[论文解读] FPT-algorithms for computing Gromov-Hausdorff and interleaving distances between trees

Elena Farahbakhsh Touli, Yusu Wang|arXiv (Cornell University)|Nov 6, 2018

Topological and Geometric Data Analysis参考文献 16被引用 5

一句话总结

本文提出了首个用于在乘法因子14内近似度量树之间Gromov-Hausdorff距离的固定参数可追踪（FPT）算法。通过建立Gromov-Hausdorff距离与合并树的交错距离之间的一种新联系，作者重新定义了交错距离，以支持动态规划，从而得到一个用于交错距离的精确FPT算法，该算法反过来实现了对Gromov-Hausdorff距离的FPT近似。

ABSTRACT

Gromov-Hausdorff (GH) distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP hard to approximate the GH distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the GH distance between two general metric trees within a factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the GH distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary, and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a FPT algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the GH distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of $O(\sqrt{n})$ even for trees with unit edge length.

研究动机与目标

• 开发一种高效算法，用于计算或近似度量空间之间自然但计算困难的度量——度量树之间的Gromov-Hausdorff距离。
• 通过利用固定参数可追踪性，解决即使在单位边树情况下，近似Gromov-Hausdorff距离因子为3的NP难问题。
• 建立度量树的Gromov-Hausdorff距离与合并树的交错距离之间的联系，实现两种距离之间的算法转移。
• 设计一种重新定义的、支持动态规划的交错距离形式，以简化计算并支持精确FPT算法。

提出的方法

• 使用单个映射而非成对映射重新定义交错距离，简化了算法设计。
• 在具有参数化度数界τ的合并树上使用动态规划，以控制时间复杂度。
• 在候选δ值和τ值上引入双重二分查找过程，以高效找到最小交错距离。
• 利用度量树上Gromov-Hausdorff距离与相关合并树上交错距离之间的等价性，实现算法结果的转移。
• 对所有合并树根节点对的O(n⁴)个临界δ值进行排序并枚举，以指导对δ的二分查找。
• 通过τδ和τ₂δ之间的不等式链，对最终参数τ′进行界定，确保时间复杂度在参数τ上保持FPT。

实验结果

研究问题

• RQ1 能否使用固定参数可追踪算法，在常数乘法因子内近似度量树之间的Gromov-Hausdorff距离？
• RQ2 Gromov-Hausdorff距离与合并树的交错距离之间是否存在有意义的联系，从而支持算法转移？
• RQ3 交错距离能否以一种支持动态规划并导致精确FPT计算的方式重新定义？
• RQ4 该交错距离的新形式是否允许时间复杂度在参数τ上保持FPT，尽管其近似问题本身是NP难的？
• RQ5 能否将用于交错距离的FPT算法用于推导出Gromov-Hausdorff距离的FPT近似算法，且近似因子有界？

主要发现

• 本文提出了一种FPT算法，可在乘法因子14内近似两个度量树之间的Gromov-Hausdorff距离。
• 该算法基于对交错距离的全新重新定义，支持动态规划，并导致了计算合并树之间交错距离的精确FPT算法。
• 该算法的时间复杂度为O(n² 2^τ̂ τ̂^{τ̂+2} log³n)，其中τ̂ = 2τ̂₂₈δ*，τ̂₂₈δ*是源自最优交错距离的参数。
• 作者证明了两个合并树T₁^{fu}和T₂^{gw}之间的交错距离被限制在14δ*以内，其中δ*为最优Gromov-Hausdorff距离。
• 该算法通过在临界δ值和τ值上进行双重二分查找，避免了对所有O(n²)对根节点的暴力计算，显著减少了搜索空间。
• 本工作在拓扑数据分析中两个重要距离——Gromov-Hausdorff距离与交错距离之间建立了理论桥梁，推动了新的算法发展。