Local computation algorithms for spanners

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 best-known existential size-stretch trade-off 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 (sparse) spanner or not. Such LCAs give the user the “illusion” that a specific sparse spanner for the graph is maintained, without ever fully computing it. We present several results for this setting, including: ▬ For general n-vertex graphs and for parameter r ∊ {2, 3}, there exists an LCA for (2r – 1)-spanners with Õ(n^1+1/r) edges and sublinear probe complexity of Õ(n^1−1/2r). These size/stretch trade-offs are best possible (up to polylogarithmic factors). ▬ For every k ≥ 1 and n-vertex graph with maximum degree ∆, there exists an LCA for O(k²) spanners with Õ(n^1+1/k) edges, probe complexity of Õ(∆⁴n^2/3), and random seed of size polylog(n). This improves upon, and extends the work of [Lenzen-Levi, ICALP’18]. We also complement these constructions 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. To the best of our knowledge, our results on 3 and 5-spanners are the first LCAs with sublinear (in ∆) probe-complexity for ∆ = n^Ω(1).