TY - JOUR

T1 - A fully dynamic algorithm for recognizing and representing proper interval graphs

AU - Hell, Pavol

AU - Shamir, Ron

AU - Sharan, Roded

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Fully dynamic algorithms

KW - Graph algorithms

KW - Lower bounds

KW - Proper interval graphs

UR - http://www.scopus.com/inward/record.url?scp=0036215514&partnerID=8YFLogxK

U2 - 10.1137/S0097539700372216

DO - 10.1137/S0097539700372216

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AN - SCOPUS:0036215514

SN - 0097-5397

VL - 31

SP - 289

EP - 305

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 1

ER -