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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ε and a set P of n points in ℝ^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range γ, we can efficiently find a list of approximate heavy hitters in γ∩P, i.e., colors that appear at least ε |γ∩P| times in γ∩P, as well as their frequencies with an additive error of ε |γ∩P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/ε weights such that the i-th weight in S has approximate rank iε|γ∩P|, meaning, rank iε|γ∩P| up to an additive error of ε|γ∩P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012].
We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = Θ(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/ε). Note that as the output size is 1/ε, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/ε) (resp. log log_w(1/ε)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/ε) factors in both time and space, we can also support quantile queries.
We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations.

Peyman Afshani, Pingan Cheng, Aniket Basu Roy, and Zhewei Wei. On Range Summary Queries. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{afshani_et_al:LIPIcs.ICALP.2023.7, author = {Afshani, Peyman and Cheng, Pingan and Basu Roy, Aniket and Wei, Zhewei}, title = {{On Range Summary Queries}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel 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.2023.7}, URN = {urn:nbn:de:0030-drops-180590}, doi = {10.4230/LIPIcs.ICALP.2023.7}, annote = {Keywords: Computational Geometry, Range Searching, Random Sampling, Geometric Approximation, Data Structures and Algorithms} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Recently, Ezra and Sharir [Esther Ezra and Micha Sharir, 2022] showed an O(n^{3/2+σ}) space and O(n^{1/2+σ}) query time data structure for ray shooting among triangles in ℝ³. This improves the upper bound given by the classical S(n)Q(n)⁴ = O(n^{4+σ}) space-time tradeoff for the first time in almost 25 years and in fact lies on the tradeoff curve of S(n)Q(n)³ = O(n^{3+σ}). However, it seems difficult to apply their techniques beyond this specific space and time combination. This pheonomenon appears persistently in almost all recent advances of flat object intersection searching, e.g., line-tetrahedron intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], triangle-triangle intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], or even among flat semialgebraic objects [Agarwal et al., 2022].
We give a timely explanation to this phenomenon from a lower bound perspective. We prove that given a set 𝒮 of (d-1)-dimensional simplicies in ℝ^d, any data structure that can report all intersections with a query line in small (n^o(1)) query time must use Ω(n^{2(d-1)-o(1)}) space. This dashes the hope of any significant improvement to the tradeoff curves for small query time and almost matches the classical upper bound. We also obtain an almost matching space lower bound of Ω(n^{6-o(1)}) for triangle-triangle intersection reporting in ℝ⁴ when the query time is small. Along the way, we further develop the previous lower bound techniques by Afshani and Cheng [Afshani and Cheng, 2021; Afshani and Cheng, 2022].

Peyman Afshani and Pingan Cheng. Lower Bounds for Intersection Reporting Among Flat Objects. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2023.3, author = {Afshani, Peyman and Cheng, Pingan}, title = {{Lower Bounds for Intersection Reporting Among Flat Objects}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {3:1--3:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.3}, URN = {urn:nbn:de:0030-drops-178536}, doi = {10.4230/LIPIcs.SoCG.2023.3}, annote = {Keywords: Computational Geometry, Intersection Searching, Data Structure Lower Bounds} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

In colored range counting (CRC), the input is a set of points where each point is assigned a "color" (or a "category") and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.
However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through reductions from the orthogonal vectors problem.

Peyman Afshani, Rasmus Killmann, and Kasper Green Larsen. Hierarchical Categories in Colored Searching. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{afshani_et_al:LIPIcs.ISAAC.2022.25, author = {Afshani, Peyman and Killmann, Rasmus and Larsen, Kasper Green}, title = {{Hierarchical Categories in Colored Searching}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.25}, URN = {urn:nbn:de:0030-drops-173100}, doi = {10.4230/LIPIcs.ISAAC.2022.25}, annote = {Keywords: Categorical Data, Computational Geometry} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s.
When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case.
We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well.
The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2, author = {Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung}, title = {{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {2:1--2:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2}, URN = {urn:nbn:de:0030-drops-160109}, doi = {10.4230/LIPIcs.SoCG.2022.2}, annote = {Keywords: Approximation, Motion Planning, Scheduling} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Semialgebraic range searching, arguably the most general version of range searching, is a fundamental problem in computational geometry. In the problem, we are to preprocess a set of points in ℝ^D such that the subset of points inside a semialgebraic region described by a constant number of polynomial inequalities of degree Δ can be found efficiently.
Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" data structures [Agarwal et al., 2013; Matoušek and Patáková, 2015] with almost optimal query time of Q(n) = O(n^{1-1/D+o(1)}) were obtained. For "fast query" data structures (i.e., when Q(n) = n^{o(1)}), it was conjectured that a similar improvement is possible, i.e., it is possible to achieve space S(n) = O(n^{D+o(1)}). The conjecture was refuted very recently by Afshani and Cheng [Afshani and Cheng, 2021]. In the plane, i.e., D = 2, they proved that S(n) = Ω(n^{Δ+1 - o(1)}/Q(n)^{(Δ+3)Δ/2}) which shows Ω(n^{Δ+1-o(1)}) space is needed for Q(n) = n^{o(1)}. While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of n or Q(n) seem to be tight even for D = 2, as the best known upper bounds have S(n) = O(n^{m+o(1)}/Q(n)^{(m-1)D/(D-1)}) where m = binom(D+Δ,D)-1 = Ω(Δ^D) is the maximum number of parameters to define a monic degree-Δ D-variate polynomial, for any constant dimension D and degree Δ.
In this paper, we resolve two of the issues: we prove a lower bound in D-dimensions, for constant D, and show that when the query time is n^{o(1)}+O(k), the space usage is Ω(n^{m-o(1)}), which almost matches the Õ(n^{m}) upper bound and essentially closes the problem for the fast-query case, as far as the exponent of n is considered in the pointer machine model. When considering the exponent of Q(n), we show that the analysis in [Afshani and Cheng, 2021] is tight for D = 2, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or to obtain better lower bounds a new fundamentally different input set needs to be constructed.

Peyman Afshani and Pingan Cheng. On Semialgebraic Range Reporting. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.3, author = {Afshani, Peyman and Cheng, Pingan}, title = {{On Semialgebraic Range Reporting}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.3}, URN = {urn:nbn:de:0030-drops-160117}, doi = {10.4230/LIPIcs.SoCG.2022.3}, annote = {Keywords: Computational Geometry, Range Searching, Data Structures and Algorithms, Lower Bounds} }

Document

**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

In the semialgebraic range searching problem, we are given a set of n points in ℝ^d and we want to preprocess the points such that for any query range belonging to a family of constant complexity semialgebraic sets (Tarski cells), all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, then the problem is well-understood: it can be solved using S(n) space and with Q(n) query time with S(n)Q^d(n) = Õ(n^d) where the Õ(⋅) notation hides polylogarithmic factors and this trade-off is tight (up to n^o(1) factors). Consequently, there exists "low space" structures that use O(n) space with O(n^{1-1/d}) query time and "fast query" structures that use O(n^d) space with O(log^{d+1} n) query time. However, for the general semialgebraic ranges, only "low space" solutions are known, but the best solutions match the same trade-off curve as the simplex queries, with O(n) space and Õ(n^{1-1/d}) query time. It has been conjectured that the same could be done for the "fast query" case but this open problem has stayed unresolved.
Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and other related problems. More precisely, we show that any data structure for reporting the points between two concentric circles, a problem that we call 2D annulus reporting problem, with Q(n) query time must use S(n) = Ω^o(n³/Q(n)⁵) space where the Ω^o(⋅) notation hides n^o(1) factors, meaning, for Q(n) = O(log^{O(1)}n), Ω^o(n³) space must be used. In addition, we study the problem of reporting the subset of input points between two polynomials of the form Y = ∑_{i=0}^Δ a_i Xⁱ where values a_0,⋯,a_Δ are given at the query time, a problem that we call polynomial slab reporting. For this, we show a space lower bound of Ω^o(n^{Δ+1}/Q(n)^{Δ²+Δ}), which shows for Q(n) = O(log^{O(1)}n), we must use Ω^o(n^{Δ+1}) space. We also consider the dual problems of semialgebraic range searching, semialgebraic stabbing problems, and present lower bounds for them. In particular, we show that in linear space, any data structure that solves 2D annulus stabbing problems must use Ω(n^{2/3}) query time. Note that this almost matches the upper bound obtained by lifting 2D annuli to 3D. Like semialgebraic range searching, we also present lower bounds for general semialgebraic slab stabbing problems. Again, our lower bounds are almost tight for linear size data structures in this case.

Peyman Afshani and Pingan Cheng. Lower Bounds for Semialgebraic Range Searching and Stabbing Problems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2021.8, author = {Afshani, Peyman and Cheng, Pingan}, title = {{Lower Bounds for Semialgebraic Range Searching and Stabbing Problems}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.8}, URN = {urn:nbn:de:0030-drops-138072}, doi = {10.4230/LIPIcs.SoCG.2021.8}, annote = {Keywords: Computational Geometry, Data Structures and Algorithms} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible.

Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava. Fragile Complexity of Comparison-Based Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2019.2, author = {Afshani, Peyman and Fagerberg, Rolf and Hammer, David and Jacob, Riko and Kostitsyna, Irina and Meyer, Ulrich and Penschuck, Manuel and Sitchinava, Nodari}, title = {{Fragile Complexity of Comparison-Based Algorithms}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {2:1--2:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.2}, URN = {urn:nbn:de:0030-drops-111235}, doi = {10.4230/LIPIcs.ESA.2019.2}, annote = {Keywords: Algorithms, comparison based algorithms, lower bounds} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant’s conjectures.

Peyman Afshani, Casper Benjamin Freksen, Lior Kamma, and Kasper Green Larsen. Lower Bounds for Multiplication via Network Coding. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 10:1-10:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani_et_al:LIPIcs.ICALP.2019.10, author = {Afshani, Peyman and Freksen, Casper Benjamin and Kamma, Lior and Larsen, Kasper Green}, title = {{Lower Bounds for Multiplication via Network Coding}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {10:1--10:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.10}, URN = {urn:nbn:de:0030-drops-105861}, doi = {10.4230/LIPIcs.ICALP.2019.10}, annote = {Keywords: Circuit Complexity, Circuit Lower Bounds, Multiplication, Network Coding, Fine-Grained Complexity} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We report the first improvement in the space-time trade-off of lower bounds for the orthogonal range searching problem in the semigroup model, since Chazelle’s result from 1990. This is one of the very fundamental problems in range searching with a long history. Previously, Andrew Yao’s influential result had shown that the problem is already non-trivial in one dimension [Yao, 1982]: using m units of space, the query time Q(n) must be Omega(alpha(m,n) + n/(m-n+1)) where alpha(*,*) is the inverse Ackermann’s function, a very slowly growing function. In d dimensions, Bernard Chazelle [Chazelle, 1990] proved that the query time must be Q(n) = Omega((log_beta n)^{d-1}) where beta = 2m/n. Chazelle’s lower bound is known to be tight for when space consumption is "high" i.e., m = Omega(n log^{d+epsilon}n).
We have two main results. The first is a lower bound that shows Chazelle’s lower bound was not tight for "low space": we prove that we must have m Q(n) = Omega(n (log n log log n)^{d-1}). Our lower bound does not close the gap to the existing data structures, however, our second result is that our analysis is tight. Thus, we believe the gap is in fact natural since lower bounds are proven for idempotent semigroups while the data structures are built for general semigroups and thus they cannot assume (and use) the properties of an idempotent semigroup. As a result, we believe to close the gap one must study lower bounds for non-idempotent semigroups or building data structures for idempotent semigroups. We develope significantly new ideas for both of our results that could be useful in pursuing either of these directions.

Peyman Afshani. A New Lower Bound for Semigroup Orthogonal Range Searching. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani:LIPIcs.SoCG.2019.3, author = {Afshani, Peyman}, title = {{A New Lower Bound for Semigroup Orthogonal Range Searching}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.3}, URN = {urn:nbn:de:0030-drops-104075}, doi = {10.4230/LIPIcs.SoCG.2019.3}, annote = {Keywords: Data Structures, Range Searching, Lower bounds} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We revisit the range sampling problem: the input is a set of points where each point is associated with a real-valued weight. The goal is to store them in a structure such that given a query range and an integer k, we can extract k independent random samples from the points inside the query range, where the probability of sampling a point is proportional to its weight.
This line of work was initiated in 2014 by Hu, Qiao, and Tao and it was later followed up by Afshani and Wei. The first line of work mostly studied unweighted but dynamic version of the problem in one dimension whereas the second result considered the static weighted problem in one dimension as well as the unweighted problem in 3D for halfspace queries.
We offer three main results and some interesting insights that were missed by the previous work: We show that it is possible to build efficient data structures for range sampling queries if we allow the query time to hold in expectation (the first result), or obtain efficient worst-case query bounds by allowing the sampling probability to be approximately proportional to the weight (the second result). The third result is a conditional lower bound that shows essentially one of the previous two concessions is needed. For instance, for the 3D range sampling queries, the first two results give efficient data structures with near-linear space and polylogarithmic query time whereas the lower bound shows with near-linear space the worst-case query time must be close to n^{2/3}, ignoring polylogarithmic factors. Up to our knowledge, this is the first such major gap between the expected and worst-case query time of a range searching problem.

Peyman Afshani and Jeff M. Phillips. Independent Range Sampling, Revisited Again. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2019.4, author = {Afshani, Peyman and Phillips, Jeff M.}, title = {{Independent Range Sampling, Revisited Again}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {4:1--4:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.4}, URN = {urn:nbn:de:0030-drops-104088}, doi = {10.4230/LIPIcs.SoCG.2019.4}, annote = {Keywords: Range Searching, Data Structures, Sampling} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

We study permuting and batched orthogonal geometric reporting problems in the External Memory Model (EM), assuming indivisibility of the input records.
Our main results are twofold. First, we prove a general simulation result that essentially shows that any permutation algorithm (resp. duplicate removal algorithm) that does alpha*N/B I/Os (resp. to remove a fraction of the existing duplicates) can be simulated with an algorithm that does alpha phases where each phase reads and writes each element once, but using a factor alpha smaller block size.
Second, we prove two lower bounds for batched rectangle stabbing and batched orthogonal range reporting queries. Assuming a short cache, we prove very high lower bounds that currently are not possible with the existing techniques under the tall cache assumption.

Peyman Afshani and Ingo van Duijn. Permuting and Batched Geometric Lower Bounds in the I/O Model. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2017.2, author = {Afshani, Peyman and van Duijn, Ingo}, title = {{Permuting and Batched Geometric Lower Bounds in the I/O Model}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {2:1--2:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.2}, URN = {urn:nbn:de:0030-drops-78695}, doi = {10.4230/LIPIcs.ESA.2017.2}, annote = {Keywords: I/O Model, Batched Geometric Queries, Lower Bounds, Permuting} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

In the independent range sampling (IRS) problem, given an input set P of n points in R^d, the task is to build a data structure, such that given a range R and an integer t >= 1, it returns t points that are uniformly and independently drawn from P cap R. The samples must satisfy inter-query independence, that is, the samples returned by every query must be independent of the samples returned by all the previous queries. This problem was first tackled by Hu, Qiao and Tao in 2014, who proposed optimal structures for one-dimensional dynamic IRS problem in internal memory and one-dimensional static IRS problem in external memory.
In this paper, we study two natural extensions of the independent range sampling problem. In the first extension, we consider the static IRS problem in two and three dimensions in internal memory. We obtain data structures with optimal space-query tradeoffs for 3D halfspace, 3D dominance, and 2D three-sided queries. The second extension considers weighted IRS problem. Each point is associated with a real-valued weight, and given a query range R, a sample is drawn independently such that each point in P cap R is selected with probability proportional to its weight. Walker's alias method is a classic solution to this problem when no query range is specified. We obtain optimal data structure for one dimensional weighted range sampling problem, thereby extending the alias method to allow range queries.

Peyman Afshani and Zhewei Wei. Independent Range Sampling, Revisited. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{afshani_et_al:LIPIcs.ESA.2017.3, author = {Afshani, Peyman and Wei, Zhewei}, title = {{Independent Range Sampling, Revisited}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.3}, URN = {urn:nbn:de:0030-drops-78592}, doi = {10.4230/LIPIcs.ESA.2017.3}, annote = {Keywords: data structures, range searching, range sampling, random sampling} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower bounds that cannot be obtained in any other known model of computation with the current techniques. Often our lower bounds match the known space-query time trade-off curve and in fact for all the problems considered, there is a very good and reasonable match between our lower bounds and the known upper bounds, at least for some choice of input parameters.
The problems that we consider are set intersection queries (both the reporting variant and the semi-group counting variant), indexing a set of documents for two-pattern queries, or forbidden-pattern queries, or queries with wild-cards, and indexing an input set of gapped-patterns (or two-patterns) to find those matching a document given at the query time.

Peyman Afshani and Jesper Sindahl Nielsen. Data Structure Lower Bounds for Document Indexing Problems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 93:1-93:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{afshani_et_al:LIPIcs.ICALP.2016.93, author = {Afshani, Peyman and Nielsen, Jesper Sindahl}, title = {{Data Structure Lower Bounds for Document Indexing Problems}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {93:1--93:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.93}, URN = {urn:nbn:de:0030-drops-61923}, doi = {10.4230/LIPIcs.ICALP.2016.93}, annote = {Keywords: Data Structure Lower Bounds, Pointer Machine, Set Intersection, Pattern Matching} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed.
First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).

Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen. Applications of Incidence Bounds in Point Covering Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2016.60, author = {Afshani, Peyman and Berglin, Edvin and van Duijn, Ingo and Sindahl Nielsen, Jesper}, title = {{Applications of Incidence Bounds in Point Covering Problems}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {60:1--60:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.60}, URN = {urn:nbn:de:0030-drops-59527}, doi = {10.4230/LIPIcs.SoCG.2016.60}, annote = {Keywords: Point Cover, Incidence Bounds, Inclusion Exclusion, Exponential Algorithm} }

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