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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.144

Undirected Replacement Paths: Dual Fault Reduces to Single Source

Authors: Jakob Nogler and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

Given a graph and two vertices s and t, the Replacement Path Problem (RP) is to compute for every edge e, the distance between s and t when e is removed. There are two natural extensions to RP: - Single Source Replacement Paths (SSRP): Given a graph 𝐆 and a source node s, compute for every vertex v and every edge e the s-v distance in 𝐆⧵e. That is, we do not fix the target anymore. - 2-Fault Replacement Paths (2-FRP): Given a graph 𝐆 and two nodes s and t, compute for every pair of edges e,e' the s-t distance in 𝐆⧵e,e'. That is, there are two failures instead of one. Previously, there was no known formal reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected 2-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we get new algorithms for 2-FRP that match the best-known runtimes for SSRP: (a) 𝒪̃(M n^ω) for weights in [1..M] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving upon 𝒪(Mn^{2.87}) [Chechik, Zhang, ICALP 2024]; (b) n³/2^Ω(√{log n}) for weights in [1..poly(n)] [Grandoni and Vassilevska Williams, FOCS 2012 & TALG 2019], improving over the previous n³polylog(n) running time [Vassilevska W., Woldeghebriel and Xu, FOCS 2022]; (c) 𝒪̃(mn^{1/2} + n²) combinatorial time for unweighted graphs [Chechik and Cohen, SODA 2019], and more generally for rational weights in [1,2] [Chechik and Magen, ICALP 2020], improving upon 𝒪̃(n^{3-1/18}) [Chechik, Zhang ICALP 2024]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

Cite as

Jakob Nogler and Virginia Vassilevska Williams. Undirected Replacement Paths: Dual Fault Reduces to Single Source. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 144:1-144:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{nogler_et_al:LIPIcs.ICALP.2026.144,
  author =	{Nogler, Jakob and Vassilevska Williams, Virginia},
  title =	{{Undirected Replacement Paths: Dual Fault Reduces to Single Source}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{144:1--144:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.144},
  URN =		{urn:nbn:de:0030-drops-265332},
  doi =		{10.4230/LIPIcs.ICALP.2026.144},
  annote =	{Keywords: Single Source Replacement Paths, Dualt Fault Replacement Paths, Fine-Grained Complexity}
}

Document

DOI: 10.4230/LIPIcs.CPM.2026.26

The Communication Complexity of Pattern Matching with Edits Revisited

Authors: Tomasz Kociumaka, Jakob Nogler, and Philip Wellnitz

Published in: LIPIcs, Volume 369, 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)

Abstract

The decades-old Pattern Matching with Edits problem, given a length-n string T (the text), a length-m string P (the pattern), and a positive integer k (the threshold), asks to list the k-error occurrences of P in T, that is, all fragments of T whose edit distance to P is at most k. The one-way communication complexity of this problem is the minimum number of bits that Alice, given an instance (P,T,k) of the problem, must send to Bob so that Bob can reconstruct the answer solely from that message. In recent work [STOC'24], we showed that, in the natural parameter regime 0 < k < m < n/2, Ω(n/m ⋅ k log(m/k)) bits are necessary and 𝒪(n/m ⋅ k log² m) bits are sufficient for this problem. More generally, for strings over an alphabet Σ, we gave an 𝒪(n/m ⋅ k log m log(m|Σ|))-bit encoding that allows one to recover a shortest sequence of edits for every k-error occurrence of P in T. In this paper, we revisit the original proof and improve the encoding size to 𝒪(n/m ⋅ k log (m|Σ|/k)), which matches the lower bound for constant-sized alphabets. We further establish a new tight lower bound of Ω(n/m ⋅ k log(m|Σ|/k)) for the edit sequence reporting variant we solve. Our encoding size also matches the communication complexity established for the simpler Pattern Matching with Mismatches problem in the context of streaming algorithms [Clifford, Kociumaka, Porat; SODA'19].

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Tomasz Kociumaka, Jakob Nogler, and Philip Wellnitz. The Communication Complexity of Pattern Matching with Edits Revisited. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kociumaka_et_al:LIPIcs.CPM.2026.26,
  author =	{Kociumaka, Tomasz and Nogler, Jakob and Wellnitz, Philip},
  title =	{{The Communication Complexity of Pattern Matching with Edits Revisited}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-420-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{369},
  editor =	{Bille, Philip and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.26},
  URN =		{urn:nbn:de:0030-drops-259525},
  doi =		{10.4230/LIPIcs.CPM.2026.26},
  annote =	{Keywords: Edit distance, Pattern matching, Communication complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.78

Hardness of Dynamic Tree Edit Distance and Friends

Authors: Bingbing Hu, Jakob Nogler, and Barna Saha

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

String Edit Distance is a more-than-classical problem whose behavior in the dynamic setting, where the strings are updated over time, is well studied. A single-character substitution, insertion, or deletion can be processed in time 𝒪̃(n w) when operation costs are positive integers bounded by w [Charalampopoulos, Kociumaka, Mozes, CPM 2020][Gorbachev, Kociumaka, STOC 2025]. If the weights are further uniform (insertions and deletions have equal cost), also an 𝒪̃(n √n)-update time algorithm exists [Charalampopoulos, Kociumaka, Mozes, CPM 2020]. This is a substantial improvement over the static 𝒪(n²) algorithm when w ≪ n or when we are dealing with uniform weights. In contrast, for inherently related problems such as Tree Edit Distance, Dyck Edit Distance, and RNA Folding, it has remained unknown whether it is possible to devise dynamic algorithms with an advantage over the static algorithm. In this paper, we resolve this question by showing that (weighted) Tree Edit Distance, Dyck Edit Distance, and RNA Folding admit no dynamic speedup: under well-known fine-grained assumptions, we show that the best possible algorithm recomputes the solution from scratch after each update. Furthermore, we prove a quadratic per-update lower bound for unweighted Tree Edit Distance under the k-Clique Conjecture. This provides the first separation between dynamic unweighted String Edit Distance and unweighted Tree Edit Distance, problems whose relative difficulty in the static setting is still open.

Cite as

Bingbing Hu, Jakob Nogler, and Barna Saha. Hardness of Dynamic Tree Edit Distance and Friends. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 78:1-78:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hu_et_al:LIPIcs.ITCS.2026.78,
  author =	{Hu, Bingbing and Nogler, Jakob and Saha, Barna},
  title =	{{Hardness of Dynamic Tree Edit Distance and Friends}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{78:1--78:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.78},
  URN =		{urn:nbn:de:0030-drops-253653},
  doi =		{10.4230/LIPIcs.ITCS.2026.78},
  annote =	{Keywords: fine-grained complexity, dynamic lower bounds, pattern matching}
}

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Documents authored by Nogler, Jakob

Undirected Replacement Paths: Dual Fault Reduces to Single Source

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The Communication Complexity of Pattern Matching with Edits Revisited

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Hardness of Dynamic Tree Edit Distance and Friends

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