1 The Disjoint Set ADT CS146 Chapter 8 Yan Qing Lei

2 Issues: The equivalence problem The equivalence problem The first algorithm The first algorithm Smart union algorithms Smart union algorithms Union and Find Union and Find

3 Equivalence Relations A relation R is defined on a set S if for every pair of elements (a,b), a,b  S, a R b is either true or false. If a R b is true, then we say that a is related to b. A relation R is defined on a set S if for every pair of elements (a,b), a,b  S, a R b is either true or false. If a R b is true, then we say that a is related to b. An equivalence relation is a relation R that satisfy three properties: An equivalence relation is a relation R that satisfy three properties: –(reflexive) a R a, for all a  S. –(symmetric) a R b if and only if b R a. –(transitive) a R b and b R c implies that a R c.

4 The Dynamic Equivalence Problem The equivalence class of an element a  S is the subset of S that contains all the elements that are related to a. The equivalence class of an element a  S is the subset of S that contains all the elements that are related to a. Equivalence classes form a partition of S: every member of S appears in exactly one equivalence class. Equivalence classes form a partition of S: every member of S appears in exactly one equivalence class. To decide if two member are related, only need to check whether the two are in the same equivalence class. To decide if two member are related, only need to check whether the two are in the same equivalence class.

5 Disjoint Sets Make the input a collection of N sets, each with one element. Make the input a collection of N sets, each with one element. –All relations (except reflexive) are false; –Each set has a different element: S i  S j =  => it makes the sets disjoint. Find operation: returns the name of the set containing a given element. Find operation: returns the name of the set containing a given element. Add operation (e.g., add relation a~b) Add operation (e.g., add relation a~b) –Check if a and b are already related: if they are in the same equivalence class. –If not, apply “union”: merge the two equivalence classes containing a and b into a new equivalence class.

6 Example: On a set of subsets, the three operations amount to On a set of subsets, the three operations amount to Create a set of n disjoint subsets with every node in its own subset Create a set of n disjoint subsets with every node in its own subset Test wheter A and B are in the same subset Test wheter A and B are in the same subset If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong

7 The most efficient way to implement the operation is: The most efficient way to implement the operation is: For every A it is possible to ask for the index of the subset to which A belongs For every A it is possible to ask for the index of the subset to which A belongs This will be denoted as find(A). And we have A ~ B find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, for union(A, B) one performs This will be denoted as find(A). And we have A ~ B find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, for union(A, B) one performs

8 coding void union(Node A, Node B) { if (find(A) != find(B)) if (find(A) != find(B)) unify(A, B); unify(A, B); }

9 The first algorithm All the elements in S are numbered sequentially from 0 to N-1—the numbering can be determined by hashing —we have S i ={i} for i=0 through N-1. All the elements in S are numbered sequentially from 0 to N-1—the numbering can be determined by hashing —we have S i ={i} for i=0 through N-1. Maintain, in an array, the name of the equivalence class for each element. Maintain, in an array, the name of the equivalence class for each element. “find” is just a simple O(1) lookup. “find” is just a simple O(1) lookup. “union(a,b)”: suppose that a is in equivalence class i and b is in equivalence class j. Scan through the array, changing each i to j. It takes O(N) for one operation and O(N 2 ) for N number of union “union(a,b)”: suppose that a is in equivalence class i and b is in equivalence class j. Scan through the array, changing each i to j. It takes O(N) for one operation and O(N 2 ) for N number of union

10 Optimizations Keep all the elements that are in the same equivalence class in a linked list. Keep all the elements that are in the same equivalence class in a linked list. By tracking the size of each equivalence class, we, when “union”, change the name of the smaller equivalence class to the larger. Thus the total time spent for N “union” is O(NlogN). Using this strategy, any sequence of M finds and up to N-1unions takes at most O(M+NlogN) time. By tracking the size of each equivalence class, we, when “union”, change the name of the smaller equivalence class to the larger. Thus the total time spent for N “union” is O(NlogN). Using this strategy, any sequence of M finds and up to N-1unions takes at most O(M+NlogN) time.

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14 The O(M+N) algorithm Class DisjSets {pubic: explicit DisjSets(int numElements); int find(int x) const; int find(int x); void unionSets(int root1,int root2); private: vector s; }

15 The O(M+N) algorithm DisjSets::DisjSets(int numElements):s(numElements) { for(int j=0; j<s.size(); j++) s[j]=-1;} Void DisjSets::unionSets(root1, root2) {s[root2]=root1;}

16 The O(M+N) algorithm DisjSets:: find(int x) const {if(s[x]<0) return x; else return find(s[x]); }

17 Smart Union Algorithms Union-by-size: make the smaller tree a subtree of the larger. Union-by-size: make the smaller tree a subtree of the larger. If union-by-size, the depth of any node is never more than logN: a find operation is O(logN), and O(MlogN) for a sequence of M. The worse-case trees are binomial trees. If union-by-size, the depth of any node is never more than logN: a find operation is O(logN), and O(MlogN) for a sequence of M. The worse-case trees are binomial trees.

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20 Path Compression for faster find Path compression for find(x): every node on the path from x to the root has its parent changed to the root: make the tree shallow. Path compression for find(x): every node on the path from x to the root has its parent changed to the root: make the tree shallow.

21 Smart Union Algorithms /* Union-by-height: make the shallow tree a subtree of the deeper. */ void DisjSets::unionSets(root1, root2) { if(s[root2]<s[root1]) //root2 is deeper s[root1]=root2; else { //update height if same if(s[root1]==s[root2]) s[root1]--; s[root1]--;s[root2]=root1;}}

22 DisjSets:: find(int x) {if(s[x]<0) return x; else return s[x]=find(s[x]); }

23 Union and Find The Union-Find problem starts with n elements (numbered 1 to n), each one representing a singleton set. We allow two operations to be performed:

24 Find (i) returns the ID number of the set that i currently is in. We assume the ID number of each set is the number of one of the elements in it. Union (i; j) combines the elements in the sets with ID numbers i and j into a single set. The ID number of the new set will be either i or j. This is a destructive operation in the sense that the original sets i and j are lost when they are combined to form the new set.

25 We can implement Union-Find by maintaining an array A[1 : : : n] of integers. If i is the ID number of a set, then A[i] will be the negative of the number of elements in this set. Otherwise, A[i] will the number of another element in this set. -1 We begin with each A[i] equal to -1The Union operation is easily implemented as follows:

26 void Union (int i, int j) { if (A [i] < A [j]) { A [i] += A [j]; A [j] = i; } else { A [j] += A [i]; A [i] = j; } return; } This simply makes the \leader" of the smaller set point to the leader of the larger set, and adjusts the size of the leader. The time complexity is O(1)

27 To implement Find (i) we just follow the pointers to the leader: int Find (int i) { while (A [i] > 0) i = A [i]; return i; } This yields O(lg n) time. We can do better, though, with path compression. After we find the leader, we make all the nodes we've passed through point directly to it:

28 int Find (int i) { int j = i, k; if (A [i] < 0) return i; while (A [j] > 0) j = A [j]; while (A [i] != j) { k = A [i]; A [i] = j; i = k; } return j; }

29 Using Union by size (rank) and Find with path compression, we get the following result: Starting with n singleton sets, any sequence of M Union and/or Find operations takes O(M logN) time.