Complexity theory typically studies the complexity of computing a function $h(x) : zo^m to zo^n$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits. Our main results are: (1) Any function $f : zo^ell to zon$ such that (i) each output bit $f_i$ depends on $o(log n)$ input bits, and (ii) $ell le log_2 binom{n}{alpha n} + n^{0.99}$, has output distribution $f(U)$ at statistical distance $ge 1 - 1/n^{0.49}$ from the uniform distribution over $n$-bit strings of hamming weight $alpha n$. We also prove lower bounds for generating $(X,b(X))$ for boolean $b$, and in the case in which each bit $f_i$ is a small-depth decision tree. These lower bounds seem to be the first of their kind, the proofs use anti-concentration results for the sum of random variables. (2) Lower bounds for generating distributions imply succinct data structures lower bounds. As a corollary of (1), we obtain the first lower bound for the membership problem of representing a set $S subseteq [n]$ of size $alpha n$, in the case where $1/alpha$ is a power of $2$: If queries ``$i in S$?'' are answered by non-adaptively probing $o(log n)$ bits, then the representation uses $ge log_2 binom{n}{alpha n} + Omega(log n)$ bits. (3) Upper bounds complementing the bounds in (1) for various settings of parameters. (4) Uniform randomized $acz$ circuits of $poly(n)$ size and depth $d = O(1)$ with error $e$ can be simulated by uniform randomized $acz$ circuits of $poly(n)$ size and depth $d+1$ with error $e + o(1)$ using $le (log n)^{O( log log n)}$ random bits. Previous derandomizations [Ajtai and Wigderson '85, Nisan '91] increase the depth by a constant factor, or else have poor seed length.
复杂性理论通常研究计算给定输入$x$的函数$h(x) : zo^m to zo^n$的复杂性。我们提倡研究对于给定随机比特的均匀$x$生成分布$h(x)$的复杂性。我们的主要结果是:(1)任何函数$f : zo^ell to zon$这样(i)每个输出位$f_i$依赖于$o(log n)$输入位，以及(ii) $ell le log_2 binom{n}{alpha n} + n^{0.99}$，在统计距离$ge 1 - 1/n^{0.49}$上的输出分布$f(U)$与汉明权值$alpha n$的$n$位串的均匀分布。我们还证明了布尔$b$生成$(X,b(X))$的下界，其中每个位$f_i$是一个小深度决策树。这些下界似乎是同类中的第一个，证明使用了随机变量和的反集中结果。(2)生成分布的下界意味着简洁的数据结构下界。作为(1)的推论，我们得到了表示大小为$alpha n$的集合$S subseteq [n]$的隶属性问题的第一个下界，在$1/alpha$是$2$的幂次的情况下:如果查询“$i in S$ ?”由非自适应探测$o(log n)$位回答，则表示使用$ge log_2 binom{n}{alpha n} + Omega(log n)$位。(3)各参数设置的上界与(1)的上界互补。(4)均匀随机化$acz$电路的$poly(n)$大小和深度$d = O(1)$有误差$e$可以通过均匀随机化$acz$电路的$poly(n)$大小和深度$d+1$有误差$e + o(1)$使用$le (log n)^{O( log log n)}$随机位来模拟。以前的非随机化[Ajtai和Wigderson '85, Nisan '91]以常数因子增加深度，否则种子长度就很差。
{"title":"The Complexity of Distributions","authors":"Emanuele Viola","doi":"10.1137/100814998","DOIUrl":"https://doi.org/10.1137/100814998","url":null,"abstract":"Complexity theory typically studies the complexity of computing a function $h(x) : zo^m to zo^n$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits. Our main results are: (1) Any function $f : zo^ell to zon$ such that (i) each output bit $f_i$ depends on $o(log n)$ input bits, and (ii) $ell le log_2 binom{n}{alpha n} + n^{0.99}$, has output distribution $f(U)$ at statistical distance $ge 1 - 1/n^{0.49}$ from the uniform distribution over $n$-bit strings of hamming weight $alpha n$. We also prove lower bounds for generating $(X,b(X))$ for boolean $b$, and in the case in which each bit $f_i$ is a small-depth decision tree. These lower bounds seem to be the first of their kind, the proofs use anti-concentration results for the sum of random variables. (2) Lower bounds for generating distributions imply succinct data structures lower bounds. As a corollary of (1), we obtain the first lower bound for the membership problem of representing a set $S subseteq [n]$ of size $alpha n$, in the case where $1/alpha$ is a power of $2$: If queries ``$i in S$?'' are answered by non-adaptively probing $o(log n)$ bits, then the representation uses $ge log_2 binom{n}{alpha n} + Omega(log n)$ bits. (3) Upper bounds complementing the bounds in (1) for various settings of parameters. (4) Uniform randomized $acz$ circuits of $poly(n)$ size and depth $d = O(1)$ with error $e$ can be simulated by uniform randomized $acz$ circuits of $poly(n)$ size and depth $d+1$ with error $e + o(1)$ using $le (log n)^{O( log log n)}$ random bits. Previous derandomizations [Ajtai and Wigderson '85, Nisan '91] increase the depth by a constant factor, or else have poor seed length.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117007385","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}
We generalize algorithms from computational learning theory that are successful under the uniform distribution on the Boolean hypercube ${0,1}^n$ to algorithms successful on permutation invariant distributions. A permutation invariant distribution is a distribution where the probability mass remains constant upon permutations in the instances. While the tools in our generalization mimic those used for the Boolean hypercube, the fact that permutation invariant distributions are not product distributions presents a significant obstacle. Under the uniform distribution, half spaces can be agnostically learned in polynomial time for constant $eps$. The main tools used are a theorem of Peres~cite{Peres04} bounding the {it noise sensitivity} of a half space, a result of~cite{KOS04} that this theorem implies Fourier concentration, and a modification of the Low-Degree algorithm of Linial, Man sour, Nisan~cite{LMN:93} made by Kalai et. al.~cite{KKMS08}. These results are extended to arbitrary product distributions in~cite{BOWi08}. We prove analogous results for permutation invariant distributions, more generally, we work in the domain of the symmetric group. We define noise sensitivity in this setting, and show that noise sensitivity has a nice combinatorial interpretation in terms of Young tableaux. The main technical innovations involve techniques from the representation theory of the symmetric group, especially the combinatorics of Young tableaux. We show that low noise sensitivity implies concentration on “simple'' components of the Fourier spectrum, and that this fact will allow us to agnostically learn half spaces under permutation invariant distributions to constant accuracy in roughly the same time as in the uniform distribution over the Boolean hypercube case.
我们将计算学习理论中在布尔超立方${0,1}^n$均匀分布下成功的算法推广到在置换不变分布下成功的算法。排列不变分布是指实例中发生排列后概率质量保持不变的分布。虽然我们泛化中的工具模拟了用于布尔超立方体的工具，但排列不变分布不是乘积分布这一事实构成了一个重大障碍。在均匀分布下，对于常数$eps$，半空间可以在多项式时间内进行不可知论学习。使用的主要工具是Peres的一个定理cite{Peres04}限定了半空间的{it噪声灵敏度}，cite{KOS04}的结果表明该定理意味着傅里叶浓度，以及Kalai等人cite{KKMS08}对Linial, Man sour, Nisan cite{LMN:93}的Low-Degree算法的修改。这些结果推广到cite{BOWi08}中的任意乘积分布。我们证明了置换不变分布的类似结果，更一般地说，我们是在对称群的定义域上工作的。在这种情况下，我们定义了噪声敏感性，并表明噪声敏感性在杨氏场景中有一个很好的组合解释。主要的技术创新涉及对称群的表示理论，特别是杨格表的组合学。我们表明，低噪声灵敏度意味着集中在傅立叶谱的“简单”分量上，这一事实将使我们能够在排列不变分布下以恒定精度不可知论地学习半空间，其时间与布尔超立方体情况下的均匀分布大致相同。
{"title":"Agnostically Learning under Permutation Invariant Distributions","authors":"K. Wimmer","doi":"10.1109/FOCS.2010.17","DOIUrl":"https://doi.org/10.1109/FOCS.2010.17","url":null,"abstract":"We generalize algorithms from computational learning theory that are successful under the uniform distribution on the Boolean hypercube ${0,1}^n$ to algorithms successful on permutation invariant distributions. A permutation invariant distribution is a distribution where the probability mass remains constant upon permutations in the instances. While the tools in our generalization mimic those used for the Boolean hypercube, the fact that permutation invariant distributions are not product distributions presents a significant obstacle. Under the uniform distribution, half spaces can be agnostically learned in polynomial time for constant $eps$. The main tools used are a theorem of Peres~cite{Peres04} bounding the {it noise sensitivity} of a half space, a result of~cite{KOS04} that this theorem implies Fourier concentration, and a modification of the Low-Degree algorithm of Linial, Man sour, Nisan~cite{LMN:93} made by Kalai et. al.~cite{KKMS08}. These results are extended to arbitrary product distributions in~cite{BOWi08}. We prove analogous results for permutation invariant distributions, more generally, we work in the domain of the symmetric group. We define noise sensitivity in this setting, and show that noise sensitivity has a nice combinatorial interpretation in terms of Young tableaux. The main technical innovations involve techniques from the representation theory of the symmetric group, especially the combinatorics of Young tableaux. We show that low noise sensitivity implies concentration on “simple'' components of the Fourier spectrum, and that this fact will allow us to agnostically learn half spaces under permutation invariant distributions to constant accuracy in roughly the same time as in the uniform distribution over the Boolean hypercube case.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114879144","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}
We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality'' between linear perturbations of the objective function of an optimization problem and of its feasible set, and use the “primal'' and “dual'' viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.
{"title":"Black-Box Randomized Reductions in Algorithmic Mechanism Design","authors":"S. Dughmi, T. Roughgarden","doi":"10.1137/110843654","DOIUrl":"https://doi.org/10.1137/110843654","url":null,"abstract":"We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multi-parameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality'' between linear perturbations of the objective function of an optimization problem and of its feasible set, and use the “primal'' and “dual'' viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124514371","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}
We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search. Our first main result is a simple deterministic algorithm running in time $2^{n - Omega(n)}$ for satisfiability of formulae of linear size in $n$, where $n$ is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound. Our second main result is a deterministic algorithm running in time $2^{n - Omega(n/log(n))}$ for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are ``structured'', in a certain precise sense, the algorithm can be modified to run in time $2^{n - Omega(n)}$. To the best of our knowledge, no non-trivial algorithms were known for these problems before. As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size $2^{n - Omega(n)}$. As a consequence, we get strong super linear {it average-case} formula size lower bounds for the Parity function.
{"title":"Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability","authors":"R. Santhanam","doi":"10.1109/FOCS.2010.25","DOIUrl":"https://doi.org/10.1109/FOCS.2010.25","url":null,"abstract":"We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search. Our first main result is a simple deterministic algorithm running in time $2^{n - Omega(n)}$ for satisfiability of formulae of linear size in $n$, where $n$ is the number of variables in the formula. This algorithm extends to exactly counting the number of satisfying assignments, within the same time bound. Our second main result is a deterministic algorithm running in time $2^{n - Omega(n/log(n))}$ for solving QBFs in which the number of occurrences of any variable is bounded by a constant. For instances which are ``structured'', in a certain precise sense, the algorithm can be modified to run in time $2^{n - Omega(n)}$. To the best of our knowledge, no non-trivial algorithms were known for these problems before. As a byproduct of the technique used to establish our first main result, we show that every function computable by linear-size formulae can be represented by decision trees of size $2^{n - Omega(n)}$. As a consequence, we get strong super linear {it average-case} formula size lower bounds for the Parity function.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126343448","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}
Zvika Brakerski, Y. Kalai, Jonathan Katz, V. Vaikuntanathan
In recent years, there has been a major effort to design cryptographic schemes that remain secure even when arbitrary information about the secret key is leaked (e.g., via side-channel attacks). We explore the possibility of achieving security under emph{continual} leakage from the emph{entire} secret key by designing schemes in which the secret key is updated over time. In this model, we construct public-key encryption schemes, digital signatures, and identity-based encryption schemes that remain secure even if an attacker can leak a constant fraction of the secret memory (including the secret key) in each time period between key updates. We also consider attackers who may probe the secret memory during the updates themselves. We stress that we allow unrestricted leakage, without the assumption that ``only computation leaks information''. Prior to this work, constructions of public-key encryption schemes secure under continual leakage were not known even under this assumption.
{"title":"Overcoming the Hole in the Bucket: Public-Key Cryptography Resilient to Continual Memory Leakage","authors":"Zvika Brakerski, Y. Kalai, Jonathan Katz, V. Vaikuntanathan","doi":"10.1109/FOCS.2010.55","DOIUrl":"https://doi.org/10.1109/FOCS.2010.55","url":null,"abstract":"In recent years, there has been a major effort to design cryptographic schemes that remain secure even when arbitrary information about the secret key is leaked (e.g., via side-channel attacks). We explore the possibility of achieving security under emph{continual} leakage from the emph{entire} secret key by designing schemes in which the secret key is updated over time. In this model, we construct public-key encryption schemes, digital signatures, and identity-based encryption schemes that remain secure even if an attacker can leak a constant fraction of the secret memory (including the secret key) in each time period between key updates. We also consider attackers who may probe the secret memory during the updates themselves. We stress that we allow unrestricted leakage, without the assumption that ``only computation leaks information''. Prior to this work, constructions of public-key encryption schemes secure under continual leakage were not known even under this assumption.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131036183","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}
We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present a $(polylog(n), poly(loglog n))$-approximation, which means that if there exists a solution that routes $X$ demands integrally on edge-disjoint paths (i.e. with congestion $1$), then the approximation algorithm can route $X/polylog(n)$ demands with congestion $poly(loglog n)$. The best previous result for this problem was a $(n^{1/beta}, beta)$-approximation for $beta
{"title":"Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions","authors":"M. Andrews","doi":"10.1109/FOCS.2010.33","DOIUrl":"https://doi.org/10.1109/FOCS.2010.33","url":null,"abstract":"We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present a $(polylog(n), poly(loglog n))$-approximation, which means that if there exists a solution that routes $X$ demands integrally on edge-disjoint paths (i.e. with congestion $1$), then the approximation algorithm can route $X/polylog(n)$ demands with congestion $poly(loglog n)$. The best previous result for this problem was a $(n^{1/beta}, beta)$-approximation for $beta","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128720073","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}
We consider statistical data analysis in the interactive setting. In this setting a trusted curator maintains a database of sensitive information about individual participants, and releases privacy-preserving answers to queries as they arrive. Our primary contribution is a new differentially private multiplicative weights mechanism for answering a large number of interactive counting (or linear) queries that arrive online and may be adaptively chosen. This is the first mechanism with worst-case accuracy guarantees that can answer large numbers of interactive queries and is {em efficient} (in terms of the runtime's dependence on the data universe size). The error is asymptotically emph{optimal} in its dependence on the number of participants, and depends only logarithmically on the number of queries being answered. The running time is nearly {em linear} in the size of the data universe. As a further contribution, when we relax the utility requirement and require accuracy only for databases drawn from a rich class of databases, we obtain exponential improvements in running time. Even in this relaxed setting we continue to guarantee privacy for {em any} input database. Only the utility requirement is relaxed. Specifically, we show that when the input database is drawn from a {em smooth} distribution — a distribution that does not place too much weight on any single data item — accuracy remains as above, and the running time becomes {em poly-logarithmic} in the data universe size. The main technical contributions are the application of multiplicative weights techniques to the differential privacy setting, a new privacy analysis for the interactive setting, and a technique for reducing data dimensionality for databases drawn from smooth distributions.
{"title":"A Multiplicative Weights Mechanism for Privacy-Preserving Data Analysis","authors":"Moritz Hardt, G. Rothblum","doi":"10.1109/FOCS.2010.85","DOIUrl":"https://doi.org/10.1109/FOCS.2010.85","url":null,"abstract":"We consider statistical data analysis in the interactive setting. In this setting a trusted curator maintains a database of sensitive information about individual participants, and releases privacy-preserving answers to queries as they arrive. Our primary contribution is a new differentially private multiplicative weights mechanism for answering a large number of interactive counting (or linear) queries that arrive online and may be adaptively chosen. This is the first mechanism with worst-case accuracy guarantees that can answer large numbers of interactive queries and is {em efficient} (in terms of the runtime's dependence on the data universe size). The error is asymptotically emph{optimal} in its dependence on the number of participants, and depends only logarithmically on the number of queries being answered. The running time is nearly {em linear} in the size of the data universe. As a further contribution, when we relax the utility requirement and require accuracy only for databases drawn from a rich class of databases, we obtain exponential improvements in running time. Even in this relaxed setting we continue to guarantee privacy for {em any} input database. Only the utility requirement is relaxed. Specifically, we show that when the input database is drawn from a {em smooth} distribution — a distribution that does not place too much weight on any single data item — accuracy remains as above, and the running time becomes {em poly-logarithmic} in the data universe size. The main technical contributions are the application of multiplicative weights techniques to the differential privacy setting, a new privacy analysis for the interactive setting, and a technique for reducing data dimensionality for databases drawn from smooth distributions.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131696287","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}
We prove that planar graphs have poly-logarithmic queue number, thus improving upon the previous polynomial upper bound. Consequently, planar graphs admit 3D straight-line crossing-free grid drawings in small volume.
{"title":"On the Queue Number of Planar Graphs","authors":"G. Battista, Fabrizio Frati, J. Pach","doi":"10.1109/FOCS.2010.42","DOIUrl":"https://doi.org/10.1109/FOCS.2010.42","url":null,"abstract":"We prove that planar graphs have poly-logarithmic queue number, thus improving upon the previous polynomial upper bound. Consequently, planar graphs admit 3D straight-line crossing-free grid drawings in small volume.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124267378","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}
We present an all-pairs shortest path algorithm whose running time on a complete directed graph on $n$ vertices whose edge weights are chosen independently and uniformly at random from $[0,1]$ is~$O(n^2)$, in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in $O(log^{2}n)$ expected time.
{"title":"All-Pairs Shortest Paths in O(n²) Time with High Probability","authors":"Y. Peres, D. Sotnikov, B. Sudakov, Uri Zwick","doi":"10.1145/2508028.2505988","DOIUrl":"https://doi.org/10.1145/2508028.2505988","url":null,"abstract":"We present an all-pairs shortest path algorithm whose running time on a complete directed graph on $n$ vertices whose edge weights are chosen independently and uniformly at random from $[0,1]$ is~$O(n^2)$, in expectation and with high probability. This resolves a long standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano. The analysis relies on a proof that the number of emph{locally shortest paths} in such randomly weighted graphs is $O(n^2)$, in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in $O(log^{2}n)$ expected time.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123083873","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}
Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when "linear time" is replaced by "logarithmic space." The transfer of the powerful theoretical framework of monadic second-order logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the log space world.
{"title":"Logspace Versions of the Theorems of Bodlaender and Courcelle","authors":"Michael Elberfeld, A. Jakoby, Till Tantau","doi":"10.1109/FOCS.2010.21","DOIUrl":"https://doi.org/10.1109/FOCS.2010.21","url":null,"abstract":"Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when \"linear time\" is replaced by \"logarithmic space.\" The transfer of the powerful theoretical framework of monadic second-order logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the log space world.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129760191","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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