Algorithms - Priceton - 01 - Dynamic Connectivity & Union Find

Connected components : Maximal set of objects that are mutually connected.

Find query : Check if two objects are in the same component.

Union command : Replace components containing two objects with their union.

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Quick-find [eager approach]

Data structure :

• Integer array id[] of size N.
• Interpretation: p and q are connected if and only if they have the same id.

Find : Check if p and q have the same id. O(1)

Union : To merge components containing p and q, change all entries whose id equals id[p] to id[q]. O(N)

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Quick-union [lazy approach]

Data structure ：

• Integer array id[] of size N.
• Interpretation : id[i] is parent of i.
• Root of i is id[id[id[...id[i]...]]].

Find : Check if p and q have the same root. O(height)

Union : To merge components containing p and q, set the id of p's root to the id of q's root. O(height)

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Improvement 1 to quick-union : weighting

Weighted quick-union :

• Modify quick-union to avoid tall trees.
• Keep track of size of each tree(number of objects).
• Balance by linking root of smaller tree to root of larger tree.

Data structure : 

Same as quick-union, but maintain extra array sz[i] to count number of objects in the tree rooted at i.

Find : Identical to quick-union. O(lgN)

Union : Modify quick-union to O(lgN)

• Link root of smaller tree to root of large tree.
• Update the sz[] array.

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Improvement 2 : path compression

Quick-union with path compression : Just after computing the root of p,

set the id of each examined node to point to that root. much closer to linear