A graph spanner is a fundamental graph structure that faithfully preserves the pairwise distances in the input graph up to a small multiplicative stretch. The common objective in the computation of spanners is to achieve the bestknown existential sizestretch tradeoff efficiently. Classical models and algorithmic analysis of graph spanners essentially assume that the algorithm can read the input graph, construct the desired spanner, and write the answer to the output tape. However, when considering massive graphs containing millions or even billions of nodes not only the input graph, but also the output spanner might be too large for a single processor to store. To tackle this challenge, we initiate the study of local computation algorithms (LCAs) for graph spanners in general graphs, where the algorithm should locally decide whether a given edge (u,v)∈E belongs to the output spanner. Such LCAs give the user the `illusion' that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present the following results: For general nvertex graphs and r∈{2,3}, there exists an LCA for (2r−1)spanners with O˜(n1+1/r) edges and sublinear probe complexity of O˜(n1−1/2r). These size/stretch tradeoffs are best possible (up to polylogarithmic factors). For every k≥1 andmore »
Local Computation Algorithms for Graphs of Nonconstant Degrees
In the model of local computation algorithms (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n, the number of vertices. Nonetheless, these complexities are generally at least exponential in d, the upper bound on the degree of the input graph. Instead, we consider the case where parameter d can be moderately dependent on n, and aim for complexities with subexponential dependence on d, while maintaining polylogarithmic dependence on n. We present:
a randomized LCA for computing maximal independent sets whose time and space complexities are quasipolynomial in d and polylogarithmic in n;
for constant ε>0, a randomized LCA that provides a (1−ε)approximation to maximum matching with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n.
 Award ID(s):
 1650733
 Publication Date:
 NSFPAR ID:
 10026349
 Journal Name:
 Algorithmica
 Volume:
 77
 Issue:
 4
 Page Range or eLocationID:
 971994
 ISSN:
 14320541
 Sponsoring Org:
 National Science Foundation
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