CS 157 : Assignment 2 Douglas R . Lanman

The union-find data structure, utilizing the “union-by-rank” and “path compression” heuristics, is one of the most efficient disjoint-set data structures known [1]. In general, union-find can be applied to any problem in which one must determine membership within a set. For instance, in class we studied its application within Kruskal’s algorithm, where it was used to efficiently detect connected component of a graph. In general, union-find supports three primary operations: insertion, union, and find. As discussed by Kaplan et al., in many applications it is also desirable to support deletion [3]. In this write-up we present “lazy-deletion”: a straightforward modification of the classic unionfind data structure which will allow deletions. In particular, lazy deletion will not alter the asymptotic worst-case running times of either the union or find procedures. Recall that the union-find data structure maintains a number of rooted trees. If a node has no children, then we can delete it without incident (i.e., free any memory or other resources associated with it). If a node has children then we cannot eliminate it immediately – otherwise its children would become isolated from the root node. As a result, we adopt the “lazy deletion” policy, in which nodes with children are marked for deletion and, as their children are removed over time, eventually deleted. This write-up is organized into four sections. In Section 1 we will discuss the basic modifications that must be made to Make-Set and Union to implement lazy deletion. In Section 2, the pseudocode for insertion and deletion will be presented. In Section 3, the necessary modifications for Find-Set will be presented. Finally, in Section 4, the relative merits of lazy deletion will be discussed in the context of alternate implementations.