.len()],
-        };
-        dijkstra.distances[from as usize] = 0;
-
-        // TODO: could use a binary heap here to store the distances, or a btreemap
-        while let Some(cur_node) =
-            dijkstra.unvisited.iter().min_by_key(|&n| dijkstra.distances[n as usize])
-        {
-            let cur_node_dist = dijkstra.distances[cur_node as usize];
-            if cur_node_dist == u64::MAX {
-                return None;
-            }
-            if cur_node == self.query_graph.end_node {
-                break;
-            }
-
-            let succ_cur_node = &self.successors[cur_node as usize];
-            let unvisited_succ_cur_node = succ_cur_node & &dijkstra.unvisited;
-            for succ in unvisited_succ_cur_node {
-                let Some((cheapest_edge, cheapest_edge_cost)) = self.cheapest_edge(cur_node, succ) else {
-                    continue
-                };
-
-                let old_dist_succ = &mut dijkstra.distances[succ as usize];
-                let new_potential_distance = cur_node_dist + cheapest_edge_cost as u64;
-                if new_potential_distance < *old_dist_succ {
-                    *old_dist_succ = new_potential_distance;
-                    dijkstra.edges[succ as usize] = cheapest_edge;
-                    dijkstra.edge_costs[succ as usize] = cheapest_edge_cost;
-                    dijkstra.paths[succ as usize] = Some(cur_node);
-                }
-            }
-            dijkstra.unvisited.remove(cur_node);
-        }
-
-        let mut cur = self.query_graph.end_node;
-        let mut path_edges = vec![];
-        while let Some(n) = dijkstra.paths[cur as usize] {
-            path_edges.push(dijkstra.edges[cur as usize]);
-            cur = n;
-        }
-        path_edges.reverse();
-        Some(Path {
-            edges: path_edges,
-            cost: dijkstra.distances[self.query_graph.end_node as usize],
-        })
-    }
-
-    pub fn cheapest_edge(&self, cur_node: u32, succ: u32) -> Option<(u32, u8)> {
-        self.visit_edges(cur_node, succ, |edge_idx, edge| {
-            std::ops::ControlFlow::Break((edge_idx, edge.cost))
-        })
+        distances_to_end
     }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/edge_docids_cache.rs b/milli/src/search/new/ranking_rule_graph/edge_docids_cache.rs
index cb3e3da38..ef2eba895 100644
--- a/milli/src/search/new/ranking_rule_graph/edge_docids_cache.rs
+++ b/milli/src/search/new/ranking_rule_graph/edge_docids_cache.rs
@@ -32,16 +32,19 @@ impl<G: RankingRuleGraphTrait> EdgeDocidsCache<G> {
         db_cache: &mut DatabaseCache<'transaction>,
         edge_index: u32,
         graph: &RankingRuleGraph<G>,
+        // TODO: maybe universe doesn't belong here
+        universe: &RoaringBitmap,
     ) -> Result<BitmapOrAllRef<'s>> {
-        if self.cache.contains_key(&edge_index) {
-            return Ok(BitmapOrAllRef::Bitmap(&self.cache[&edge_index]));
-        }
         let edge = graph.all_edges[edge_index as usize].as_ref().unwrap();
 
         match &edge.details {
             EdgeDetails::Unconditional => Ok(BitmapOrAllRef::All),
             EdgeDetails::Data(details) => {
-                let docids = G::compute_docids(index, txn, db_cache, details)?;
+                if self.cache.contains_key(&edge_index) {
+                    return Ok(BitmapOrAllRef::Bitmap(&self.cache[&edge_index]));
+                }
+                // TODO: maybe universe doesn't belong here
+                let docids = universe & G::compute_docids(index, txn, db_cache, details)?;
 
                 let _ = self.cache.insert(edge_index, docids);
                 let docids = &self.cache[&edge_index];
diff --git a/milli/src/search/new/ranking_rule_graph/empty_paths_cache.rs b/milli/src/search/new/ranking_rule_graph/empty_paths_cache.rs
index db68838b5..bbfe2eedd 100644
--- a/milli/src/search/new/ranking_rule_graph/empty_paths_cache.rs
+++ b/milli/src/search/new/ranking_rule_graph/empty_paths_cache.rs
@@ -1,26 +1,60 @@
-use roaring::RoaringBitmap;
-
 use super::paths_map::PathsMap;
 
-#[derive(Default, Clone)]
+#[derive(Clone)]
 pub struct EmptyPathsCache {
-    pub empty_edges: RoaringBitmap,
+    pub empty_edges: Vec<bool>,
     pub empty_prefixes: PathsMap<()>,
+    pub empty_couple_edges: Vec<Vec<u32>>,
 }
 impl EmptyPathsCache {
+    pub fn new(all_edges_len: usize) -> Self {
+        Self {
+            empty_edges: vec![false; all_edges_len],
+            empty_prefixes: PathsMap::default(),
+            empty_couple_edges: vec![vec![]; all_edges_len],
+        }
+    }
     pub fn forbid_edge(&mut self, edge_idx: u32) {
-        self.empty_edges.insert(edge_idx);
+        self.empty_edges[edge_idx as usize] = true;
+        self.empty_couple_edges[edge_idx as usize] = vec![];
         self.empty_prefixes.remove_edge(&edge_idx);
+        for edges2 in self.empty_couple_edges.iter_mut() {
+            if let Some(edge2_pos) = edges2.iter().position(|e| *e == edge_idx) {
+                edges2.swap_remove(edge2_pos);
+            }
+        }
+    }
+    pub fn forbid_prefix(&mut self, prefix: &[u32]) {
+        self.empty_prefixes.insert(prefix.iter().copied(), ());
+    }
+    pub fn forbid_couple_edges(&mut self, edge1: u32, edge2: u32) {
+        assert!(!self.empty_couple_edges[edge1 as usize].contains(&edge2));
+        self.empty_couple_edges[edge1 as usize].push(edge2);
     }
     pub fn path_is_empty(&self, path: &[u32]) -> bool {
         for edge in path {
-            if self.empty_edges.contains(*edge) {
+            if self.empty_edges[*edge as usize] {
                 return true;
             }
         }
         if self.empty_prefixes.contains_prefix_of_path(path) {
             return true;
         }
+        for (edge1, edges2) in self.empty_couple_edges.iter().enumerate() {
+            if let Some(pos_edge1) = path.iter().position(|e| *e == edge1 as u32) {
+                if path[pos_edge1..].iter().any(|e| edges2.contains(e)) {
+                    return true;
+                }
+            }
+        }
+        // for (edge1, edge2) in self.empty_couple_edges.iter() {
+        //     if path.contains(edge1) && path.contains(edge2) {
+        //         return true;
+        //     }
+        // }
+        // if self.empty_prefixes.contains_prefix_of_path(path) {
+        //     return true;
+        // }
         false
     }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/mod.rs b/milli/src/search/new/ranking_rule_graph/mod.rs
index b1adb80fc..ac5e1f46b 100644
--- a/milli/src/search/new/ranking_rule_graph/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/mod.rs
@@ -13,7 +13,6 @@ use heed::RoTxn;
 use roaring::RoaringBitmap;
 
 use self::empty_paths_cache::EmptyPathsCache;
-use self::paths_map::PathsMap;
 
 use super::db_cache::DatabaseCache;
 use super::logger::SearchLogger;
@@ -83,8 +82,11 @@ pub trait RankingRuleGraphTrait: Sized {
 
     fn log_state(
         graph: &RankingRuleGraph<Self>,
-        paths: &PathsMap<u64>,
+        paths: &[Vec<u32>],
         empty_paths_cache: &EmptyPathsCache,
+        universe: &RoaringBitmap,
+        distances: &[Vec<u64>],
+        cost: u64,
         logger: &mut dyn SearchLogger<QueryGraph>,
     );
 }
@@ -135,7 +137,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         None
     }
 
-    fn remove_edge(&mut self, edge_index: u32) {
+    pub fn remove_edge(&mut self, edge_index: u32) {
         let edge_opt = &mut self.all_edges[edge_index as usize];
         let Some(edge) = &edge_opt else { return };
         let (from_node, _to_node) = (edge.from_node, edge.to_node);
@@ -151,44 +153,4 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         }
         self.successors[from_node as usize] = new_successors_from_node;
     }
-
-    pub fn graphviz(&self) -> String {
-        let mut desc = String::new();
-        desc.push_str("digraph G {\nrankdir = LR;\nnode [shape = \"record\"]\n");
-
-        for (node_idx, node) in self.query_graph.nodes.iter().enumerate() {
-            if matches!(node, QueryNode::Deleted) {
-                continue;
-            }
-            desc.push_str(&format!("{node_idx} [label = {:?}]", node));
-            if node_idx == self.query_graph.root_node as usize {
-                desc.push_str("[color = blue]");
-            } else if node_idx == self.query_graph.end_node as usize {
-                desc.push_str("[color = red]");
-            }
-            desc.push_str(";\n");
-        }
-        for edge in self.all_edges.iter().flatten() {
-            let Edge { from_node, to_node, details, .. } = edge;
-
-            match &details {
-                EdgeDetails::Unconditional => {
-                    desc.push_str(&format!(
-                        "{from_node} -> {to_node} [label = \"always cost {cost}\"];\n",
-                        cost = edge.cost,
-                    ));
-                }
-                EdgeDetails::Data(details) => {
-                    desc.push_str(&format!(
-                        "{from_node} -> {to_node} [label = \"cost {cost} {edge_label}\"];\n",
-                        cost = edge.cost,
-                        edge_label = G::graphviz_edge_details_label(details)
-                    ));
-                }
-            }
-        }
-
-        desc.push('}');
-        desc
-    }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/paths_map.rs b/milli/src/search/new/ranking_rule_graph/paths_map.rs
index 111b55140..3b01508c9 100644
--- a/milli/src/search/new/ranking_rule_graph/paths_map.rs
+++ b/milli/src/search/new/ranking_rule_graph/paths_map.rs
@@ -1,12 +1,11 @@
-use std::collections::hash_map::DefaultHasher;
-use std::fmt::Write;
-use std::hash::{Hash, Hasher};
+
+
+
 
 use roaring::RoaringBitmap;
 
 use super::cheapest_paths::Path;
-use super::{Edge, EdgeDetails, RankingRuleGraph, RankingRuleGraphTrait};
-use crate::new::QueryNode;
+
 
 #[derive(Debug, Clone)]
 pub struct PathsMap<V> {
@@ -157,6 +156,24 @@ impl<V> PathsMap<V> {
         }
     }
 
+    pub fn final_edges_ater_prefix(&self, prefix: &[u32]) -> Vec<u32> {
+        let [first_edge, remaining_prefix @ ..] = prefix else {
+            return self.nodes.iter().filter_map(|n| {
+                if n.1.value.is_some() {
+                    Some(n.0)
+                } else {
+                    None
+                }
+            }).collect();
+        };
+        for (edge, rest) in self.nodes.iter() {
+            if edge == first_edge {
+                return rest.final_edges_ater_prefix(remaining_prefix);
+            }
+        }
+        vec![]
+    }
+
     pub fn edge_indices_after_prefix(&self, prefix: &[u32]) -> Vec<u32> {
         let [first_edge, remaining_prefix @ ..] = prefix else {
             return self.nodes.iter().map(|n| n.0).collect();
@@ -185,88 +202,4 @@ impl<V> PathsMap<V> {
             }
         }
     }
-
-    pub fn graphviz<G: RankingRuleGraphTrait>(&self, graph: &RankingRuleGraph<G>) -> String {
-        let mut desc = String::new();
-        desc.push_str("digraph G {\n");
-        self.graphviz_rec(&mut desc, vec![], graph);
-        desc.push_str("\n}\n");
-        desc
-    }
-    fn graphviz_rec<G: RankingRuleGraphTrait>(
-        &self,
-        desc: &mut String,
-        path_from: Vec<u64>,
-        graph: &RankingRuleGraph<G>,
-    ) {
-        let id_from = {
-            let mut h = DefaultHasher::new();
-            path_from.hash(&mut h);
-            h.finish()
-        };
-        for (edge_idx, rest) in self.nodes.iter() {
-            let Some(Edge { from_node, to_node, cost, .. }) = graph.all_edges[*edge_idx as usize].as_ref() else {
-                continue;
-            };
-            let mut path_to = path_from.clone();
-            path_to.push({
-                let mut h = DefaultHasher::new();
-                edge_idx.hash(&mut h);
-                h.finish()
-            });
-            let id_to = {
-                let mut h = DefaultHasher::new();
-                path_to.hash(&mut h);
-                h.finish()
-            };
-            writeln!(desc, "{id_to} [label = \"{from_node}→{to_node} [{cost}]\"];").unwrap();
-            writeln!(desc, "{id_from} -> {id_to};").unwrap();
-
-            rest.graphviz_rec(desc, path_to, graph);
-        }
-    }
-}
-
-impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
-    pub fn graphviz_with_path(&self, path: &Path) -> String {
-        let mut desc = String::new();
-        desc.push_str("digraph G {\nrankdir = LR;\nnode [shape = \"record\"]\n");
-
-        for (node_idx, node) in self.query_graph.nodes.iter().enumerate() {
-            if matches!(node, QueryNode::Deleted) {
-                continue;
-            }
-            desc.push_str(&format!("{node_idx} [label = {:?}]", node));
-            if node_idx == self.query_graph.root_node as usize {
-                desc.push_str("[color = blue]");
-            } else if node_idx == self.query_graph.end_node as usize {
-                desc.push_str("[color = red]");
-            }
-            desc.push_str(";\n");
-        }
-
-        for (edge_idx, edge) in self.all_edges.iter().enumerate() {
-            let Some(edge) = edge else { continue };
-            let Edge { from_node, to_node, .. } = edge;
-            let color = if path.edges.contains(&(edge_idx as u32)) { "red" } else { "green" };
-            match &edge.details {
-                EdgeDetails::Unconditional => {
-                    desc.push_str(&format!(
-                        "{from_node} -> {to_node} [label = \"cost {cost}\", color = {color}];\n",
-                        cost = edge.cost,
-                    ));
-                }
-                EdgeDetails::Data(details) => {
-                    desc.push_str(&format!(
-                        "{from_node} -> {to_node} [label = \"cost {cost} {edge_label}\", color = {color}];\n",
-                        cost = edge.cost,
-                        edge_label = G::graphviz_edge_details_label(details),
-                    ));
-                }
-            }
-        }
-
-        desc.push('}');
-        desc
-    }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/proximity/build.rs b/milli/src/search/new/ranking_rule_graph/proximity/build.rs
index 6d2fefa65..9b4fa8edf 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/build.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/build.rs
@@ -16,9 +16,9 @@ pub fn visit_from_node(from_node: &QueryNode) -> Result<Option<(WordDerivations,
         QueryNode::Term(LocatedQueryTerm { value: value1, positions: pos1 }) => {
             match value1 {
                 QueryTerm::Word { derivations } => (derivations.clone(), *pos1.end()),
-                QueryTerm::Phrase(phrase1) => {
+                QueryTerm::Phrase { phrase: phrase1 } => {
                     // TODO: remove second unwrap
-                    let original = phrase1.last().unwrap().as_ref().unwrap().clone();
+                    let original = phrase1.words.last().unwrap().as_ref().unwrap().clone();
                     (
                         WordDerivations {
                             original: original.clone(),
@@ -26,6 +26,8 @@ pub fn visit_from_node(from_node: &QueryNode) -> Result<Option<(WordDerivations,
                             one_typo: vec![],
                             two_typos: vec![],
                             use_prefix_db: false,
+                            synonyms: vec![],
+                            split_words: None,
                         },
                         *pos1.end(),
                     )
@@ -39,6 +41,8 @@ pub fn visit_from_node(from_node: &QueryNode) -> Result<Option<(WordDerivations,
                 one_typo: vec![],
                 two_typos: vec![],
                 use_prefix_db: false,
+                synonyms: vec![],
+                split_words: None,
             },
             -100,
         ),
@@ -63,9 +67,9 @@ pub fn visit_to_node<'transaction, 'from_data>(
 
     let (derivations2, pos2, ngram_len2) = match value2 {
         QueryTerm::Word { derivations } => (derivations.clone(), *pos2.start(), pos2.len()),
-        QueryTerm::Phrase(phrase2) => {
+        QueryTerm::Phrase { phrase: phrase2 } => {
             // TODO: remove second unwrap
-            let original = phrase2.last().unwrap().as_ref().unwrap().clone();
+            let original = phrase2.words.last().unwrap().as_ref().unwrap().clone();
             (
                 WordDerivations {
                     original: original.clone(),
@@ -73,6 +77,8 @@ pub fn visit_to_node<'transaction, 'from_data>(
                     one_typo: vec![],
                     two_typos: vec![],
                     use_prefix_db: false,
+                    synonyms: vec![],
+                    split_words: None,
                 },
                 *pos2.start(),
                 1,
diff --git a/milli/src/search/new/ranking_rule_graph/proximity/mod.rs b/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
index fc1a44310..5b3869ea8 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
@@ -2,18 +2,21 @@ pub mod build;
 pub mod compute_docids;
 
 use heed::RoTxn;
+use roaring::RoaringBitmap;
 
 use super::empty_paths_cache::EmptyPathsCache;
-use super::paths_map::PathsMap;
+
 use super::{EdgeDetails, RankingRuleGraphTrait};
 use crate::new::db_cache::DatabaseCache;
+use crate::new::logger::SearchLogger;
 use crate::new::query_term::WordDerivations;
-use crate::new::QueryNode;
+use crate::new::{QueryGraph, QueryNode};
 use crate::{Index, Result};
 
+// TODO: intern the strings, refer to them by their pointer?
+
 #[derive(Debug, Clone)]
 pub enum WordPair {
-    // TODO: add WordsSwapped and WordPrefixSwapped case
     Words { left: String, right: String },
     WordsSwapped { left: String, right: String },
     WordPrefix { left: String, right_prefix: String },
@@ -22,6 +25,7 @@ pub enum WordPair {
 
 #[derive(Clone)]
 pub struct ProximityEdge {
+    // TODO: use a list of pointers to the word pairs instead?
     pairs: Vec<WordPair>,
     proximity: u8,
 }
@@ -67,10 +71,20 @@ impl RankingRuleGraphTrait for ProximityGraph {
 
     fn log_state(
         graph: &super::RankingRuleGraph<Self>,
-        paths: &PathsMap<u64>,
+        paths: &[Vec<u32>],
         empty_paths_cache: &EmptyPathsCache,
-        logger: &mut dyn crate::new::logger::SearchLogger<crate::new::QueryGraph>,
+        universe: &RoaringBitmap,
+        distances: &[Vec<u64>],
+        cost: u64,
+        logger: &mut dyn SearchLogger<QueryGraph>,
     ) {
-        logger.log_proximity_state(graph, paths, empty_paths_cache);
+        logger.log_proximity_state(
+            graph,
+            paths,
+            empty_paths_cache,
+            universe,
+            distances.to_vec(),
+            cost,
+        );
     }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/resolve_paths.rs b/milli/src/search/new/ranking_rule_graph/resolve_paths.rs
index 90650340f..f3394206b 100644
--- a/milli/src/search/new/ranking_rule_graph/resolve_paths.rs
+++ b/milli/src/search/new/ranking_rule_graph/resolve_paths.rs
@@ -5,7 +5,7 @@ use roaring::{MultiOps, RoaringBitmap};
 
 use super::edge_docids_cache::EdgeDocidsCache;
 use super::empty_paths_cache::EmptyPathsCache;
-use super::paths_map::PathsMap;
+
 use super::{RankingRuleGraph, RankingRuleGraphTrait};
 use crate::new::db_cache::DatabaseCache;
 
@@ -21,44 +21,65 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         edge_docids_cache: &mut EdgeDocidsCache<G>,
         empty_paths_cache: &mut EmptyPathsCache,
         universe: &RoaringBitmap,
-        mut paths: PathsMap<u64>,
+        mut paths: Vec<Vec<u32>>,
     ) -> Result<RoaringBitmap> {
+        paths.sort_unstable();
+        let mut needs_filtering = false;
         let mut path_bitmaps = vec![];
+        'path_loop: loop {
+            if needs_filtering {
+                for path in paths.iter_mut() {
+                    if empty_paths_cache.path_is_empty(path) {
+                        path.clear();
+                    }
+                }
+                needs_filtering = false;
+            }
+            let Some(edge_indexes) = paths.pop() else {
+                break;
+            };
 
-        paths.remove_edges(&empty_paths_cache.empty_edges);
-        paths.remove_prefixes(&empty_paths_cache.empty_prefixes);
+            if edge_indexes.is_empty() {
+                continue;
+            }
 
-        'path_loop: while let Some((edge_indexes, _)) = paths.remove_first() {
-            // if path is excluded, continue...
-            let mut processed_edges = vec![];
             let mut path_bitmap = universe.clone();
+            let mut visited_edges = vec![];
+            let mut cached_edge_docids = vec![];
             'edge_loop: for edge_index in edge_indexes {
-                processed_edges.push(edge_index);
-                let edge_docids =
-                    edge_docids_cache.get_edge_docids(index, txn, db_cache, edge_index, self)?;
+                visited_edges.push(edge_index);
+                let edge_docids = edge_docids_cache
+                    .get_edge_docids(index, txn, db_cache, edge_index, self, universe)?;
                 match edge_docids {
                     BitmapOrAllRef::Bitmap(edge_docids) => {
+                        cached_edge_docids.push((edge_index, edge_docids.clone()));
+                        let (_, edge_docids) = cached_edge_docids.last().unwrap();
                         if edge_docids.is_disjoint(universe) {
                             // 1. Store in the cache that this edge is empty for this universe
                             empty_paths_cache.forbid_edge(edge_index);
-                            // 2. remove all the paths that contain this edge for this universe
-                            paths.remove_edge(&edge_index);
-                            // 3. remove this edge from the proximity graph
-
+                            // 2. remove this edge from the proximity graph
                             self.remove_edge(edge_index);
-
-                            // 4. continue executing this function again on the remaining paths
+                            edge_docids_cache.cache.remove(&edge_index);
+                            needs_filtering = true;
+                            // 3. continue executing this function again on the remaining paths
                             continue 'path_loop;
                         } else {
                             path_bitmap &= edge_docids;
                             if path_bitmap.is_disjoint(universe) {
-                                // 1. Store in the cache that this prefix is empty for this universe
-                                empty_paths_cache
-                                    .empty_prefixes
-                                    .insert(processed_edges.iter().copied(), ());
-                                // 2. remove all the paths beginning with this prefix
-                                paths.remove_prefix(&processed_edges);
-                                // 3. continue executing this function again on the remaining paths?
+                                needs_filtering = true;
+                                empty_paths_cache.forbid_prefix(&visited_edges);
+                                // if the intersection between this edge and any
+                                // previous one is disjoint with the universe,
+                                // then we add these two edges to the empty_path_cache
+                                for (edge_index2, edge_docids2) in
+                                    cached_edge_docids[..cached_edge_docids.len() - 1].iter()
+                                {
+                                    let intersection = edge_docids & edge_docids2;
+                                    if intersection.is_disjoint(universe) {
+                                        empty_paths_cache
+                                            .forbid_couple_edges(*edge_index2, edge_index);
+                                    }
+                                }
                                 continue 'path_loop;
                             }
                         }
@@ -68,6 +89,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
             }
             path_bitmaps.push(path_bitmap);
         }
+
         Ok(MultiOps::union(path_bitmaps))
     }
 }
diff --git a/milli/src/search/new/ranking_rule_graph/typo/mod.rs b/milli/src/search/new/ranking_rule_graph/typo/mod.rs
index 55a45e3c3..c9ca7c229 100644
--- a/milli/src/search/new/ranking_rule_graph/typo/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/typo/mod.rs
@@ -2,16 +2,18 @@ use heed::{BytesDecode, RoTxn};
 use roaring::RoaringBitmap;
 
 use super::empty_paths_cache::EmptyPathsCache;
-use super::paths_map::PathsMap;
-use super::{EdgeDetails, RankingRuleGraphTrait};
+
+use super::{EdgeDetails, RankingRuleGraph, RankingRuleGraphTrait};
 use crate::new::db_cache::DatabaseCache;
-use crate::new::query_term::{LocatedQueryTerm, QueryTerm, WordDerivations};
-use crate::new::QueryNode;
+use crate::new::logger::SearchLogger;
+use crate::new::query_term::{LocatedQueryTerm, Phrase, QueryTerm, WordDerivations};
+use crate::new::resolve_query_graph::resolve_phrase;
+use crate::new::{QueryGraph, QueryNode};
 use crate::{Index, Result, RoaringBitmapCodec};
 
 #[derive(Clone)]
 pub enum TypoEdge {
-    Phrase,
+    Phrase { phrase: Phrase },
     Word { derivations: WordDerivations, nbr_typos: u8 },
 }
 
@@ -23,7 +25,7 @@ impl RankingRuleGraphTrait for TypoGraph {
 
     fn graphviz_edge_details_label(edge: &Self::EdgeDetails) -> String {
         match edge {
-            TypoEdge::Phrase => format!(", 0 typos"),
+            TypoEdge::Phrase { .. } => ", 0 typos".to_owned(),
             TypoEdge::Word { nbr_typos, .. } => format!(", {nbr_typos} typos"),
         }
     }
@@ -33,9 +35,9 @@ impl RankingRuleGraphTrait for TypoGraph {
         txn: &'transaction RoTxn,
         db_cache: &mut DatabaseCache<'transaction>,
         edge: &Self::EdgeDetails,
-    ) -> Result<roaring::RoaringBitmap> {
+    ) -> Result<RoaringBitmap> {
         match edge {
-            TypoEdge::Phrase => todo!(),
+            TypoEdge::Phrase { phrase } => resolve_phrase(index, txn, db_cache, phrase),
             TypoEdge::Word { derivations, nbr_typos } => {
                 let words = match nbr_typos {
                     0 => &derivations.zero_typo,
@@ -68,21 +70,23 @@ impl RankingRuleGraphTrait for TypoGraph {
         _index: &Index,
         _txn: &'transaction RoTxn,
         _db_cache: &mut DatabaseCache<'transaction>,
-        from_node: &QueryNode,
+        _from_node: &QueryNode,
     ) -> Result<Option<Self::BuildVisitedFromNode>> {
         Ok(Some(()))
     }
 
     fn build_visit_to_node<'from_data, 'transaction: 'from_data>(
-        index: &Index,
-        txn: &'transaction RoTxn,
-        db_cache: &mut DatabaseCache<'transaction>,
+        _index: &Index,
+        _txn: &'transaction RoTxn,
+        _db_cache: &mut DatabaseCache<'transaction>,
         to_node: &QueryNode,
-        from_node_data: &'from_data Self::BuildVisitedFromNode,
+        _from_node_data: &'from_data Self::BuildVisitedFromNode,
     ) -> Result<Vec<(u8, EdgeDetails<Self::EdgeDetails>)>> {
         match to_node {
             QueryNode::Term(LocatedQueryTerm { value, .. }) => match value {
-                QueryTerm::Phrase(_) => Ok(vec![(0, EdgeDetails::Data(TypoEdge::Phrase))]),
+                QueryTerm::Phrase { phrase } => {
+                    Ok(vec![(0, EdgeDetails::Data(TypoEdge::Phrase { phrase: phrase.clone() }))])
+                }
                 QueryTerm::Word { derivations } => {
                     let mut edges = vec![];
                     if !derivations.zero_typo.is_empty() || derivations.use_prefix_db {
@@ -121,11 +125,14 @@ impl RankingRuleGraphTrait for TypoGraph {
     }
 
     fn log_state(
-        graph: &super::RankingRuleGraph<Self>,
-        paths: &PathsMap<u64>,
+        graph: &RankingRuleGraph<Self>,
+        paths: &[Vec<u32>],
         empty_paths_cache: &EmptyPathsCache,
-        logger: &mut dyn crate::new::logger::SearchLogger<crate::new::QueryGraph>,
+        universe: &RoaringBitmap,
+        distances: &[Vec<u64>],
+        cost: u64,
+        logger: &mut dyn SearchLogger<QueryGraph>,
     ) {
-        logger.log_typo_state(graph, paths, empty_paths_cache);
+        logger.log_typo_state(graph, paths, empty_paths_cache, universe, distances.to_vec(), cost);
     }
 }
diff --git a/milli/src/search/new/ranking_rules.rs b/milli/src/search/new/ranking_rules.rs
index f023f94d1..9b3bcb38c 100644
--- a/milli/src/search/new/ranking_rules.rs
+++ b/milli/src/search/new/ranking_rules.rs
@@ -1,3 +1,5 @@
+use std::time::Instant;
+
 use heed::RoTxn;
 use roaring::RoaringBitmap;
 
@@ -9,7 +11,7 @@ use crate::new::graph_based_ranking_rule::GraphBasedRankingRule;
 use crate::new::ranking_rule_graph::proximity::ProximityGraph;
 use crate::new::ranking_rule_graph::typo::TypoGraph;
 use crate::new::words::Words;
-use crate::search::new::sort::Sort;
+// use crate::search::new::sort::Sort;
 use crate::{Filter, Index, Result, TermsMatchingStrategy};
 
 pub trait RankingRuleOutputIter<'transaction, Query> {
@@ -123,13 +125,14 @@ pub fn execute_search<'transaction>(
     length: usize,
     logger: &mut dyn SearchLogger<QueryGraph>,
 ) -> Result<Vec<u32>> {
+    logger.initial_query(query_graph, Instant::now());
     let words = &mut Words::new(TermsMatchingStrategy::Last);
-    let sort = &mut Sort::new(index, txn, "release_date".to_owned(), true)?;
+    // let sort = &mut Sort::new(index, txn, "release_date".to_owned(), true)?;
     let proximity = &mut GraphBasedRankingRule::<ProximityGraph>::new("proximity".to_owned());
     let typo = &mut GraphBasedRankingRule::<TypoGraph>::new("typo".to_owned());
     // TODO: ranking rules given as argument
     let mut ranking_rules: Vec<&mut dyn RankingRule<'transaction, QueryGraph>> =
-        vec![words, typo, proximity, sort];
+        vec![words, typo, proximity /*sort*/];
 
     logger.ranking_rules(&ranking_rules);
 
@@ -144,7 +147,13 @@ pub fn execute_search<'transaction>(
     }
 
     let ranking_rules_len = ranking_rules.len();
-    logger.start_iteration_ranking_rule(0, ranking_rules[0], query_graph, &universe);
+    logger.start_iteration_ranking_rule(
+        0,
+        ranking_rules[0],
+        query_graph,
+        &universe,
+        Instant::now(),
+    );
     ranking_rules[0].start_iteration(index, txn, db_cache, logger, &universe, query_graph)?;
 
     let mut candidates = vec![RoaringBitmap::default(); ranking_rules_len];
@@ -154,11 +163,12 @@ pub fn execute_search<'transaction>(
 
     macro_rules! back {
         () => {
-            // assert!(candidates[cur_ranking_rule_index].is_empty());
+            assert!(candidates[cur_ranking_rule_index].is_empty());
             logger.end_iteration_ranking_rule(
                 cur_ranking_rule_index,
                 ranking_rules[cur_ranking_rule_index],
                 &candidates[cur_ranking_rule_index],
+                Instant::now(),
             );
             candidates[cur_ranking_rule_index].clear();
             ranking_rules[cur_ranking_rule_index].end_iteration(index, txn, db_cache, logger);
@@ -187,6 +197,7 @@ pub fn execute_search<'transaction>(
                             cur_ranking_rule_index,
                             ranking_rules[cur_ranking_rule_index],
                             &candidates,
+                            Instant::now(),
                         );
                     } else {
                         let all_candidates = candidates.iter().collect::<Vec<_>>();
@@ -196,6 +207,7 @@ pub fn execute_search<'transaction>(
                             cur_ranking_rule_index,
                             ranking_rules[cur_ranking_rule_index],
                             &skipped_candidates.into_iter().collect(),
+                            Instant::now(),
                         );
                         let candidates = candidates
                             .iter()
@@ -219,24 +231,26 @@ pub fn execute_search<'transaction>(
         // The universe for this bucket is zero or one element, so we don't need to sort
         // anything, just extend the results and go back to the parent ranking rule.
         if candidates[cur_ranking_rule_index].len() <= 1 {
-            candidates[cur_ranking_rule_index].clear();
             maybe_add_to_results!(&candidates[cur_ranking_rule_index]);
+            candidates[cur_ranking_rule_index].clear();
             back!();
             continue;
         }
 
-        logger.next_bucket_ranking_rule(
-            cur_ranking_rule_index,
-            ranking_rules[cur_ranking_rule_index],
-            &candidates[cur_ranking_rule_index],
-        );
-
         let Some(next_bucket) = ranking_rules[cur_ranking_rule_index].next_bucket(index, txn, db_cache, logger, &candidates[cur_ranking_rule_index])? else {
             // TODO: add remaining candidates automatically here?
             back!();
             continue;
         };
 
+        logger.next_bucket_ranking_rule(
+            cur_ranking_rule_index,
+            ranking_rules[cur_ranking_rule_index],
+            &candidates[cur_ranking_rule_index],
+            &next_bucket.candidates,
+            Instant::now(),
+        );
+
         assert!(candidates[cur_ranking_rule_index].is_superset(&next_bucket.candidates));
         candidates[cur_ranking_rule_index] -= &next_bucket.candidates;
 
@@ -255,6 +269,7 @@ pub fn execute_search<'transaction>(
             ranking_rules[cur_ranking_rule_index],
             &next_bucket.query,
             &candidates[cur_ranking_rule_index],
+            Instant::now(),
         );
         ranking_rules[cur_ranking_rule_index].start_iteration(
             index,
@@ -271,17 +286,18 @@ pub fn execute_search<'transaction>(
 
 #[cfg(test)]
 mod tests {
-    use std::fs::File;
-    use std::io::{BufRead, BufReader, Cursor, Seek};
-    use std::time::Instant;
-
-    use heed::EnvOpenOptions;
-
     use super::execute_search;
+    // use crate::allocator::ALLOC;
     use crate::documents::{DocumentsBatchBuilder, DocumentsBatchReader};
     use crate::index::tests::TempIndex;
     use crate::new::db_cache::DatabaseCache;
-    use crate::new::logger::detailed::DetailedSearchLogger;
+    use big_s::S;
+    use heed::EnvOpenOptions;
+    use maplit::hashset;
+    use std::fs::File;
+    use std::io::{BufRead, BufReader, Cursor, Seek};
+    use std::time::Instant;
+    // use crate::new::logger::detailed::DetailedSearchLogger;
     use crate::new::logger::{DefaultSearchLogger, SearchLogger};
     use crate::new::make_query_graph;
     use crate::update::{IndexDocuments, IndexDocumentsConfig, IndexerConfig, Settings};
@@ -323,16 +339,119 @@ mod tests {
         let mut db_cache = DatabaseCache::default();
 
         let query_graph =
-            make_query_graph(&index, &txn, &mut db_cache, "b b b b b b b b b b").unwrap();
-        println!("{}", query_graph.graphviz());
-        logger.initial_query(&query_graph);
+            make_query_graph(&index, &txn, &mut db_cache, "releases from poison by the government")
+                .unwrap();
+        logger.initial_query(&query_graph, Instant::now());
 
         let results =
-            execute_search(&index, &txn, &mut db_cache, &query_graph, None, 0, 20, &mut logger)
+            execute_search(&index, &txn, &mut db_cache, &query_graph, None, 0, 50, &mut logger)
                 .unwrap();
         println!("{results:?}")
     }
 
+    #[test]
+    fn search_wiki_new() {
+        let mut options = EnvOpenOptions::new();
+        options.map_size(100 * 1024 * 1024 * 1024); // 100 GB
+
+        let index = Index::new(options, "data_wiki").unwrap();
+        let txn = index.read_txn().unwrap();
+
+        println!("nbr docids: {}", index.documents_ids(&txn).unwrap().len());
+
+        let primary_key = index.primary_key(&txn).unwrap().unwrap();
+        let primary_key = index.fields_ids_map(&txn).unwrap().id(primary_key).unwrap();
+        // loop {
+        let start = Instant::now();
+
+        let mut db_cache = DatabaseCache::default();
+
+        let query_graph = make_query_graph(
+            &index,
+            &txn,
+            &mut db_cache,
+            "which a the releases from poison by the government",
+        )
+        .unwrap();
+
+        // let mut logger = crate::new::logger::detailed::DetailedSearchLogger::new("log");
+
+        let results = execute_search(
+            &index,
+            &txn,
+            &mut db_cache,
+            &query_graph,
+            None,
+            0,
+            20,
+            &mut DefaultSearchLogger,
+            // &mut logger,
+        )
+        .unwrap();
+
+        // logger.write_d2_description();
+
+        let elapsed = start.elapsed();
+
+        let ids = index
+            .documents(&txn, results.iter().copied())
+            .unwrap()
+            .into_iter()
+            .map(|x| {
+                let obkv = &x.1;
+                let id = obkv.get(primary_key).unwrap();
+                let id: serde_json::Value = serde_json::from_slice(id).unwrap();
+                id.as_str().unwrap().to_owned()
+            })
+            .collect::<Vec<_>>();
+
+        println!("{}us: {results:?}", elapsed.as_micros());
+        println!("external ids: {ids:?}");
+        // println!("max_resident: {}", ALLOC.max_resident.load(std::sync::atomic::Ordering::SeqCst));
+        // println!("allocated: {}", ALLOC.allocated.load(std::sync::atomic::Ordering::SeqCst));
+        // }
+    }
+
+    #[test]
+    fn search_wiki_old() {
+        let mut options = EnvOpenOptions::new();
+        options.map_size(100 * 1024 * 1024 * 1024); // 100 GB
+
+        let index = Index::new(options, "data_wiki").unwrap();
+
+        let txn = index.read_txn().unwrap();
+
+        let rr = index.criteria(&txn).unwrap();
+        println!("{rr:?}");
+
+        let primary_key = index.primary_key(&txn).unwrap().unwrap();
+        let primary_key = index.fields_ids_map(&txn).unwrap().id(primary_key).unwrap();
+
+        let start = Instant::now();
+
+        let mut s = Search::new(&txn, &index);
+        s.query("releases from poison by the government");
+        s.terms_matching_strategy(TermsMatchingStrategy::Last);
+        s.criterion_implementation_strategy(crate::CriterionImplementationStrategy::OnlySetBased);
+        let docs = s.execute().unwrap();
+
+        let elapsed = start.elapsed();
+
+        let ids = index
+            .documents(&txn, docs.documents_ids.iter().copied())
+            .unwrap()
+            .into_iter()
+            .map(|x| {
+                let obkv = &x.1;
+                let id = obkv.get(primary_key).unwrap();
+                let id: serde_json::Value = serde_json::from_slice(id).unwrap();
+                id.as_str().unwrap().to_owned()
+            })
+            .collect::<Vec<_>>();
+
+        println!("{}us: {:?}", elapsed.as_micros(), docs.documents_ids);
+        println!("external ids: {ids:?}");
+    }
     #[test]
     fn search_movies_new() {
         let mut options = EnvOpenOptions::new();
@@ -343,7 +462,7 @@ mod tests {
 
         let primary_key = index.primary_key(&txn).unwrap().unwrap();
         let primary_key = index.fields_ids_map(&txn).unwrap().id(primary_key).unwrap();
-
+        // loop {
         let start = Instant::now();
 
         let mut db_cache = DatabaseCache::default();
@@ -352,7 +471,7 @@ mod tests {
             make_query_graph(&index, &txn, &mut db_cache, "releases from poison by the government")
                 .unwrap();
 
-        let mut logger = DetailedSearchLogger::new("log");
+        let mut logger = crate::new::logger::detailed::DetailedSearchLogger::new("log");
 
         let results = execute_search(
             &index,
@@ -360,9 +479,10 @@ mod tests {
             &mut db_cache,
             &query_graph,
             None,
-            5,
+            0,
             20,
-            &mut logger, //&mut DefaultSearchLogger,
+            // &mut DefaultSearchLogger,
+            &mut logger,
         )
         .unwrap();
 
@@ -384,6 +504,7 @@ mod tests {
 
         println!("{}us: {results:?}", elapsed.as_micros());
         println!("external ids: {ids:?}");
+        // }
     }
 
     #[test]
@@ -392,19 +513,39 @@ mod tests {
         options.map_size(100 * 1024 * 1024 * 1024); // 100 GB
 
         let index = Index::new(options, "data_movies").unwrap();
+
         let txn = index.read_txn().unwrap();
 
+        let rr = index.criteria(&txn).unwrap();
+        println!("{rr:?}");
+
+        let primary_key = index.primary_key(&txn).unwrap().unwrap();
+        let primary_key = index.fields_ids_map(&txn).unwrap().id(primary_key).unwrap();
+
         let start = Instant::now();
 
         let mut s = Search::new(&txn, &index);
-        s.query("b b b b b b b b b b");
+        s.query("releases from poison by the government");
         s.terms_matching_strategy(TermsMatchingStrategy::Last);
         s.criterion_implementation_strategy(crate::CriterionImplementationStrategy::OnlySetBased);
         let docs = s.execute().unwrap();
 
         let elapsed = start.elapsed();
 
+        let ids = index
+            .documents(&txn, docs.documents_ids.iter().copied())
+            .unwrap()
+            .into_iter()
+            .map(|x| {
+                let obkv = &x.1;
+                let id = obkv.get(primary_key).unwrap();
+                let id: serde_json::Value = serde_json::from_slice(id).unwrap();
+                id.as_str().unwrap().to_owned()
+            })
+            .collect::<Vec<_>>();
+
         println!("{}us: {:?}", elapsed.as_micros(), docs.documents_ids);
+        println!("external ids: {ids:?}");
     }
 
     #[test]
@@ -420,10 +561,16 @@ mod tests {
 
         builder.set_min_word_len_one_typo(5);
         builder.set_min_word_len_two_typos(100);
-
-        builder.set_criteria(vec![Criterion::Words, Criterion::Proximity]);
+        builder.set_sortable_fields(hashset! { S("release_date") });
+        builder.set_criteria(vec![
+            Criterion::Words,
+            Criterion::Typo,
+            Criterion::Proximity,
+            Criterion::Asc("release_date".to_owned()),
+        ]);
 
         builder.execute(|_| (), || false).unwrap();
+        wtxn.commit().unwrap();
     }
 
     #[test]
@@ -445,6 +592,7 @@ mod tests {
         builder.set_searchable_fields(searchable_fields);
         let filterable_fields = filterable_fields.iter().map(|s| s.to_string()).collect();
         builder.set_filterable_fields(filterable_fields);
+
         builder.set_min_word_len_one_typo(5);
         builder.set_min_word_len_two_typos(100);
         builder.set_criteria(vec![Criterion::Words, Criterion::Proximity]);
@@ -467,6 +615,48 @@ mod tests {
 
         index.prepare_for_closing().wait();
     }
+    #[test]
+    fn _index_wiki() {
+        let mut options = EnvOpenOptions::new();
+        options.map_size(100 * 1024 * 1024 * 1024); // 100 GB
+
+        let index = Index::new(options, "data_wiki").unwrap();
+        let mut wtxn = index.write_txn().unwrap();
+
+        // let primary_key = "id";
+        let searchable_fields = vec!["body", "title", "url"];
+        // let filterable_fields = vec![];
+        let config = IndexerConfig::default();
+        let mut builder = Settings::new(&mut wtxn, &index, &config);
+        // builder.set_primary_key(primary_key.to_owned());
+        let searchable_fields = searchable_fields.iter().map(|s| s.to_string()).collect();
+        builder.set_searchable_fields(searchable_fields);
+        // let filterable_fields = filterable_fields.iter().map(|s| s.to_string()).collect();
+        // builder.set_filterable_fields(filterable_fields);
+
+        // builder.set_min_word_len_one_typo(5);
+        // builder.set_min_word_len_two_typos(100);
+        builder.set_criteria(vec![Criterion::Words, Criterion::Typo, Criterion::Proximity]);
+        builder.execute(|_| (), || false).unwrap();
+
+        let config = IndexerConfig::default();
+        let indexing_config =
+            IndexDocumentsConfig { autogenerate_docids: true, ..Default::default() };
+        let builder =
+            IndexDocuments::new(&mut wtxn, &index, &config, indexing_config, |_| (), || false)
+                .unwrap();
+
+        let documents = documents_from(
+            "/Users/meilisearch/Documents/milli2/benchmarks/datasets/smol-wiki-articles.csv",
+            "csv",
+        );
+        let (builder, user_error) = builder.add_documents(documents).unwrap();
+        user_error.unwrap();
+        builder.execute().unwrap();
+        wtxn.commit().unwrap();
+
+        index.prepare_for_closing().wait();
+    }
 
     fn documents_from(filename: &str, filetype: &str) -> DocumentsBatchReader<impl BufRead + Seek> {
         let reader = File::open(filename)