Partitions (partition)

Definition

An instance P of the data type partition consists of a finite set of items (partition_item) and a partition of this set into blocks.

Creation

partition	P;	creates an instance P of type partition and initializes it to the empty partition.

Operations

partition_item	P.make_block()	returns a new partition_item it and adds the block it to partition P.
partition_item	P.find(partition_item p)	returns a canonical item of the block that contains item p, i.e., iff P.same_block(p,q) then P.find(p) and P.find(q) return the same item. Precondition: p is an item in P.
int	P.size(partition_item p)	returns the size of the block containing p.
bool	P.same_block(partition_item p, partition_item q)
		returns true if p and q belong to the same block of partition P. Precondition: p and q are items in P.
void	P.union_blocks(partition_item p, partition_item q)
		unites the blocks of partition P containing items p and q. Precondition: p and q are items in P.
void	P.split(list<partition_item> L)
		turns all items in L to singleton blocks. Precondition: L is a union of blocks.

Implementation

Partitions are implemented by the union find algorithm with weighted union and path compression (cf. [76]). Any sequence of n make_block and m >= n other operations (except for split) takes time O(m alpha(m,n)). The cost of a split is proportional to the size of the blocks dismantled.

Example

Spanning Tree Algorithms (cf. section Graph Algorithms).