TY - JOUR

T1 - Constructing light spanners deterministically in near-linear time

AU - Alstrup, Stephen

AU - Dahlgaard, Søren

AU - Filtser, Arnold

AU - Stöckel, Morten

AU - Wulff-Nilsen, Christian

N1 - Publisher Copyright: © 2022 Elsevier B.V.

PY - 2022/3/12

Y1 - 2022/3/12

N2 - Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [10] 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 [18] 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 [10]). 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+nlog⁡n) 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=log⁡n, for every constant ε>0, we provide an O(m+n1+ε) time construction that produces an O(log⁡n)-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.

AB - Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [10] 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 [18] 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 [10]). 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+nlog⁡n) 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=log⁡n, for every constant ε>0, we provide an O(m+n1+ε) time construction that produces an O(log⁡n)-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.

KW - Deterministic dynamic distance oracle

KW - Efficient construction

KW - Light spanners

KW - Spanners

U2 - 10.1016/j.tcs.2022.01.021

DO - 10.1016/j.tcs.2022.01.021

M3 - Journal article

AN - SCOPUS:85123635666

VL - 907

SP - 82

EP - 112

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -