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 and nvertex graph with maximum degree Δ, there exists an LCA for O(k2) spanners with O˜(n1+1/k) edges, probe complexity of O˜(Δ4n2/3), and random seed of size polylog(n). This improves upon, and extends the work of [LenzenLevi, 2018].
We also complement our results by providing a polynomial lower bound on the probe complexity of LCAs for graph spanners that holds even for the simpler task of computing a sparse connected subgraph with o(m) edges.
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Optimal Vertex FaultTolerant Spanners in Polynomial Time
Recent work has pinned down the existentially optimal size bounds for vertex faulttolerant spanners: for any positive integer k, every nnode graph has a (2k – 1)spanner on O(f^{1–1/k} n^{1+1/k}) edges resilient to f vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponentialtime greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized algorithm which takes Õ(f^{1–1/k} n^{2+1/k} + mf2) time. We also derandomize our algorithm to give a deterministic algorithm with similar bounds. This reflects an exponential improvement in runtime over [BodwinPatel PODC '19], the only previously known algorithm for constructing optimal vertex faulttolerant spanners.
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 Award ID(s):
 1909111
 NSFPAR ID:
 10295389
 Date Published:
 Journal Name:
 Proceedings of the 2021 ACMSIAM Symposium on Discrete Algorithms (SODA)
 Page Range / eLocation ID:
 29242938
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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