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DOI: 10.4230/LIPIcs.ITCS.2026.87

Range Longest Increasing Subsequence and Its Relatives

Authors: Karthik C. S. and Saladi Rahul

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

Abstract

Longest increasing subsequence (LIS) is a classical textbook problem which is still actively studied in various computational models. In this work, we present a few results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to Range-LIS is a sequence 𝒮 of n real numbers and a collection 𝒬 of m query ranges and for each query in 𝒬, the goal is to report the LIS of the sequence 𝒮 restricted to that query. Our two main results are for the following generalizations of the Range-LIS problem: 2D Range Queries: In this variant of the Range-LIS problem, each query is a pair of ranges, one of indices and the other of values, and we provide a randomized algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k), where k is the cumulative length of the m output subsequences. This improves on the elementary Õ(mn) runtime algorithm when m = Ω(√n). Previously, the only known result breaking the quadratic barrier was of Tiskin [SODA'10] which could only handle 1D range queries (i.e., each query was a range of indices) and also just outputted the length of the LIS (instead of reporting the subsequence achieving that length). Subsequent to our paper, Gawrychowski, Gorbachev, and Kociumaka in a preprint have extended Tiskin’s approach to handle reporting 1D range queries in O(n(log n)³+m+k) time. Colored Sequences: In this variant of the Range-LIS problem, each element in 𝒮 is colored and for each query in 𝒬, the goal is to report a monochromatic LIS contained in the sequence 𝒮 restricted to that query. For 2D queries, we provide a randomized algorithm for this colored version with running time Õ(mn^{2/3}+ n^{5/3})+O(k). Moreover, for 1D queries, we provide an improved algorithm with running time Õ(mn^{1/2}+ n^{3/2})+O(k). Thus, we again improve on the elementary Õ(mn) runtime algorithm. Additionally, we prove that assuming the well-known Combinatorial Boolean Matrix Multiplication Hypothesis, that the runtime for 1D queries is essentially tight for combinatorial algorithms. Our algorithms combine several tools such as dynamic programming (to precompute increasing subsequences with some desirable properties), geometric data structures (to efficiently compute the dynamic programming entries), random sampling (to capture elements which are part of the LIS), classification of query ranges into large LIS and small LIS, and classification of colors into light and heavy. We believe that our techniques will be of interest to tackle other variants of LIS problem and other range-searching problems.

Cite as

Karthik C. S. and Saladi Rahul. Range Longest Increasing Subsequence and Its Relatives. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 87:1-87:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{karthikc.s._et_al:LIPIcs.ITCS.2026.87,
  author =	{Karthik C. S. and Rahul, Saladi},
  title =	{{Range Longest Increasing Subsequence and Its Relatives}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{87:1--87:20},
  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.2026.87},
  URN =		{urn:nbn:de:0030-drops-253740},
  doi =		{10.4230/LIPIcs.ITCS.2026.87},
  annote =	{Keywords: Longest Increasing Subsequence, Range Query, Fine-Grained Complexity}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.4

Pseudodeterministic Algorithms for Minimum Cut Problems

Authors: Aryan Agarwala and Nithin Varma

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

Abstract

In this paper we present efficient pseudodeterministic algorithms for both the global minimum cut and minimum s-t cut problems. The running time of our algorithm for the global minimum cut problem is asymptotically better than the fastest sequential deterministic global minimum cut algorithm (Henzinger, Li, Rao, Wang; SODA 2024). Furthermore, we implement our algorithm in streaming, PRAM, and cut-query models, where no efficient deterministic global minimum cut algorithms are known.

Cite as

Aryan Agarwala and Nithin Varma. Pseudodeterministic Algorithms for Minimum Cut Problems. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{agarwala_et_al:LIPIcs.ITCS.2026.4,
  author =	{Agarwala, Aryan and Varma, Nithin},
  title =	{{Pseudodeterministic Algorithms for Minimum Cut Problems}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{4:1--4:15},
  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.2026.4},
  URN =		{urn:nbn:de:0030-drops-252917},
  doi =		{10.4230/LIPIcs.ITCS.2026.4},
  annote =	{Keywords: Minimum Cut, Pseudodeterministic Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.53

Average Sensitivity of Geometric Algorithms

Authors: Matthijs Ebbens and Yuichi Yoshida

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

Abstract

In modern applications of geometric algorithms, it is often unrealistic to assume that the input representation fully captures all relevant aspects of the problem, because the input data is often large and dynamic. To address this challenge, we consider the notion of average sensitivity, which is defined as the average earth mover’s distance between the output distributions of the algorithm when run on an input and the same input with one point removed, where the average is over removed points and the distance between two outputs is measured using the symmetric difference size. We start by showing that a number of classical problems from computational geometry, in particular the convex hull, Delaunay triangulation, and Voronoi diagram problems, are "simple" from the viewpoint of average sensitivity by proving tight bounds for the average sensitivity of any algorithm for these problems. Then, we continue by constructing an algorithm with low average sensitivity that computes, for any ε > 0, a set of (1/3+ε)n guards for the art gallery problem. This is the main technical contribution of this work, which combines algorithms from computational geometry with results from the theory of local computation algorithms (LCAs) and property testing.

Cite as

Matthijs Ebbens and Yuichi Yoshida. Average Sensitivity of Geometric Algorithms. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ebbens_et_al:LIPIcs.ITCS.2026.53,
  author =	{Ebbens, Matthijs and Yoshida, Yuichi},
  title =	{{Average Sensitivity of Geometric Algorithms}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{53:1--53:16},
  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.2026.53},
  URN =		{urn:nbn:de:0030-drops-253409},
  doi =		{10.4230/LIPIcs.ITCS.2026.53},
  annote =	{Keywords: Average Sensitivity, Convex Hull, Delaunay Triangulation, Voronoi Diagram, Art Gallery}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.15

Extending EFX Allocations to Further Multi-Graph Classes

Authors: Umang Bhaskar and Yeshwant Pandit

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

The existence of EFX allocations is one of the most significant open questions in fair division. Recent work by Christodoulou, Fiat, Koutsoupias, and Sgouritsa ("Fair allocation in graphs," EC 2023) establishes the existence of EFX allocations for graphical valuations, when agents are vertices in a graph, items are edges, and each item has zero value for all agents other than those at its endpoints. Thus, in this setting, each good has non-zero value for at most two agents, and there is at most one good valued by any pair of agents. This marks one of the few cases when an exact and complete EFX allocation is known to exist for more than three agents. In this work, we partially extend these results to multi-graphs, when each pair of vertices can have more than one edge between them. The existence of EFX allocations in multi-graphs is a natural open question given their existence in simple graphs. We show that EFX allocations exist, and can be computed in polynomial time, for agents with cancelable valuations in the following cases: (i) bipartite multi-graphs, (ii) multi-trees with monotone valuations, and (iii) multi-graphs with girth (2t-1), where t is the chromatic number of the multi-graph. The existence of EFX in cycle multi-graphs follows from (i), (iii), and the known existence of EFX for three agents.

Cite as

Umang Bhaskar and Yeshwant Pandit. Extending EFX Allocations to Further Multi-Graph Classes. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhaskar_et_al:LIPIcs.FSTTCS.2025.15,
  author =	{Bhaskar, Umang and Pandit, Yeshwant},
  title =	{{Extending EFX Allocations to Further Multi-Graph Classes}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.15},
  URN =		{urn:nbn:de:0030-drops-250958},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.15},
  annote =	{Keywords: Fair Division, EFX, Multi-graphs}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.11

Courcelle’s Theorem for Lipschitz Continuity

Authors: Tatsuya Gima, Soh Kumabe, and Yuichi Yoshida

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Lipschitz continuity of algorithms, introduced by Kumabe and Yoshida (FOCS'23), measures the stability of an algorithm against small input perturbations. Algorithms with small Lipschitz continuity are desirable, as they ensure reliable decision-making and reproducible scientific research. Several studies have proposed Lipschitz continuous algorithms for various combinatorial optimization problems, but these algorithms are problem-specific, requiring a separate design for each problem. To address this issue, we provide the first algorithmic meta-theorem in the field of Lipschitz continuous algorithms. Our result can be seen as a Lipschitz continuous analogue of Courcelle’s theorem, which offers Lipschitz continuous algorithms for problems on bounded-treewidth graphs. Specifically, we consider the problem of finding a vertex set in a graph that maximizes or minimizes the total weight, subject to constraints expressed in monadic second-order logic (MSO₂). We show that for any ε > 0, there exists a (1±ε)-approximation algorithm for the problem with a polylogarithmic Lipschitz constant on bounded treewidth graphs. On such graphs, our result outperforms most existing Lipschitz continuous algorithms in terms of approximability and/or Lipschitz continuity. Further, we provide similar results for problems on bounded-clique-width graphs subject to constraints expressed in MSO₁. Additionally, we construct a Lipschitz continuous version of Baker’s decomposition using our meta-theorem as a subroutine.

Cite as

Tatsuya Gima, Soh Kumabe, and Yuichi Yoshida. Courcelle’s Theorem for Lipschitz Continuity. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gima_et_al:LIPIcs.ESA.2025.11,
  author =	{Gima, Tatsuya and Kumabe, Soh and Yoshida, Yuichi},
  title =	{{Courcelle’s Theorem for Lipschitz Continuity}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.11},
  URN =		{urn:nbn:de:0030-drops-244793},
  doi =		{10.4230/LIPIcs.ESA.2025.11},
  annote =	{Keywords: Fixed-Parameter Tractability, Algorithmic Meta-Theorem, Lipschitz Continuity}
}

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Invited Talk

DOI: 10.4230/LIPIcs.ICALP.2025.2

Let’s Try to Be More Tolerant: On Tolerant Property Testing and Distance Approximation (Invited Talk)

Authors: Dana Ron

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

This short paper accompanies an invited talk given at ICALP2025. It is an informal, high-level presentation of tolerant testing and distance approximation. It includes some general results as well as a few specific ones, with the aim of providing a taste of this research direction within the area of sublinear algorithms.

Cite as

Dana Ron. Let’s Try to Be More Tolerant: On Tolerant Property Testing and Distance Approximation (Invited Talk). In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 2:1-2:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ron:LIPIcs.ICALP.2025.2,
  author =	{Ron, Dana},
  title =	{{Let’s Try to Be More Tolerant: On Tolerant Property Testing and Distance Approximation}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{2:1--2:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.2},
  URN =		{urn:nbn:de:0030-drops-233798},
  doi =		{10.4230/LIPIcs.ICALP.2025.2},
  annote =	{Keywords: Sublinear Algorithms, Tolerant Property Testing, Distance Approximation}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2025.56

Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions

Authors: Martin G. Herold, Danupon Nanongkai, Joachim Spoerhase, Nithin Varma, and Zihang Wu

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Geometric data structures have been extensively studied in the regime where the dimension is much smaller than the number of input points. But in many scenarios in Machine Learning, the dimension can be much higher than the number of points and can be so high that the data structure might be unable to read and store all coordinates of the input and query points. Inspired by these scenarios and related studies in feature selection and explainable clustering, we initiate the study of geometric data structures in this ultra-high dimensional regime. Our focus is the approximate nearest neighbor problem. In this problem, we are given a set of n points C ⊆ ℝ^d and have to produce a small data structure that can quickly answer the following query: given q ∈ ℝ^d, return a point c ∈ C that is approximately nearest to q, where the distance is under 𝓁₁, 𝓁₂, or other norms. Many groundbreaking (1+ε)-approximation algorithms have recently been discovered for 𝓁₁- and 𝓁₂-norm distances in the regime where d≪ n. The main question in this paper is: Is there a data structure with sublinear (o(nd)) space and sublinear (o(d)) query time when d≫ n? This question can be partially answered from the machine-learning literature: - For 𝓁₁-norm distances, an Õ(log(n))-approximation data structure with Õ(n log d) space and O(n) query time can be obtained from explainable clustering techniques [Dasgupta et al. ICML'20; Makarychev and Shan ICML'21; Esfandiari, Mirrokni, and Narayanan SODA'22; Gamlath et al. NeurIPS'21; Charikar and Hu SODA'22]. - For 𝓁₂-norm distances, a (√3+ε)-approximation data structure with Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time can be obtained from feature selection techniques [Boutsidis, Drineas, and Mahoney NeurIPS'09; Boutsidis et al. IEEE Trans. Inf. Theory'15; Cohen et al. STOC'15]. - For 𝓁_p-norm distances, a O(n^{p-1}log²(n))-approximation data structure with O(nlog(n) + nlog(d)) space and O(n) query time can be obtained from the explainable clustering algorithms of [Gamlath et al. NeurIPS'21]. An important open problem is whether a (1+ε)-approximation data structure exists. This is not known for any norm, even with higher (e.g. poly(n)⋅ o(d)) space and query time. In this paper, we answer this question affirmatively. We present (1+ε)-approximation data structures with the following guarantees. - For 𝓁₁- and 𝓁₂-norm distances: Õ(n log(d)/poly(ε)) space and Õ(n/poly(ε)) query time. We show that these space and time bounds are tight up to poly (log n/ε) factors. - For 𝓁_p-norm distances: Õ(n² log(d) (log log(n)/ε)^p) space and Õ (n(log log(n)/ε)^p) query time. Via simple reductions, our data structures imply sublinear-in-d data structures for some other geometric problems; e.g. approximate orthogonal range search (in the style of [Arya and Mount SoCG'95]), furthest neighbor, and give rise to a sublinear O(1)-approximate representation of k-median and k-means clustering. We hope that this paper inspires future work on sublinear geometric data structures.

Cite as

Martin G. Herold, Danupon Nanongkai, Joachim Spoerhase, Nithin Varma, and Zihang Wu. Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{herold_et_al:LIPIcs.SoCG.2025.56,
  author =	{Herold, Martin G. and Nanongkai, Danupon and Spoerhase, Joachim and Varma, Nithin and Wu, Zihang},
  title =	{{Sublinear Data Structures for Nearest Neighbor in Ultra High Dimensions}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{56:1--56:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.56},
  URN =		{urn:nbn:de:0030-drops-232087},
  doi =		{10.4230/LIPIcs.SoCG.2025.56},
  annote =	{Keywords: sublinear data structure, approximate nearest neighbor}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.94

New Direct Sum Tests

Authors: Alek Westover, Edward Yu, and Kai Zhe Zheng

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

A function f:[n]^{d} → 𝔽₂ is a direct sum if there are functions L_i:[n] → 𝔽₂ such that f(x) = ∑_i L_i(x_i). In this work we give multiple results related to the property testing of direct sums. Our first result concerns a test proposed by Dinur and Golubev in [Dinur and Golubev, 2019]. We call their test the Diamond test and show that it is indeed a direct sum tester. More specifically, we show that if a function f is ε-far from being a direct sum function, then the Diamond test rejects f with probability at least Ω_{n,ε}(1). Even in the case of n = 2, the Diamond test is, to the best of our knowledge, novel and yields a new tester for the classic property of affinity. Apart from the Diamond test, we also analyze a broad family of direct sum tests, which at a high level, run an arbitrary affinity test on the restriction of f to a random hypercube inside of [n]^d. This family of tests includes the direct sum test analyzed in [Dinur and Golubev, 2019], but does not include the Diamond test. As an application of our result, we obtain a direct sum test which works in the online adversary model of [Iden Kalemaj et al., 2022]. Finally, we also discuss a Fourier analytic interpretation of the diamond tester in the n = 2 case, as well as prove local correction results for direct sum as conjectured by [Dinur and Golubev, 2019].

Cite as

Alek Westover, Edward Yu, and Kai Zhe Zheng. New Direct Sum Tests. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 94:1-94:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{westover_et_al:LIPIcs.ITCS.2025.94,
  author =	{Westover, Alek and Yu, Edward and Zheng, Kai Zhe},
  title =	{{New Direct Sum Tests}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{94:1--94:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.94},
  URN =		{urn:nbn:de:0030-drops-227229},
  doi =		{10.4230/LIPIcs.ITCS.2025.94},
  annote =	{Keywords: Linearity testing, Direct sum, Grids}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.65

Online Versus Offline Adversaries in Property Testing

Authors: Esty Kelman, Ephraim Linder, and Sofya Raskhodnikova

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

We study property testing with incomplete or noisy inputs. The models we consider allow for adversarial manipulation of the input, but differ in whether the manipulation can be done only offline, i.e., before the execution of the algorithm, or online, i.e., as the algorithm runs. The manipulations by an adversary can come in the form of erasures or corruptions. We compare the query complexity and the randomness complexity of property testing in the offline and online models. Kalemaj, Raskhodnikova, and Varma (Theory Comput. `23) provide properties that can be tested with a small number of queries with offline erasures, but cannot be tested at all with online erasures. We demonstrate that the two models are incomparable in terms of query complexity: we construct properties that can be tested with a constant number of queries in the online corruption model, but require querying a significant fraction of the input in the offline erasure model. We also construct properties that exhibit a strong separation between the randomness complexity of testing in the presence of offline and online adversaries: testing these properties in the online model requires exponentially more random bits than in the offline model, even when they are tested with nearly the same number of queries in both models. Our randomness separation relies on a novel reduction from randomness-efficient testers in the adversarial online model to query-efficient testers in the standard model.

Cite as

Esty Kelman, Ephraim Linder, and Sofya Raskhodnikova. Online Versus Offline Adversaries in Property Testing. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 65:1-65:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kelman_et_al:LIPIcs.ITCS.2025.65,
  author =	{Kelman, Esty and Linder, Ephraim and Raskhodnikova, Sofya},
  title =	{{Online Versus Offline Adversaries in Property Testing}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{65:1--65:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.65},
  URN =		{urn:nbn:de:0030-drops-226933},
  doi =		{10.4230/LIPIcs.ITCS.2025.65},
  annote =	{Keywords: Property Testing, Online Adversary, Offline Adversary, Query Complexity, Randomness Complexity, Separations}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.98

Strongly Sublinear Algorithms for Testing Pattern Freeness

Authors: Ilan Newman and Nithin Varma

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract

For a permutation π:[k] → [k], a function f:[n] → ℝ contains a π-appearance if there exists 1 ≤ i₁ < i₂ < … < i_k ≤ n such that for all s,t ∈ [k], f(i_s) < f(i_t) if and only if π(s) < π(t). The function is π-free if it has no π-appearances. In this paper, we investigate the problem of testing whether an input function f is π-free or whether f differs on at least ε n values from every π-free function. This is a generalization of the well-studied monotonicity testing and was first studied by Newman, Rabinovich, Rajendraprasad and Sohler [Ilan Newman et al., 2019]. We show that for all constants k ∈ ℕ, ε ∈ (0,1), and permutation π:[k] → [k], there is a one-sided error ε-testing algorithm for π-freeness of functions f:[n] → ℝ that makes Õ(n^o(1)) queries. We improve significantly upon the previous best upper bound O(n^{1 - 1/(k-1)}) by Ben-Eliezer and Canonne [Omri Ben-Eliezer and Clément L. Canonne, 2018]. Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.

Cite as

Ilan Newman and Nithin Varma. Strongly Sublinear Algorithms for Testing Pattern Freeness. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 98:1-98:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{newman_et_al:LIPIcs.ICALP.2022.98,
  author =	{Newman, Ilan and Varma, Nithin},
  title =	{{Strongly Sublinear Algorithms for Testing Pattern Freeness}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{98:1--98:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.98},
  URN =		{urn:nbn:de:0030-drops-164390},
  doi =		{10.4230/LIPIcs.ICALP.2022.98},
  annote =	{Keywords: Property testing, Pattern freeness, Sublinear algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2022.90

Sublinear-Time Computation in the Presence of Online Erasures

Authors: Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

We initiate the study of sublinear-time algorithms that access their input via an online adversarial erasure oracle. After answering each query to the input object, such an oracle can erase t input values. Our goal is to understand the complexity of basic computational tasks in extremely adversarial situations, where the algorithm’s access to data is blocked during the execution of the algorithm in response to its actions. Specifically, we focus on property testing in the model with online erasures. We show that two fundamental properties of functions, linearity and quadraticity, can be tested for constant t with asymptotically the same complexity as in the standard property testing model. For linearity testing, we prove tight bounds in terms of t, showing that the query complexity is Θ(log t). In contrast to linearity and quadraticity, some other properties, including sortedness and the Lipschitz property of sequences, cannot be tested at all, even for t = 1. Our investigation leads to a deeper understanding of the structure of violations of linearity and other widely studied properties.

Cite as

Iden Kalemaj, Sofya Raskhodnikova, and Nithin Varma. Sublinear-Time Computation in the Presence of Online Erasures. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 90:1-90:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kalemaj_et_al:LIPIcs.ITCS.2022.90,
  author =	{Kalemaj, Iden and Raskhodnikova, Sofya and Varma, Nithin},
  title =	{{Sublinear-Time Computation in the Presence of Online Erasures}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{90:1--90:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.90},
  URN =		{urn:nbn:de:0030-drops-156867},
  doi =		{10.4230/LIPIcs.ITCS.2022.90},
  annote =	{Keywords: Randomized algorithms, property testing, Fourier analysis, linear functions, quadratic functions, Lipschitz and monotone functions, sorted sequences}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2021.14

Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four

Authors: Suryajith Chillara

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d^{O(1)}-variate and degree d polynomial P_{d} ∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to P_d, then C must have size 2^Ω(√dlog{d}). The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC⁰ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC⁰ can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form - sums of powers of polynomials) of 2^(log^O(1) n) size. Thus they argued that a 2^{ω(polylog n)} "functional" lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the "corresponding Boolean function" of Q against non-uniform ACC⁰. In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log²n) ≤ d ≤ n^{0.01}, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O(d/k²) that functionally computes Iterated Matrix Multiplication polynomial IMM_{n,d} (∈ VP) over {0,1}^{n²d} must have size n^Ω(k). Since Iterated Matrix Multiplication IMM_{n,d} over {0,1}^{n²d} is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC⁰ from GapL. For the sake of completeness, we also show a syntactic size lower bound against any Σ∧ΣΠ circuit computing IMM_{n,d} (for the same regime of d) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute IMM_{n,d}, for a slightly larger range of individual degree.

Cite as

Suryajith Chillara. Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chillara:LIPIcs.FSTTCS.2021.14,
  author =	{Chillara, Suryajith},
  title =	{{Functional Lower Bounds for Restricted Arithmetic Circuits of Depth Four}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.14},
  URN =		{urn:nbn:de:0030-drops-155251},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.14},
  annote =	{Keywords: Functional Lower Bounds, Boolean Circuit Lower Bounds, Depth Four, Connections to Boolean Complexity, Iterated Matrix Multiplication}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2021.100

New Sublinear Algorithms and Lower Bounds for LIS Estimation

Authors: Ilan Newman and Nithin Varma

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract

Estimating the length of the longest increasing subsequence (LIS) in an array is a problem of fundamental importance. Despite the significance of the LIS estimation problem and the amount of attention it has received, there are important aspects of the problem that are not yet fully understood. There are no better lower bounds for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, and also provide novel algorithms that complement our lower bound. Specifically, we show that for every ε ∈ (0,1), every nonadaptive algorithm that outputs an estimate of the LIS length in an array of length n to within an additive error of ε n has to make log^{Ω(log (1/ε))} n queries. Next, we design nonadaptive LIS estimation algorithms whose complexity decreases as the number of distinct values, r, in the array decreases. We first present a simple algorithm that makes Õ(r/ε³) queries and approximates the LIS length with an additive error bounded by ε n. This algorithm has better complexity than the best previously known adaptive algorithm (Saks and Seshadhri; 2017) for the same problem when r ≪ polylog (n). We use our algorithm to construct a nonadaptive algorithm with query complexity Õ(√r⋅ poly(1/λ)) that, when the LIS is of length at least λ n, outputs a multiplicative Ω(λ)-approximation to the LIS length. Our algorithm improves upon the state of the art nonadaptive LIS estimation algorithm (Rubinstein, Seddighin, Song, and Sun; 2019) in terms of the approximation guarantee. Finally, we present a O(log n)-query nonadaptive erasure-resilient tester for monotonicity. Our result implies that lower bounds on erasure-resilient testing of monotonicity does not give good lower bounds for LIS estimation. It also implies that nonadaptive tolerant testing is strictly harder than nonadaptive erasure-resilient testing for the natural property of monotonicity.

Cite as

Ilan Newman and Nithin Varma. New Sublinear Algorithms and Lower Bounds for LIS Estimation. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 100:1-100:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{newman_et_al:LIPIcs.ICALP.2021.100,
  author =	{Newman, Ilan and Varma, Nithin},
  title =	{{New Sublinear Algorithms and Lower Bounds for LIS Estimation}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{100:1--100:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.100},
  URN =		{urn:nbn:de:0030-drops-141699},
  doi =		{10.4230/LIPIcs.ICALP.2021.100},
  annote =	{Keywords: longest increasing subsequence, monotonicity, distance estimation, sublinear algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.80

Erasure-Resilient Sublinear-Time Graph Algorithms

Authors: Amit Levi, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. `06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms `08) and Eden et al. (ICALP `17). We conclude with a discussion of our model and open questions raised by our work.

Cite as

Amit Levi, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Erasure-Resilient Sublinear-Time Graph Algorithms. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 80:1-80:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{levi_et_al:LIPIcs.ITCS.2021.80,
  author =	{Levi, Amit and Pallavoor, Ramesh Krishnan S. and Raskhodnikova, Sofya and Varma, Nithin},
  title =	{{Erasure-Resilient Sublinear-Time Graph Algorithms}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{80:1--80:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.80},
  URN =		{urn:nbn:de:0030-drops-136192},
  doi =		{10.4230/LIPIcs.ITCS.2021.80},
  annote =	{Keywords: Graph property testing, Computing with incomplete information, Approximating graph parameters}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.33

Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists)

Authors: Noga Ron-Zewi, Ronen Shaltiel, and Nithin Varma

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

A binary code Enc:{0,1}^k → {0,1}ⁿ is (1/2-ε,L)-list decodable if for every w ∈ {0,1}ⁿ, there exists a set List(w) of size at most L, containing all messages m ∈ {0,1}^k such that the relative Hamming distance between Enc(m) and w is at most 1/2-ε. A q-query local list-decoder for Enc is a randomized procedure Dec that when given oracle access to a string w, makes at most q oracle calls, and for every message m ∈ List(w), with high probability, there exists j ∈ [L] such that for every i ∈ [k], with high probability, Dec^w(i,j) = m_i. We prove lower bounds on q, that apply even if L is huge (say L = 2^{k^{0.9}}) and the rate of Enc is small (meaning that n ≥ 2^{k}): - For ε = 1/k^{ν} for some constant 0 < ν < 1, we prove a lower bound of q = Ω(log(1/δ)/ε²), where δ is the error probability of the local list-decoder. This bound is tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of q = O(log(1/δ)/ε²) for the Hadamard code (which has n = 2^k). This bound extends an earlier work of Grinberg, Shaltiel and Viola (FOCS 2018) which only works if n ≤ 2^{k^ν} and the number of coins tossed by Dec is small (and therefore does not apply to the Hadamard code, or other codes with low rate). - For smaller ε, we prove a lower bound of roughly q = Ω(1/(√ε)). To the best of our knowledge, this is the first lower bound on the number of queries of local list-decoders that gives q ≥ k for small ε. Local list-decoders with small ε form the key component in the celebrated theorem of Goldreich and Levin that extracts a hard-core predicate from a one-way function. We show that black-box proofs cannot improve the Goldreich-Levin theorem and produce a hard-core predicate that is hard to predict with probability 1/2 + 1/𝓁^ω(1) when provided with a one-way function f:{0,1}^𝓁 → {0,1}^𝓁, where f is such that circuits of size poly(𝓁) cannot invert f with probability ρ = 1/2^√𝓁 (or even ρ = 1/2^Ω(𝓁)). This limitation applies to any proof by black-box reduction (even if the reduction is allowed to use nonuniformity and has oracle access to f).

Cite as

Noga Ron-Zewi, Ronen Shaltiel, and Nithin Varma. Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ronzewi_et_al:LIPIcs.ITCS.2021.33,
  author =	{Ron-Zewi, Noga and Shaltiel, Ronen and Varma, Nithin},
  title =	{{Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists)}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{33:1--33:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.33},
  URN =		{urn:nbn:de:0030-drops-135724},
  doi =		{10.4230/LIPIcs.ITCS.2021.33},
  annote =	{Keywords: Local list-decoding, Hard-core predicates, Black-box reduction, Hadamard code}
}

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