Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.04685
Sepehr Assadi, Vihan Shah
We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $widetilde{O}(k n)$ space. Our upper bound matches the known $Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a"certificate"of $k$-vertex-connectivity as a union of $O(k^2log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.
{"title":"Tight Bounds for Vertex Connectivity in Dynamic Streams","authors":"Sepehr Assadi, Vihan Shah","doi":"10.48550/arXiv.2211.04685","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04685","url":null,"abstract":"We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $widetilde{O}(k n)$ space. Our upper bound matches the known $Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a\"certificate\"of $k$-vertex-connectivity as a union of $O(k^2log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"50 1","pages":"213-227"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86106798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.04643
Josh Alman
We use lookup tables to design faster algorithms for important algebraic problems over finite fields. These faster algorithms, which only use arithmetic operations and lookup table operations, may help to explain the difficulty of determining the complexities of these important problems. Our results over a constant-sized finite field are as follows. The Walsh-Hadamard transform of a vector of length $N$ can be computed using $O(N log N / log log N)$ bit operations. This generalizes to any transform defined as a Kronecker power of a fixed matrix. By comparison, the Fast Walsh-Hadamard transform (similar to the Fast Fourier transform) uses $O(N log N)$ arithmetic operations, which is believed to be optimal up to constant factors. Any algebraic algorithm for multiplying two $N times N$ matrices using $O(N^omega)$ operations can be converted into an algorithm using $O(N^omega / (log N)^{omega/2 - 1})$ bit operations. For example, Strassen's algorithm can be converted into an algorithm using $O(N^{2.81} / (log N)^{0.4})$ bit operations. It remains an open problem with practical implications to determine the smallest constant $c$ such that Strassen's algorithm can be implemented to use $c cdot N^{2.81} + o(N^{2.81})$ arithmetic operations; using a lookup table allows one to save a super-constant factor in bit operations.
{"title":"Faster Walsh-Hadamard Transform and Matrix Multiplication over Finite Fields using Lookup Tables","authors":"Josh Alman","doi":"10.48550/arXiv.2211.04643","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04643","url":null,"abstract":"We use lookup tables to design faster algorithms for important algebraic problems over finite fields. These faster algorithms, which only use arithmetic operations and lookup table operations, may help to explain the difficulty of determining the complexities of these important problems. Our results over a constant-sized finite field are as follows. The Walsh-Hadamard transform of a vector of length $N$ can be computed using $O(N log N / log log N)$ bit operations. This generalizes to any transform defined as a Kronecker power of a fixed matrix. By comparison, the Fast Walsh-Hadamard transform (similar to the Fast Fourier transform) uses $O(N log N)$ arithmetic operations, which is believed to be optimal up to constant factors. Any algebraic algorithm for multiplying two $N times N$ matrices using $O(N^omega)$ operations can be converted into an algorithm using $O(N^omega / (log N)^{omega/2 - 1})$ bit operations. For example, Strassen's algorithm can be converted into an algorithm using $O(N^{2.81} / (log N)^{0.4})$ bit operations. It remains an open problem with practical implications to determine the smallest constant $c$ such that Strassen's algorithm can be implemented to use $c cdot N^{2.81} + o(N^{2.81})$ arithmetic operations; using a lookup table allows one to save a super-constant factor in bit operations.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"246 1","pages":"137-144"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86705348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.05178
Binghui Peng, A. Rubinstein
Dynamic algorithms come in three main flavors: $mathit{incremental}$ (insertions-only), $mathit{decremental}$ (deletions-only), or $mathit{fully}$ $mathit{dynamic}$ (both insertions and deletions). Fully dynamic is the holy grail of dynamic algorithm design; it is obviously more general than the other two, but is it strictly harder? Several works managed to reduce fully dynamic to the incremental or decremental models by taking advantage of either specific structure of the incremental/decremental algorithms (e.g. [HK99, HLT01, BKS12, ADKKP16, BS80, OL81, OvL81]), or specific order of insertions/deletions (e.g. [AW14,HKNS15,KPP16]). Our goal in this work is to get a black-box fully-to-incremental reduction that is as general as possible. We find that the following conditions are necessary: $bullet$ The incremental algorithm must have a worst-case (rather than amortized) running time guarantee. $bullet$ The reduction must work in what we call the $mathit{deletions}$-$mathit{look}$-$mathit{ahead}$ $mathit{model}$, where the order of deletions among current elements is known in advance. A notable practical example is the"sliding window"(FIFO) order of updates. Under those conditions, we design: $bullet$ A simple, practical, amortized-fully-dynamic to worst-case-incremental reduction with a $log(T)$-factor overhead on the running time, where $T$ is the total number of updates. $bullet$ A theoretical worst-case-fully-dynamic to worst-case-incremental reduction with a $mathsf{polylog}(T)$-factor overhead on the running time.
{"title":"Fully-dynamic-to-incremental reductions with known deletion order (e.g. sliding window)","authors":"Binghui Peng, A. Rubinstein","doi":"10.48550/arXiv.2211.05178","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05178","url":null,"abstract":"Dynamic algorithms come in three main flavors: $mathit{incremental}$ (insertions-only), $mathit{decremental}$ (deletions-only), or $mathit{fully}$ $mathit{dynamic}$ (both insertions and deletions). Fully dynamic is the holy grail of dynamic algorithm design; it is obviously more general than the other two, but is it strictly harder? Several works managed to reduce fully dynamic to the incremental or decremental models by taking advantage of either specific structure of the incremental/decremental algorithms (e.g. [HK99, HLT01, BKS12, ADKKP16, BS80, OL81, OvL81]), or specific order of insertions/deletions (e.g. [AW14,HKNS15,KPP16]). Our goal in this work is to get a black-box fully-to-incremental reduction that is as general as possible. We find that the following conditions are necessary: $bullet$ The incremental algorithm must have a worst-case (rather than amortized) running time guarantee. $bullet$ The reduction must work in what we call the $mathit{deletions}$-$mathit{look}$-$mathit{ahead}$ $mathit{model}$, where the order of deletions among current elements is known in advance. A notable practical example is the\"sliding window\"(FIFO) order of updates. Under those conditions, we design: $bullet$ A simple, practical, amortized-fully-dynamic to worst-case-incremental reduction with a $log(T)$-factor overhead on the running time, where $T$ is the total number of updates. $bullet$ A theoretical worst-case-fully-dynamic to worst-case-incremental reduction with a $mathsf{polylog}(T)$-factor overhead on the running time.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"42 1","pages":"261-271"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82522320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.04981
T. Eden, Shyam Narayanan, Jakub Tvetek
Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the number of edges. An algorithm for pointwise $varepsilon$-approximate edge sampling with complexity $O(n/sqrt{varepsilon m})$ has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by Tv{e}tek and Thorup [STOC 2022] to $O(n log(varepsilon^{-1})/sqrt{m})$. At the same time, $Omega(n/sqrt{m})$ time is necessary. We close the problem, by giving an algorithm with complexity $O(n/sqrt{m})$ for the task of sampling an edge exactly uniformly.
{"title":"Sampling an Edge in Sublinear Time Exactly and Optimally","authors":"T. Eden, Shyam Narayanan, Jakub Tvetek","doi":"10.48550/arXiv.2211.04981","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04981","url":null,"abstract":"Sampling edges from a graph in sublinear time is a fundamental problem and a powerful subroutine for designing sublinear-time algorithms. Suppose we have access to the vertices of the graph and know a constant-factor approximation to the number of edges. An algorithm for pointwise $varepsilon$-approximate edge sampling with complexity $O(n/sqrt{varepsilon m})$ has been given by Eden and Rosenbaum [SOSA 2018]. This has been later improved by Tv{e}tek and Thorup [STOC 2022] to $O(n log(varepsilon^{-1})/sqrt{m})$. At the same time, $Omega(n/sqrt{m})$ time is necessary. We close the problem, by giving an algorithm with complexity $O(n/sqrt{m})$ for the task of sampling an edge exactly uniformly.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"24 1","pages":"253-260"},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84024407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-08DOI: 10.48550/arXiv.2211.04444
Anupam Gupta, Euiwoong Lee, Jason Li
In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a $H_k approx ln k + O(1)$ approximation for this problem, where the size of each set is bounded by $k$. Moreover, getting a $ln k - O(ln ln k)$ approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of $k$, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first $H_k$-approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an $(H_k - Omega(log^2 k)/k)$-approximation, improving on the previous bound of $(H_k - Omega(1/k^8))$ (emph{R. Hassin and A. Levin, SICOMP '05}) based on a modified greedy algorithm.
{"title":"A Local Search-Based Approach for Set Covering","authors":"Anupam Gupta, Euiwoong Lee, Jason Li","doi":"10.48550/arXiv.2211.04444","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04444","url":null,"abstract":"In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a $H_k approx ln k + O(1)$ approximation for this problem, where the size of each set is bounded by $k$. Moreover, getting a $ln k - O(ln ln k)$ approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of $k$, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first $H_k$-approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an $(H_k - Omega(log^2 k)/k)$-approximation, improving on the previous bound of $(H_k - Omega(1/k^8))$ (emph{R. Hassin and A. Levin, SICOMP '05}) based on a modified greedy algorithm.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"97 1","pages":"1-11"},"PeriodicalIF":0.0,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76972287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03601
Georg Anegg, Laura Vargas Koch, R. Zenklusen
Recent progress on robust clustering led to constant-factor approximations for Robust Matroid Center. After a first combinatorial $7$-approximation that is based on a matroid intersection approach, two tight LP-based $3$-approximations were discovered, both relying on the Ellipsoid Method. In this paper, we show how a carefully designed, yet very simple, greedy selection algorithm gives a $5$-approximation. An important ingredient of our approach is a well-chosen use of Rado matroids. This enables us to capture with a single matroid a relaxed version of the original matroid, which, as we show, is amenable to straightforward greedy selections.
{"title":"A Simple Combinatorial Algorithm for Robust Matroid Center","authors":"Georg Anegg, Laura Vargas Koch, R. Zenklusen","doi":"10.48550/arXiv.2211.03601","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03601","url":null,"abstract":"Recent progress on robust clustering led to constant-factor approximations for Robust Matroid Center. After a first combinatorial $7$-approximation that is based on a matroid intersection approach, two tight LP-based $3$-approximations were discovered, both relying on the Ellipsoid Method. In this paper, we show how a carefully designed, yet very simple, greedy selection algorithm gives a $5$-approximation. An important ingredient of our approach is a well-chosen use of Rado matroids. This enables us to capture with a single matroid a relaxed version of the original matroid, which, as we show, is amenable to straightforward greedy selections.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"31 1","pages":"96-102"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82989557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03683
Jakob Baek Tejs Houen, R. Pagh, Stefan Walzer
Imagine handling collisions in a hash table by storing, in each cell, the bit-wise exclusive-or of the set of keys hashing there. This appears to be a terrible idea: For $alpha n$ keys and $n$ buckets, where $alpha$ is constant, we expect that a constant fraction of the keys will be unrecoverable due to collisions. We show that if this collision resolution strategy is repeated three times independently the situation reverses: If $alpha$ is below a threshold of $approx 0.81$ then we can recover the set of all inserted keys in linear time with high probability. Even though the description of our data structure is simple, its analysis is nontrivial. Our approach can be seen as a variant of the Invertible Bloom Filter (IBF) of Eppstein and Goodrich. While IBFs involve an explicit checksum per bucket to decide whether the bucket stores a single key, we exploit the idea of quotienting, namely that some bits of the key are implicit in the location where it is stored. We let those serve as an implicit checksum. These bits are not quite enough to ensure that no errors occur and the main technical challenge is to show that decoding can recover from these errors.
{"title":"Simple Set Sketching","authors":"Jakob Baek Tejs Houen, R. Pagh, Stefan Walzer","doi":"10.48550/arXiv.2211.03683","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03683","url":null,"abstract":"Imagine handling collisions in a hash table by storing, in each cell, the bit-wise exclusive-or of the set of keys hashing there. This appears to be a terrible idea: For $alpha n$ keys and $n$ buckets, where $alpha$ is constant, we expect that a constant fraction of the keys will be unrecoverable due to collisions. We show that if this collision resolution strategy is repeated three times independently the situation reverses: If $alpha$ is below a threshold of $approx 0.81$ then we can recover the set of all inserted keys in linear time with high probability. Even though the description of our data structure is simple, its analysis is nontrivial. Our approach can be seen as a variant of the Invertible Bloom Filter (IBF) of Eppstein and Goodrich. While IBFs involve an explicit checksum per bucket to decide whether the bucket stores a single key, we exploit the idea of quotienting, namely that some bits of the key are implicit in the location where it is stored. We let those serve as an implicit checksum. These bits are not quite enough to ensure that no errors occur and the main technical challenge is to show that decoding can recover from these errors.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"25 1 1","pages":"228-241"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75419019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.48550/arXiv.2211.00707
Alexander Braun, Thomas Kesselheim
We consider prophet inequalities for XOS and MPH-$k$ combinatorial auctions and give a simplified proof for the existence of static and anonymous item prices which recover the state-of-the-art competitive ratios. Our proofs make use of a linear programming formulation which has a non-negative objective value if there are prices which admit a given competitive ratio $alpha geq 1$. Changing our perspective to dual space by an application of strong LP duality, we use an interpretation of the dual variables as probabilities to directly obtain our result. In contrast to previous work, our proofs do not require to argue about specific values of buyers for bundles, but only about the presence or absence of items. As a side remark, for any $k geq 2$, this simplification also leads to a tiny improvement in the best competitive ratio for MPH-$k$ combinatorial auctions from $4k-2$ to $2k + 2 sqrt{k(k-1)} -1$.
{"title":"Simplified Prophet Inequalities for Combinatorial Auctions","authors":"Alexander Braun, Thomas Kesselheim","doi":"10.48550/arXiv.2211.00707","DOIUrl":"https://doi.org/10.48550/arXiv.2211.00707","url":null,"abstract":"We consider prophet inequalities for XOS and MPH-$k$ combinatorial auctions and give a simplified proof for the existence of static and anonymous item prices which recover the state-of-the-art competitive ratios. Our proofs make use of a linear programming formulation which has a non-negative objective value if there are prices which admit a given competitive ratio $alpha geq 1$. Changing our perspective to dual space by an application of strong LP duality, we use an interpretation of the dual variables as probabilities to directly obtain our result. In contrast to previous work, our proofs do not require to argue about specific values of buyers for bundles, but only about the presence or absence of items. As a side remark, for any $k geq 2$, this simplification also leads to a tiny improvement in the best competitive ratio for MPH-$k$ combinatorial auctions from $4k-2$ to $2k + 2 sqrt{k(k-1)} -1$.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"38 1","pages":"381-389"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74495332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-26DOI: 10.48550/arXiv.2210.14736
P. Afshani, P. Cheng
We give a simplified and improved lower bound for the simplex range reporting problem. We show that given a set $P$ of $n$ points in $mathbb{R}^d$, any data structure that uses $S(n)$ space to answer such queries must have $Q(n)=Omega((n^2/S(n))^{(d-1)/d}+k)$ query time, where $k$ is the output size. For near-linear space data structures, i.e., $S(n)=O(nlog^{O(1)}n)$, this improves the previous lower bounds by Chazelle and Rosenberg [CR96] and Afshani [A12] but perhaps more importantly, it is the first ever tight lower bound for any variant of simplex range searching for $dge 3$ dimensions. We obtain our lower bound by making a simple connection to well-studied problems in incident geometry which allows us to use known constructions in the area. We observe that a small modification of a simple already existing construction can lead to our lower bound. We believe that our proof is accessible to a much wider audience, at least compared to the previous intricate probabilistic proofs based on measure arguments by Chazelle and Rosenberg [CR96] and Afshani [A12]. The lack of tight or almost-tight (up to polylogarithmic factor) lower bounds for near-linear space data structures is a major bottleneck in making progress on problems such as proving lower bounds for multilevel data structures. It is our hope that this new line of attack based on incidence geometry can lead to further progress in this area.
{"title":"An Optimal Lower Bound for Simplex Range Reporting","authors":"P. Afshani, P. Cheng","doi":"10.48550/arXiv.2210.14736","DOIUrl":"https://doi.org/10.48550/arXiv.2210.14736","url":null,"abstract":"We give a simplified and improved lower bound for the simplex range reporting problem. We show that given a set $P$ of $n$ points in $mathbb{R}^d$, any data structure that uses $S(n)$ space to answer such queries must have $Q(n)=Omega((n^2/S(n))^{(d-1)/d}+k)$ query time, where $k$ is the output size. For near-linear space data structures, i.e., $S(n)=O(nlog^{O(1)}n)$, this improves the previous lower bounds by Chazelle and Rosenberg [CR96] and Afshani [A12] but perhaps more importantly, it is the first ever tight lower bound for any variant of simplex range searching for $dge 3$ dimensions. We obtain our lower bound by making a simple connection to well-studied problems in incident geometry which allows us to use known constructions in the area. We observe that a small modification of a simple already existing construction can lead to our lower bound. We believe that our proof is accessible to a much wider audience, at least compared to the previous intricate probabilistic proofs based on measure arguments by Chazelle and Rosenberg [CR96] and Afshani [A12]. The lack of tight or almost-tight (up to polylogarithmic factor) lower bounds for near-linear space data structures is a major bottleneck in making progress on problems such as proving lower bounds for multilevel data structures. It is our hope that this new line of attack based on incidence geometry can lead to further progress in this area.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"15 1","pages":"272-277"},"PeriodicalIF":0.0,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75242395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-21DOI: 10.48550/arXiv.2210.11784
Václav Rozhoň, Bernhard Haeupler, C. Grunau
We give a simple, local process for nodes in an undirected graph to form non-adjacent clusters that (1) have at most a polylogarithmic diameter and (2) contain at least half of all vertices. Efficient deterministic distributed clustering algorithms for computing strong-diameter network decompositions and other key tools follow immediately. Overall, our process is a direct and drastically simplified way for computing these fundamental objects.
{"title":"A Simple Deterministic Distributed Low-Diameter Clustering","authors":"Václav Rozhoň, Bernhard Haeupler, C. Grunau","doi":"10.48550/arXiv.2210.11784","DOIUrl":"https://doi.org/10.48550/arXiv.2210.11784","url":null,"abstract":"We give a simple, local process for nodes in an undirected graph to form non-adjacent clusters that (1) have at most a polylogarithmic diameter and (2) contain at least half of all vertices. Efficient deterministic distributed clustering algorithms for computing strong-diameter network decompositions and other key tools follow immediately. Overall, our process is a direct and drastically simplified way for computing these fundamental objects.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"74 1","pages":"166-174"},"PeriodicalIF":0.0,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73681785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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