N objects numbered 0, 1,, N 1, grouped into disjoint sets One member of a set chosen to be the label

Transcription

1 CS 0 Theory of Algorithms Supplement Disjoint Set Structures Disjoint Set Structures N objects numbered 0, 1,, N 1, grouped into disjoint sets One member of a set chosen to be the label of the set Example: If we choose the smallest number as the label, set {, 5, 8, 10} is referred to as set Initially, N objects are in N different sets Each set contains one member

2 Disjoint Set Structures Two operations: find() and union() find(x) Returns the label of the set that contains object x union(a, b) Unions the two sets labeled a and b Problem Find efficient data structure and algorithms for dijit disjoint set structures t Disjoint Set Structures Approach 1 Use an integer array set of size N The label of a set is the smallest number in the set set[x] is the label of x, i.e., x is in set set[x] Initially, set[x] = x find1(x) return set[x] union1(a, b) i = min(a, b) ) j = max(a, b) for (k = 0; k < N; k++) if (set[k] k == j) set[k] = i

3 Disjoint Set Structures Approach 1 Example sets: {0, }, {1,, 6, 9}, and {, 5, 7, 8} set[] = [ ] After union1(, 0) sets: {0,,, 5, 7, 8} and {1,, 6, 9} set[] = [ ] 5 Analysis of Approach 1 Suppose a sequence of n find and (N 1) union operations are executed Note that: only one set exists after N-1 unions performed find1(): O(1) union1(): O(N) n find1(): O(n) (N 1) union1: O(N ) Total: O(n + N ) If N and n are of the same order, the time the sequence of operations takes is O(n ) 6

4 Disjoint Set Structures Approach Use an integer array set of size N Each set is a rooted tree, the label of a set is the smallest number in the set Each non-root node in a tree points to its parent, and the root points to itself Ifset[x] = x, x is the label and the root of a tree If set[x] ] = y and x y, then y is x s parentin a tree Initially, set[x] = x find(x) r = x while (set[r] r) r = set[r] return r union(a, b) if (a < b) ) set[b] = a else set[a] = b 7 Disjoint Set Structures Approach Example sets: set[] = [ ] After union(0, 1) sets: {0,,, 5, 7, 8} and {1,, 6, 9} set[] = [ ]

5 Analysis of Approach Suppose a sequence of n find and (N 1) union operations are executed find(): O(N), worst case union(): O(1) n find(): O(nN) (N 1) union: O(N) Total: O(nN) If N and n are of the same order, the time the sequence of operations takes is O(n ) Problem: a union operation might increase the height of a tree 9 Disjoint Set Structures Approach Solution to problem of Approach Make the shorter tree a subtree of the other when two trees merge Additional array height[], where height[x] [ ] is the height of node x in the tree If x is the root (i.e., label) of a tree, height[x] is the height of the tree Initially, height[x] = 1 for all x find(x) // same find function in Approach union(a, b) if (height[a] == height[b]) { height[a] += 1 set[b] = a } else if(height[a]>height[b]) ht[ ]>hiht[b]) set[b] = a else set[a] = b 10

6 Disjoint Set Structures Approach Example sets: set[] = [ ] After union(0, 1) sets: {0,,, 5, 7, 8} and {1,, 6, 9} set[] = [ ] 11 Analysis of Approach Suppose a sequence of n find and(n 1) union operations are executed We can show that the height of a tree of k nodes is O(log k) find(): O(log N), worst case union(): O(1) n find(): O(n log N) (N 1) union: O(N) Total: O(N + n log N) If N and n are of the same order, the time the sequence of operations takes is O(n log n) Can we do even better? 1

7 Disjoint Set Structures Approach Improvement to Approach Path compression When finding the root for x, make root become the parent of all the nodes on the path from the root to x find(x) r = x while (set[r] r) ) r = set[r] i = x while (i!= r) { j = set[i] //j is i s parent set[i]=r // make root r the parent of i i = j // continue all the way up } return r 1 Disjoint Set Structures Approach findrec(x) // recursive version offind, it s really neat if (set[x] x) set[x] = findrec(set[x]) return set[x] union(a, b) // same union function in Approach 1

8 Disjoint Set Structures Approach Example Before 9 find(8) After Analysis of Approach Suppose a sequence of n find and (N 1) union operations are executed Let c = n + (N 1) The time the sequence of operations takes is O(c log * N), which is very close to O(c), i.e., near linear time. The function log * N = the number of s in the equation.. = N. 16

9 Application Kruskal s MST Algorithm Repeatedly select the edge w/ min weight that is not yet processed andwon t form a cycleuntil MST found Sort the edges Initialize disjoint sets While #edges in the solution V - 1 Select the edge (u, v) w/ min weight not processed a = find(u); b = find(v) If (a!= b) {union(a, b); add (u, v) to the solution} Analysis: Given a graph G = (V, E)lett m = E and n = V o O(m) to make a MIN heap of the edges o O(n) to initialize the n disjoint sets o O(log m) to process each of the edges (i.e., perform delete from heap and union/find) which is equivalent to O(log n) O(m log n) to process all edges in the worst case Kruskal s algorithm takes a time in O(m log n) 17 Kruskal s Minimum Spanning Tree Algorithm - Example 1 G T, initially