Sets and disjoint sets union123
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The document discusses disjoint set data structures and union-find algorithms. Disjoint set data structures track partitions of elements into separate, non-overlapping sets. Union-find algorithms perform two operations on these data structures: find, to determine which set an element belongs to; and union, to combine two sets into a single set. The document describes array-based representations of disjoint sets and algorithms for the union and find operations, including a weighted union algorithm that aims to keep trees relatively balanced by favoring attaching the smaller tree to the root of the larger tree.
![Array representation of S1, S2 and S3 i [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] p -1 5 -1 3 -1 3 1 1 1 5](https://image.slidesharecdn.com/setsanddisjointsetsunion123-110611085116-phpapp02/85/Sets-and-disjoint-sets-union123-7-320.jpg)
![Simple algorithms for union and find Algorithm SimpleUnion(i,j) { P[i]:=j; Algorithm SimpleFind(i) { While(p[i]>=0) do i:=p[i]; Return i; }](https://image.slidesharecdn.com/setsanddisjointsetsunion123-110611085116-phpapp02/85/Sets-and-disjoint-sets-union123-8-320.jpg)
![Union algorithm with weighting rule Algorithm WeightedUnion(i,j) // Union sets with roots I and j, i != j, using the // weighting rule. p[i]= -count[i] and p[j]= -count[j] { temp:=p[i]+p[j]; if (p[i]>p[j]) then { // i has fewer nodes. p[i]:=j; p[j]:=temp; } else { // j has fewer or equal nodes. P[j]:=i; p[i]:=temp; } }](https://image.slidesharecdn.com/setsanddisjointsetsunion123-110611085116-phpapp02/85/Sets-and-disjoint-sets-union123-11-320.jpg)
Sets and disjoint sets union123
- 1. Sets And Disjoint Sets Union By: Ankita Goyal
- 2. Definition Given a set of elements, it is often useful to break them up or partition them into a number of separate, nonoverlapping sets . A disjoint-set data structure is a data structure that keeps track of such a partitioning.
- 3. Operation we can perform: A union-find algorithm is an algorithm that performs two useful operations on such a data structure: Find : Determine which set a particular element is in. Also useful for determining if two elements are in the same set. Union : Combine or merge two sets into a single set.
- 4. Possible representations of sets S1 S2 S3 1 7 8 9 5 2 10 3 4 6
- 5. Possible Representations of S1 U S2 1 7 9 8 2 5 10 7 10 2 1 5 9 8 S1 ∪ S2 S1 ∪ S2
- 6. Data representation for s1, s2 and s3 1 6 3 5 4 2 10 7 8 9 S1 S2 S3
- 7. Array representation of S1, S2 and S3 i [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] p -1 5 -1 3 -1 3 1 1 1 5
- 8. Simple algorithms for union and find Algorithm SimpleUnion(i,j) { P[i]:=j; Algorithm SimpleFind(i) { While(p[i]>=0) do i:=p[i]; Return i; }
- 9. Weighting rule for Union(i,j) If the number of nodes in the tree with root i is less than the number in the tree with root j, then make j the parent of i; otherwise make i the parent of j.
- 10. Trees obtained using the weighted rule Initial union(1,2) union(1,3) Union(1,4) union(1,n) 1 2 n 3 2 1 5 4 n n 3 2 1 4 n 4 3 2 1 2 n 3 1
- 11. Union algorithm with weighting rule Algorithm WeightedUnion(i,j) // Union sets with roots I and j, i != j, using the // weighting rule. p[i]= -count[i] and p[j]= -count[j] { temp:=p[i]+p[j]; if (p[i]>p[j]) then { // i has fewer nodes. p[i]:=j; p[j]:=temp; } else { // j has fewer or equal nodes. P[j]:=i; p[i]:=temp; } }
- 12. Thank You