A fully dynamic algorithm for recognizing and representing proper interval graphs

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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 Ω(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.

Original languageEnglish
Pages (from-to)289-305
Number of pages17
JournalSIAM Journal on Computing
Volume31
Issue number1
DOIs
StatePublished - 2001

Keywords

  • Fully dynamic algorithms
  • Graph algorithms
  • Lower bounds
  • Proper interval graphs

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