Summary: Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, S. Chechik and C. Wulff-Nilsen [ACM Trans. Algorithms 14, No. 3, Article No. 33, 15 p. (2018; Zbl 1457.05029)] improved the state-of-the-art for light spanners by constructing a \((2k-1)(1+\varepsilon)\)-spanner with \(O(n^{1+\frac{1}{k}})\) edges and \(O_\varepsilon(n^{\frac{1}{k}})\) lightness. Soon after, A. Filtser and S. Solomon [SIAM J. Comput. 49, No. 2, 429–447 (2020; Zbl 1437.05221)] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of \(O(mn^{1+\frac{1}{k}})\) (which is faster than [Chechik and Wulff-Nilsen, loc. cit.]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness \(\Omega_\varepsilon(kn^{\frac{1}{k}})\), 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 \(O_\varepsilon(n^{2+\frac{1}{k}+\varepsilon^\prime})\) time spanner construction which achieves the state-of-the-art bounds. Our second result is an \(O_\varepsilon(m+n\log n)\) time construction of a spanner with \((2k-1)(1+\varepsilon)\) stretch, \(O(\log k\cdot n^{1+\frac{1}{k}})\) edges and \(O_\varepsilon(\log k\cdot n^{\frac{1}{k}})\) 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 \(\varepsilon > 0\), we provide an \(O(m + n^{1 + \varepsilon})\) 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 \(k=\omega(1)\).
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.