TY - JOUR

T1 - A note on labeling schemes for graph connectivity

AU - Izsak, Rani

AU - Nutov, Zeev

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2012/1/15

Y1 - 2012/1/15

N2 - Let G=(V,E) be an undirected graph and let S⊆V. The S-connectivity λGS(u,v) of u,v ε V is the maximum number of uv-paths that no two of them have an edge or a node in S{u,v} in common. Edge-connectivity is the case S=ø and node-connectivity is the case S=V. Given an integer k and a subset T⊆V of terminals, we consider the problem of assigning small "labels" (binary strings) to the terminals, such that given the labels of two terminals u,v ε T, one can decide whether λGS(u,v)≥k (k-partial labeling scheme) or to return min{λGS(u,v),k} (k-full labeling scheme). We observe that the best known labeling schemes for edge-connectivity (the case S=ø) extend to element-connectivity (the case S⊆VT), and use it to obtain a simple k-full labeling scheme for node-connectivity (the case S=V). If q distinct queries are expected, our k-full scheme has max-label size O(klog2|T|logq), with success probability 1-1q for all queries. We also give a deterministic k-full labeling scheme with max-label size O(k log3|T|). Recently, Hsu and Lu (2009) [6] gave an optimal O(klog|T|) labeling scheme for the k-partial case. This implies an O(k2log|T|) labeling scheme for the k-full case. Our deterministic k-full labeling scheme is better for k=Ω(log2|T|).

AB - Let G=(V,E) be an undirected graph and let S⊆V. The S-connectivity λGS(u,v) of u,v ε V is the maximum number of uv-paths that no two of them have an edge or a node in S{u,v} in common. Edge-connectivity is the case S=ø and node-connectivity is the case S=V. Given an integer k and a subset T⊆V of terminals, we consider the problem of assigning small "labels" (binary strings) to the terminals, such that given the labels of two terminals u,v ε T, one can decide whether λGS(u,v)≥k (k-partial labeling scheme) or to return min{λGS(u,v),k} (k-full labeling scheme). We observe that the best known labeling schemes for edge-connectivity (the case S=ø) extend to element-connectivity (the case S⊆VT), and use it to obtain a simple k-full labeling scheme for node-connectivity (the case S=V). If q distinct queries are expected, our k-full scheme has max-label size O(klog2|T|logq), with success probability 1-1q for all queries. We also give a deterministic k-full labeling scheme with max-label size O(k log3|T|). Recently, Hsu and Lu (2009) [6] gave an optimal O(klog|T|) labeling scheme for the k-partial case. This implies an O(k2log|T|) labeling scheme for the k-full case. Our deterministic k-full labeling scheme is better for k=Ω(log2|T|).

KW - Data structures

KW - Graph connectivity

KW - Labeling scheme

UR - http://www.scopus.com/inward/record.url?scp=80054124090&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2011.10.001

DO - 10.1016/j.ipl.2011.10.001

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AN - SCOPUS:80054124090

SN - 0020-0190

VL - 112

SP - 39

EP - 43

JO - Information Processing Letters

JF - Information Processing Letters

IS - 1-2

ER -