A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs

Abstract

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA. We give a near-optimal fully dynamic algorithm for this problem. It operates in O(log n) worst-case time per edge insertion or deletion. We prove a close lower bound of $\Omega(\log n/(\log\log n+\log b))$ amortized time per operation in the cell probe model with word-size b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time. As a byproduct of our algorithm, we solve in O(log n) worst-case time the problem of maintaining connectivity in a dynamically changing proper interval graph.