Constructing light spanners deterministically in near-linear time
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Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen  improved the state-of-the-art for light spanners by constructing a (2k−1)(1+ε)-spanner with [Formula presented] edges and [Formula presented] lightness. Soon after, Filtser and Solomon  showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of [Formula presented] (which is faster than ). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness [Formula presented], even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an [Formula presented] time spanner construction which achieves the state-of-the-art bounds. Our second result is an Oε(m+nlogn) time construction of a spanner with (2k−1)(1+ε) stretch, [Formula presented] edges and [Formula presented] lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=logn, for every constant ε>0, we provide an O(m+n1+ε) time construction that produces an O(logn)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k=ω(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest.
|Journal||Theoretical Computer Science|
|Publication status||Published - 12 Mar 2022|
© 2022 Elsevier B.V.
- Deterministic dynamic distance oracle, Efficient construction, Light spanners, Spanners