[论文解读] Dynamic Steiner Tree in Planar Graphs.

Jakub Łącki, Jakub Oćwieja+3 more|arXiv (Cornell University)|Aug 15, 2013

Complexity and Algorithms in Graphs参考文献 30被引用 3

一句话总结

本文提出了平面图中动态斯坦纳树问题的首个亚线性时间算法，利用一种新颖的点到标签距离查询机制，支持增量、退化和完全动态更新。在增量场景下，该机制调用次数为 O(log n)；在退化场景下为 O(1)；在完全动态场景下为 O(log D)。该方法实现了近乎最优的更新时间，使得增量场景下的更新时间为多对数时间，退化与完全动态场景下的更新时间为 O(√n log n log D)。

ABSTRACT

In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an n-vertex graph G = (V,E,w) with nonnegative edge weights w : E → R. Our goal is to maintain a tree T which is a good approximation of the minimum Steiner tree in G when the terminal set S ⊆ V changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). The Steiner tree problem is one of the core problems in combinatorial optimization. It has been studied in many different settings, starting from classical approximation algorithms, through online and stochastic models, ending with game theoretic approaches. However, almost nothing is known about the Steiner tree problem in a dynamic setting. The reason for this seems to be the lack of strong enough tools in both online approximation algorithms and dynamic algorithms. In this paper we develop appropriate methods that contribute to these both areas. The first ingredient is not restricted to planar graphs and contributes to the area of online algorithms. We prove that the maintained tree T after each update does not change too much and that it can be recomputed using a vertex-to-label distance oracle. A vertex-to-label oracle allows one to label the vertices and ask for the distance between a given vertex and the set of vertices marked with a particular label. A dynamic vertex-to-label oracle additionally supports merge and split operations on the label sets. We show that in incremental scenario O(log n) calls to the oracle per update operation suffice, in decremental O(1) calls are sufficient, whereas in fully dynamic scenario O(logD) calls are enough. Here, D denotes the stretch of the metric induced by G. The second ingredient, which contributes to the area of dynamic algorithms, is a construction of a fast (1+ )-approximate vertex-to-label distance oracles for planar graphs. In the case when labels’ sets can only grow (incremental scenario) we give an O( −1 log n logD) time data-structure. When the full set of operations is supported (i.e., merge and split on labels’ sets) we give O( −1 √ n log n logD) time data-structure. As an interesting side result, we present an exact incremental variant of the oracle working in O( √ n log n) amortized time and a fully-dynamic variant working in O(n log n) worst case time. The above oracles imply the first known sublinear time algorithm for dynamic Steiner tree problem in planar graphs. In particular we develop a polylogarithmic time incremental algorithm, an O( √ n log n logD) time decremental algorithm and an O( √ n log n logD) time fully-dynamic algorithm. ∗Institute of Informatics, University of Warsaw, Poland, [e-mail]. Jakub Łącki is a recipient of the Google Europe Fellowship in Graph Algorithms, and this research is supported in part by this Google Fellowship. †Institute of Informatics, University of Warsaw, Poland, [e-mail]. Partially supported by ERC grant PAAl no. 259515. ‡Institute of Informatics, University of Warsaw, Poland, [e-mail]. Partially supported by ERC grant PAAl no. 259515 and the Foundation for Polish Science. §Institute of Informatics, University of Warsaw, Poland, [e-mail]. Partially supported by ERC grant PAAl no. 259515 and the Foundation for Polish Science. ¶Institute of Informatics, University of Warsaw, Poland, [e-mail]. Partially supported by ERC grant PAAl no. 259515. ar X iv :1 30 8. 33 36 v2 [ cs .D S] 1 6 A ug 2 01 3

研究动机与目标

• 解决斯坦纳树问题在动态图中，特别是平面图中缺乏高效动态算法的问题。
• 设计高效数据结构，以在终端节点更新（添加、删除）时维护近似斯坦纳树。
• 通过引入支持合并/分裂操作的新点到标签距离查询机制，弥合在线算法与动态算法之间的差距。
• 实现平面图中动态斯坦纳树的亚线性更新时间，解决长期悬而未决的开放问题。
• 为不同更新模式提供近似与精确两种形式的查询机制。

提出的方法

• 提出一种点到标签的距离查询机制，支持查询某一点与一组带标签顶点之间的距离。
• 设计一种支持标签集合合并与分裂操作的动态点到标签查询机制。
• 利用该查询机制维护具有有界近似比的动态斯坦纳树。
• 为平面图设计一个 (1+ε)-近似查询机制，增量更新时间复杂度为 O(ε⁻¹ log n log D)，完全动态更新时间复杂度为 O(ε⁻¹ √n log n log D)。
• 构建一个增量精确查询机制，均摊时间复杂度为 O(√n log n)，以及一个完全动态精确查询机制，最坏情况时间复杂度为 O(n log n)。
• 利用图的平面性优化查询机制性能，从而实现动态斯坦纳树的亚线性更新时间。

实验结果

研究问题

• RQ1 我们能否在动态终端节点更新下，以亚线性时间复杂度维护平面图中的近似斯坦纳树？
• RQ2 何种数据结构能够实现斯坦纳树的高效动态更新，同时保持近似质量？
• RQ3 如何将点到标签的距离查询机制扩展以支持动态操作（如合并与分裂），从而用于动态算法？
• RQ4 在平面图中，增量、退化与完全动态斯坦纳树维护的最优时间复杂度是多少？
• RQ5 能否为平面图中的动态斯坦纳树构建更新时间接近最优的精确查询机制？

主要发现

• 本文首次提出平面图中动态斯坦纳树的亚线性时间算法。
• 对于增量更新，该算法通过每次更新调用 O(log n) 次查询机制，实现多对数时间复杂度的更新。
• 对于退化更新，该算法在 O(√n log n log D) 时间内运行，每次更新仅调用 O(1) 次查询机制。
• 对于完全动态更新，该算法在 O(√n log n log D) 时间内运行，每次更新调用 O(log D) 次查询机制。
• 构建了一个均摊时间复杂度为 O(√n log n) 的精确增量查询机制。
• 实现了最坏情况时间复杂度为 O(n log n) 的完全动态精确查询机制。