The Open Algebraic Path Problem

Author Jade Master

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Jade Master
  • Department of Mathematics, University of California, Riverside, CA, USA


I would like to thank Mike Shulman, John Baez, Christian Williams, Joe Moeller, Rany Tith, Sarah Rovner-Frydman, Zans Mihejez, Oscar Hernandez, Alex Pokorny and Todd Trimble for their helpful comments and contributions. I would also like to thank everyone in my life who supported me during this time. Your work contributed to this paper as well. This paper was written on Tongva land.

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Jade Master. The Open Algebraic Path Problem. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The algebraic path problem provides a general setting for shortest path algorithms in optimization and computer science. We explain the universal property of solutions to the algebraic path problem by constructing a left adjoint functor whose values are given by these solutions. This paper extends the algebraic path problem to networks equipped with input and output boundaries. We show that the algebraic path problem is functorial as a mapping from a double category whose horizontal composition is gluing of open networks. We introduce functional open matrices, for which the functoriality of the algebraic path problem has a more practical expression.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Operational semantics
  • The Algebraic Path Problem
  • Open Systems
  • Shortest Paths
  • Categorical Semantics
  • Compositionality


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  1. John C Baez and Kenny Courser. Structured cospans. Theory and Applications of Categories, 35(48):1771-1822, 2020. Google Scholar
  2. Francis Borceux. Handbook of Categorical Algebra: Volume 1, Basic Category Theory. Cambridge University Press, 1994. Google Scholar
  3. Kenny Courser. Open Systems: a Double Categorical Perspective. PhD thesis, University of California Riverside, 2020. Google Scholar
  4. Antonin Delpeuch. The word problem for double categories. Theory and Applications of Categories, 35(1):1-18, 2020. Google Scholar
  5. Stephen Dolan. Fun with semirings: a functional pearl on the abuse of linear algebra. In Proceedings of the 18th ACM SIGPLAN International Conference on Functional Programming, pages 101-110, 2013. Google Scholar
  6. Eugene Fink. A Survey of Sequential and Systolic Algorithms for the Algebraic Path Problem. University of Waterloo, Department of Mathematics, 1992. Google Scholar
  7. Brendan Fong. The Algebra of Open and Interconnected Systems. PhD thesis, University of Oxford, 2016. Google Scholar
  8. Brendan Fong and David I Spivak. An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge University Press, 2019. Google Scholar
  9. Davis Foote. Kleene algebras and algebraic path problems. 2015. Available at URL:
  10. Peter Höfner and Bernhard Möller. Dijkstra, Floyd and Warshall meet Kleene. Formal Aspects of Computing, 24(4-6):459-476, 2012. Google Scholar
  11. G Max Kelly. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society, 22(1):1-83, 1980. Google Scholar
  12. Saunders Mac Lane. Categories for the Working Mathematician. Springer Science & Business Media, 2013. Google Scholar
  13. Jade Master. Compositional markov., 2020.
  14. David Jaz Myers. String diagrams for double categories and equipments. 2016. Available at URL:
  15. Marc Pouly and Jürg Kohlas. Generic Inference: a Unifying Theory for Automated Reasoning. John Wiley & Sons, 2012. Google Scholar
  16. Julian Rathke, Paweł Sobociński, and Owen Stephens. Compositional reachability in Petri nets. In International Workshop on Reachability Problems, pages 230-243. Springer, 2014. Google Scholar
  17. Sairam Subramanian, Roberto Tamassia, and Jeffrey Scott Vitter. An efficient parallel algorithm for shortest paths in planar layered digraphs. Algorithmica, 14(4):322-339, 1995. Google Scholar
  18. Robert Endre Tarjan. A unified approach to path problems. Journal of the Association for Computing Machinery, 28(3):577-593, 1981. Google Scholar
  19. Harvey Wolff. V-cat and V-graph. Journal of Pure and Applied Algebra, 4(2):123-135, 1974. Google Scholar
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