TY - JOUR

T1 - Distance labeling in graphs

AU - Gavoille, Cyril

AU - Peleg, David

AU - Pérennes, Stéphane

AU - Raz, Ran

N1 - Funding Information:
✩ This work has been supported in part by AFIRST. * Corresponding author. E-mail addresses: [email protected] (C. Gavoille), [email protected] (D. Peleg), [email protected] (S. Pérennes), [email protected] (R. Raz). 1 Supported in part by grants from the Israel Science Foundation and the Israel Ministry of Science and Art.

PY - 2004/10

Y1 - 2004/10

N2 - We consider the problem of labeling the nodes of a graph in a way that will allow one to compute the distance between any two nodes directly from their labels (without using any additional information). Our main interest is in the minimal length of labels needed in different cases. We obtain upper and lower bounds for several interesting families of graphs. In particular, our main results are the following. For general graphs, we show that the length needed is Θ(n). For trees, we show that the length needed is Θ(log 2n). For planar graphs, we show an upper bound of O(nlogn) and a lower bound of Ω(n1/3). For bounded degree graphs, we show a lower bound of Ω(n). The upper bounds for planar graphs and for trees follow by a more general upper bound for graphs with a r(n)-separator. The two lower bounds, however, are obtained by two different arguments that may be interesting in their own right. We also show some lower bounds on the length of the labels, even if it is only required that distances be approximated to a multiplicative factor s. For example, we show that for general graphs the required length is Ω(n) for every s<3. We also consider the problem of the time complexity of the distance function once the labels are computed. We show that there are graphs with optimal labels of length 3logn, such that if we use any labels with fewer than n bits per label, computing the distance function requires exponential time. A similar result is obtained for planar and bounded degree graphs.

AB - We consider the problem of labeling the nodes of a graph in a way that will allow one to compute the distance between any two nodes directly from their labels (without using any additional information). Our main interest is in the minimal length of labels needed in different cases. We obtain upper and lower bounds for several interesting families of graphs. In particular, our main results are the following. For general graphs, we show that the length needed is Θ(n). For trees, we show that the length needed is Θ(log 2n). For planar graphs, we show an upper bound of O(nlogn) and a lower bound of Ω(n1/3). For bounded degree graphs, we show a lower bound of Ω(n). The upper bounds for planar graphs and for trees follow by a more general upper bound for graphs with a r(n)-separator. The two lower bounds, however, are obtained by two different arguments that may be interesting in their own right. We also show some lower bounds on the length of the labels, even if it is only required that distances be approximated to a multiplicative factor s. For example, we show that for general graphs the required length is Ω(n) for every s<3. We also consider the problem of the time complexity of the distance function once the labels are computed. We show that there are graphs with optimal labels of length 3logn, such that if we use any labels with fewer than n bits per label, computing the distance function requires exponential time. A similar result is obtained for planar and bounded degree graphs.

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U2 - 10.1016/j.jalgor.2004.05.002

DO - 10.1016/j.jalgor.2004.05.002

M3 - Article

AN - SCOPUS:3943059465

SN - 0196-6774

VL - 53

SP - 85

EP - 112

JO - Journal of Algorithms

JF - Journal of Algorithms

IS - 1

ER -