fix: added path compression to UnionFind (#734)

* fix: implemented path compression in Find & removed unnecessary return value of Union

* feat: added a few test cases

* fix: modified kruskal implementation to conform to the updated Union method

* fix: changed to pointer receivers

---------

Co-authored-by: Rak Laptudirm <[email protected]>

Author: soondubu137

graph/kruskal.go

@@ -42,7 +42,7 @@ func KruskalMST(n int, edges []Edge) ([]Edge, int) {
 			// Add the weight of the edge to the total cost
 			cost += edge.Weight
 			// Merge the sets containing the start and end vertices of the current edge
-			u = u.Union(int(edge.Start), int(edge.End))
+			u.Union(int(edge.Start), int(edge.End))
 		}
 	}
 

graph/unionfind.go

@@ -3,12 +3,13 @@
 // is used to efficiently maintain connected components in a graph that undergoes dynamic changes,
 // such as edges being added or removed over time
 // Worst Case Time Complexity: The time complexity of find operation is nearly constant or
-//O(α(n)), where where α(n) is the inverse Ackermann function
+//O(α(n)), where α(n) is the inverse Ackermann function
 // practically, this is a very slowly growing function making the time complexity for find
 //operation nearly constant.
 // The time complexity of the union operation is also nearly constant or O(α(n))
 // Worst Case Space Complexity: O(n), where n is the number of nodes or element in the structure
 // Reference: https://www.scaler.com/topics/data-structures/disjoint-set/
+// https://en.wikipedia.org/wiki/Disjoint-set_data_structure
 // Author: Mugdha Behere[https://github.com/MugdhaBehere]
 // see: unionfind.go, unionfind_test.go
 
@@ -17,43 +18,45 @@ package graph
 // Defining the union-find data structure
 type UnionFind struct {
 	parent []int
-	size   []int
+	rank   []int
 }
 
 // Initialise a new union find data structure with s nodes
 func NewUnionFind(s int) UnionFind {
 	parent := make([]int, s)
-	size := make([]int, s)
-	for k := 0; k < s; k++ {
-		parent[k] = k
-		size[k] = 1
+	rank := make([]int, s)
+	for i := 0; i < s; i++ {
+		parent[i] = i
+		rank[i] = 1
 	}
-	return UnionFind{parent, size}
+	return UnionFind{parent, rank}
 }
 
-// to find the root of the set to which the given element belongs, the Find function serves the purpose
-func (u UnionFind) Find(q int) int {
-	for q != u.parent[q] {
-		q = u.parent[q]
+// Find finds the root of the set to which the given element belongs.
+// It performs path compression to make future Find operations faster.
+func (u *UnionFind) Find(q int) int {
+	if q != u.parent[q] {
+		u.parent[q] = u.Find(u.parent[q])
 	}
-	return q
+	return u.parent[q]
 }
 
-// to merge two sets to which the given elements belong, the Union function serves the purpose
-func (u UnionFind) Union(a, b int) UnionFind {
-	rootP := u.Find(a)
-	rootQ := u.Find(b)
+// Union merges the sets, if not already merged, to which the given elements belong.
+// It performs union by rank to keep the tree as flat as possible.
+func (u *UnionFind) Union(p, q int) {
+	rootP := u.Find(p)
+	rootQ := u.Find(q)
 
 	if rootP == rootQ {
-		return u
+		return
 	}
 
-	if u.size[rootP] < u.size[rootQ] {
+	if u.rank[rootP] < u.rank[rootQ] {
 		u.parent[rootP] = rootQ
-		u.size[rootQ] += u.size[rootP]
+	} else if u.rank[rootP] > u.rank[rootQ] {
+		u.parent[rootQ] = rootP
 	} else {
 		u.parent[rootQ] = rootP
-		u.size[rootP] += u.size[rootQ]
+		u.rank[rootP]++
 	}
-	return u
 }

graph/unionfind_test.go

@@ -8,10 +8,10 @@ func TestUnionFind(t *testing.T) {
 	u := NewUnionFind(10) // Creating a Union-Find data structure with 10 elements
 
 	//union operations
-	u = u.Union(0, 1)
-	u = u.Union(2, 3)
-	u = u.Union(4, 5)
-	u = u.Union(6, 7)
+	u.Union(0, 1)
+	u.Union(2, 3)
+	u.Union(4, 5)
+	u.Union(6, 7)
 
 	// Testing the parent of specific elements
 	t.Run("Test Find", func(t *testing.T) {
@@ -20,12 +20,21 @@ func TestUnionFind(t *testing.T) {
 		}
 	})
 
-	u = u.Union(1, 5) // Additional union operation
-	u = u.Union(3, 7) // Additional union operation
+	u.Union(1, 5) // Additional union operation
+	u.Union(3, 7) // Additional union operation
 
 	// Testing the parent of specific elements after more union operations
 	t.Run("Test Find after Union", func(t *testing.T) {
-		if u.Find(0) != u.Find(5) || u.Find(2) != u.Find(7) {
+		if u.Find(0) != u.Find(5) || u.Find(1) != u.Find(4) || u.Find(2) != u.Find(7) || u.Find(3) != u.Find(6) {
+			t.Error("Union operation not functioning correctly")
+		}
+	})
+
+	u.Union(3, 7) // Repeated union operation
+
+	// Testing that repeated union operations are idempotent
+	t.Run("Test Find after repeated Union", func(t *testing.T) {
+		if u.Find(2) != u.Find(6) || u.Find(2) != u.Find(7) || u.Find(3) != u.Find(6) || u.Find(3) != u.Find(7) {
 			t.Error("Union operation not functioning correctly")
 		}
 	})