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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We first prove that there are uncountably many objects with distinct computational powers. More precisely, we show that there is an uncountable set of objects such that for any two of them, at least one cannot be implemented from the other (and registers) in a wait-free manner. We then strengthen this result by showing that there are uncountably many linearizable objects with distinct computational powers. To do so, we prove that for all positive integers n and k, there is a linearizable object that is computationally equivalent to the k-set agreement task among n processes. To the best of our knowledge, these are the first linearizable objects proven to be computationally equivalent to set agreement tasks.

David Yu Cheng Chan, Vassos Hadzilacos, and Sam Toueg. On the Number of Objects with Distinct Power and the Linearizability of Set Agreement Objects. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chan_et_al:LIPIcs.DISC.2017.12, author = {Chan, David Yu Cheng and Hadzilacos, Vassos and Toueg, Sam}, title = {{On the Number of Objects with Distinct Power and the Linearizability of Set Agreement Objects}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.12}, URN = {urn:nbn:de:0030-drops-79973}, doi = {10.4230/LIPIcs.DISC.2017.12}, annote = {Keywords: Set Agreement, Asynchronous System, Shared Memory} }

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Track A: Algorithms, Complexity and Games

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

In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczúr and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).

Ruoxu Cen, Yu Cheng, Debmalya Panigrahi, and Kevin Sun. Sparsification of Directed Graphs via Cut Balance. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 45:1-45:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cen_et_al:LIPIcs.ICALP.2021.45, author = {Cen, Ruoxu and Cheng, Yu and Panigrahi, Debmalya and Sun, Kevin}, title = {{Sparsification of Directed Graphs via Cut Balance}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {45:1--45:21}, 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.2021.45}, URN = {urn:nbn:de:0030-drops-141143}, doi = {10.4230/LIPIcs.ICALP.2021.45}, annote = {Keywords: Graph sparsification, directed graphs, cut sketches, space complexity} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

In this paper we present a generic reduction from the problem of finding an \epsilon-well-supported Nash equilibrium (WSNE) to that of finding an \Theta(\epsilon)-approximate Nash equilibrium (ANE), in large games with n players and a bounded number of strategies for each player.
Our reduction complements the existing literature on relations between WSNE and ANE, and can be applied to extend hardness results on WSNE to similar results on ANE.
This allows one to focus on WSNE first, which is in general easier to analyze and control in hardness constructions.
As an application we prove a 2^{\Omega(n/\log n)} lower bound on the randomized query complexity of finding an \epsilon-ANE in binary-action n-player games, for some constant \epsilon>0.
This answers an open problem posed by Hart and Nisan and Babichenko, and is very close to the trivial upper bound of 2^n.
Previously for WSNE, Babichenko showed a 2^{\Omega(n)} lower bound on the randomized query complexity of finding an \epsilon-WSNE for some constant epsilon>0.
Our result follows directly from combining Babichenko's result and our new reduction from WSNE to ANE.

Xi Chen, Yu Cheng, and Bo Tang. Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 57:1-57:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2017.57, author = {Chen, Xi and Cheng, Yu and Tang, Bo}, title = {{Well-Supported vs. Approximate Nash Equilibria: Query Complexity of Large Games}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {57:1--57:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.57}, URN = {urn:nbn:de:0030-drops-81636}, doi = {10.4230/LIPIcs.ITCS.2017.57}, annote = {Keywords: Equilibrium Computation, Query Complexity, Large Games} }

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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 time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results:
- We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time.
- We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs.
- Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}).
- We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2.
- We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2.
- We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound.

Timothy M. Chan, Qizheng He, and Yuancheng Yu. On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chan_et_al:LIPIcs.ICALP.2023.34, author = {Chan, Timothy M. and He, Qizheng and Yu, Yuancheng}, title = {{On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {34:1--34:19}, 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.2023.34}, URN = {urn:nbn:de:0030-drops-180868}, doi = {10.4230/LIPIcs.ICALP.2023.34}, annote = {Keywords: Geometric set cover, discrete k-center, conditional 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)

We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between u and v is the minimum of the shortest path distances from u to v and from v to u. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help.
By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in O~(mn) time for directed graphs on n vertices, m edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in O(mn^{1-epsilon}) time for constant epsilon>0. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off.

Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu. Approximation Algorithms for Min-Distance Problems. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dalirrooyfard_et_al:LIPIcs.ICALP.2019.46, author = {Dalirrooyfard, Mina and Williams, Virginia Vassilevska and Vyas, Nikhil and Wein, Nicole and Xu, Yinzhan and Yu, Yuancheng}, title = {{Approximation Algorithms for Min-Distance Problems}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {46:1--46:14}, 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.2019.46}, URN = {urn:nbn:de:0030-drops-106223}, doi = {10.4230/LIPIcs.ICALP.2019.46}, annote = {Keywords: fine-grained complexity, graph algorithms, diameter, radius, eccentricities} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Since the introduction of retroactive data structures at SODA 2004, a major unsolved problem has been to bound the gap between the best partially retroactive data structure (where changes can be made to the past, but only the present can be queried) and the best fully retroactive data structure (where the past can also be queried) for any problem. It was proved in 2004 that any partially retroactive data structure with operation time T_{op}(n,m) can be transformed into a fully retroactive data structure with operation time O(sqrt{m} * T_{op}(n,m)), where n is the size of the data structure and m is the number of operations in the timeline [Demaine et al., 2004]. But it has been open for 14 years whether such a gap is necessary.
In this paper, we prove nearly matching upper and lower bounds on this gap for all n and m. We improve the upper bound for n << sqrt m by showing a new transformation with multiplicative overhead n log m. We then prove a lower bound of min {n log m, sqrt m}^{1-o(1)} assuming any of the following conjectures:
- Conjecture I: Circuit SAT requires 2^{n - o(n)} time on n-input circuits of size 2^{o(n)}. This conjecture is far weaker than the well-believed SETH conjecture from complexity theory, which asserts that CNF SAT with n variables and O(n) clauses already requires 2^{n-o(n)} time.
- Conjecture II: Online (min,+) product between an integer n x n matrix and n vectors requires n^{3 - o(1)} time. This conjecture is weaker than the APSP conjectures widely used in fine-grained complexity.
- Conjecture III (3-SUM Conjecture): Given three sets A,B,C of integers, each of size n, deciding whether there exist a in A, b in B, c in C such that a + b + c = 0 requires n^{2 - o(1)} time. This 1995 conjecture [Anka Gajentaan and Mark H. Overmars, 1995] was the first conjecture in fine-grained complexity.
Our lower bound construction illustrates an interesting power of fully retroactive queries: they can be used to quickly solve batched pair evaluation. We believe this technique can prove useful for other data structure lower bounds, especially dynamic ones.

Lijie Chen, Erik D. Demaine, Yuzhou Gu, Virginia Vassilevska Williams, Yinzhan Xu, and Yuancheng Yu. Nearly Optimal Separation Between Partially and Fully Retroactive Data Structures. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.SWAT.2018.33, author = {Chen, Lijie and Demaine, Erik D. and Gu, Yuzhou and Williams, Virginia Vassilevska and Xu, Yinzhan and Yu, Yuancheng}, title = {{Nearly Optimal Separation Between Partially and Fully Retroactive Data Structures}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {33:1--33:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.33}, URN = {urn:nbn:de:0030-drops-88593}, doi = {10.4230/LIPIcs.SWAT.2018.33}, annote = {Keywords: retroactive data structure, conditional lower bound} }

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RANDOM

**Published in:** LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Naively storing a counter up to value n would require Ω(log n) bits of memory. Nelson and Yu [Jelani Nelson and Huacheng Yu, 2022], following work of Morris [Robert H. Morris, 1978], showed that if the query answers need only be (1+ε)-approximate with probability at least 1 - δ, then O(log log n + log log(1/δ) + log(1/ε)) bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather k counters for k large. This motivates the following question: for k large, can k counters be simultaneously maintained using asymptotically less memory than k times the cost of an individual counter? That is to say, does this problem benefit from an improved amortized space complexity bound?
We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff [Mark Braverman et al., 2020] to prove lower bounds for streaming algorithms.

Ishaq Aden-Ali, Yanjun Han, Jelani Nelson, and Huacheng Yu. On the Amortized Complexity of Approximate Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{adenali_et_al:LIPIcs.APPROX/RANDOM.2024.33, author = {Aden-Ali, Ishaq and Han, Yanjun and Nelson, Jelani and Yu, Huacheng}, title = {{On the Amortized Complexity of Approximate Counting}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.33}, URN = {urn:nbn:de:0030-drops-210264}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.33}, annote = {Keywords: streaming, approximate counting, information complexity, lower bounds} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of n elements in [n], outputs O(n) elements, such that:
- There exists a randomized oblivious algorithm with space O(log n), time O(nlog n) and one-way access to randomness, that computes the function with probability 1-O(1/n);
- Any deterministic oblivious branching program with space S and time T that computes the function must satisfy T²S ≥ Ω(n^{2.5}/log n). This implies that logspace randomized algorithms for multi-output functions cannot be black-box derandomized without an Ω̃(n^{1/4}) overhead in time.
Since previously all the polynomial time-space tradeoffs of multi-output functions are proved via the Borodin-Cook method, which is a probabilistic method that inherently gives the same lower bound for randomized and deterministic branching programs, our lower bound proof is intrinsically different from previous works.
We also examine other natural candidates for proving such separations, and show that any polynomial separation for these problems would resolve the long-standing open problem of proving n^{1+Ω(1)} time lower bound for decision problems with polylog(n) space.

Huacheng Yu and Wei Zhan. Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 99:1-99:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{yu_et_al:LIPIcs.ITCS.2024.99, author = {Yu, Huacheng and Zhan, Wei}, title = {{Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {99:1--99:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.99}, URN = {urn:nbn:de:0030-drops-196270}, doi = {10.4230/LIPIcs.ITCS.2024.99}, annote = {Keywords: Time-space tradeoffs, Randomness, Borodin-Cook method} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Given a distribution over [n]ⁿ such that any k coordinates need k/log^{O(1)}n bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality Ω(log(n/k)/log log(n/k)). In particular, we show that for any constant δ > 0, there exists ε = 2^{-Ω(n^{1-δ})} such that Ω(log n/log log n) non-adaptive cell probes on uniform cells are required to:
- Sample a uniformly random permutation on n elements with error 1-ε. This provides an exponential improvement on the Ω(log log n) cell probe lower bound by Viola.
- Sample an n-vector with each element independently drawn from a random n^{1-δ}-vector, with error 1-ε. This provides the first adaptive vs non-adaptive cell probe separation for sampling.
The major technical component in our proof is a new combinatorial theorem about flower with small kernel, i.e. a collection of sets where few elements appear more than once. We show that in a family of n sets, each with size O(log n/log log n), there must be k = poly(n) sets where at most k/log^{O(1)}n elements appear more than once.
To show the lower bound on sampling permutation, we also prove a new Ω(k) communication lower bound on sampling uniformly distributed disjoint subsets of [n] of size k, with error 1-2^{-Ω(k²/n)}. This result unifies and subsumes the lower bound for k = Θ(√n) by Ambainis et al., and the lower bound for k = Θ(n) by Göös and Watson.

Huacheng Yu and Wei Zhan. Sampling, Flowers and Communication. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 100:1-100:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{yu_et_al:LIPIcs.ITCS.2024.100, author = {Yu, Huacheng and Zhan, Wei}, title = {{Sampling, Flowers and Communication}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {100:1--100:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.100}, URN = {urn:nbn:de:0030-drops-196288}, doi = {10.4230/LIPIcs.ITCS.2024.100}, annote = {Keywords: Flower, Sampling, Cell probe, Communcation complexity} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area.
We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA'21), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work.

Sepehr Assadi, Michael Kapralov, and Huacheng Yu. On Constructing Spanners from Random Gaussian Projections. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2023.57, author = {Assadi, Sepehr and Kapralov, Michael and Yu, Huacheng}, title = {{On Constructing Spanners from Random Gaussian Projections}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {57:1--57:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.57}, URN = {urn:nbn:de:0030-drops-188821}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.57}, annote = {Keywords: sketching algorithm, lower bound, graph spanner} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study boolean constraint satisfaction problems (CSPs) Max-CSP^f_n for all predicates f: {0,1}^k → {0,1}. In these problems, given an integer v and a list of constraints over n boolean variables, each obtained by applying f to a sequence of literals, we wish to decide if there is an assignment to the variables that satisfies at least v constraints. We consider these problems in the streaming model, where the algorithm makes a small number of passes over the list of constraints.
Our first and main result is the following complete characterization: For every predicate f, the streaming space complexity of the Max-CSP^f_n problem is Θ̃(n^deg(f)), where deg(f) is the degree of f when viewed as a multilinear polynomial. While the upper bound is obtained by a (very simple) one-pass streaming algorithm, our lower bound shows that a better space complexity is impossible even with constant-pass streaming algorithms.
Building on our techniques, we are also able to get an optimal Ω(n²) lower bound on the space complexity of constant-pass streaming algorithms for the well studied Max-CUT problem, even though it is not technically a Max-CSP^f_n problem as, e.g., negations of variables and repeated constraints are not allowed.

Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, and Huacheng Yu. Characterizing the Multi-Pass Streaming Complexity for Solving Boolean CSPs Exactly. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 80:1-80:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kol_et_al:LIPIcs.ITCS.2023.80, author = {Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R. and Yu, Huacheng}, title = {{Characterizing the Multi-Pass Streaming Complexity for Solving Boolean CSPs Exactly}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {80:1--80:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.80}, URN = {urn:nbn:de:0030-drops-175837}, doi = {10.4230/LIPIcs.ITCS.2023.80}, annote = {Keywords: Streaming algorithms, Constraint Satisfaction Problems} }

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Track A: Algorithms, Complexity and Games

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

For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be Θ̃(n ⋅ L). Prior to our work, a better space complexity was only known with Õ(√L) passes.
We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is Θ̃(n ⋅ √L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses Ω̃(n ⋅ L^{1/p}) space. In addition, we show a similar Θ̃(n ⋅ √L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph.

Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2021.52, author = {Chen, Lijie and Kol, Gillat and Paramonov, Dmitry and Saxena, Raghuvansh R. and Song, Zhao and Yu, Huacheng}, title = {{Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {52:1--52:19}, 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.2021.52}, URN = {urn:nbn:de:0030-drops-141218}, doi = {10.4230/LIPIcs.ICALP.2021.52}, annote = {Keywords: streaming algorithms, random walk sampling} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

The membership problem asks to maintain a set S ⊆ [u], supporting insertions and membership queries, i.e., testing if a given element is in the set. A data structure that computes exact answers is called a dictionary. When a (small) false positive rate ε is allowed, the data structure is called a filter.
The space usages of the standard dictionaries or filters usually depend on the upper bound on the size of S, while the actual set can be much smaller.
Pagh, Segev and Wieder [Pagh et al., 2013] were the first to study filters with varying space usage based on the current |S|. They showed in order to match the space with the current set size n = |S|, any filter data structure must use (1-o(1))n(log(1/ε)+(1-O(ε))log log n) bits, in contrast to the well-known lower bound of N log(1/ε) bits, where N is an upper bound on |S|. They also presented a data structure with almost optimal space of (1+o(1))n(log(1/ε)+O(log log n)) bits provided that n > u^0.001, with expected amortized constant insertion time and worst-case constant lookup time.
In this work, we present a filter data structure with improvements in two aspects:
- it has constant worst-case time for all insertions and lookups with high probability;
- it uses space (1+o(1))n(log (1/ε)+log log n) bits when n > u^0.001, achieving optimal leading constant for all ε = o(1). We also present a dictionary that uses (1+o(1))nlog(u/n) bits of space, matching the optimal space in terms of the current size, and performs all operations in constant time with high probability.

Mingmou Liu, Yitong Yin, and Huacheng Yu. Succinct Filters for Sets of Unknown Sizes. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 79:1-79:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{liu_et_al:LIPIcs.ICALP.2020.79, author = {Liu, Mingmou and Yin, Yitong and Yu, Huacheng}, title = {{Succinct Filters for Sets of Unknown Sizes}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {79:1--79:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.79}, URN = {urn:nbn:de:0030-drops-124867}, doi = {10.4230/LIPIcs.ICALP.2020.79}, annote = {Keywords: Bloom filters, Data structures, Approximate set membership, Dictionaries} }

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