diff --git a/src/iop/witness.rs b/src/iop/witness.rs
index 8a0f2652..3e884891 100644
--- a/src/iop/witness.rs
+++ b/src/iop/witness.rs
@@ -190,7 +190,7 @@ impl<F: Field> Witness<F> for PartialWitness<F> {
 /// The value of a target is defined to be the value of its root in the forest.
 #[derive(Clone)]
 pub struct PartitionWitness<F: Field> {
-    pub forest: Vec<ForestNode<Target, F>>,
+    pub forest: Vec<ForestNode<F>>,
     pub num_wires: usize,
     pub num_routed_wires: usize,
     pub degree: usize,
diff --git a/src/plonk/circuit_builder.rs b/src/plonk/circuit_builder.rs
index 415c011e..db9e855b 100644
--- a/src/plonk/circuit_builder.rs
+++ b/src/plonk/circuit_builder.rs
@@ -484,6 +484,8 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
             partition_witness.merge(a, b);
         }
 
+        partition_witness.compress_paths();
+
         let wire_partition = partition_witness.wire_partition();
         (
             wire_partition.get_sigma_polys(degree_log, k_is, subgroup),
diff --git a/src/plonk/permutation_argument.rs b/src/plonk/permutation_argument.rs
index 3d7d68b8..761c1bd8 100644
--- a/src/plonk/permutation_argument.rs
+++ b/src/plonk/permutation_argument.rs
@@ -11,11 +11,9 @@ use crate::polynomial::polynomial::PolynomialValues;
 
 /// Node in the Disjoint Set Forest.
 #[derive(Debug, Copy, Clone, Eq, PartialEq)]
-pub struct ForestNode<T: Debug + Copy + Eq + PartialEq, V: Field> {
-    pub t: T,
+pub struct ForestNode<V: Field> {
     pub parent: usize,
     pub size: usize,
-    pub index: usize,
     pub value: Option<V>,
 }
 
@@ -40,69 +38,70 @@ impl<F: Field> PartitionWitness<F> {
         let index = self.forest.len();
         debug_assert_eq!(self.target_index(t), index);
         self.forest.push(ForestNode {
-            t,
             parent: index,
             size: 1,
-            index,
             value: None,
         });
     }
 
     /// Path compression method, see https://en.wikipedia.org/wiki/Disjoint-set_data_structure#Finding_set_representatives.
-    pub fn find(&mut self, x: ForestNode<Target, F>) -> ForestNode<Target, F> {
-        if x.parent != x.index {
-            let root = self.find(self.forest[x.parent]);
-            self.forest[x.index].parent = root.index;
-            root
+    pub fn find(&mut self, x_index: usize) -> usize {
+        let x = self.forest[x_index];
+        if x.parent != x_index {
+            let root_index = self.find(x.parent);
+            self.forest[x_index].parent = root_index;
+            root_index
         } else {
-            x
+            x_index
         }
     }
 
     /// Merge two sets.
     pub fn merge(&mut self, tx: Target, ty: Target) {
-        let mut x = self.forest[self.target_index(tx)];
-        let mut y = self.forest[self.target_index(ty)];
+        let x_index = self.find(self.target_index(tx));
+        let y_index = self.find(self.target_index(ty));
 
-        x = self.find(x);
-        y = self.find(y);
-
-        if x == y {
+        if x_index == y_index {
             return;
         }
 
+        let mut x = self.forest[x_index];
+        let mut y = self.forest[y_index];
+
         if x.size >= y.size {
-            y.parent = x.index;
+            y.parent = x_index;
             x.size += y.size;
         } else {
-            x.parent = y.index;
+            x.parent = y_index;
             y.size += x.size;
         }
 
-        self.forest[x.index] = x;
-        self.forest[y.index] = y;
+        self.forest[x_index] = x;
+        self.forest[y_index] = y;
+    }
+
+    /// Compress all paths. After calling this, every `parent` value will point to the node's
+    /// representative.
+    pub(crate) fn compress_paths(&mut self) {
+        for i in 0..self.forest.len() {
+            self.find(i);
+        }
     }
 
     pub fn wire_partition(&mut self) -> WirePartition {
         let mut partition = HashMap::<_, Vec<_>>::new();
+
+        // Here we keep just the Wire targets, filtering out everything else.
         for gate in 0..self.degree {
             for input in 0..self.num_routed_wires {
                 let w = Wire { gate, input };
                 let t = Target::Wire(w);
                 let x = self.forest[self.target_index(t)];
-                partition.entry(self.find(x).t).or_default().push(w);
+                partition.entry(x.parent).or_default().push(w);
             }
         }
-        // I'm not 100% sure this loop is needed, but I'm afraid removing it might lead to subtle bugs.
-        for index in 0..self.forest.len() - self.degree * self.num_wires {
-            let t = Target::VirtualTarget { index };
-            let x = self.forest[self.target_index(t)];
-            self.find(x);
-        }
-
-        // Here we keep just the Wire targets, filtering out everything else.
-        let partition = partition.into_values().collect::<Vec<_>>();
 
+        let partition = partition.into_values().collect();
         WirePartition { partition }
     }
 }