30 Search Results for "M�ndez, Gonzalo"

Document
DOI: 10.4230/LIPIcs.CPM.2023.22
Merging Sorted Lists of Similar Strings

Authors: Gene Myers

Published in: LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

Abstract
Merging T sorted, non-redundant lists containing M elements into a single sorted, non-redundant result of size N ≥ M/T is a classic problem typically solved practically in O(M log T) time with a priority-queue data structure the most basic of which is the simple heap. We revisit this problem in the situation where the list elements are strings and the lists contain many identical or nearly identical elements. By keeping simple auxiliary information with each heap node, we devise an O(M log T+S) worst-case method that performs no more character comparisons than the sum of the lengths of all the strings S, and another O(M log (T/e¯)+S) method that becomes progressively more efficient as a function of the fraction of equal elements e¯ = M/N between input lists, reaching linear time when the lists are all identical. The methods perform favorably in practice versus an alternate formulation based on a trie.
Cite as

Gene Myers. Merging Sorted Lists of Similar Strings. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{myers:LIPIcs.CPM.2023.22,
  author =	{Myers, Gene},
  title =	{{Merging Sorted Lists of Similar Strings}},
  booktitle =	{34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-276-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{259},
  editor =	{Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.22},
  URN =		{urn:nbn:de:0030-drops-179763},
  doi =		{10.4230/LIPIcs.CPM.2023.22},
  annote =	{Keywords: heap, trie, longest common prefix}
}
Document
DOI: 10.4230/LIPIcs.CPM.2023.24
Computing MEMs on Repetitive Text Collections

Authors: Gonzalo Navarro

Published in: LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

Abstract
We consider the problem of computing the Maximal Exact Matches (MEMs) of a given pattern P[1..m] on a large repetitive text collection T[1..n], which is represented as a (hopefully much smaller) run-length context-free grammar of size g_{rl}. We show that the problem can be solved in time O(m² log^ε n), for any constant ε > 0, on a data structure of size O(g_{rl}). Further, on a locally consistent grammar of size O(δ log n/δ), the time decreases to O(m log m(log m + log^ε n)). The value δ is a function of the substring complexity of T and Ω(δ log n/δ) is a tight lower bound on the compressibility of repetitive texts T, so our structure has optimal size in terms of n and δ.
Cite as

Gonzalo Navarro. Computing MEMs on Repetitive Text Collections. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{navarro:LIPIcs.CPM.2023.24,
  author =	{Navarro, Gonzalo},
  title =	{{Computing MEMs on Repetitive Text Collections}},
  booktitle =	{34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-276-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{259},
  editor =	{Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.24},
  URN =		{urn:nbn:de:0030-drops-179787},
  doi =		{10.4230/LIPIcs.CPM.2023.24},
  annote =	{Keywords: grammar-based indices, maximal exact matches, locally consistent grammars, substring complexity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.5
TSP in a Simple Polygon

Authors: Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We study the Traveling Salesman Problem inside a simple polygon. In this problem, which we call tsp in a simple polygon, we wish to compute a shortest tour that visits a given set S of n sites inside a simple polygon P with m edges while staying inside the polygon. This natural problem has, to the best of our knowledge, not been studied so far from a theoretical perspective. It can be solved exactly in poly(n,m) + 2^O(√nlog n) time, using an algorithm by Marx, Pilipczuk, and Pilipczuk (FOCS 2018) for subset tsp as a subroutine. We present a much simpler algorithm that solves tsp in a simple polygon directly and that has the same running time.
Cite as

Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous. TSP in a Simple Polygon. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alkema_et_al:LIPIcs.ESA.2022.5,
  author =	{Alkema, Henk and de Berg, Mark and Monemizadeh, Morteza and Theocharous, Leonidas},
  title =	{{TSP in a Simple Polygon}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.5},
  URN =		{urn:nbn:de:0030-drops-169434},
  doi =		{10.4230/LIPIcs.ESA.2022.5},
  annote =	{Keywords: Traveling Salesman Problem, Subexponential algorithms, TSP with obstacles}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.6
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Authors: Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.
Cite as

Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6,
  author =	{Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos},
  title =	{{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.6},
  URN =		{urn:nbn:de:0030-drops-169444},
  doi =		{10.4230/LIPIcs.ESA.2022.6},
  annote =	{Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.13
Online Metric Allocation and Time-Varying Regularization

Authors: Nikhil Bansal and Christian Coester

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We introduce a general online allocation problem that connects several of the most fundamental problems in online optimization. Let M be an n-point metric space. Consider a resource that can be allocated in arbitrary fractions to the points of M. At each time t, a convex monotone cost function c_t: [0,1] → ℝ_+ appears at some point r_t ∈ M. In response, an algorithm may change the allocation of the resource, paying movement cost as determined by the metric and service cost c_t(x_{r_t}), where x_{r_t} is the fraction of the resource at r_t at the end of time t. For example, when the cost functions are c_t(x) = α x, this is equivalent to randomized MTS, and when the cost functions are c_t(x) = ∞⋅1_{x < 1/k}, this is equivalent to fractional k-server. Because of an inherent scale-freeness property of the problem, existing techniques for MTS and k-server fail to achieve similar guarantees for metric allocation. To handle this, we consider a generalization of the online multiplicative update method where we decouple the rate at which a variable is updated from its value, resulting in interesting new dynamics. We use this to give an O(log n)-competitive algorithm for weighted star metrics. We then show how this corresponds to an extension of the online mirror descent framework to a setting where the regularizer is time-varying. Using this perspective, we further refine the guarantees of our algorithm. We also consider the case of non-convex cost functions. Using a simple 𝓁₂²-regularizer, we give tight bounds of Θ(n) on tree metrics, which imply deterministic and randomized competitive ratios of O(n²) and O(nlog n) respectively on arbitrary metrics.
Cite as

Nikhil Bansal and Christian Coester. Online Metric Allocation and Time-Varying Regularization. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bansal_et_al:LIPIcs.ESA.2022.13,
  author =	{Bansal, Nikhil and Coester, Christian},
  title =	{{Online Metric Allocation and Time-Varying Regularization}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.13},
  URN =		{urn:nbn:de:0030-drops-169515},
  doi =		{10.4230/LIPIcs.ESA.2022.13},
  annote =	{Keywords: Online algorithms, competitive analysis, k-server, metrical task systems, mirror descent, regularization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.30
Search-Space Reduction via Essential Vertices

Authors: Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.
Cite as

Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-Space Reduction via Essential Vertices. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bumpus_et_al:LIPIcs.ESA.2022.30,
  author =	{Bumpus, Benjamin Merlin and Jansen, Bart M. P. and de Kroon, Jari J. H.},
  title =	{{Search-Space Reduction via Essential Vertices}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.30},
  URN =		{urn:nbn:de:0030-drops-169687},
  doi =		{10.4230/LIPIcs.ESA.2022.30},
  annote =	{Keywords: fixed-parameter tractability, essential vertices, covering versus packing}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.31
Width Helps and Hinders Splitting Flows

Authors: Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.
Cite as

Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{caceres_et_al:LIPIcs.ESA.2022.31,
  author =	{C\'{a}ceres, Manuel and Cairo, Massimo and Grigorjew, Andreas and Khan, Shahbaz and Mumey, Brendan and Rizzi, Romeo and Tomescu, Alexandru I. and Williams, Lucia},
  title =	{{Width Helps and Hinders Splitting Flows}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.31},
  URN =		{urn:nbn:de:0030-drops-169695},
  doi =		{10.4230/LIPIcs.ESA.2022.31},
  annote =	{Keywords: Flow decomposition, approximation algorithms, graph width}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.32
Counting Simplices in Hypergraph Streams

Authors: Amit Chakrabarti and Themistoklis Haris

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H. We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs.
Cite as

Amit Chakrabarti and Themistoklis Haris. Counting Simplices in Hypergraph Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.ESA.2022.32,
  author =	{Chakrabarti, Amit and Haris, Themistoklis},
  title =	{{Counting Simplices in Hypergraph Streams}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.32},
  URN =		{urn:nbn:de:0030-drops-169705},
  doi =		{10.4230/LIPIcs.ESA.2022.32},
  annote =	{Keywords: data streaming, graph algorithms, hypergraphs, sub-linear algorithms, triangle counting}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.34
Distinct Elements in Streams: An Algorithm for the (Text) Book

Authors: Sourav Chakraborty, N. V. Vinodchandran¹, and Kuldeep S. Meel

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Given a data stream 𝒟 = ⟨ a₁, a₂, …, a_m ⟩ of m elements where each a_i ∈ [n], the Distinct Elements problem is to estimate the number of distinct elements in 𝒟. Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it. All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.
Cite as

Sourav Chakraborty, N. V. Vinodchandran¹, and Kuldeep S. Meel. Distinct Elements in Streams: An Algorithm for the (Text) Book. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 34:1-34:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2022.34,
  author =	{Chakraborty, Sourav and Vinodchandran¹, N. V. and Meel, Kuldeep S.},
  title =	{{Distinct Elements in Streams: An Algorithm for the (Text) Book}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{34:1--34:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.34},
  URN =		{urn:nbn:de:0030-drops-169725},
  doi =		{10.4230/LIPIcs.ESA.2022.34},
  annote =	{Keywords: F₀ Estimation, Streaming, Sampling}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.35
Approximate Circular Pattern Matching

Authors: Panagiotis Charalampopoulos, Tomasz Kociumaka, Jakub Radoszewski, Solon P. Pissis, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We investigate the complexity of approximate circular pattern matching (CPM, in short) under the Hamming and edit distance. Under each of these two basic metrics, we are given a length-n text T, a length-m pattern P, and a positive integer threshold k, and we are to report all starting positions (called occurrences) of fragments of T that are at distance at most k from some cyclic rotation of P. In the decision version of the problem, we are to check if there is any such occurrence. All previous results for approximate CPM were either average-case upper bounds or heuristics, with the exception of the work of Charalampopoulos et al. [CKP^+, JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP^+, JCSS'21] from 𝒪(n+(n/m) ⋅ k⁴) to 𝒪(n+(n/m) ⋅ k³ log log k) time; for the edit distance, we give an 𝒪(nk²)-time algorithm. Notably, for the decision versions and wide parameter-ranges, we give algorithms whose complexities are almost identical to the state-of-the-art for standard (i.e., non-circular) approximate pattern matching: - For the decision version of the approximate CPM problem under the Hamming distance, we obtain an 𝒪(n+(n/m) ⋅ k² log k / log log k)-time algorithm, which works in 𝒪(n) time whenever k = 𝒪(√{m log log m / log m}). In comparison, the fastest algorithm for the standard counterpart of the problem, by Chan et al. [CGKKP, STOC’20], runs in 𝒪(n) time only for k = 𝒪(√m). We achieve this result via a reduction to a geometric problem by building on ideas from [CKP^+, JCSS'21] and Charalampopoulos et al. [CKW, FOCS'20]. - For the decision version of the approximate CPM problem under the edit distance, the 𝒪(nklog³ k) runtime of our algorithm near matches the 𝒪(nk) runtime of the Landau-Vishkin algorithm [LV, J. Algorithms'89] for approximate pattern matching under edit distance; the latter algorithm remains the fastest known for k = Ω(m^{2/5}). As a stepping stone, we propose an 𝒪(nklog³ k)-time algorithm for solving the Longest Prefix k'-Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all k' ∈ {1,…,k}. Our algorithm is based on Tiskin’s theory of seaweeds [Tiskin, Math. Comput. Sci.'08], with recent advancements (see Charalampopoulos et al. [CKW, FOCS'22]), and on exploiting the seaweeds' relation to Monge matrices. In contrast, we obtain a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute the Strong Exponential Time Hypothesis.
Cite as

Panagiotis Charalampopoulos, Tomasz Kociumaka, Jakub Radoszewski, Solon P. Pissis, Wojciech Rytter, Tomasz Waleń, and Wiktor Zuba. Approximate Circular Pattern Matching. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{charalampopoulos_et_al:LIPIcs.ESA.2022.35,
  author =	{Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Radoszewski, Jakub and Pissis, Solon P. and Rytter, Wojciech and Wale\'{n}, Tomasz and Zuba, Wiktor},
  title =	{{Approximate Circular Pattern Matching}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{35:1--35:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.35},
  URN =		{urn:nbn:de:0030-drops-169738},
  doi =		{10.4230/LIPIcs.ESA.2022.35},
  annote =	{Keywords: approximate circular pattern matching, Hamming distance, edit distance}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.40
A Simpler QPTAS for Scheduling Jobs with Precedence Constraints

Authors: Syamantak Das and Andreas Wiese

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We study the classical scheduling problem of minimizing the makespan of a set of unit size jobs with precedence constraints on parallel identical machines. Research on the problem dates back to the landmark paper by Graham from 1966 who showed that the simple List Scheduling algorithm is a (2-1/m)-approximation. Interestingly, it is open whether the problem is NP-hard if m = 3 which is one of the few remaining open problems in the seminal book by Garey and Johnson. Recently, quite some progress has been made for the setting that m is a constant. In a break-through paper, Levey and Rothvoss presented a (1+ε)-approximation with a running time of n^{(log n)^{O((m²/ε²)log log n)}} [STOC 2016, SICOMP 2019] and this running time was improved to quasi-polynomial by Garg [ICALP 2018] and to even n^O_{m,ε}(log³log n) by Li [SODA 2021]. These results use techniques like LP-hierarchies, conditioning on certain well-selected jobs, and abstractions like (partial) dyadic systems and virtually valid schedules. In this paper, we present a QPTAS for the problem which is arguably simpler than the previous algorithms. We just guess the positions of certain jobs in the optimal solution, recurse on a set of guessed subintervals, and fill in the remaining jobs with greedy routines. We believe that also our analysis is more accessible, in particular since we do not use (LP-)hierarchies or abstractions of the problem like the ones above, but we guess properties of the optimal solution directly.
Cite as

Syamantak Das and Andreas Wiese. A Simpler QPTAS for Scheduling Jobs with Precedence Constraints. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 40:1-40:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{das_et_al:LIPIcs.ESA.2022.40,
  author =	{Das, Syamantak and Wiese, Andreas},
  title =	{{A Simpler QPTAS for Scheduling Jobs with Precedence Constraints}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{40:1--40:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.40},
  URN =		{urn:nbn:de:0030-drops-169782},
  doi =		{10.4230/LIPIcs.ESA.2022.40},
  annote =	{Keywords: makespan minimization, precedence constraints, QPTAS}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.42
Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem

Authors: Dipan Dey and Manoj Gupta

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
In a graph G with a source s, we design a distance oracle that can answer the following query: Query(s,t,e) - find the length of shortest path from a fixed source s to any destination vertex t while avoiding any edge e. We design a deterministic algorithm that builds such an oracle in Õ(m √n) time. Our oracle uses Õ(n √n) space and can answer queries in Õ(1) time. Our oracle is an improvement of the work of Bilò et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in Õ(m √n+n²) time. Using our distance oracle, we also solve the single source replacement path problem (Ssrp problem). Chechik and Cohen (SODA 2019) designed a randomized combinatorial algorithm to solve the Ssrp problem. The running time of their algorithm is Õ(m √n + n²). In this paper, we show that the Ssrp problem can be solved in Õ(m √n + |ℛ|) time, where ℛ is the output set of the Ssrp problem in G. Our Ssrp algorithm is optimal (upto polylogarithmic factor) as there is a conditional lower bound of Ω(m √n) for any combinatorial algorithm that solves this problem.
Cite as

Dipan Dey and Manoj Gupta. Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dey_et_al:LIPIcs.ESA.2022.42,
  author =	{Dey, Dipan and Gupta, Manoj},
  title =	{{Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path Problem}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{42:1--42:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.42},
  URN =		{urn:nbn:de:0030-drops-169800},
  doi =		{10.4230/LIPIcs.ESA.2022.42},
  annote =	{Keywords: distance sensitivity oracle, single-source replacement paths}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.65
Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs

Authors: Monika Henzinger, Ami Paz, and A. R. Sricharan

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ε}) update time and O(m^{1 -ε}) query time, for any small ε > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.
Cite as

Monika Henzinger, Ami Paz, and A. R. Sricharan. Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{henzinger_et_al:LIPIcs.ESA.2022.65,
  author =	{Henzinger, Monika and Paz, Ami and Sricharan, A. R.},
  title =	{{Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{65:1--65:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.65},
  URN =		{urn:nbn:de:0030-drops-170035},
  doi =		{10.4230/LIPIcs.ESA.2022.65},
  annote =	{Keywords: Dynamic graph algorithms, Expander graphs, Power-law graphs}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.68
Maximum Weight b-Matchings in Random-Order Streams

Authors: Chien-Chung Huang and François Sellier

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We consider the maximum weight b-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from [1,W], we present a 2 - 1/(2W) + ε approximation algorithm, using a memory of O(max(|M_G|, n) ⋅ poly(log(m),W,1/ε)), where |M_G| denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Aaron Bernstein, 2020], which achieves a 3/2 + ε approximation for the maximum cardinality simple matching. When W is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a 2 - δ approximation (for some small constant δ ∼ 10^{-17}) for the maximum weight simple matching. In particular, for the weighted b-matching problem, ours is the first result beating the approximation ratio of 2. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Aaron Bernstein and Cliff Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a 2 - 1/(2W) + ε approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry’s duality theorem.
Cite as

Chien-Chung Huang and François Sellier. Maximum Weight b-Matchings in Random-Order Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{huang_et_al:LIPIcs.ESA.2022.68,
  author =	{Huang, Chien-Chung and Sellier, Fran\c{c}ois},
  title =	{{Maximum Weight b-Matchings in Random-Order Streams}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{68:1--68:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.68},
  URN =		{urn:nbn:de:0030-drops-170062},
  doi =		{10.4230/LIPIcs.ESA.2022.68},
  annote =	{Keywords: Maximum weight matching, b-matching, streaming, random order}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.70
Vertex Sparsifiers for Hyperedge Connectivity

Authors: Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Recently, Chalermsook et al. {[}SODA'21{]} introduces a notion of vertex sparsifiers for c-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for c-edge st-connectivity {[}Jin and Sun FOCS'22{]}. We study a natural extension called vertex sparsifiers for c-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph G = (V,E) with n vertices and m hyperedges with k terminal vertices and a parameter c, there exists a hypergraph H containing only O(kc³) hyperedges that preserves all minimum cuts (up to value c) between all subset of terminals. This matches the best bound of O(kc³) edges for normal graphs by [Liu'20]. Moreover, H can be constructed in almost-linear O(p^{1+o(1)} + n(rclog n)^{O(rc)}log m) time where r = max_{e ∈ E}|e| is the rank of G and p = ∑_{e ∈ E}|e| is the total size of G, or in poly(m, n) time if we slightly relax the size to O(kc³log^{1.5}(kc)) hyperedges.
Cite as

Han Jiang, Shang-En Huang, Thatchaphol Saranurak, and Tian Zhang. Vertex Sparsifiers for Hyperedge Connectivity. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 70:1-70:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{jiang_et_al:LIPIcs.ESA.2022.70,
  author =	{Jiang, Han and Huang, Shang-En and Saranurak, Thatchaphol and Zhang, Tian},
  title =	{{Vertex Sparsifiers for Hyperedge Connectivity}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{70:1--70:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.70},
  URN =		{urn:nbn:de:0030-drops-170081},
  doi =		{10.4230/LIPIcs.ESA.2022.70},
  annote =	{Keywords: Vertex sparsifier, hypergraph, connectivity}
}
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