[Paper Review] Efficient Graph Algorithms for Network Analysis

Tong-Wook Shinn, Tadao Takaoka|arXiv (Cornell University)|Sep 16, 2013

Complexity and Algorithms in Graphs2 references3 citations

TL;DR

This paper presents efficient algorithms for the 1-center graph center (GC) and graph bottleneck (GB) problems on directed graphs with unit edge costs, using binary search over Boolean matrix powers to avoid full APSP computation. It achieves a time complexity of Õ(n².³⁷³), improving upon prior bounds of O(n².⁵⁷⁵) and O(n².⁶⁸⁸) for GC and GB, respectively, and introduces the SP-AF problem with a distance/flow semi-ring to compute shortest paths for all flow values using Floyd's algorithm over this structure.

ABSTRACT

The GC problem is to identify a pre-determined number of center vertices such that the distances or costs from (or to) the centers to (or from) other vertices is minimized. The bottleneck of a path is the minimum capacity of edges on the path. The Bottleneck Paths (BP) problem is to compute the paths that give us the maximum bottleneck values between pairs of vertices. The Graph Bottleneck (GB) problem is to find the minimum bottleneck value out of bottleneck paths for all possible pairs of vertices. We give two similar algorithms that are based on binary search to solve the 1-center GC problem and the GB problem on directed graphs with unit edge costs. We achieve $ ilde{O}(n^{2.373})$ worst case time complexity for both the 1-center GC problem and the GB problem, where $n$ is the number of vertices in the graph. This is better than the straightforward methods of solving the two problems in $O(n^{2.575})$ and $O(n^{2.688})$ time bounds, respectively. We then combine the Bottleneck Paths (BP) problem with the well known Shortest Paths (SP) problem to compute the shortest paths for all possible flow values. We call this problem the Shortest Paths for All Flows (SP-AF) problem. We show that if the flow demand is uncertain, but between two consecutive capacity values, the unique shortest path can be computed to push that flow. If the uncertainty stretches over two intervals, we need to prepare two shortest paths to accommodate the uncertainty, etc. In introducing this new problem, we define a new semi-ring called the distance/flow semi-ring, and show that the well known algorithm by Floyd can be used over the distance/flow semi-ring to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem.

Motivation & Objective

• To develop faster algorithms for the 1-center graph center (GC) and graph bottleneck (GB) problems in network analysis.
• To avoid the high cost of computing all-pairs shortest paths (APSP) and all-pairs bottleneck paths (APBP) by using binary search over matrix powers.
• To introduce and solve the Shortest Paths for All Flows (SP-AF) problem by defining a distance/flow semi-ring and generalizing Floyd's algorithm.
• To achieve improved asymptotic time complexity for GC and GB problems compared to existing methods.
• To enable efficient computation of shortest paths under varying flow demands by handling incomparable distance-flow pairs.

Proposed method

• Uses repeated squaring of the adjacency matrix B to compute reachability up to increasing path lengths, forming the basis for binary search on the diameter Δ.
• Employs a Boolean matrix multiplication framework where B^ℓ captures all pairs reachable within ℓ edges, enabling binary search on the optimal center radius.
• Applies binary search on the threshold t to determine the minimal Δ such that P(C) = 1, where P(C) checks if any vertex can reach all others within t steps.
• Introduces a distance/flow semi-ring where each matrix element stores sets of incomparable (distance, flow) pairs, sorted in natural order.
• Generalizes Floyd’s all-pairs shortest paths algorithm to compute the closure A* over the distance/flow semi-ring, solving the APSP-AF problem.
• Uses a custom multiplication procedure for sets of (distance, flow) pairs that merges pairs in O(|x| + |y|) time by comparing flow values.

Experimental results

Research questions

• RQ1 Can the 1-center graph center problem be solved faster than O(n².⁵⁷⁵) without computing the full APSP?
• RQ2 Can the graph bottleneck problem be solved in less than O(n².⁶⁸⁸) time by avoiding explicit APBP computation?
• RQ3 How can shortest paths be computed efficiently for all possible flow values between vertex pairs in a network?
• RQ4 Can a semi-ring structure be defined to combine distance and flow information in a way that supports efficient all-pairs computation?
• RQ5 Is it possible to use binary search over matrix powers to determine the minimal radius for the 1-center without full APSP?

Key findings

• The 1-center GC problem is solved in Õ(n².³⁷³) time, improving upon the previous best bound of O(n².⁵⁷⁵) by Zwick (2002).
• The graph bottleneck (GB) problem is solved in Õ(n².³⁷³) time, improving upon the prior O(n².⁶⁸⁸) bound by Duan and Pettie (2009).
• The SP-AF problem is formalized using a distance/flow semi-ring, enabling the computation of shortest paths for all flow values.
• The APSP-AF problem is solved in O(tn³) time using a generalized Floyd’s algorithm over the distance/flow semi-ring, where t is the number of distinct capacities.
• The multiplication of sets of incomparable (distance, flow) pairs is implemented in O(|x| + |y|) time, enabling efficient semi-ring operations.
• The algorithm avoids full APSP and APBP computation by using binary search on matrix powers, significantly reducing time complexity.