Weighted quick-union (without path compression)

The WeightedQuickUnionUF class represents a union–find data type(also known as thedisjoint-sets data type). It supports the unionandfindoperations, along with aconnectedoperation for determining whether two sites are in the same component and acount` operation that returns the total number of components.

The union–find data type models connectivity among a set of n sites, named 0 through n–1. The is-connected-to relation must be an equivalence relation:

• Reflexive: p is connected to p.
• Symmetric: If p is connected to q, then q is connected to p.
• Transitive: If p is connected to q and q is connected to r, then p is connected to r.

An equivalence relation partitions the sites into equivalence classes (or components). In this case, two sites are in the same component if and only if they are connected. Both sites and components are identified with integers between 0 and n–1. Initially, there are n components, with each site in its own component. The component identifier of a component (also known as the root, canonical element, leader, or set representative) is one of the sites in the component: two sites have the same component identifier if and only if they are in the same component.

-union(p, q) adds a connection between the two sites p and q. If p and q are in different components, then it replaces these two components with a new component that is the union of the two. -find(p) returns the component identifier of the component containing p. -connected(p, q) returns true if both p and q are in the same component, and false otherwise. -count() returns the number of components.

The component identifier of a component can change only when the component itself changes during a call to union—it cannot change during a call to find, connected, or count.

This implementation uses weighted quick union by size (without path compression). Initializing a data structure with n sites takes linear time. Afterwards, the union, find, and connected operations take logarithmic time (in the worst case) and the count operation takes constant time.

For additional documentation, see Section 1.5 of Algorithms, 4th Edition by Robert Sedgewick and Kevin Wayne.