Path Minima in Incremental Unrooted Trees

Abstract

Consider a dynamic forest of unrooted trees over a set of n vertices which we update by link operations: Each link operation adds a new edge adjacent to vertices in two different trees. Every edge in the forest has a weight associated with it, and at any time we want to be able to answer a path-min query which returns that edge of minimum weight along the path between two given vertices.

For the case where the weights are integers we give an algorithm that performs n − 1 link operations and m pathmin queries in O(n + mα(m,n)) time. This extends well known results of Tarjan [11] and Yao [12] to a more general dynamic setting at the cost of restricting the weights to be integers. We also suggest a simpler data structures for the case where trees are rooted and the link operation always adds an edge between the root of one tree and an arbitrary vertex of another tree.

This work is partially supported by United States - Israel Binational Science Foundation, project number 2006204.