TY - GEN

T1 - Constructing labeling schemes through universal matrices

AU - Korman, Amos

AU - Peleg, David

AU - Rodeh, Yoav

PY - 2006

Y1 - 2006

N2 - Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs labels the vertices of all graphs in such that for every graph and every two vertices u,v G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in . This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let be a family of connected graphs of size at most n and let denote the collection of graphs of size at most n, such that each graph in is composed of a disjoint union of some graphs in . We first investigate methods for translating f-labeling schemes for to f-labeling schemes for . In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family , one can construct an f-labeling scheme of size g(n)+loglogn+O(1) for . We also show that in several cases, the above mentioned extra additive term of loglogn+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size logn+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5logn+O(1) and a lower bound of 4/3logn-O(1) for the size of any such labeling scheme. Keywords: Labeling schemes, Universal graphs, Universal matrices.

AB - Let f be a function on pairs of vertices. An f-labeling scheme for a family of graphs labels the vertices of all graphs in such that for every graph and every two vertices u,v G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in . This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let be a family of connected graphs of size at most n and let denote the collection of graphs of size at most n, such that each graph in is composed of a disjoint union of some graphs in . We first investigate methods for translating f-labeling schemes for to f-labeling schemes for . In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family , one can construct an f-labeling scheme of size g(n)+loglogn+O(1) for . We also show that in several cases, the above mentioned extra additive term of loglogn+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size logn+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5logn+O(1) and a lower bound of 4/3logn-O(1) for the size of any such labeling scheme. Keywords: Labeling schemes, Universal graphs, Universal matrices.

KW - Labeling schemes

KW - Universal graphs

KW - Universal matrices

UR - http://www.scopus.com/inward/record.url?scp=77249174873&partnerID=8YFLogxK

U2 - 10.1007/11940128_42

DO - 10.1007/11940128_42

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AN - SCOPUS:77249174873

SN - 3540496947

SN - 9783540496946

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 409

EP - 418

BT - Algorithms and Computation - 17th International Symposium, ISAAC 2006, Proceedings

T2 - 17th International Symposium on Algorithms and Computation, ISAAC 2006

Y2 - 18 December 2006 through 20 December 2006

ER -