[论文解读] Constructing Light Spanners Deterministically in Near-Linear Time

Stephen Alstrup, Søren Dahlgaard+3 more|arXiv (Cornell University)|Jan 1, 2019

Optical Polarization and Ellipsometry被引用 11

一句话总结

该论文提出了首个确定性、近乎线性时间的算法，用于构建 (2k−1)(1+ε)-稀疏图，其稀疏度和轻度接近最优。通过引入一种新型的确定性增量近似距离查询机制，作者实现了 Oε(m + n log n) 的时间复杂度——比以往的确定性方法快了指数级——同时在拉伸度、规模和轻度方面达到当前最先进的界限。当 k = log n 时，进一步提出了 O(m + n^{1+ε}) 时间复杂度的 O(log n)-稀疏图构造方法，其边数为 O(n)，轻度为 O(1)，该结果在渐近意义上是最优的。

ABSTRACT

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] 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+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(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_epsilon(n^(2+1/k+epsilon')) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(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 epsilon>0, we provide an O(m+n^(1+epsilon)) 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.

研究动机与目标

• 弥合贪婪稀疏图在理论最优性与其实用性之间的差距，后者构造时间高达 O(mn^{1+1/k})。
• 开发快速、确定性的稀疏图算法，使其在拉伸度、规模和轻度方面达到与最优随机化或较慢确定化构造方法相同的近乎最优边界。
• 通过实现近乎线性时间复杂度，使轻量级稀疏图在大规模图中的实际部署成为可能。
• 克服随机化动态距离查询机制的局限性，后者依赖于非自适应对手假设，而该假设在自适应稀疏图构造中不成立。

提出的方法

• 设计一种新型的确定性增量近似距离查询机制，支持动态边插入和距离查询，且无需固定插入顺序。
• 将距离查询机制作为核心子程序，应用于修改后的贪婪稀疏图算法中，其中边的插入决策依赖于实时距离查询。
• 基于簇直径和边权构建分层聚类框架，利用势函数追踪进展，以控制轻度和拉伸度。
• 应用修改后的 Halperin-Zwick 稀疏图作为子程序，并通过基于度的顶点选择机制增强，以提升稀疏度和运行时间。
• 使用预言机并查集数据结构维护连通性，并在稀疏图构造过程中支持高效更新。
• 利用势函数论证，限制所有层级中添加边的总权重，从而确保轻度和稀疏度的保证。

实验结果

研究问题

• RQ1 确定性稀疏图构造能否在近乎线性时间内实现与最优随机化或较慢构造方法相同的近乎最优拉伸度、规模和轻度边界？
• RQ2 是否可能设计一种确定性动态距离查询机制，支持自适应边插入顺序，而无需假设非自适应对手？
• RQ3 对于确定性稀疏图算法，构造时间、拉伸度、规模和轻度之间的最优权衡是什么？
• RQ4 对于 k = ω(1)，特别是 k = log n 的情况，能否实现子二次时间构造，同时达到最优轻度和稀疏度？
• RQ5 如何利用分层聚类和势函数分析来控制确定性稀疏图构造中添加边的总权重？

主要发现

• 该论文提出了一种确定性 Oε(m + n log n)-时间算法，可构造出 (2k−1)(1+ε)-稀疏图，其边数为 O(log k · n^{1+1/k})，轻度为 Oε(log k · n^{1/k})，相较于以往近乎线性时间方法，其在 k 相关依赖上实现了指数级改进。
• 在特殊情形 k = log n 下，论文实现了 O(m + n^{1+ε})-时间复杂度的 O(log n)-稀疏图构造，其边数为 O(n)，轻度为 O(1)，该结果在渐近意义上是最优的，且是首个已知的针对任意 k = ω(1) 的子二次时间构造方法。
• 作者提出了一种新型的确定性增量近似距离查询机制，支持自适应边插入顺序，克服了以往随机化查询机制所依赖的非自适应对手假设。
• 该距离查询机制使得贪婪式稀疏图算法能够高效实现，并具备拉伸度、稀疏度和轻度的可证明边界。
• 稀疏图的总轻度被限制在 Oε(n^{1/k} · log k) 以内，与最优随机化构造方法的已知边界一致。
• 该方法在所有指标——拉伸度、规模、轻度和运行时间——上均实现了近乎最优的权衡，同时保持完全确定性，并适用于大规模图的实际应用。