CS2420: Lecture 42 Vladimir Kulyukin Computer Science Department Utah State University

Outline Graph Algorithms (Chapter 9)

Initial Graph BC A D F E

Kruskal’s Algorithm: Basic Insight A cycle is created if and only if the new edge connects two vertices already connected by a path, i.e., belonging to the same tree.

Kruskal’s: Basic Ideas Initially, all vertices are in separate trees. Subsequently, when an edge (u, v) is examined, we look for the tree containing u and the tree containing v. If the trees are NOT the same, we unite them into a larger tree containing both u, v and (u, v).

Kruskal’s Algorithm: UnionFind Kruskal’s Algorithm can be implemented with the UnionFind data structure. The UnionFind data structure was designed to solve the problem of partitioning a given set into a collection of disjoint subsets. The set partitioning problem has many applications in data mining, information retrieval, molecular biology, bioinformatics, etc.

Disjoint Subsets of a Set Let S be a set of n elements. Partition S into a collection of k disjoint subsets S 1, S 2, S 3,..., S k.

Disjoint Subsets of a Set Set S of n elements

Disjoint Subsets of a Set Partitioning of S into 5 disjoint subsets

Example: Partition S into 6 Disjoint Subsets S = {1, 2, 3, 4, 5, 6}, k = 6 –S 1 = {1} –S 2 = {2} –S 3 = {3} –S 4 = {4} –S 5 = {5} –S 6 = {6}

Example: Partition S into 5 Disjoint Subsets S = {1, 2, 3, 4, 5, 6}, k = 5 –S 1 = {1, 2} –S 2 = {3} –S 3 = {4} –S 4 = {5} –S 5 = {6}

UnionFind Data Structure: Operations The UnionFind data structure has the following operations: –make a set out of a single element (singleton). –find a subset that contains some element x. –compute the union of two disjoint subsets, the first of which contains x and the second of which contains y.

UnionFind: Operations MakeSet(x) - creates a one-element set {x}. Find(x) - returns a subset containing x. Union(x, y) - constructs the union of the disjoint sets S x and S y such that S x contains x and S y contains y. S x U S y replaces both S x and S y in the collection of subsets.

MakeSet: Example S = {1, 2, 3, 4, 5, 6}. MakeSet(1); MakeSet(2); MakeSet(3); MakeSet(4); MakeSet(5); MakeSet(6); The above sequence of MakeSet operations gives us: {{1}, {2}, {3}, {4}, {5}, {6}}.

Union: Examples {{1}, {2}, {3}, {4}, {5}, {6}} Union(1, 4)  {{1, 4}, {2}, {3}, {5}, {6}} Union(5, 2)  {{1, 4}, {5, 2}, {3}, {6}}

Union: Examples {{1, 4}, {5, 2}, {3}, {6}} Union(4, 5)  {{1, 4, 5, 2}, {3}, {6}} Union(3, 6)  {{1, 4, 5, 2}, {3, 6}}

UnionFind: Implementation Use one element of each disjoint set as a representative. Two implementation alternatives: –QuickFind: Find = O(1); Union = O(N). –QuickUnion: Union = O(1); Find = O(N).

QuickFind Two data structures: –An array of representatives; this array maps each element into its representative. –An array of subsets; this array contains each subset implemented as a linked list.

{{1}, {2}, {3}, {4}, {5}, {6}} Representatives Subsets

Union(1, 4) => {{1, 4}, {2}, {3}, {5}, {6}} Representatives Subsets Null

Union(5, 2) => {{1, 4}, {5, 2}, {3}, {6}} Representatives Subsets Null

Union(4, 5) => {{1, 4, 5, 2}, {3}, {6}} Representatives Subsets Null

union(3, 6) => {{1, 4, 5, 2}, {3, 6}} Representatives Subsets Null

QuickFind: Asymptotic Analysis MakeSet(x) = O(1) Why? All we need to do is to create a list and add x to it.

QuickFind: Asymptotic Analysis Find(x) = O(1) Why? All we need to do is look up x’s representative and then return the list to which the representative points.

QuickFind: Asymptotic Analysis Union(x, y) = O(n) Here is what we need to do to compute Union(x, y): –Do Find(x) and Find(y); // O(1) –Append Find(y) to the end of Find(x); // O(1) –Set the y-subset in the subsets array to NULL; // O(1) –Update the representatives for each element in Find(y); // O(N).

QuickFind: Asymptotic Analysis Consider the following sequence of unions: –union(2, 1), union(3, 2), union(4, 3),..., union(n, n-1). The first union requires 1 step, the second - 2 steps, the third - 3 steps,..., the n-th - n-1 steps. The sequence of n unions is asymptotically quadratic: (n - 1) = O(n 2 ).

QuickFind: Optimizations When performing the union operation, always append the shorter list to the longer one (union by size). The worst case run time of any legitimate sequence of unions is O(nlogn). The worst case run time of a sequence of n-1 unions and m finds is O(nlogn + m).

QuickUnion Two data structures: –An array of nodes, each containing exactly one element. –An array of trees, each containing one subset. –The root of each tree is the representative for the subset represented by that tree. –Each node has a pointer to its parent so that we can get to the root.

{{1}, {2}, {3}, {4}, {5}, {6}} Nodes Trees

Union(1, 4) => {{1, 4}, {2}, {3}, {5}, {6}} Nodes Trees Null

Union(5, 2) => {{1, 4}, {5, 2}, {3}, {6}} Nodes Trees Null

Union(4, 5) => {{1, 4, 5, 2}, {3}, {6}} Nodes Trees Null

Union(3, 6) => {{1, 4, 5, 2}, {3, 6}} Nodes Trees Null

Union(5, 6) => {1, 4, 5, 2, 3, 6} Nodes Trees Null

Quick Union: Asymptotic Analysis MakeSet(x) = O(1) Why? All we need to do is to create a tree node and add x to it.

QuickUnion: Asymptotic Analysis Find(x) = O(n) Why? We need to look up x’s node and then chase the upward pointers to the root of the tree that contains x.

QuickUnion: Asymptotic Analysis Union(x, y) = O(1) Why? –Do Find(x) and Find(y); // O(1) –Attach the root of Find(y) to the root of Find(x); // O(1) –Set the pointer in the tree array that used to point to Find(y) to NULL; // O(1)

QuickUnion: Optimization We can optimize the union operation by always attaching the root of a larger tree to the root of the smaller tree. Two alternatives: –Union by size - the size of the tree is the number of nodes in the tree. –Union by rank - the rank of the tree is the tree’s height.

QuickUnion: Optimization If union by size or union by rank is used, the height of the tree is logn. The time efficiency of a sequence of n-1 unions and m finds is O(n + mlogn).