9.1.1.1. Disjoint Sets and Equivalence Classes¶

Sometimes we have a collection of objects that we want to divide into separate sets.

9.1.1.2. Approach¶

Each object initially is a separate node in its own tree.

When two objects are "equivalent", then add them to the same tree.

Key question: Given two nodes, are they in the same tree?

9.1.1.3. Parent Pointer Implementation¶

9.1.1.4. Union/FIND¶

// General Tree implementation for UNION/FIND
class ParPtrTree {
  private int[] array;     // Node array

  ParPtrTree(int size) {
    array = new int[size]; // Create node array
    for (int i=0; i<size; i++)
      array[i] = -1;       // Each node is its own root to start
  }

  // Merge two subtrees if they are different
  void UNION(int a, int b) {
    int root1 = FIND(a);     // Find root of node a
    int root2 = FIND(b);     // Find root of node b
    if (root1 != root2)          // Merge two trees
      array[root1] = root2;
  }

  // Return the root of curr's tree
  int FIND(int curr) {
    if (array[curr] == -1) return curr; // At root
    while (array[curr] != -1) curr = array[curr];
    return curr;
  }
}

9.1.1.5. Weighted Union¶

A key goal is to keep the depth of nodes as shallow as possible (consistent with efficient processing).

Weighted Union rule:

• When two trees are union'ed, add one with fewer nodes as a child of the root of the tree with more nodes.
• Depth is \(O(\log n)\)

9.1.1.6. Algorithm Visualization¶

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9.1.1.7. .¶

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9.1.1.8. Path Compression¶

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