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Disjoint Sets

Disjoint Sets. CS2 -- 9/8/2010. Disjoint Sets. A disjoint set contains a set of sets such that in each set, an element is designated as a marker for the set. A simple Disjoint Set: {1}, {2}, {3}, {4}, {5} There is only one marker for each of these sets. The element itself.

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Disjoint Sets

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  1. Disjoint Sets CS2 -- 9/8/2010

  2. Disjoint Sets • A disjoint set contains a set of sets such that in each set, an element is designated as a marker for the set. • A simple Disjoint Set: {1}, {2}, {3}, {4}, {5} • There is only one marker for each of these sets. • The element itself.

  3. Disjoint Sets • Given the original Disjoint Set: {1}, {2}, {3}, {4}, {5} • Union(1,3) would make our structure look like: {1,3}, {2}, {4}, {5} • we can choose 1 or 3 as the marker for the set. Let’s choose 1. • Union(1,4): {1,3,4}, {2}, {5} • Union(2,5): {1,3,4}, {2, 5} • where we choose 2 as the marker.

  4. Disjoint Sets • Given the last Disjoint Set: {1,3,4}, {2, 5} • We can do a findsetoperation. • findset(3) should return 1, since 1 is the marked element in the set.

  5. Disjoint Set Implementation • A set with disjoint sets can be represented in several ways. • Given {2,4,5,8} • with 5 as the marked element. • Here are a few ways it could be stored: 5 5 5 2 4 8 2 8 8 4 2 4

  6. Disjoint Set Implementation • We can also store a disjoint set in an array. • Given: {2,4,5,8} , {1} , {3,6,7} • Could be stored as: • The 5 stored in array[2], signifies that 5 is 2’s parent. • The 2 in array[8] signifies that 2 is 8’s parent, etc. • The 5 in array[5] signifies that 5 is a marker for its set. • Based on this storage scheme, how could we implement the initial makeset algorithm and how could we implement a findset algorithm? 7 1 5 2 4 3 6 8

  7. Disjoint Set Implementation • We can also store a disjoint set in an array. • Given: {2,4,5,8} , {1} , {3,6,7} • Could be stored as: • The 5 stored in array[2], signifies that 5 is 2’s parent. • The 2 in array[8] signifies that 2 is 8’s parent, etc. • The 5 in array[5] signifies that 5 is a marker for its set. • Based on this storage scheme, how could we implement the initial makeset algorithm and how could we implement a findset algorithm?

  8. Union Operation • Given two values, we must 1st find the markers for those two values, then merge those two trees into one. • Given the Disjoint Set from before: • If we perform union(5,1)we could do either of the following: • We prefer the right one, since it minimizes the height of the tree. So we should probably keep track of the height of our tree to do our merges efficiently. 1 5 7 2 4 3 6 8 1 5 5 2 4 1 2 4 8 8

  9. Union Operation • Minimizing the height of the tree • We choose which tree to merge with based on which tree has a small height. • If they are equal we are forced to add 1 to the height of the new tree. • Given the Disjoint set from before: • We have 2 options: Option 1: Option 2:

  10. Path Compression • One last enhancement! Every time we are forced to do a findset operation, we can directly connect each node on the path from the original node to the root. • First, we find the root of this tree which is 1. • Then you go through the path again, starting at 8, changing the parent of each of the nodes on that path to 1. • Then you take the 2 that was previously stored in index 8, and then change the value in that index to 1: 1 1 Path Compression 5 2 8 5 2 4 4 8

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