arxiv: 2605.03225 · v1 · submitted 2026-05-04 · 💻 cs.DS

Recognition: unknown

Dynamic Detours

Amadeus Reinald, Daniel Dadush, Marek Soko{\l}owski, Micha{\l} Pilipczuk, Micha{\l} W{\l}odarczyk

Pith reviewed 2026-05-08 17:01 UTC · model grok-4.3

classification 💻 cs.DS

keywords dynamic graphspath queriesbiconnected componentsdata structuresedge insertionslong pathsparity

0 comments

The pith

Dynamic data structures support queries for long paths, detours, and parity in fully dynamic graphs.

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

The paper establishes dynamic data structures for undirected graphs that change through insertions and deletions of edges. These structures can determine if two vertices u and v are connected by a path of length at least a fixed k, or by a path that is at least k longer than the shortest path, or by a path of even or odd length. The time for each update or query is amortized and equals 2 to the O of k cubed times log n plus some log log factors for the length queries, while parity queries take only polylog time. This stands in contrast to the known hardness of querying short paths. The structures achieve this through a local delayed edge insertion technique applied at the level of biconnected components.

Core claim

We present dynamic data structures for a fully dynamic undirected graph G that support long (u,v)-path queries, long (u,v)-detour queries, and even/odd (u,v)-path queries. For a fixed parameter k, the amortized time per update or query is 2^{O(k^3)} log n plus O(log^2 n log^2 log n) for the first two query types and O(log^2 n log^2 log n) for the parity queries. The main technical contribution is the mechanism of delayed edge insertion that works locally on the level of biconnected components.

What carries the argument

The delayed edge insertion mechanism, which processes new edges with delay to maintain correct information about long paths and parities locally within each biconnected component.

If this is right

Long-path and detour queries become feasible with time depending exponentially on k but only logarithmically on graph size.
Parity queries admit faster polylogarithmic amortized times without any k dependence.
The results provide an efficient alternative to short-path queries, which face conditional lower bounds preventing subpolynomial times.
The local operation on biconnected components enables efficient handling of updates without full graph recomputation.

Where Pith is reading between the lines

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

The delayed insertion idea may apply to maintaining other path properties such as the existence of paths through specific regions.
Implementation for small fixed k could be tested on large dynamic networks to check if the theoretical times translate to practice.
Similar local maintenance strategies might help in other dynamic graph problems like connectivity or flow computations.

Load-bearing premise

The delayed edge insertion mechanism maintains accurate path length and parity information when operating locally on biconnected components under arbitrary insertions and deletions.

What would settle it

An adversary sequence of edge insertions and deletions on a graph with a known long path or specific parity, followed by a query that the structure answers incorrectly.

read the original abstract

Fix a parameter $k\in \mathbf{N}$. We give dynamic data structures that for a fully dynamic undirected graph $G$, updated over time by edge insertions and edge deletions, can answer the following queries: - Long $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $k$? - Long $(u,v)$-detour: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $\text{dist}_G(u,v)+k$? - Even/odd $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of even/odd length? The amortized time of executing an update or answering a query is $2^{O(k^3)} \log n + O(\log^2 n \log^2 \log n)$ in the first two cases, and $O(\log^2 n \log^2 \log n)$ in the last, where $n$ is the number of vertices of $G$. The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most $k$. Specifically, there is no data structure supporting queries about $(u,v)$-paths of length at most two in time $n^{o(1)}$ unless the Triangle Conjecture fails. Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.

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.

Desk Editor's Note private letter to a colleague

The paper gives dynamic algorithms for long-path and detour queries via delayed edge insertion on biconnected components, with times 2^{O(k^3)} polylog n that contrast with short-path hardness.

read the letter

The main thing to know is that this work supplies the first explicit dynamic data structures for long (u,v)-path and long-detour queries in fully dynamic undirected graphs, plus a faster variant for even/odd parity paths. The amortized update and query time is 2^{O(k^3)} log n plus lower-order log factors for the first two, and just O(log^2 n log^2 log n) for parity. The central new mechanism is delayed edge insertion that runs locally on biconnected components, which lets them avoid the conditional lower bounds that block fast short-path reporting. They make the contrast with triangle-conjecture hardness clear and give concrete bounds rather than just existence claims. The construction is algorithmic and avoids circularity or fitted parameters. The soft spot is the handling of biconnected-component splits on deletions. The abstract does not detail an invariant showing that local delayed insertions still produce correct long-path answers after a split changes the shortest-path distances inside the new pieces; if a previously delayed edge no longer meets the length threshold, queries could flip without detection. The full proofs need to supply a charging argument or maintenance rule that keeps the amortized bounds intact under arbitrary deletion sequences. The citation pattern follows standard dynamic-connectivity references and does not rely on self-referential results. This paper is for researchers in dynamic graph algorithms who care about length-constrained connectivity. A reader working on bypassing short-path lower bounds or extending dynamic connectivity techniques would get direct value from the mechanism, provided the deletion case holds up. It deserves a serious referee because the results are new, the time bounds are explicit, and the approach is technically distinct from prior work.

Referee Report

2 major / 1 minor

Summary. The paper presents dynamic data structures for fully dynamic undirected graphs supporting three types of queries: existence of a path of length at least k between u and v, existence of a detour of length at least dist(u,v) + k, and existence of an even- or odd-length path. The amortized time per update or query is 2^{O(k^3)} log n + O(log² n log² log n) for the first two queries and O(log² n log² log n) for parity queries. The central technical contribution is a delayed edge insertion mechanism that maintains the required information locally on biconnected components.

Significance. If the correctness of the delayed insertion mechanism holds under all update sequences, the results provide a sharp contrast to known conditional lower bounds for short-path queries and introduce a potentially reusable technique for handling long-path and parity information in dynamic settings. The parameter-dependent exponential dependence on k is acceptable given the problem's nature, and the work strengthens the toolkit for dynamic graph algorithms.

major comments (2)

[mechanism of delayed edge insertion] The description of the delayed edge insertion mechanism (abstract and technical sections on biconnected-component maintenance) does not provide an explicit invariant or charging argument showing that local updates remain correct after an edge deletion splits a biconnected component. In such cases, a previously delayed edge may connect vertices whose shortest path in the new component structure falls below the length threshold, potentially causing query answers to flip without detection by the mechanism.
[analysis of update times] The amortized time bounds are stated to hold for arbitrary sequences of insertions and deletions, yet the analysis of how splits and merges of biconnected components are charged (particularly the 2^{O(k^3)} term) is not detailed enough to confirm that the bounds survive component fragmentation.

minor comments (1)

[abstract] Notation for dist_G(u,v) and the precise definition of 'long detour' should be introduced earlier and used consistently throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and will revise the paper to strengthen the presentation of the technical details.

read point-by-point responses

Referee: [mechanism of delayed edge insertion] The description of the delayed edge insertion mechanism (abstract and technical sections on biconnected-component maintenance) does not provide an explicit invariant or charging argument showing that local updates remain correct after an edge deletion splits a biconnected component. In such cases, a previously delayed edge may connect vertices whose shortest path in the new component structure falls below the length threshold, potentially causing query answers to flip without detection by the mechanism.

Authors: We agree that an explicit invariant would improve clarity. In the revised version we will add a dedicated paragraph in the biconnected-component maintenance section that states the invariant maintained by the delayed-insertion mechanism: a delayed edge {x,y} is kept only while the current biconnected component containing x and y has no path of length at least k between them. Upon an edge deletion that splits the component, the mechanism recomputes the biconnected components locally and immediately re-evaluates each delayed edge against the new component boundaries; any edge whose endpoints now lie in different components or whose shortest path in the new component already meets the threshold is inserted or discarded at that moment. A simple charging argument shows that each such local re-evaluation is charged to the deleted edge, preserving correctness without undetected flips. We will also include a short proof sketch that the invariant is re-established after every split. revision: yes
Referee: [analysis of update times] The amortized time bounds are stated to hold for arbitrary sequences of insertions and deletions, yet the analysis of how splits and merges of biconnected components are charged (particularly the 2^{O(k^3)} term) is not detailed enough to confirm that the bounds survive component fragmentation.

Authors: We acknowledge that the charging scheme for component splits and merges can be made more explicit. In the revision we will expand the amortized-analysis subsection to include a precise potential-function argument. The potential is defined per biconnected component and charges the 2^{O(k^3)} cost of local recomputation to the edges that caused the split or merge; because each edge participates in O(log n) splits over its lifetime (by the standard biconnected-component charging), the exponential term is amortized correctly across arbitrary update sequences. We will also add a short lemma showing that fragmentation does not increase the number of components beyond what the potential already accounts for. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper presents an algorithmic construction for dynamic data structures supporting long-path, detour, and parity queries via a 'delayed edge insertion' mechanism localized to biconnected components. This is a direct technical contribution with amortized time bounds derived from the construction itself, without any fitted parameters, self-referential definitions, or load-bearing self-citations that reduce the central claim to prior unverified results by the same authors. The abstract and described mechanism stand as an independent algorithmic argument; no step equates a prediction or theorem to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard graph-theoretic properties of undirected graphs and biconnected components together with the correctness of the newly introduced delayed edge insertion technique; no numerical parameters are fitted and no new physical entities are postulated.

axioms (1)

standard math Standard properties of undirected graphs, distances, and biconnected components hold and can be maintained under edge insertions and deletions.
Invoked implicitly when the data structure operates locally on biconnected components.

invented entities (1)

delayed edge insertion mechanism no independent evidence
purpose: To process edge updates locally within biconnected components while preserving correctness for long-path and detour queries.
This is the main technical contribution described in the abstract; it is an algorithmic technique rather than an independent physical or mathematical object.

pith-pipeline@v0.9.0 · 5611 in / 1407 out tokens · 73782 ms · 2026-05-08T17:01:54.868876+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 3 canonical work pages

[1]

Dynamic parameterized problems on unit disk graphs

[ACJ+24] Shinwoo An, Kyungjin Cho, Leo Jang, Byeonghyeon Jung, Yudam Lee, Eunjin Oh, Donghun Shin, Hyeonjun Shin, and Chanho Song. Dynamic parameterized problems on unit disk graphs. In35th International Symposium on Algorithms and Computation, ISAAC 2024, LIPIcs, pages 6:1–6:15. Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Informatik,

2024
[2]

Popular conjectures imply strong lower bounds for dynamic problems

[AV14] Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pages 434–443. IEEE Computer Society,

2014
[3]

Faster Detours in Undirected Graphs

[AVWWX23] Shyan Akmal, Virginia Vassilevska Williams, Ryan Williams, and Zixuan Xu. Faster Detours in Undirected Graphs. In31st Annual European Symposium on Algorithms, ESA 2023, volume 274 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 7:1–7:17, Dagstuhl, Germany,

2023
[4]

Dy- namic meta-kernelization.CoRR, abs/2511.03461,

[BHJK25] Christian Bertram, Deborah Haun, Mads Vestergaard Jensen, and Tuukka Korhonen. Dy- namic meta-kernelization.CoRR, abs/2511.03461,

work page arXiv
[5]

Bodlaender and Ton Kloks

[BK91] Hans L. Bodlaender and Ton Kloks. Better algorithms for the pathwidth and treewidth of graphs. In18th International Colloquium on Automata, Languages and Programming, ICALP 1991, Lecture Notes in Computer Science, pages 544–555. Springer,

1991
[6]

Efficient fully dynamic elimination forests with applications to detecting long paths and cycles

[CCD+21] Jiehua Chen, Wojciech Czerwiński, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, and Anna Zych-Pawlewicz. Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In2021 ACM-SIAM Symposium on Discrete Algorithms,...

2021
[7]

Demaine, Mohammad Taghi Hajiaghayi, and Dimitrios M

[DHT02] Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos.1.5-approximation for treewidth of graphs excluding a graph with one crossing as a minor. In5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX 2002, Lecture Notes in Computer Science, pages 67–80. Springer,

2002
[8]

A dynamic data structure for MSO properties in graphs with bounded tree-depth

[DKT14] Zdenek Dvoˇ r´ ak, Martin Kupec, and Vojtech T˚ uma. A dynamic data structure for MSO properties in graphs with bounded tree-depth. In22nd Annual European Symposium on Algorithms, ESA 2014, volume 8737 ofLecture Notes in Computer Science, pages 334–345. Springer,

2014
[9]

A dynamic data structure for counting subgraphs in sparse graphs

[DT13] Zdenek Dvoˇ r´ ak and Vojtech T˚ uma. A dynamic data structure for counting subgraphs in sparse graphs. In13th International Symposium on Algorithms and Data Structures, WADS 2013, volume 8037 ofLecture Notes in Computer Science, pages 304–315. Springer,

2013
[10]

New algo- rithms and hardness for incremental single-source shortest paths in directed graphs

[GVW20] Maximilian Probst Gutenberg, Virginia Vassilevska Williams, and Nicole Wein. New algo- rithms and hardness for incremental single-source shortest paths in directed graphs. In52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 153–166. ACM,

2020
[11]

Fully dynamic bicon- nectivity in eO(log2 n)time

[HNRS25a] Jacob Holm, Wojciech Nadara, Eva Rotenberg, and Marek Sokołowski. Fully dynamic bicon- nectivity in eO(log2 n)time. In57th Annual ACM Symposium on Theory of Computing, STOC 2025, pages 156–165. ACM,

2025
[12]

Fully dynamic bicon- nectivity in eO(log2 n)time.CoRR, abs/2503.21733,

[HNRS25b] Jacob Holm, Wojciech Nadara, Eva Rotenberg, and Marek Sokołowski. Fully dynamic bicon- nectivity in eO(log2 n)time.CoRR, abs/2503.21733,

work page arXiv
[13]

[HR20] Jacob Holm and Eva Rotenberg

Full version of [HNRS25a]. [HR20] Jacob Holm and Eva Rotenberg. Fully-dynamic planarity testing in polylogarithmic time. In 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 167–180, New York, NY, USA,

2020
[14]

[IO14] Yoichi Iwata and Keigo Oka

ACM. [IO14] Yoichi Iwata and Keigo Oka. Fast dynamic graph algorithms for parameterized problems. In14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2014, volume 8503 ofLecture Notes in Computer Science, pages 241–252. Springer,

2014
[15]

Dynamic treewidth

[KMN+23a] Tuukka Korhonen, Konrad Majewski, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Dynamic treewidth. In64th IEEE Annual Symposium on Foundations of Com- puter Science, FOCS 2023, pages 1734–1744,

2023
[16]

Dynamic treewidth.CoRR, abs/2304.01744,

[KMN+23b] Tuukka Korhonen, Konrad Majewski, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Dynamic treewidth.CoRR, abs/2304.01744,

work page arXiv
[17]

Fully dynamic approximation schemes on planar and apex-minor-free graphs

[KNPS24] Tuukka Korhonen, Wojciech Nadara, Michał Pilipczuk, and Marek Sokołowski. Fully dynamic approximation schemes on planar and apex-minor-free graphs. In David P. Woodruff, editor, 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, pages 296–313. SIAM,

2024
[18]

Dynamic treewidth in logarithmic time

[Kor25] Tuukka Korhonen. Dynamic treewidth in logarithmic time. In66th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2025, pages 787–796. IEEE,

2025
[19]

Almost-linear time parameterized algorithm for rankwidth via dynamic rankwidth

12 [KS24] Tuukka Korhonen and Marek Sokołowski. Almost-linear time parameterized algorithm for rankwidth via dynamic rankwidth. In56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1538–1549. ACM,

2024
[20]

Maintaining CMSO 2 properties on dynamic structures with bounded feedback vertex number

[MPS23] Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski. Maintaining CMSO 2 properties on dynamic structures with bounded feedback vertex number. In40th International Sympo- sium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 ofLIPIcs, pages 46:1–46:13. Schloss Dagstuhl — Leibniz-Zentrum f¨ ur Informatik,

2023
[21]

Parameterized dynamic data structure for split completion

[MPZ24] Konrad Majewski, Michał Pilipczuk, and Anna Zych-Pawlewicz. Parameterized dynamic data structure for split completion. In32nd Annual European Symposium on Algorithms, ESA 2024, LIPIcs, pages 87:1–87:17. Schloss Dagstuhl – Leibniz-Zentrum f¨ ur Informatik,

2024
[22]

Dynamic data structures for parameterized string problems

[OPR+23] Jędrzęj Olkowski, Michał Pilipczuk, Mateusz Rychlicki, Karol Węgrzycki, and Anna Zych- Pawlewicz. Dynamic data structures for parameterized string problems. In40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, volume 254 ofLIPIcs, pages 50:1–50:22. Schloss Dagstuhl — Leibniz-Zentrum f¨ ur Informatik,

2023
[23]

[PD04] Mihai P˘ atra¸ scu and Erik D. Demaine. Lower bounds for dynamic connectivity. In L´ aszló Babai, editor,Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 546–553. ACM,

2004