pith. sign in

arxiv: 2606.01809 · v1 · pith:NA2GWSUKnew · submitted 2026-06-01 · 💻 cs.DS

A Near-Optimal Offline Algorithm for Dynamic All-Pairs Shortest Paths in Planar Digraphs

Debarati Das , Maximilian Probst Gutenberg , Christian Wulff-Nilsen This is my paper

Pith reviewed 2026-06-28 12:37 UTC · model grok-4.3

classification 💻 cs.DS
keywords dynamic shortest pathsplanar digraphsall-pairs shortest pathsoffline dynamic algorithmsdense distance graphsupdate and query time
0
0 comments X

The pith

An offline algorithm achieves Õ(√n) worst-case time for dynamic all-pairs shortest paths in planar digraphs.

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

The paper establishes an algorithm that maintains all-pairs shortest paths in planar weighted digraphs under a known sequence of edge weight updates. It runs in Õ(√n) worst-case time per update and per query, which is near-optimal given the conditional lower bounds. This improves upon the prior Õ(n^{2/3}) bound by introducing the first offline dynamic method for maintaining dense distance graphs without full recomputation after each update. The work also includes an online incremental APSP algorithm with the same time bound that reduces the general online problem to its decremental counterpart.

Core claim

The central discovery is the first offline algorithm for dynamic planar all-pairs shortest paths with Õ(√n) worst-case update and query time. It is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs faster than recomputing from scratch after each update. An online algorithm for incremental APSP with the same time bound is also presented, allowing a reduction of the online dynamic APSP to the decremental version.

What carries the argument

Offline dynamic maintenance of dense distance graphs (DDGs) that supports faster updates than recomputing the DDG after each edge-weight change.

If this is right

  • The offline version of dynamic planar APSP now has near-optimal Õ(√n) time bounds.
  • An online incremental APSP algorithm exists with Õ(√n) time.
  • The online dynamic APSP problem reduces to the online decremental APSP problem.
  • The Õ(n^{2/3}) barrier for this problem is surpassed in the offline setting.

Where Pith is reading between the lines

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

  • Techniques for offline DDG maintenance might inspire online versions if the sequence can be approximated or handled in batches.
  • The reduction suggests that progress on decremental planar APSP would immediately yield progress on the full dynamic problem.
  • Similar DDG ideas could apply to other dynamic problems on planar graphs such as minimum cuts or flows.
  • If the graph changes in a way that violates planarity, the algorithm no longer applies, pointing to the need for robustness to planarity breaks.

Load-bearing premise

The update sequence is provided in advance and the graph stays planar throughout the updates.

What would settle it

A planar digraph and a long sequence of edge updates where the algorithm's total running time for updates and queries exceeds O(√n polylog n) per operation in the worst case.

Figures

Figures reproduced from arXiv: 2606.01809 by Christian Wulff-Nilsen, Debarati Das, Maximilian Probst Gutenberg.

Figure 1
Figure 1. Figure 1: (a): Illustration of Definitions 5.4 and 5.5. In this example, the nine shortest paths from v to h induce nine faces (in addition to h) where f is the face containing h ′ . The indices i of vertices vi(h) are clockwise around h in this drawing. (b): The contradiction reached in the proof of Lemma 5.6. The bold paths are Tv[vi(h)] and Tv[vj (h)] and the dashed path is a fictitious shortest path from b to vk… view at source ↗
read the original abstract

In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph $G$ undergoes a sequence of edge weight updates and the goal is to maintain a data structure on $G$, that can quickly answer distance queries between any two vertices $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query time $n^{0.5-\delta}$ is needed for any constant $\delta > 0$. In this article, we present the first algorithm with near-optimal $\tilde{O}(\sqrt{n})$ worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an \emph{online} algorithm for the incremental APSP problem with $\tilde{O}(\sqrt{n})$ worst-case update/ query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.

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

0 major / 2 minor

Summary. The paper claims the first near-optimal offline algorithm for dynamic all-pairs shortest paths (APSP) on planar weighted digraphs, achieving Õ(√n) worst-case update and query time when the entire update sequence is given in advance. The result is obtained via a new offline dynamic algorithm for maintaining dense distance graphs (DDGs) that improves on per-update recomputation from scratch; the paper also gives an online incremental APSP algorithm with the same bound that reduces the general online dynamic APSP problem to the online decremental case.

Significance. If the central claims hold, the work closes the gap to the known conditional lower bounds for the offline planar dynamic APSP problem and supplies the first sub-n^{2/3} offline DDG maintenance routine. The reduction from online dynamic to online decremental APSP constitutes partial progress on the harder online setting. The offline model is explicitly acknowledged and the planarity invariant is maintained throughout.

minor comments (2)
  1. [Abstract] Abstract: the high-level description of the offline DDG routine does not indicate whether the Õ(√n) bound is achieved by a single global pass over the update sequence or by a more involved batching technique; a one-sentence high-level idea would improve readability.
  2. [Abstract] The distinction between the offline result (matching lower bounds) and the online incremental result (partial progress) is clear, but the manuscript should explicitly state whether the online algorithm reuses any of the offline DDG machinery or is independent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the offline nature of the main result, the DDG maintenance contribution, and the reduction to the decremental case.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents an algorithmic construction for an offline dynamic APSP data structure in planar digraphs, achieving the claimed Õ(√n) bounds via a new offline DDG maintenance routine that processes the full update sequence at once. No derivation chain reduces any claimed result to its own inputs by construction, fitted parameters, or self-citation load-bearing steps; the offline model is explicitly the setting in which the bound is claimed, the planarity assumption is stated upfront, and the result is positioned against known online lower bounds without internal reduction. The contribution is a self-contained algorithmic technique rather than a mathematical derivation that collapses to prior inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard assumptions of planar graph theory and dynamic algorithm techniques; no free parameters, ad-hoc axioms, or invented entities are visible from the abstract.

axioms (2)
  • domain assumption The input graph remains planar after every update.
    Stated in the problem definition; planarity is required for the dense-distance-graph technique.
  • domain assumption The sequence of updates is known in advance (offline model).
    Explicitly required for the main result; the online case is only partially addressed.

pith-pipeline@v0.9.1-grok · 5777 in / 1275 out tokens · 19969 ms · 2026-06-28T12:37:59.991832+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 3 canonical work pages

  1. [1]

    Popular conjectures as a barrier for dynamic planar graph algorithms

    Amir Abboud and Søren Dahlgaard. Popular conjectures as a barrier for dynamic planar graph algorithms. In2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 477–486. IEEE, 2016. 1, 2, 3, 4

    2016

  2. [2]

    Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels

    Ittai Abraham, Shiri Chechik, and Cyril Gavoille. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. InProceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 1199–1218, 2012. 3

    2012

  3. [3]

    Fully dynamic all-pairs shortest paths with worst-case update-time revisited

    Ittai Abraham, Shiri Chechik, and Sebastian Krinninger. Fully dynamic all-pairs shortest paths with worst-case update-time revisited. InProceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 440–452. SIAM, 2017. 4

    2017

  4. [4]

    Incremental algorithms for minimal length paths.Journal of Algorithms, 12(4):615–638, 1991

    Giorgio Ausiello, Giuseppe F Italiano, Alberto Marchetti Spaccamela, and Umberto Nanni. Incremental algorithms for minimal length paths.Journal of Algorithms, 12(4):615–638, 1991. 4

    1991

  5. [5]

    Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths

    Surender Baswana, Ramesh Hariharan, and Sandeep Sen. Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths. InProceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 117–123. ACM, 2002. 4

    2002

  6. [6]

    The level ancestor problem simplified.Theoret- ical Computer Science, 321(1):5–12, 2004

    Michael A Bender and Martın Farach-Colton. The level ancestor problem simplified.Theoret- ical Computer Science, 321(1):5–12, 2004. 10

    2004

  7. [7]

    Almost optimal distance oracles for planar graphs

    Panagiotis Charalampopoulos, Pawe l Gawrychowski, Shay Mozes, and Oren Weimann. Almost optimal distance oracles for planar graphs. InProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 138–151, 2019. 4

    2019

  8. [8]

    Single-source shortest paths and strong connectivity in dynamic planar graphs

    Panagiotis Charalampopoulos and Adam Karczmarz. Single-source shortest paths and strong connectivity in dynamic planar graphs. In28th Annual European Symposium on Algorithms (ESA 2020). Schloss Dagstuhl-Leibniz-Zentrum f¨ ur Informatik, 2020. 4

    2020

  9. [10]

    URL:https://arxiv.org/abs/2005.02368,arXiv:2005.02368. 4

    arXiv 2005

  10. [11]

    MIT press, 2009

    Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein.Introduction to algorithms. MIT press, 2009. 15

    2009

  11. [12]

    Fully dynamic all pairs shortest paths with real edge weights

    Camil Demetrescu and Giuseppe F Italiano. Fully dynamic all pairs shortest paths with real edge weights. InProceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 260–267. IEEE, 2001. 4

    2001

  12. [13]

    A new approach to dynamic all pairs shortest paths.Journal of the ACM (JACM), 51(6):968–992, 2004

    Camil Demetrescu and Giuseppe F Italiano. A new approach to dynamic all pairs shortest paths.Journal of the ACM (JACM), 51(6):968–992, 2004. 4

    2004

  13. [14]

    Making data structures persistent.Journal of computer and system sciences, 38(1):86–124, 1989

    James R Driscoll, Neil Sarnak, Daniel D Sleator, and Robert E Tarjan. Making data structures persistent.Journal of computer and system sciences, 38(1):86–124, 1989. 11 12

    1989

  14. [15]

    Holiest minimum-cost paths and flows in surface graphs

    Jeff Erickson, Kyle Fox, and Luvsandondov Lkhamsuren. Holiest minimum-cost paths and flows in surface graphs. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1319–1332. ACM, 2018. 5

    2018

  15. [16]

    Decremental APSP in unweighted digraphs versus an adaptive adversary

    Jacob Evald, Viktor Fredslund-Hansen, Maximilian Probst Gutenberg, and Christian Wulff- Nilsen. Decremental APSP in unweighted digraphs versus an adaptive adversary. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors,48th International Colloquium on Au- tomata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtua...

    work page doi:10.4230/lipics.icalp.2021.64 2021

  16. [17]

    Planar graphs, negative weight edges, shortest paths, and near linear time

    Jittat Fakcharoenphol and Satish Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. InProceedings 42nd IEEE Symposium on Foundations of Computer Science, pages 232–241. IEEE, 2001. 1, 2, 3, 6, 7

    2001

  17. [18]

    Near-optimal (1+ϵ)-approximate fully-dynamic all-pairs shortest paths in planar graphs

    Arnold Filtser, Gramoz Goranci, Neel Patel, and Maximilian Probst Gutenberg. Near-optimal (1+ϵ)-approximate fully-dynamic all-pairs shortest paths in planar graphs. In2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 2078–2098. IEEE,

    2078

  18. [19]

    Improved bounds for shortest paths in dense distance graphs

    Pawel Gawrychowski and Adam Karczmarz. Improved bounds for shortest paths in dense distance graphs. In45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. 3

    2018

  19. [20]

    Unifying and strengthening hardness for dynamic problems via the online matrix-vector mul- tiplication conjecture

    Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector mul- tiplication conjecture. InProceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 21–30. ACM, 2015. 4

    2015

  20. [21]

    Italiano, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen

    Giuseppe F. Italiano, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen. Im- proved algorithms for min cut and max flow in undirected planar graphs. In Lance Fort- now and Salil P. Vadhan, editors,Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 313–322. ACM, 2011. doi:10.1145/19936...

    work page doi:10.1145/1993636.1993679 2011

  21. [22]

    Submatrix maximum queries in monge matrices and monge partial matrices, and their applications

    Haim Kaplan, Shay Mozes, Yahav Nussbaum, and Micha Sharir. Submatrix maximum queries in monge matrices and monge partial matrices, and their applications. InProceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 338–355. SIAM,

  22. [23]

    Decrementai transitive closure and shortest paths for planar digraphs and beyond

    Adam Karczmarz. Decrementai transitive closure and shortest paths for planar digraphs and beyond. InProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 73–92. SIAM, 2018. 4

    2018

  23. [24]

    Fast and simple connectivity in graph timelines

    Adam Karczmarz and Jakub Lacki. Fast and simple connectivity in graph timelines. pages 458–469, 2015. 4 13

    2015

  24. [25]

    Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs

    Valerie King. Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. InFoundations of Computer Science, 1999. 40th Annual Symposium on, pages 81–89. IEEE, 1999. 4

    1999

  25. [26]

    Multiple-source shortest paths in planar graphs

    Philip N Klein. Multiple-source shortest paths in planar graphs. InProceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 146–155. Society for Industrial and Applied Mathematics, 2005. 1, 2, 3, 8

    2005

  26. [27]

    Structured recursive separator decomposi- tions for planar graphs in linear time

    Philip N Klein, Shay Mozes, and Christian Sommer. Structured recursive separator decomposi- tions for planar graphs in linear time. InProceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 505–514, 2013. 1

    2013

  27. [28]

    Reachability in graph timelines

    Jakub Lacki and Pitor Sankowski. Reachability in graph timelines. pages 257–268, 2013. 4

    2013

  28. [29]

    Planar distance oracles with better time-space tradeoffs

    Yaowei Long and Seth Pettie. Planar distance oracles with better time-space tradeoffs. In D´ aniel Marx, editor,Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2517–2537. SIAM, 2021.doi: 10.1137/1.9781611976465.149. 4

    work page doi:10.1137/1.9781611976465.149 2021

  29. [30]

    Shortest paths in planar graphs with real lengths in o (nlog 2 n/loglogn) time

    Shay Mozes and Christian Wulff-Nilsen. Shortest paths in planar graphs with real lengths in o (nlog 2 n/loglogn) time. InEuropean Symposium on Algorithms, pages 206–217. Springer,

  30. [31]

    Offline dynamic higher con- nectivity.CoRR, abs/1708.03812, 2017

    Richard Peng, Bryce Sandlund, and Daniel Dominic Sleator. Offline dynamic higher con- nectivity.CoRR, abs/1708.03812, 2017. URL:http://arxiv.org/abs/1708.03812,arXiv: 1708.03812. 4

    Pith/arXiv arXiv 2017

  31. [32]

    Fully-dynamic all-pairs shortest paths: Improved worst-case time and space bounds

    Maximilian Probst Gutenberg and Christian Wulff-Nilsen. Fully-dynamic all-pairs shortest paths: Improved worst-case time and space bounds. InProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2020. 4

    2020

  32. [33]

    Worst-case update times for fully-dynamic all-pairs shortest paths

    Mikkel Thorup. Worst-case update times for fully-dynamic all-pairs shortest paths. InPro- ceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 112–119. ACM, 2005. 4 A Construction of the Last Interval Detection Data Structure Let us now show how the data structure presented in Theorem 5.2 can be constructed. We restate the the...

    2005

  33. [34]

    The data structureIthen holds the DDGs of all graphsG 1, G2, . . . , Gk′. Then, to process the next update toG, we then invokeAdvanceToNextVersion(). For queries on a particular version ofG t, we can aftertinvocations ofAdvanceToNextVersion() simply query the data structureDthat can answer DDG queries about the current graph. Algorithm 1:AdvanceToNextVers...

  34. [35]

    recompute the DDG of the piece containing the edge whose weight was changed,

  35. [36]

    recompute candidate minst-separating cycles fully contained in that piece, and

  36. [37]

    19 The first part is immediately improved from ˜O(r) to ˜O(√r) using our new data structure

    running the coarse version of Reif on the set of boundary vertices of ther-division. 19 The first part is immediately improved from ˜O(r) to ˜O(√r) using our new data structure. The second part can be solved by having a recursive max flow/minst-cut data structure for each piece. The final part remains unchanged and runs in time ˜O(n/√r) using FR-Dijkstra....