diff --git a/milli/src/search/new/query_graph.rs b/milli/src/search/new/query_graph.rs
index 726a1460c..821c1a226 100644
--- a/milli/src/search/new/query_graph.rs
+++ b/milli/src/search/new/query_graph.rs
@@ -1,7 +1,8 @@
-use std::collections::HashSet;
 use std::fmt::Debug;
+use std::{collections::HashSet, fmt};
 
 use heed::RoTxn;
+use roaring::RoaringBitmap;
 
 use super::{
     db_cache::DatabaseCache,
@@ -19,21 +20,31 @@ pub enum QueryNode {
 
 #[derive(Debug, Clone)]
 pub struct Edges {
-    pub incoming: HashSet<usize>,
-    pub outgoing: HashSet<usize>,
+    // TODO: use a tiny bitset instead
+    // something like a simple Vec<u8> where most queries will see a vector of one element
+    pub predecessors: RoaringBitmap,
+    pub successors: RoaringBitmap,
+}
+
+#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)]
+pub struct NodeIndex(pub u32);
+impl fmt::Display for NodeIndex {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        fmt::Display::fmt(&self.0, f)
+    }
 }
 
 #[derive(Debug, Clone)]
 pub struct QueryGraph {
-    pub root_node: usize,
-    pub end_node: usize,
+    pub root_node: NodeIndex,
+    pub end_node: NodeIndex,
     pub nodes: Vec<QueryNode>,
     pub edges: Vec<Edges>,
 }
 
 fn _assert_sizes() {
     let _: [u8; 112] = [0; std::mem::size_of::<QueryNode>()];
-    let _: [u8; 96] = [0; std::mem::size_of::<Edges>()];
+    let _: [u8; 48] = [0; std::mem::size_of::<Edges>()];
 }
 
 impl Default for QueryGraph {
@@ -41,32 +52,32 @@ impl Default for QueryGraph {
     fn default() -> Self {
         let nodes = vec![QueryNode::Start, QueryNode::End];
         let edges = vec![
-            Edges { incoming: HashSet::new(), outgoing: HashSet::new() },
-            Edges { incoming: HashSet::new(), outgoing: HashSet::new() },
+            Edges { predecessors: RoaringBitmap::new(), successors: RoaringBitmap::new() },
+            Edges { predecessors: RoaringBitmap::new(), successors: RoaringBitmap::new() },
         ];
 
-        Self { root_node: 0, end_node: 1, nodes, edges }
+        Self { root_node: NodeIndex(0), end_node: NodeIndex(1), nodes, edges }
     }
 }
 
 impl QueryGraph {
-    fn connect_to_node(&mut self, from_nodes: &[usize], end_node: usize) {
+    fn connect_to_node(&mut self, from_nodes: &[NodeIndex], to_node: NodeIndex) {
         for &from_node in from_nodes {
-            self.edges[from_node].outgoing.insert(end_node);
-            self.edges[end_node].incoming.insert(from_node);
+            self.edges[from_node.0 as usize].successors.insert(to_node.0);
+            self.edges[to_node.0 as usize].predecessors.insert(from_node.0);
         }
     }
-    fn add_node(&mut self, from_nodes: &[usize], node: QueryNode) -> usize {
-        let new_node_idx = self.nodes.len();
+    fn add_node(&mut self, from_nodes: &[NodeIndex], node: QueryNode) -> NodeIndex {
+        let new_node_idx = self.nodes.len() as u32;
         self.nodes.push(node);
         self.edges.push(Edges {
-            incoming: from_nodes.iter().copied().collect(),
-            outgoing: HashSet::new(),
+            predecessors: from_nodes.iter().map(|x| x.0).collect(),
+            successors: RoaringBitmap::new(),
         });
         for from_node in from_nodes {
-            self.edges[*from_node].outgoing.insert(new_node_idx);
+            self.edges[from_node.0 as usize].successors.insert(new_node_idx);
         }
-        new_node_idx
+        NodeIndex(new_node_idx)
     }
 }
 
@@ -88,7 +99,7 @@ impl QueryGraph {
         let word_set = index.words_fst(txn)?;
         let mut graph = QueryGraph::default();
 
-        let (mut prev2, mut prev1, mut prev0): (Vec<usize>, Vec<usize>, Vec<usize>) =
+        let (mut prev2, mut prev1, mut prev0): (Vec<NodeIndex>, Vec<NodeIndex>, Vec<NodeIndex>) =
             (vec![], vec![], vec![graph.root_node]);
 
         // TODO: add all the word derivations found in the fst
@@ -162,38 +173,41 @@ impl QueryGraph {
 
         Ok(graph)
     }
-    pub fn remove_nodes(&mut self, nodes: &[usize]) {
+    pub fn remove_nodes(&mut self, nodes: &[NodeIndex]) {
         for &node in nodes {
-            self.nodes[node] = QueryNode::Deleted;
-            let edges = self.edges[node].clone();
-            for &pred in edges.incoming.iter() {
-                self.edges[pred].outgoing.remove(&node);
+            self.nodes[node.0 as usize] = QueryNode::Deleted;
+            let edges = self.edges[node.0 as usize].clone();
+            for pred in edges.predecessors.iter() {
+                self.edges[pred as usize].successors.remove(node.0);
             }
-            for succ in edges.outgoing {
-                self.edges[succ].incoming.remove(&node);
+            for succ in edges.successors {
+                self.edges[succ as usize].predecessors.remove(node.0);
             }
-            self.edges[node] = Edges { incoming: HashSet::new(), outgoing: HashSet::new() };
+            self.edges[node.0 as usize] =
+                Edges { predecessors: RoaringBitmap::new(), successors: RoaringBitmap::new() };
         }
     }
-    pub fn remove_nodes_keep_edges(&mut self, nodes: &[usize]) {
+    pub fn remove_nodes_keep_edges(&mut self, nodes: &[NodeIndex]) {
         for &node in nodes {
-            self.nodes[node] = QueryNode::Deleted;
-            let edges = self.edges[node].clone();
-            for &pred in edges.incoming.iter() {
-                self.edges[pred].outgoing.remove(&node);
-                self.edges[pred].outgoing.extend(edges.outgoing.iter());
+            self.nodes[node.0 as usize] = QueryNode::Deleted;
+            let edges = self.edges[node.0 as usize].clone();
+            for pred in edges.predecessors.iter() {
+                self.edges[pred as usize].successors.remove(node.0);
+                self.edges[pred as usize].successors |= &edges.successors;
             }
-            for succ in edges.outgoing {
-                self.edges[succ].incoming.remove(&node);
-                self.edges[succ].incoming.extend(edges.incoming.iter());
+            for succ in edges.successors {
+                self.edges[succ as usize].predecessors.remove(node.0);
+                self.edges[succ as usize].predecessors |= &edges.predecessors;
             }
-            self.edges[node] = Edges { incoming: HashSet::new(), outgoing: HashSet::new() };
+            self.edges[node.0 as usize] =
+                Edges { predecessors: RoaringBitmap::new(), successors: RoaringBitmap::new() };
         }
     }
     pub fn remove_words_at_position(&mut self, position: i8) {
         let mut nodes_to_remove_keeping_edges = vec![];
         let mut nodes_to_remove = vec![];
         for (node_idx, node) in self.nodes.iter().enumerate() {
+            let node_idx = NodeIndex(node_idx as u32);
             let QueryNode::Term(LocatedQueryTerm { value: _, positions }) = node else { continue };
             if positions.contains(&position) {
                 nodes_to_remove_keeping_edges.push(node_idx)
@@ -213,11 +227,11 @@ impl QueryGraph {
             let mut nodes_to_remove = vec![];
             for (node_idx, node) in self.nodes.iter().enumerate() {
                 if (!matches!(node, QueryNode::End | QueryNode::Deleted)
-                    && self.edges[node_idx].outgoing.is_empty())
+                    && self.edges[node_idx].successors.is_empty())
                     || (!matches!(node, QueryNode::Start | QueryNode::Deleted)
-                        && self.edges[node_idx].incoming.is_empty())
+                        && self.edges[node_idx].predecessors.is_empty())
                 {
-                    nodes_to_remove.push(node_idx);
+                    nodes_to_remove.push(NodeIndex(node_idx as u32));
                 }
             }
             if nodes_to_remove.is_empty() {
@@ -301,14 +315,14 @@ node [shape = "record"]
                 continue;
             }
             desc.push_str(&format!("{node} [label = {:?}]", &self.nodes[node],));
-            if node == self.root_node {
+            if node == self.root_node.0 as usize {
                 desc.push_str("[color = blue]");
-            } else if node == self.end_node {
+            } else if node == self.end_node.0 as usize {
                 desc.push_str("[color = red]");
             }
             desc.push_str(";\n");
 
-            for edge in self.edges[node].outgoing.iter() {
+            for edge in self.edges[node].successors.iter() {
                 desc.push_str(&format!("{node} -> {edge};\n"));
             }
             // for edge in self.edges[node].incoming.iter() {
diff --git a/milli/src/search/new/ranking_rule_graph/build.rs b/milli/src/search/new/ranking_rule_graph/build.rs
index 605fe82d1..45dda3c1f 100644
--- a/milli/src/search/new/ranking_rule_graph/build.rs
+++ b/milli/src/search/new/ranking_rule_graph/build.rs
@@ -1,10 +1,11 @@
 use std::collections::{BTreeSet, HashMap, HashSet};
 
 use heed::RoTxn;
+use roaring::RoaringBitmap;
 
 use super::{Edge, RankingRuleGraph, RankingRuleGraphTrait};
 use crate::new::db_cache::DatabaseCache;
-use crate::new::QueryGraph;
+use crate::new::{NodeIndex, QueryGraph};
 use crate::{Index, Result};
 
 impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
@@ -14,29 +15,38 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         db_cache: &mut DatabaseCache<'transaction>,
         query_graph: QueryGraph,
     ) -> Result<Self> {
-        let mut ranking_rule_graph = Self { query_graph, all_edges: vec![], node_edges: vec![] };
+        let mut ranking_rule_graph =
+            Self { query_graph, all_edges: vec![], node_edges: vec![], successors: vec![] };
 
         for (node_idx, node) in ranking_rule_graph.query_graph.nodes.iter().enumerate() {
-            ranking_rule_graph.node_edges.push(BTreeSet::new());
+            ranking_rule_graph.node_edges.push(RoaringBitmap::new());
+            ranking_rule_graph.successors.push(RoaringBitmap::new());
             let new_edges = ranking_rule_graph.node_edges.last_mut().unwrap();
+            let new_successors = ranking_rule_graph.successors.last_mut().unwrap();
 
             let Some(from_node_data) = G::build_visit_from_node(index, txn, db_cache, node)? else { continue };
 
-            for &successor_idx in ranking_rule_graph.query_graph.edges[node_idx].outgoing.iter() {
-                let to_node = &ranking_rule_graph.query_graph.nodes[successor_idx];
-                let Some(edges) = G::build_visit_to_node(index, txn, db_cache, to_node, &from_node_data)? else { continue };
+            for successor_idx in ranking_rule_graph.query_graph.edges[node_idx].successors.iter() {
+                let to_node = &ranking_rule_graph.query_graph.nodes[successor_idx as usize];
+                let mut edges =
+                    G::build_visit_to_node(index, txn, db_cache, to_node, &from_node_data)?;
+                if edges.is_empty() {
+                    continue;
+                }
+                edges.sort_by_key(|e| e.0);
                 for (cost, details) in edges {
                     ranking_rule_graph.all_edges.push(Some(Edge {
-                        from_node: node_idx,
-                        to_node: successor_idx,
+                        from_node: NodeIndex(node_idx as u32),
+                        to_node: NodeIndex(successor_idx),
                         cost,
                         details,
                     }));
-                    new_edges.insert(ranking_rule_graph.all_edges.len() - 1);
+                    new_edges.insert(ranking_rule_graph.all_edges.len() as u32 - 1);
+                    new_successors.insert(successor_idx);
                 }
             }
         }
-        ranking_rule_graph.simplify();
+        // ranking_rule_graph.simplify();
 
         Ok(ranking_rule_graph)
     }
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 3bd43fd6f..f1c1035a3 100644
--- a/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
+++ b/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
@@ -1,6 +1,9 @@
 use std::collections::{BTreeMap, HashSet};
 
 use itertools::Itertools;
+use roaring::RoaringBitmap;
+
+use crate::new::NodeIndex;
 
 use super::{
     empty_paths_cache::EmptyPathsCache, paths_map::PathsMap, Edge, EdgeIndex, RankingRuleGraph,
@@ -14,18 +17,11 @@ pub struct Path {
 }
 
 struct DijkstraState {
-    unvisited: HashSet<usize>, // should be a small bitset
-    distances: Vec<u64>,       // or binary heap (f64, usize)
+    unvisited: RoaringBitmap, // should be a small bitset?
+    distances: Vec<u64>,      // or binary heap, or btreemap? (f64, usize)
     edges: Vec<EdgeIndex>,
     edge_costs: Vec<u8>,
-    paths: Vec<Option<usize>>,
-}
-
-#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)]
-pub struct PathEdgeId<Id> {
-    pub from: usize,
-    pub to: usize,
-    pub id: Id,
+    paths: Vec<Option<NodeIndex>>,
 }
 
 pub struct KCheapestPathsState {
@@ -127,9 +123,10 @@ impl KCheapestPathsState {
             // 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].remove(&edge_index_to_remove.0);
+                let was_removed =
+                    graph.node_edges[spur_node.0 as usize].remove(edge_index_to_remove.0 as u32);
                 if was_removed {
-                    tmp_removed_edges.push(edge_index_to_remove.0);
+                    tmp_removed_edges.push(edge_index_to_remove.0 as u32);
                 }
             }
 
@@ -137,7 +134,7 @@ impl KCheapestPathsState {
             // 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].extend(tmp_removed_edges);
+            graph.node_edges[spur_node.0 as usize].extend(tmp_removed_edges);
 
             let Some(spur_path) = spur_path else { continue; };
             let total_cost = root_cost + spur_path.cost;
@@ -182,68 +179,73 @@ impl KCheapestPathsState {
 }
 
 impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
-    fn cheapest_path_to_end(&self, from: usize) -> Option<Path> {
+    fn cheapest_path_to_end(&self, from: NodeIndex) -> Option<Path> {
         let mut dijkstra = DijkstraState {
-            unvisited: (0..self.query_graph.nodes.len()).collect(),
+            unvisited: (0..self.query_graph.nodes.len() as u32).collect(),
             distances: vec![u64::MAX; self.query_graph.nodes.len()],
             edges: vec![EdgeIndex(usize::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] = 0;
+        dijkstra.distances[from.0 as usize] = 0;
 
-        // TODO: could use a binary heap here to store the distances
-        while let Some(&cur_node) =
-            dijkstra.unvisited.iter().min_by_key(|&&n| dijkstra.distances[n])
+        // 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];
+            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 {
+            if cur_node == self.query_graph.end_node.0 {
                 break;
             }
 
-            let succ_cur_node: HashSet<_> = self.node_edges[cur_node]
-                .iter()
-                .map(|e| self.all_edges[*e].as_ref().unwrap().to_node)
-                .collect();
+            // this is expensive, but shouldn't
+            // ideally I could quickly get a bitmap of all a node's successors
+            // then take the intersection with unvisited
+            let succ_cur_node: &RoaringBitmap = &self.successors[cur_node as usize];
+            // .iter()
+            // .map(|e| self.all_edges[e as usize].as_ref().unwrap().to_node.0)
+            // .collect();
             // TODO: this intersection may be slow but shouldn't be,
             // can use a bitmap intersection instead
-            let unvisited_succ_cur_node = succ_cur_node.intersection(&dijkstra.unvisited);
-            for &succ in unvisited_succ_cur_node {
-                let Some((cheapest_edge, cheapest_edge_cost)) = self.cheapest_edge(cur_node, succ) else {
+            let unvisited_succ_cur_node = succ_cur_node & &dijkstra.unvisited;
+            for succ in unvisited_succ_cur_node {
+                // cheapest_edge() is also potentially too expensive
+                let Some((cheapest_edge, cheapest_edge_cost)) = self.cheapest_edge(NodeIndex(cur_node), NodeIndex(succ)) else {
                     continue
                 };
 
                 // println!("cur node dist {cur_node_dist}");
-                let old_dist_succ = &mut dijkstra.distances[succ];
+                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] = cheapest_edge;
-                    dijkstra.edge_costs[succ] = cheapest_edge_cost;
-                    dijkstra.paths[succ] = Some(cur_node);
+                    dijkstra.edges[succ as usize] = cheapest_edge;
+                    dijkstra.edge_costs[succ as usize] = cheapest_edge_cost;
+                    dijkstra.paths[succ as usize] = Some(NodeIndex(cur_node));
                 }
             }
-            dijkstra.unvisited.remove(&cur_node);
+            dijkstra.unvisited.remove(cur_node);
         }
 
         let mut cur = self.query_graph.end_node;
         // let mut edge_costs = vec![];
         // let mut distances = vec![];
         let mut path_edges = vec![];
-        while let Some(n) = dijkstra.paths[cur] {
-            path_edges.push(dijkstra.edges[cur]);
+        while let Some(n) = dijkstra.paths[cur.0 as usize] {
+            path_edges.push(dijkstra.edges[cur.0 as usize]);
             cur = n;
         }
         path_edges.reverse();
-        Some(Path { edges: path_edges, cost: dijkstra.distances[self.query_graph.end_node] })
+        Some(Path {
+            edges: path_edges,
+            cost: dijkstra.distances[self.query_graph.end_node.0 as usize],
+        })
     }
 
-    // TODO: this implementation is VERY fragile, as we assume that the edges are ordered by cost
-    // already. Change it.
-    pub fn cheapest_edge(&self, cur_node: usize, succ: usize) -> Option<(EdgeIndex, u8)> {
+    pub fn cheapest_edge(&self, cur_node: NodeIndex, succ: NodeIndex) -> Option<(EdgeIndex, u8)> {
         self.visit_edges(cur_node, succ, |edge_idx, edge| {
             std::ops::ControlFlow::Break((edge_idx, edge.cost))
         })
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 301810847..0c9768f04 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
@@ -9,6 +9,12 @@ use crate::new::db_cache::DatabaseCache;
 use crate::new::BitmapOrAllRef;
 use crate::{Index, Result};
 
+// TODO: the cache should have a G::EdgeDetails as key
+// but then it means that we should have a quick way of
+// computing their hash and comparing them
+// which can be done...
+// by using a pointer (real, Rc, bumpalo, or in a vector)???
+
 pub struct EdgeDocidsCache<G: RankingRuleGraphTrait> {
     pub cache: HashMap<EdgeIndex, RoaringBitmap>,
 
diff --git a/milli/src/search/new/ranking_rule_graph/mod.rs b/milli/src/search/new/ranking_rule_graph/mod.rs
index 12f397df3..f7a312240 100644
--- a/milli/src/search/new/ranking_rule_graph/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/mod.rs
@@ -13,7 +13,7 @@ use heed::RoTxn;
 use roaring::RoaringBitmap;
 
 use super::db_cache::DatabaseCache;
-use super::{QueryGraph, QueryNode};
+use super::{NodeIndex, QueryGraph, QueryNode};
 use crate::{Index, Result};
 
 #[derive(Debug, Clone)]
@@ -24,8 +24,8 @@ pub enum EdgeDetails<E> {
 
 #[derive(Debug, Clone)]
 pub struct Edge<E> {
-    from_node: usize,
-    to_node: usize,
+    from_node: NodeIndex,
+    to_node: NodeIndex,
     cost: u8,
     details: EdgeDetails<E>,
 }
@@ -38,22 +38,20 @@ pub struct EdgePointer<'graph, E> {
 
 #[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)]
 pub struct EdgeIndex(pub usize);
-//  {
-//     // TODO: they could all be u16 instead
-//     // There may be a way to store all the edge indices in a u32 as well,
-//     // if the edges are in a vector
-//     // then we can store sets of edges in a bitmap efficiently
-//     pub from: usize,
-//     pub to: usize,
-//     pub edge_idx: usize,
-// }
 
 pub trait RankingRuleGraphTrait {
+    /// The details of an edge connecting two query nodes. These details
+    /// should be sufficient to compute the edge's cost and associated document ids
+    /// in [`compute_docids`](RankingRuleGraphTrait).
     type EdgeDetails: Sized;
+
     type BuildVisitedFromNode;
 
-    fn edge_details_dot_label(edge: &Self::EdgeDetails) -> String;
+    /// Return the label of the given edge details, to be used when visualising
+    /// the ranking rule graph using GraphViz.
+    fn graphviz_edge_details_label(edge: &Self::EdgeDetails) -> String;
 
+    /// Compute the document ids associated with the given edge.
     fn compute_docids<'transaction>(
         index: &Index,
         txn: &'transaction RoTxn,
@@ -61,6 +59,10 @@ pub trait RankingRuleGraphTrait {
         edge_details: &Self::EdgeDetails,
     ) -> Result<RoaringBitmap>;
 
+    /// Prepare to build the edges outgoing from `from_node`.
+    ///
+    /// This call is followed by zero, one or more calls to [`build_visit_to_node`](RankingRuleGraphTrait::build_visit_to_node),
+    /// which builds the actual edges.
     fn build_visit_from_node<'transaction>(
         index: &Index,
         txn: &'transaction RoTxn,
@@ -68,39 +70,59 @@ pub trait RankingRuleGraphTrait {
         from_node: &QueryNode,
     ) -> Result<Option<Self::BuildVisitedFromNode>>;
 
+    /// Return the cost and details of the edges going from the previously visited node
+    /// (with [`build_visit_from_node`](RankingRuleGraphTrait::build_visit_from_node)) to `to_node`.
     fn build_visit_to_node<'from_data, 'transaction: 'from_data>(
         index: &Index,
         txn: &'transaction RoTxn,
         db_cache: &mut DatabaseCache<'transaction>,
         to_node: &QueryNode,
         from_node_data: &'from_data Self::BuildVisitedFromNode,
-    ) -> Result<Option<Vec<(u8, EdgeDetails<Self::EdgeDetails>)>>>;
+    ) -> Result<Vec<(u8, EdgeDetails<Self::EdgeDetails>)>>;
 }
 
 pub struct RankingRuleGraph<G: RankingRuleGraphTrait> {
     pub query_graph: QueryGraph,
     // pub edges: Vec<HashMap<usize, Vec<Edge<G::EdgeDetails>>>>,
     pub all_edges: Vec<Option<Edge<G::EdgeDetails>>>,
-    pub node_edges: Vec<BTreeSet<usize>>,
+
+    pub node_edges: Vec<RoaringBitmap>,
+
+    pub successors: Vec<RoaringBitmap>,
+    // to get the edges between two nodes:
+    // 1. get node_outgoing_edges[from]
+    // 2. get node_incoming_edges[to]
+    // 3. take intersection betweem the two
+
+    // TODO: node edges could be different I guess
+    // something like:
+    // pub node_edges: Vec<BitSet>
+    // where each index is the result of:
+    // the successor index in the top 16 bits, the edge index in the bottom 16 bits
+
+    // TODO:
+    // node_successors?
+
     // pub removed_edges: HashSet<EdgeIndex>,
     // pub tmp_removed_edges: HashSet<EdgeIndex>,
 }
 impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
-    // NOTE: returns the edge even if it was removed
     pub fn get_edge(&self, edge_index: EdgeIndex) -> &Option<Edge<G::EdgeDetails>> {
         &self.all_edges[edge_index.0]
     }
+
+    // Visit all edges between the two given nodes in order of increasing cost.
     pub fn visit_edges<'graph, O>(
         &'graph self,
-        from: usize,
-        to: usize,
+        from: NodeIndex,
+        to: NodeIndex,
         mut visit: impl FnMut(EdgeIndex, &'graph Edge<G::EdgeDetails>) -> ControlFlow<O>,
     ) -> Option<O> {
-        let from_edges = &self.node_edges[from];
-        for &edge_idx in from_edges {
-            let edge = self.all_edges[edge_idx].as_ref().unwrap();
+        let from_edges = &self.node_edges[from.0 as usize];
+        for edge_idx in from_edges {
+            let edge = self.all_edges[edge_idx as usize].as_ref().unwrap();
             if edge.to_node == to {
-                let cf = visit(EdgeIndex(edge_idx), edge);
+                let cf = visit(EdgeIndex(edge_idx as usize), edge);
                 match cf {
                     ControlFlow::Continue(_) => continue,
                     ControlFlow::Break(o) => return Some(o),
@@ -113,54 +135,61 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
 
     fn remove_edge(&mut self, edge_index: EdgeIndex) {
         let edge_opt = &mut self.all_edges[edge_index.0];
-        let Some(Edge { from_node, to_node, cost, details }) = &edge_opt else { return };
-
-        let node_edges = &mut self.node_edges[*from_node];
-        node_edges.remove(&edge_index.0);
-
+        let Some(edge) = &edge_opt else { return };
+        let (from_node, to_node) = (edge.from_node, edge.to_node);
         *edge_opt = None;
-    }
-    pub fn remove_nodes(&mut self, nodes: &[usize]) {
-        for &node in nodes {
-            let edge_indices = &mut self.node_edges[node];
-            for edge_index in edge_indices.iter() {
-                self.all_edges[*edge_index] = None;
-            }
-            edge_indices.clear();
 
-            let preds = &self.query_graph.edges[node].incoming;
-            for pred in preds {
-                let edge_indices = &mut self.node_edges[*pred];
-                for edge_index in edge_indices.iter() {
-                    let edge_opt = &mut self.all_edges[*edge_index];
-                    let Some(edge) = edge_opt else { continue; };
-                    if edge.to_node == node {
-                        *edge_opt = None;
-                    }
-                }
-                panic!("remove nodes is incorrect at the moment");
-                edge_indices.clear();
-            }
-        }
-        self.query_graph.remove_nodes(nodes);
-    }
-    pub fn simplify(&mut self) {
-        loop {
-            let mut nodes_to_remove = vec![];
-            for (node_idx, node) in self.query_graph.nodes.iter().enumerate() {
-                if !matches!(node, QueryNode::End | QueryNode::Deleted)
-                    && self.node_edges[node_idx].is_empty()
-                {
-                    nodes_to_remove.push(node_idx);
-                }
-            }
-            if nodes_to_remove.is_empty() {
-                break;
-            } else {
-                self.remove_nodes(&nodes_to_remove);
-            }
+        let from_node_edges = &mut self.node_edges[from_node.0 as usize];
+        from_node_edges.remove(edge_index.0 as u32);
+
+        let mut new_successors_from_node = RoaringBitmap::new();
+        for edge in from_node_edges.iter() {
+            let Edge { to_node, .. } = &self.all_edges[edge as usize].as_ref().unwrap();
+            new_successors_from_node.insert(to_node.0);
         }
+        self.successors[from_node.0 as usize] = new_successors_from_node;
     }
+    // pub fn remove_nodes(&mut self, nodes: &[usize]) {
+    //     for &node in nodes {
+    //         let edge_indices = &mut self.node_edges[node];
+    //         for edge_index in edge_indices.iter() {
+    //             self.all_edges[*edge_index] = None;
+    //         }
+    //         edge_indices.clear();
+
+    //         let preds = &self.query_graph.edges[node].incoming;
+    //         for pred in preds {
+    //             let edge_indices = &mut self.node_edges[*pred];
+    //             for edge_index in edge_indices.iter() {
+    //                 let edge_opt = &mut self.all_edges[*edge_index];
+    //                 let Some(edge) = edge_opt else { continue; };
+    //                 if edge.to_node == node {
+    //                     *edge_opt = None;
+    //                 }
+    //             }
+    //             panic!("remove nodes is incorrect at the moment");
+    //             edge_indices.clear();
+    //         }
+    //     }
+    //     self.query_graph.remove_nodes(nodes);
+    // }
+    // pub fn simplify(&mut self) {
+    //     loop {
+    //         let mut nodes_to_remove = vec![];
+    //         for (node_idx, node) in self.query_graph.nodes.iter().enumerate() {
+    //             if !matches!(node, QueryNode::End | QueryNode::Deleted)
+    //                 && self.node_edges[node_idx].is_empty()
+    //             {
+    //                 nodes_to_remove.push(node_idx);
+    //             }
+    //         }
+    //         if nodes_to_remove.is_empty() {
+    //             break;
+    //         } else {
+    //             self.remove_nodes(&nodes_to_remove);
+    //         }
+    //     }
+    // }
     // fn is_removed_edge(&self, edge: EdgeIndex) -> bool {
     //     self.removed_edges.contains(&edge) || self.tmp_removed_edges.contains(&edge)
     // }
@@ -174,9 +203,9 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
                 continue;
             }
             desc.push_str(&format!("{node_idx} [label = {:?}]", node));
-            if node_idx == self.query_graph.root_node {
+            if node_idx == self.query_graph.root_node.0 as usize {
                 desc.push_str("[color = blue]");
-            } else if node_idx == self.query_graph.end_node {
+            } else if node_idx == self.query_graph.end_node.0 as usize {
                 desc.push_str("[color = red]");
             }
             desc.push_str(";\n");
@@ -195,7 +224,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
                     desc.push_str(&format!(
                         "{from_node} -> {to_node} [label = \"cost {cost} {edge_label}\"];\n",
                         cost = edge.cost,
-                        edge_label = G::edge_details_dot_label(details)
+                        edge_label = G::graphviz_edge_details_label(details)
                     ));
                 }
             }
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 589a1a52f..b1e4bb451 100644
--- a/milli/src/search/new/ranking_rule_graph/paths_map.rs
+++ b/milli/src/search/new/ranking_rule_graph/paths_map.rs
@@ -235,9 +235,9 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
                 continue;
             }
             desc.push_str(&format!("{node_idx} [label = {:?}]", node));
-            if node_idx == self.query_graph.root_node {
+            if node_idx == self.query_graph.root_node.0 as usize {
                 desc.push_str("[color = blue]");
-            } else if node_idx == self.query_graph.end_node {
+            } else if node_idx == self.query_graph.end_node.0 as usize {
                 desc.push_str("[color = red]");
             }
             desc.push_str(";\n");
@@ -262,7 +262,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
                     desc.push_str(&format!(
                         "{from_node} -> {to_node} [label = \"cost {cost} {edge_label}\", color = {color}];\n",
                         cost = edge.cost,
-                        edge_label = G::edge_details_dot_label(details),
+                        edge_label = G::graphviz_edge_details_label(details),
                     ));
                 }
             }
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 07ec3bb5e..7149f8bf6 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/build.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/build.rs
@@ -51,11 +51,11 @@ pub fn visit_to_node<'transaction, 'from_data>(
     db_cache: &mut DatabaseCache<'transaction>,
     to_node: &QueryNode,
     from_node_data: &'from_data (WordDerivations, i8),
-) -> Result<Option<Vec<(u8, EdgeDetails<ProximityEdge>)>>> {
+) -> Result<Vec<(u8, EdgeDetails<ProximityEdge>)>> {
     let (derivations1, pos1) = from_node_data;
     let term2 = match &to_node {
-        QueryNode::End => return Ok(Some(vec![(0, EdgeDetails::Unconditional)])),
-        QueryNode::Deleted | QueryNode::Start => return Ok(None),
+        QueryNode::End => return Ok(vec![(0, EdgeDetails::Unconditional)]),
+        QueryNode::Deleted | QueryNode::Start => return Ok(vec![]),
         QueryNode::Term(term) => term,
     };
     let LocatedQueryTerm { value: value2, positions: pos2 } = term2;
@@ -86,7 +86,7 @@ pub fn visit_to_node<'transaction, 'from_data>(
         // We want to effectively ignore this pair of terms
         // Unconditionally walk through the edge without computing the docids
         // But also what should the cost be?
-        return Ok(Some(vec![(0, EdgeDetails::Unconditional)]));
+        return Ok(vec![(0, EdgeDetails::Unconditional)]);
     }
 
     let updb1 = derivations1.use_prefix_db;
@@ -161,5 +161,5 @@ pub fn visit_to_node<'transaction, 'from_data>(
         })
         .collect::<Vec<_>>();
     new_edges.push((8 + (ngram_len2 - 1) as u8, EdgeDetails::Unconditional));
-    Ok(Some(new_edges))
+    Ok(new_edges)
 }
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 199a5eb4a..e4905ead9 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
@@ -26,7 +26,7 @@ impl RankingRuleGraphTrait for ProximityGraph {
     type EdgeDetails = ProximityEdge;
     type BuildVisitedFromNode = (WordDerivations, i8);
 
-    fn edge_details_dot_label(edge: &Self::EdgeDetails) -> String {
+    fn graphviz_edge_details_label(edge: &Self::EdgeDetails) -> String {
         let ProximityEdge { pairs, proximity } = edge;
         format!(", prox {proximity}, {} pairs", pairs.len())
     }
@@ -55,7 +55,7 @@ impl RankingRuleGraphTrait for ProximityGraph {
         db_cache: &mut DatabaseCache<'transaction>,
         to_node: &QueryNode,
         from_node_data: &'from_data Self::BuildVisitedFromNode,
-    ) -> Result<Option<Vec<(u8, EdgeDetails<Self::EdgeDetails>)>>> {
+    ) -> Result<Vec<(u8, EdgeDetails<Self::EdgeDetails>)>> {
         build::visit_to_node(index, txn, db_cache, to_node, from_node_data)
     }
 }
diff --git a/milli/src/search/new/ranking_rules.rs b/milli/src/search/new/ranking_rules.rs
index ce883ad6a..b980c1dc4 100644
--- a/milli/src/search/new/ranking_rules.rs
+++ b/milli/src/search/new/ranking_rules.rs
@@ -36,15 +36,17 @@ impl<'transaction, Query> RankingRuleOutputIter<'transaction, Query>
 }
 
 pub trait RankingRuleQueryTrait: Sized + Clone + 'static {}
+
 #[derive(Clone)]
 pub struct PlaceholderQuery;
 impl RankingRuleQueryTrait for PlaceholderQuery {}
 impl RankingRuleQueryTrait for QueryGraph {}
 
 pub trait RankingRule<'transaction, Query: RankingRuleQueryTrait> {
-    // TODO: add an update_candidates function to deal with distinct
-    // attributes?
-
+    /// Prepare the ranking rule such that it can start iterating over its
+    /// buckets using [`next_bucket`](RankingRule::next_bucket).
+    ///
+    /// The given universe is the universe that will be given to [`next_bucket`](RankingRule::next_bucket).
     fn start_iteration(
         &mut self,
         index: &Index,
@@ -54,6 +56,13 @@ pub trait RankingRule<'transaction, Query: RankingRuleQueryTrait> {
         query: &Query,
     ) -> Result<()>;
 
+    /// Return the next bucket of this ranking rule.
+    ///
+    /// The returned candidates MUST be a subset of the given universe.
+    ///
+    /// The universe given as argument is either:
+    /// - a subset of the universe given to the previous call to [`next_bucket`](RankingRule::next_bucket); OR
+    /// - the universe given to [`start_iteration`](RankingRule::start_iteration)
     fn next_bucket(
         &mut self,
         index: &Index,
@@ -62,6 +71,8 @@ pub trait RankingRule<'transaction, Query: RankingRuleQueryTrait> {
         universe: &RoaringBitmap,
     ) -> Result<Option<RankingRuleOutput<Query>>>;
 
+    /// Finish iterating over the buckets, which yields control to the parent ranking rule
+    /// The next call to this ranking rule, if any, will be [`start_iteration`](RankingRule::start_iteration).
     fn end_iteration(
         &mut self,
         index: &Index,
@@ -72,7 +83,7 @@ pub trait RankingRule<'transaction, Query: RankingRuleQueryTrait> {
 
 #[derive(Debug)]
 pub struct RankingRuleOutput<Q> {
-    /// The query tree that must be used by the child ranking rule to fetch candidates.
+    /// The query corresponding to the current bucket for the child ranking rule
     pub query: Q,
     /// The allowed candidates for the child ranking rule
     pub candidates: RoaringBitmap,
@@ -151,7 +162,6 @@ pub fn execute_search<'transaction>(
     let ranking_rules_len = ranking_rules.len();
     ranking_rules[0].start_iteration(index, txn, db_cache, universe, query_graph)?;
 
-    // TODO: parent_candidates could be used only during debugging?
     let mut candidates = vec![RoaringBitmap::default(); ranking_rules_len];
     candidates[0] = universe.clone();
 
@@ -296,43 +306,43 @@ 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();
+        loop {
+            let start = Instant::now();
 
-        //     let mut db_cache = DatabaseCache::default();
+            let mut db_cache = DatabaseCache::default();
 
-        //     let query_graph = make_query_graph(
-        //         &index,
-        //         &txn,
-        //         &mut db_cache,
-        //         "released from prison by the government",
-        //     )
-        //     .unwrap();
-        //     // println!("{}", query_graph.graphviz());
+            let query_graph = make_query_graph(
+                &index,
+                &txn,
+                &mut db_cache,
+                "released from prison by the government",
+            )
+            .unwrap();
+            // println!("{}", query_graph.graphviz());
 
-        //     // TODO: filters + maybe distinct attributes?
-        //     let universe = get_start_universe(
-        //         &index,
-        //         &txn,
-        //         &mut db_cache,
-        //         &query_graph,
-        //         TermsMatchingStrategy::Last,
-        //     )
-        //     .unwrap();
-        //     // println!("universe: {universe:?}");
+            // TODO: filters + maybe distinct attributes?
+            let universe = get_start_universe(
+                &index,
+                &txn,
+                &mut db_cache,
+                &query_graph,
+                TermsMatchingStrategy::Last,
+            )
+            .unwrap();
+            // println!("universe: {universe:?}");
 
-        //     let results = execute_search(
-        //         &index,
-        //         &txn,
-        //         &mut db_cache,
-        //         &universe,
-        //         &query_graph, /*  0, 20 */
-        //     )
-        //     .unwrap();
+            let results = execute_search(
+                &index,
+                &txn,
+                &mut db_cache,
+                &universe,
+                &query_graph, /*  0, 20 */
+            )
+            .unwrap();
 
-        //     let elapsed = start.elapsed();
-        //     println!("{}us: {results:?}", elapsed.as_micros());
-        // }
+            let elapsed = start.elapsed();
+            println!("{}us: {results:?}", elapsed.as_micros());
+        }
         let start = Instant::now();
 
         let mut db_cache = DatabaseCache::default();
@@ -388,7 +398,7 @@ mod tests {
         let mut s = Search::new(&txn, &index);
         s.query("released from prison by the government");
         s.terms_matching_strategy(TermsMatchingStrategy::Last);
-        // s.criterion_implementation_strategy(crate::CriterionImplementationStrategy::OnlySetBased);
+        s.criterion_implementation_strategy(crate::CriterionImplementationStrategy::OnlySetBased);
         let docs = s.execute().unwrap();
 
         let elapsed = start.elapsed();
@@ -431,7 +441,7 @@ mod tests {
         builder.execute(|_| (), || false).unwrap();
     }
 
-    // #[test]
+    #[test]
     fn _index_movies() {
         let mut options = EnvOpenOptions::new();
         options.map_size(100 * 1024 * 1024 * 1024); // 100 GB
@@ -446,20 +456,14 @@ mod tests {
 
         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_criteria(vec![Criterion::Words]);
-
-        // let sortable_fields = sortable_fields.iter().map(|s| s.to_string()).collect();
-        // builder.set_sortable_fields(sortable_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]);
         builder.execute(|_| (), || false).unwrap();
 
         let config = IndexerConfig::default();
diff --git a/milli/src/search/new/resolve_query_graph.rs b/milli/src/search/new/resolve_query_graph.rs
index 748524492..8bc56bb23 100644
--- a/milli/src/search/new/resolve_query_graph.rs
+++ b/milli/src/search/new/resolve_query_graph.rs
@@ -4,11 +4,12 @@ use std::collections::{HashMap, HashSet, VecDeque};
 
 use super::db_cache::DatabaseCache;
 use super::query_term::{QueryTerm, WordDerivations};
-use super::QueryGraph;
+use super::{NodeIndex, QueryGraph};
 use crate::{Index, Result, RoaringBitmapCodec};
 
 // TODO: manual performance metrics: access to DB, bitmap deserializations/operations, etc.
 
+// TODO: reuse NodeDocidsCache in between calls to resolve_query_graph
 #[derive(Default)]
 pub struct NodeDocIdsCache {
     pub cache: HashMap<usize, RoaringBitmap>,
@@ -26,7 +27,7 @@ pub fn resolve_query_graph<'transaction>(
 
     // resolve_query_graph_rec(index, txn, q, q.root_node, &mut docids, &mut cache)?;
 
-    let mut nodes_resolved = HashSet::new();
+    let mut nodes_resolved = RoaringBitmap::new();
     // TODO: should be given as an argument and kept between invocations of resolve query graph
     let mut nodes_docids = vec![RoaringBitmap::new(); q.nodes.len()];
 
@@ -34,16 +35,16 @@ pub fn resolve_query_graph<'transaction>(
     next_nodes_to_visit.push_front(q.root_node);
 
     while let Some(node) = next_nodes_to_visit.pop_front() {
-        let predecessors = &q.edges[node].incoming;
+        let predecessors = &q.edges[node.0 as usize].predecessors;
         if !predecessors.is_subset(&nodes_resolved) {
             next_nodes_to_visit.push_back(node);
             continue;
         }
         // Take union of all predecessors
-        let predecessors_iter = predecessors.iter().map(|p| &nodes_docids[*p]);
+        let predecessors_iter = predecessors.iter().map(|p| &nodes_docids[p as usize]);
         let predecessors_docids = MultiOps::union(predecessors_iter);
 
-        let n = &q.nodes[node];
+        let n = &q.nodes[node.0 as usize];
         // println!("resolving {node} {n:?}, predecessors: {predecessors:?}, their docids: {predecessors_docids:?}");
         let node_docids = match n {
             super::QueryNode::Term(located_term) => {
@@ -95,18 +96,18 @@ pub fn resolve_query_graph<'transaction>(
                 return Ok(predecessors_docids);
             }
         };
-        nodes_resolved.insert(node);
-        nodes_docids[node] = node_docids;
+        nodes_resolved.insert(node.0);
+        nodes_docids[node.0 as usize] = node_docids;
 
-        for &succ in q.edges[node].outgoing.iter() {
-            if !next_nodes_to_visit.contains(&succ) && !nodes_resolved.contains(&succ) {
-                next_nodes_to_visit.push_back(succ);
+        for succ in q.edges[node.0 as usize].successors.iter() {
+            if !next_nodes_to_visit.contains(&NodeIndex(succ)) && !nodes_resolved.contains(succ) {
+                next_nodes_to_visit.push_back(NodeIndex(succ));
             }
         }
         // This is currently slow but could easily be implemented very efficiently
-        for &prec in q.edges[node].incoming.iter() {
-            if q.edges[prec].outgoing.is_subset(&nodes_resolved) {
-                nodes_docids[prec].clear();
+        for prec in q.edges[node.0 as usize].predecessors.iter() {
+            if q.edges[prec as usize].successors.is_subset(&nodes_resolved) {
+                nodes_docids[prec as usize].clear();
             }
         }
         // println!("cached docids: {nodes_docids:?}");