[Paper Review] Dynamic Shortest Path and Transitive Closure Algorithms: A Survey

Daniel P. Martin|arXiv (Cornell University)|Sep 2, 2017

Complexity and Algorithms in Graphs61 references3 citations

TL;DR

This survey provides a comprehensive overview of dynamic shortest path and transitive closure algorithms for fully and partially dynamic graphs, covering foundational and state-of-the-art approaches with a focus on trade-offs between update and query time. It presents key algorithms, including those based on Even-Shiloach and Roddity-Zwick, and establishes connections to conditional lower bounds from OMv and BMM conjectures.

ABSTRACT

Algorithms which compute properties over graphs have always been of interest in computer science, with some of the fundamental algorithms, such as Dijkstra's algorithm, dating back to the 50s. Since the 70s there as been interest in computing over graphs which are constantly changing, in a way which is more efficient than simple recomputing after each time the graph changes. In this paper we provide a survey of both the foundational, and the state of the art, algorithms which solve either shortest path or transitive closure problems in either fully or partially dynamic graphs. We balance this with the known conditional lowerbounds.

Motivation & Objective

• To survey foundational and state-of-the-art algorithms for dynamic shortest paths and transitive closure in fully and partially dynamic graphs.
• To analyze the trade-offs between update and query time in dynamic graph algorithms, particularly in the context of fully dynamic and decremental settings.
• To present and compare known upper bounds for dynamic connectivity, shortest paths, and transitive closure problems across different graph types and constraints.
• To examine conditional lower bounds based on OMv and BMM conjectures, and assess their implications for algorithmic efficiency.
• To identify open problems, including derandomization, separation between deterministic and randomized algorithms, and tightening bounds for incremental vs. decremental settings.

Proposed method

• Surveying a broad range of dynamic graph algorithms, including Even and Shiloach’s decremental connectivity algorithm and its variants.
• Analyzing algorithms with amortized and worst-case time complexities, particularly focusing on fully dynamic all-pairs shortest paths with (1+ε)-approximation.
• Presenting trade-off mechanisms such as the use of parameter t in Roddity-Zwick’s algorithm, where update time is O(mn/t) and query time is O(t).
• Categorizing algorithms by dynamic model (incremental, decremental, fully dynamic), graph type (directed/undirected), and weight constraints (positive integers, real numbers).
• Using conditional lower bounds from the Online Matrix Vector (OMv) and Boolean Matrix Multiplication (BMM) conjectures to assess the optimality of existing algorithms.
• Comparing algorithmic performance across different problem variants, including single-source, s-t, and all-pairs settings, with and without approximation.

Experimental results

Research questions

• RQ1 What are the most efficient known algorithms for dynamic all-pairs shortest paths and transitive closure, and what trade-offs do they achieve between update and query time?
• RQ2 How do conditional lower bounds from the OMv and BMM conjectures constrain the performance of dynamic graph algorithms?
• RQ3 Can the performance gap between incremental and decremental dynamic algorithms be theoretically explained or narrowed?
• RQ4 What are the implications of derandomization for dynamic graph algorithms, particularly in the context of deterministic vs. randomized bounds?
• RQ5 To what extent can existing dynamic algorithms be improved to match or approach the conditional lower bounds?

Key findings

• The Even-Shiloach algorithm for decremental connectivity achieves O(mn) total update time and O(1) query time, serving as a foundational approach for later dynamic algorithms.
• Roddity and Zwick’s fully dynamic (1+ε)-approximation algorithm for all-pairs shortest paths achieves O(mn/t) update time and O(t) query time, enabling a tunable trade-off.
• For decremental single-source shortest paths, algorithms exist with O(n²) total update time and O(1) query time, improving upon the naive recomputation approach.
• Conditional lower bounds suggest that no algorithm can solve dynamic all-pairs shortest paths in O((mn)^{1−ε}) update time unless the OMv conjecture is false.
• The best-known decremental algorithm for strongly connected components runs in O(m log n) time, significantly improving over naive recomputation.
• The survey highlights that current algorithms for fully dynamic all-pairs shortest paths with exact distances are limited by conditional lower bounds, suggesting that approximation or amortized bounds are necessary for efficiency.