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arxiv: 2605.22149 · v1 · pith:LQUSCWMDnew · submitted 2026-05-21 · 💻 cs.DS

A Coalgebraic Dijkstra Algorithm

Takahiro Sanada , Yo\`av Montacute , Kittiphon Phalakarn , Ichiro Hasuo This is my paper

Pith reviewed 2026-05-22 02:51 UTC · model grok-4.3

classification 💻 cs.DS
keywords coalgebraic shortest path problemDijkstra algorithmshortest pathswidest pathssemiringstate-transition systemsoptimization algorithms
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The pith

A coalgebraic framework unifies shortest paths, widest paths and games so that a generalized Dijkstra algorithm solves them under one necessary and sufficient condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the coalgebraic shortest path problem as a single setting that covers classical shortest paths, widest paths, two-player games on graphs, and new tasks such as finding shortest binary trees. It supplies a coalgebraic version of Dijkstra's algorithm whose relaxation step works on any state-transition system equipped with a suitable semiring. The central result states that this algorithm returns globally optimal values exactly when a stated condition on the coalgebra and semiring holds, and that the same condition is required for correctness. The algorithm runs in the same asymptotic time as the original Dijkstra procedure. If the claim holds, many existing path and game problems become instances of one efficient routine rather than separate implementations.

Core claim

The coalgebraic shortest path problem models optimization tasks on state-transition systems by equipping each transition with an operation from a semiring. The coalgebraic Dijkstra algorithm repeatedly selects the state with the current best value and relaxes its outgoing transitions exactly as in the classical case. The paper proves that the procedure terminates with the correct optimal values for every starting state if and only if the semiring and coalgebra satisfy a specific distributivity-like condition that prevents premature commitment to suboptimal choices. The same complexity bound as the textbook Dijkstra algorithm is recovered once the condition is met.

What carries the argument

The coalgebraic shortest path problem together with its relaxation operator, which replaces the classical min-plus update with a uniform semiring operation on the coalgebraic structure.

If this is right

  • The same code solves widest-path instances without any change to the core loop.
  • Shortest binary tree problems become solvable in near-linear time under the condition.
  • Two-player game values on graphs can be computed by instantiating the same semiring.
  • Programmers obtain an exact test that tells whether Dijkstra acceleration is safe for a new problem.
  • The algorithm inherits the classical O(|E| + |V| log |V|) bound once the condition holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be used to generate correct fast solvers for new problems by checking the condition on their weight structure.
  • Implementations in functional languages could treat the coalgebra as a reusable abstraction for search routines.
  • The necessary-and-sufficient condition might be checkable automatically for common semirings arising in verification.

Load-bearing premise

The semiring and coalgebra must obey the stated condition that makes each local relaxation step preserve global optimality.

What would settle it

A concrete coalgebra and semiring that satisfy the paper's condition yet produce a suboptimal value when the coalgebraic Dijkstra algorithm is run on an explicit small transition system.

read the original abstract

The Dijkstra algorithm is a classical method for solving the shortest path problem on weighted graphs. There are several variations of the Dijkstra algorithm, including algorithms for the widest path problem and for two-player games. In this paper, we introduce the coalgebraic shortest path problem (CSPP), a unifying framework for a broad class of optimization problems on state-transition systems. This framework encompasses not only the aforementioned problems but also new ones such as the shortest binary tree problem. We further present a coalgebraic Dijkstra algorithm for solving the CSPP efficiently under a suitable condition. Our condition is necessary and sufficient for the algorithm to return correct solutions, thereby providing a precise criterion for when Dijkstra-style acceleration is possible. We also show that the proposed algorithm achieves asymptotic complexity comparable to that of the classical Dijkstra algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the coalgebraic shortest path problem (CSPP) as a unifying framework for optimization problems on state-transition systems, encompassing classical shortest-path, widest-path, two-player games, and new instances such as the shortest binary tree problem. It presents a coalgebraic Dijkstra algorithm that solves CSPP under a stated condition, which the authors claim is necessary and sufficient for the algorithm to return correct solutions, and shows that the algorithm has asymptotic complexity comparable to classical Dijkstra.

Significance. If the central claims hold, the work supplies a coalgebraic unification of several path problems together with an explicit structural criterion for when Dijkstra-style acceleration is valid. This could be useful for extending algorithmic techniques to coalgebraic settings and for identifying the precise conditions under which relaxation-based methods remain optimal. The explicit complexity comparison is a positive point.

major comments (2)
  1. [Theorem establishing necessity and sufficiency] The necessity direction of the condition (abstract and the theorem establishing necessity and sufficiency) is load-bearing for the claim of a 'precise criterion.' The argument appears to show only that the invariant fails when the condition is violated, without exhibiting a concrete CSPP instance (e.g., a small state-transition system realizing the shortest-binary-tree problem) on which the algorithm returns a suboptimal value. An explicit counterexample is required to secure the necessity claim.
  2. [Complexity analysis section] The complexity claim (abstract) states that the algorithm achieves asymptotic complexity comparable to classical Dijkstra. The analysis should explicitly bound the cost of the generalized relaxation step in terms of the underlying semiring or coalgebra operations; without this, it is unclear whether the bound remains O(|E| + |V| log |V|) or equivalent once the coalgebraic structure is instantiated.
minor comments (2)
  1. [Abstract] The abstract refers to 'new ones such as the shortest binary tree problem' without a one-sentence definition or pointer to its coalgebraic signature; adding this would improve readability for readers unfamiliar with the example.
  2. [Preliminaries] Notation for the coalgebraic structure (states, transitions, weight function) should be introduced once and used consistently; minor inconsistencies in variable names appear in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. Below we provide point-by-point responses to the major comments and indicate how we will revise the paper.

read point-by-point responses

  1. Referee: [Theorem establishing necessity and sufficiency] The necessity direction of the condition (abstract and the theorem establishing necessity and sufficiency) is load-bearing for the claim of a 'precise criterion.' The argument appears to show only that the invariant fails when the condition is violated, without exhibiting a concrete CSPP instance (e.g., a small state-transition system realizing the shortest-binary-tree problem) on which the algorithm returns a suboptimal value. An explicit counterexample is required to secure the necessity claim.

    Authors: We agree that an explicit counterexample would strengthen the necessity claim. In the revised manuscript, we will add a concrete CSPP instance based on the shortest-binary-tree problem. This small state-transition system will violate the condition and demonstrate that the algorithm returns a suboptimal value, complementing the existing argument that the invariant fails. revision: yes

  2. Referee: [Complexity analysis section] The complexity claim (abstract) states that the algorithm achieves asymptotic complexity comparable to classical Dijkstra. The analysis should explicitly bound the cost of the generalized relaxation step in terms of the underlying semiring or coalgebra operations; without this, it is unclear whether the bound remains O(|E| + |V| log |V|) or equivalent once the coalgebraic structure is instantiated.

    Authors: We thank the referee for this observation. We will revise the complexity section to explicitly bound the cost of the generalized relaxation step in terms of the coalgebra or semiring operations. Under the standard assumption that these operations take constant time (as holds for the classical shortest-path, widest-path, and game instances), the asymptotic complexity remains O(|E| + |V| log |V|). We will state this assumption clearly and derive the bound accordingly. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via independent coalgebraic condition

full rationale

The paper introduces CSPP as a new unifying framework for optimization problems on state-transition systems and derives a coalgebraic Dijkstra algorithm whose correctness holds under an explicitly stated condition on the coalgebra and semiring structure. No step in the provided abstract or described claims reduces a prediction, optimality result, or necessity claim to a fitted parameter, self-citation, or definitional renaming by construction; the necessity and sufficiency are positioned as consequences of generalized invariants on the relaxation operator rather than tautological restatements of the algorithm's success. The derivation therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework relies on standard coalgebraic definitions of state-transition systems and on algebraic structures (semirings) for weights; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math State-transition systems are modeled as coalgebras over a suitable functor
    Invoked when defining the coalgebraic shortest path problem
  • domain assumption The weight structure admits a suitable order and relaxation operation compatible with the coalgebra
    Required for the correctness condition to be well-defined

pith-pipeline@v0.9.0 · 5665 in / 1330 out tokens · 27692 ms · 2026-05-22T02:51:39.433848+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Numerische Mathematik , volume=

    A Note on Two Problems in Connexion with Graphs , author=. Numerische Mathematik , volume=

    work page

  2. [2]

    ON A ROUTING PROBLEM , urldate =

    Bellman, Richard , journal =. ON A ROUTING PROBLEM , urldate =. 1958 , ISSN =

    work page 1958

  3. [3]

    Network flow theory , author=

    work page

  4. [4]

    A walk over the shortest path: Dijkstra's Algorithm viewed as fixed-point computation , journal =

    Misra, Jayadev , keywords =. A walk over the shortest path: Dijkstra's Algorithm viewed as fixed-point computation , journal =. 2001 , note =. doi:https://doi.org/10.1016/S0020-0190(00)00202-7 , url =

    work page doi:10.1016/s0020-0190(00)00202-7 2001

  5. [5]

    Operations Research , volume =

    Pollack, Maurice , title =. Operations Research , volume =. 1960 , doi =

    work page 1960

  6. [6]

    Algorithms to Find the Most Reliable Path in a Network , year=

    Wing, Omar , journal=. Algorithms to Find the Most Reliable Path in a Network , year=

    work page

  7. [7]

    Bardi, Martino and L. A. Dyn. Games Appl. , volume =. 2016 , url =. doi:10.1007/S13235-015-0156-0 , timestamp =

    work page doi:10.1007/s13235-015-0156-0 2016

  8. [8]

    2015 , MONTH = Mar, KEYWORDS =

    Bardi, Martino and Maldonado Lopez, Juan Pablo , URL =. 2015 , MONTH = Mar, KEYWORDS =

    work page 2015

  9. [9]

    Infinite Games

    Mazala, Ren \'e. Infinite Games. Automata Logics, and Infinite Games: A Guide to Current Research. 2002. doi:10.1007/3-540-36387-4_2

    work page doi:10.1007/3-540-36387-4_2 2002

  10. [10]

    , title =

    Kahn, Arthur B. , title =. Commun. ACM , month = nov, pages =. 1962 , issue_date =. doi:10.1145/368996.369025 , abstract =

    work page doi:10.1145/368996.369025 1962

  11. [11]

    2008 , isbn =

    Baier, Christel and Katoen, Joost-Pieter , title =. 2008 , isbn =

    work page 2008

  12. [12]

    Lehmann , abstract =

    Daniel J. Lehmann , abstract =. Algebraic structures for transitive closure , journal =. 1977 , issn =. doi:https://doi.org/10.1016/0304-3975(77)90056-1 , url =

    work page doi:10.1016/0304-3975(77)90056-1 1977

  13. [13]

    2008 , isbn =

    Gondran, Michel and Minoux, Michel , title =. 2008 , isbn =

    work page 2008

  14. [14]

    Mohri, Mehryar , title =. J. Autom. Lang. Comb. , month = jan, pages =. 2002 , issue_date =

    work page 2002

  15. [15]

    Algebra and algorithms for

    Sobrinho, João Luís , journal=. Algebra and algorithms for. 2002 , volume=

    work page 2002

  16. [16]

    Jacobs, Introduction to Coalgebra: Towards Mathematics of States and Observation , ser

    Jacobs, Bart , title =. 2016 , collection =. doi:10.1017/CBO9781316823187 , publisher =

    work page doi:10.1017/cbo9781316823187 2016

  17. [17]

    2000 , note =

    Universal coalgebra: a theory of systems , journal =. 2000 , note =. doi:10.1016/S0304-3975(00)00056-6 , author =

    work page doi:10.1016/s0304-3975(00)00056-6 2000

  18. [18]

    2025 , eprint=

    Continuation Semantics for Fixpoint Modal Logic and Computation Tree Logics , author=. 2025 , eprint=

    work page 2025

  19. [19]

    2025 , eprint=

    On Coalgebraic Product Constructions for Markov Chains and Automata , author=. 2025 , eprint=

    work page 2025

  20. [20]

    Automata Learning: A Categorical Perspective

    Jacobs, Bart and Silva, Alexandra. Automata Learning: A Categorical Perspective. Horizons of the Mind. A Tribute to Prakash Panangaden: Essays Dedicated to Prakash Panangaden on the Occasion of His 60th Birthday. 2014. doi:10.1007/978-3-319-06880-0_20

    work page doi:10.1007/978-3-319-06880-0_20 2014

  21. [21]

    Coalgebra Learning via Duality

    Barlocco, Simone and Kupke, Clemens and Rot, Jurriaan. Coalgebra Learning via Duality. Foundations of Software Science and Computation Structures. 2019

    work page 2019

  22. [22]

    Generic Partition Refinement and Weighted Tree Automata

    Deifel, Hans-Peter and Milius, Stefan and Schr \"o der, Lutz and Wi mann, Thorsten. Generic Partition Refinement and Weighted Tree Automata. Formal Methods -- The Next 30 Years. 2019

    work page 2019

  23. [23]

    Explicit Hopcroft's Trick in Categorical Partition Refinement

    Sanada, Takahiro and Kojima, Ryota and Komorida, Yuichi and Muroya, Koko and Hasuo, Ichiro. Explicit Hopcroft's Trick in Categorical Partition Refinement. Coalgebraic Methods in Computer Science. 2024

    work page 2024

  24. [24]

    Fast Coalgebraic Bisimilarity Minimization , year =

    Jacobs, Jules and Wi. Fast Coalgebraic Bisimilarity Minimization , year =. Proc. ACM Program. Lang. , month = jan, articleno =. doi:10.1145/3571245 , abstract =

    work page doi:10.1145/3571245

  25. [25]

    Pacific Journal of Mathematics , number =

    Patrick Cousot and Radhia Cousot , title =. Pacific Journal of Mathematics , number =

    work page

  26. [26]

    Fibonacci heaps and their uses in improved network optimization algorithms

    Fredman, Michael L. and Tarjan, Robert Endre , title =. J. ACM , month = jul, pages =. 1987 , issue_date =. doi:10.1145/28869.28874 , abstract =

    work page doi:10.1145/28869.28874 1987

  27. [27]

    Mathematical Structures in Computer Science , author =

    Implementing collection classes with monads , volume =. Mathematical Structures in Computer Science , author =. 1998 , pages =. doi:10.1017/S0960129598002515 , number =

    work page doi:10.1017/s0960129598002515 1998

  28. [28]

    Theoretical Computer Science , volume =

    Taut Monads and. Theoretical Computer Science , volume =. 2002 , issn =. doi:https://doi.org/10.1016/S0304-3975(00)00415-1 , url =

    work page doi:10.1016/s0304-3975(00)00415-1 2002

  29. [29]

    Weakest preconditions in fibrations , journal =

    Alejandro Aguirre and Shin. Weakest preconditions in fibrations , journal =. 2022 , url =. doi:10.1017/S0960129522000330 , timestamp =

    work page doi:10.1017/s0960129522000330 2022

  30. [30]

    Ichiro Hasuo , title =. Theor. Comput. Sci. , volume =. 2015 , url =. doi:10.1016/j.tcs.2015.03.047 , timestamp =

    work page doi:10.1016/j.tcs.2015.03.047 2015

  31. [31]

    2026 , note=

    A Coalgebraic Dijkstra Algorithm , author=. 2026 , note=

    work page 2026