[Paper Review] How to Compose Shortest Paths

Jade Master|arXiv (Cornell University)|May 27, 2022

Data Management and Algorithms4 citations

TL;DR

This paper presents a categorical framework and an efficient algorithm for composing shortest paths in weighted graphs by leveraging pushouts in a min-plus semiring setting. It introduces a precompilation-based method that significantly reduces computation time for single-source, single-target shortest paths by reusing block-wise matrix computations, achieving up to 250x speedup on large graphs with small boundaries compared to Dijkstra’s algorithm.

ABSTRACT

The composition problem for shortest paths asks the following: given shortest paths on weighted graphs M and N which share a common boundary, find the shortest paths on their union. This problem is a crucial step in any algorithm which uses the divide and conquer method to find shortest paths. This extended abstract details how this problem may be understood categorically. Finding shortest paths is represented by a functor and the composition problem asks to find the value of this functor on a pushout using the values of the functor on the components. Furthermore, we present an algorithm which solves the composition problem for shortest paths. When implemented in Python, this algorithm reduces the computation time for finding shortest paths by relying on precompilation.

Motivation & Objective

• To bridge the gap between categorical compositionality and practical computation in shortest path algorithms.
• To solve the composition problem for shortest paths in weighted graphs using a functorial framework based on structured cospan double categories.
• To develop an efficient, reusable algorithm that precomputes path blocks and composes them via min-plus matrix operations.
• To demonstrate significant performance gains over Dijkstra’s algorithm in large-scale graph compositions with small boundary sets.
• To lay a foundation for generalizing the approach to other algebraic path problems, such as NFA language recognition.

Proposed method

• The paper models shortest paths as a functor F from the category of [0,∞]-graphs to [0,∞]-enriched categories, using the matrix exponential in the min-plus semiring to compute shortest paths.
• It formulates the composition problem as computing F(M +_LX N) from F(M) and F(N), where M and N are glued along a common boundary X.
• The key insight is that the pushout F(M +_LX N) is isomorphic to F(M) +_{F(LX)} F(N), which is computed via block matrix operations in the min-plus semiring.
• The algorithm precomputes F(M) and F(N), pushes them forward to matrices over the boundary, and breaks them into blocks (MM, MX, XX_M, etc.) for reuse.
• For a given source and target, it evaluates the appropriate composition symbol using min-plus operations, replacing outer terms with source and target vectors.
• The method exploits idempotency of the [0,∞] semiring to simplify repeated path terms, enabling efficient symbolic composition.

Experimental results

Research questions

• RQ1 How can the composition of shortest paths in weighted graphs be formalized using category theory and functorial semantics?
• RQ2 Can a practical algorithm be derived from the categorical composition of functors that compute shortest paths?
• RQ3 What is the computational cost of composing shortest paths via pushouts, and how can it be optimized?
• RQ4 How does precompilation of path blocks improve performance in repeated shortest path queries?
• RQ5 Can this approach be generalized to other algebraic path problems, such as those over different semirings?

Key findings

• The compositional algorithm achieves a mean computation time of 0.1602 seconds for a 2000-node graph with a 5-node boundary, compared to 39.7804 seconds for Dijkstra’s algorithm.
• The speedup is most dramatic for large graphs with small boundaries, reaching up to 250x improvement in computation time.
• Precompilation time for the compositional algorithm was 93.0590 seconds, which is a one-time cost amortized over multiple queries.
• The algorithm's performance degrades with increasing boundary size due to the growth of composition symbols, while Dijkstra’s time decreases slightly with larger boundaries.
• The method is robust to graph density and maintains high efficiency due to symbolic composition and reuse of precomputed blocks.
• The theoretical foundation is validated by a categorical isomorphism: F(M +_LX N) ≅ F(M) +_{F(LX)} F(N), which enables the practical algorithm.