Pub Date : 2010-01-16DOI: 10.1137/1.9781611973006.5
Yongwook Choi
We consider the problem of finding optimal description for general unlabeled graphs. Given a probability distribution on labeled graphs, we introduced in [4] a structural entropy as a lower bound for the lossless compression of such graphs. Specifically, we proved that the structural entropy for the Erdos--Renyi random graph, in which edges are added with probability p, is (n2)h(p) − n log n + O(n), where n is the number of vertices and h(p) = −p log p − (1 − p) log(1−p) is the entropy rate of a conventional memoryless binary source. In this paper, we prove the asymptotic equipartition property for such graphs. Then, we propose a faster compression algorithm that asymptotically achieves the structural entropy up to the first two leading terms with high probability. Our algorithm runs in O(n + e) time on average where e is the number of edges. To prove its asymptotic optimality, we introduce binary trees that one can classify as in-between tries and digital search trees. We use analytic techniques such as generating functions, Mellin transform, and poissonization to establish our findings. Our experimental results confirm theoretical results and show the usefulness of our algorithm for real-world graphs such as the Internet, biological networks, and social networks.
{"title":"Fast Algorithm for Optimal Compression of Graphs","authors":"Yongwook Choi","doi":"10.1137/1.9781611973006.5","DOIUrl":"https://doi.org/10.1137/1.9781611973006.5","url":null,"abstract":"We consider the problem of finding optimal description for general unlabeled graphs. Given a probability distribution on labeled graphs, we introduced in [4] a structural entropy as a lower bound for the lossless compression of such graphs. Specifically, we proved that the structural entropy for the Erdos--Renyi random graph, in which edges are added with probability p, is (n2)h(p) − n log n + O(n), where n is the number of vertices and h(p) = −p log p − (1 − p) log(1−p) is the entropy rate of a conventional memoryless binary source. In this paper, we prove the asymptotic equipartition property for such graphs. Then, we propose a faster compression algorithm that asymptotically achieves the structural entropy up to the first two leading terms with high probability. Our algorithm runs in O(n + e) time on average where e is the number of edges. To prove its asymptotic optimality, we introduce binary trees that one can classify as in-between tries and digital search trees. We use analytic techniques such as generating functions, Mellin transform, and poissonization to establish our findings. Our experimental results confirm theoretical results and show the usefulness of our algorithm for real-world graphs such as the Internet, biological networks, and social networks.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115323959","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 : 2010-01-16DOI: 10.1137/1.9781611973006.11
Frédérique Bassino, A. Martino, C. Nicaud, E. Ventura, P. Weil
We study and compare two natural distributions of finitely generated subgroups of free groups. One is based on the random generation of tuples of reduced words; that is the one classically used by group theorists. The other relies on Stallings' graphical representation of subgroups and in spite of its naturality, it was only recently considered. The combinatorial structures underlying both distributions are studied in this paper with methods of analytic combinatorics. We use these methods to point out the differences between these distributions. It is particularly interesting that certain important properties of subgroups that are generic in one distribution, turn out to be negligible in the other.
{"title":"On Two Distributions of Subgroups of Free Groups","authors":"Frédérique Bassino, A. Martino, C. Nicaud, E. Ventura, P. Weil","doi":"10.1137/1.9781611973006.11","DOIUrl":"https://doi.org/10.1137/1.9781611973006.11","url":null,"abstract":"We study and compare two natural distributions of finitely generated subgroups of free groups. One is based on the random generation of tuples of reduced words; that is the one classically used by group theorists. The other relies on Stallings' graphical representation of subgroups and in spite of its naturality, it was only recently considered. The combinatorial structures underlying both distributions are studied in this paper with methods of analytic combinatorics. We use these methods to point out the differences between these distributions. It is particularly interesting that certain important properties of subgroups that are generic in one distribution, turn out to be negligible in the other.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116645778","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 : 2010-01-16DOI: 10.1137/1.9781611973006.4
Amr Elmasry
Given a family F of k sets with cardinalities s1, s2, . . ., sk and N = [EQUATION], we show that the size of the partial order graph induced by the subset relation (called the subset graph) is [EQUATION], where B = log (N/ log2 N). This implies a simpler proof to the O(N2/ log2 N) bound concluded in [2]. We also give an algorithm that computes the subset graph for any family of sets F. Our algorithm requires O(nk2/ log k) time and space on a pointer machine, where n is the number of domain elements. When F is dense, i.e. N = Θ(nk), the algorithm requires O(N2/ log2 N) time and space. We give a construction for a dense family whose subset graph is of size Θ(N2/ log2 N), indicating the optimality of our algorithm for dense families. The subset graph can be dynamically maintained when F undergoes set insertion and deletion in O(nk/ log k) time per update (that is sub-linear in N for the case of dense families). If we assume words of b ≤ k bits, allow bits to be packed in words, and use bitwise operations, the above running time and space requirements can be reduced by a factor of b log (k/b + 1)/ log k and b2 log (k/b + 1)/ log k respectively.
{"title":"The Subset Partial Order: Computing and Combinatorics","authors":"Amr Elmasry","doi":"10.1137/1.9781611973006.4","DOIUrl":"https://doi.org/10.1137/1.9781611973006.4","url":null,"abstract":"Given a family F of k sets with cardinalities s1, s2, . . ., sk and N = [EQUATION], we show that the size of the partial order graph induced by the subset relation (called the subset graph) is [EQUATION], where B = log (N/ log2 N). This implies a simpler proof to the O(N2/ log2 N) bound concluded in [2]. \u0000 \u0000We also give an algorithm that computes the subset graph for any family of sets F. Our algorithm requires O(nk2/ log k) time and space on a pointer machine, where n is the number of domain elements. When F is dense, i.e. N = Θ(nk), the algorithm requires O(N2/ log2 N) time and space. We give a construction for a dense family whose subset graph is of size Θ(N2/ log2 N), indicating the optimality of our algorithm for dense families. The subset graph can be dynamically maintained when F undergoes set insertion and deletion in O(nk/ log k) time per update (that is sub-linear in N for the case of dense families). If we assume words of b ≤ k bits, allow bits to be packed in words, and use bitwise operations, the above running time and space requirements can be reduced by a factor of b log (k/b + 1)/ log k and b2 log (k/b + 1)/ log k respectively.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125961289","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 : 2010-01-16DOI: 10.1137/1.9781611973006.9
L. Fleischer, Zoya Svitkina
We introduce and study a combinatorial problem called preference-constrained oriented matching. This problem is defined on a directed graph in which each node has preferences over its out-neighbors, and the goal is to find a maximum-size matching on this graph that satisfies a certain preference constraint. One of our main results is a structural theorem showing that if the given graph is complete, then for any preference ordering there always exists a feasible matching that covers a constant fraction of the nodes. This result allows us to correct an error in a proof by Azar, Jain, and Mirrokni [1], establishing a lower bound on the price of anarchy in coordination mechanisms for scheduling. We also show that the preference-constrained oriented matching problem is APX-hard and give a constant-factor approximation algorithm for it.
{"title":"Preference-constrained Oriented Matching","authors":"L. Fleischer, Zoya Svitkina","doi":"10.1137/1.9781611973006.9","DOIUrl":"https://doi.org/10.1137/1.9781611973006.9","url":null,"abstract":"We introduce and study a combinatorial problem called preference-constrained oriented matching. This problem is defined on a directed graph in which each node has preferences over its out-neighbors, and the goal is to find a maximum-size matching on this graph that satisfies a certain preference constraint. One of our main results is a structural theorem showing that if the given graph is complete, then for any preference ordering there always exists a feasible matching that covers a constant fraction of the nodes. This result allows us to correct an error in a proof by Azar, Jain, and Mirrokni [1], establishing a lower bound on the price of anarchy in coordination mechanisms for scheduling. We also show that the preference-constrained oriented matching problem is APX-hard and give a constant-factor approximation algorithm for it.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126466924","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 : 2010-01-16DOI: 10.1137/1.9781611973006.3
D. Panario, L. Richmond, Benjamin Young
We give a new asymptotic formula for a refined enumeration of plane partitions. Specifically: color the parts πi,j of a plane partition π according to the equivalence class of i --- j (mod 2), and keep track of the sums of the 0-colored and 1-colored parts seperately. We find, for large plane partitions, that the difference between these two sums is asymptotically Gaussian (and we compute the mean and standard deviation of the distribution). Our approach is to modify a multivariate technique of Haselgrove and Temperley.
{"title":"Bivariate Asymptotics for Striped Plane Partitions","authors":"D. Panario, L. Richmond, Benjamin Young","doi":"10.1137/1.9781611973006.3","DOIUrl":"https://doi.org/10.1137/1.9781611973006.3","url":null,"abstract":"We give a new asymptotic formula for a refined enumeration of plane partitions. Specifically: color the parts πi,j of a plane partition π according to the equivalence class of i --- j (mod 2), and keep track of the sums of the 0-colored and 1-colored parts seperately. We find, for large plane partitions, that the difference between these two sums is asymptotically Gaussian (and we compute the mean and standard deviation of the distribution). Our approach is to modify a multivariate technique of Haselgrove and Temperley.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131530298","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 : 2010-01-16DOI: 10.1137/1.9781611973006.6
B. Salvy
The talk will explore a variety of old and recent algorithms whose efficiency boils down to the fast convergence of Newton iteration. Numerically, and close to the root, the number of correct digits is doubled at each iteration. When working with power series, the problem of picking a good initial point disappears and the number of coefficients is doubled at each iteration. This observation, coupled with fast multiplication, leads to fast algorithms in a variety of problems of symbolic computation, ranging from classical results on algebraic series to more recent ones on systems of differential equations.
{"title":"Newton Iteration: From Numerics to Combinatorics, and Back","authors":"B. Salvy","doi":"10.1137/1.9781611973006.6","DOIUrl":"https://doi.org/10.1137/1.9781611973006.6","url":null,"abstract":"The talk will explore a variety of old and recent algorithms whose efficiency boils down to the fast convergence of Newton iteration. Numerically, and close to the root, the number of correct digits is doubled at each iteration. When working with power series, the problem of picking a good initial point disappears and the number of coefficients is doubled at each iteration. This observation, coupled with fast multiplication, leads to fast algorithms in a variety of problems of symbolic computation, ranging from classical results on algebraic series to more recent ones on systems of differential equations.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120906148","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 : 2010-01-16DOI: 10.1137/1.9781611973006.1
Philippe Duchon, H. Larchevêque
In this paper we consider the skip graph data structure, a load balancing alternative to skip lists, designed to perform better in a distributed environment. We extend previous results of Devroye on skip lists, and prove that the maximum length of a search path in a random binary skip graph of size n is of order log n with high probability.
{"title":"On the Search Path Length of Random Binary Skip Graphs","authors":"Philippe Duchon, H. Larchevêque","doi":"10.1137/1.9781611973006.1","DOIUrl":"https://doi.org/10.1137/1.9781611973006.1","url":null,"abstract":"In this paper we consider the skip graph data structure, a load balancing alternative to skip lists, designed to perform better in a distributed environment. We extend previous results of Devroye on skip lists, and prove that the maximum length of a search path in a random binary skip graph of size n is of order log n with high probability.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127183620","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 : 2009-10-19DOI: 10.1137/1.9781611973006.13
Alexis Darrasse, Hsien-Kuei Hwang, O. Bodini, Michèle Soria
Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given.
{"title":"The Connectivity-Profile of Random Increasing k-trees","authors":"Alexis Darrasse, Hsien-Kuei Hwang, O. Bodini, Michèle Soria","doi":"10.1137/1.9781611973006.13","DOIUrl":"https://doi.org/10.1137/1.9781611973006.13","url":null,"abstract":"Random increasing k-trees represent an interesting, useful class of strongly dependent graphs for which analytic-combinatorial tools can be successfully applied. We study in this paper a notion called connectivity-profile and derive asymptotic estimates for it; some interesting consequences will also be given.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125379689","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 : 2009-01-03DOI: 10.1137/1.9781611972993.12
M. Lladser
Imagine an urn with colored balls but with an unknown composition i.e. you do not know what are the specific colors in the urn nor their relative proportions. The urn could be composed by infinitely many colors and to learn about its composition you have been sampling balls with replacement from it. In this extended abstract we will construct exact confidence intervals for the proportion in the urn of the so far unobserved colors when there is an upper-bound m for the additional number of samples permitted from the urn. The research is motivated by a variety of situations of practical interest. For instance, the different colors in the urn could represent different solutions to a particular binding site problem in a random RNA pool, or the number of different species of bacteria present in a sample of soil or the gut of a person with a digestive disorder.
{"title":"Prediction of Unseen Proportions in Urn Models with Restricted Sampling","authors":"M. Lladser","doi":"10.1137/1.9781611972993.12","DOIUrl":"https://doi.org/10.1137/1.9781611972993.12","url":null,"abstract":"Imagine an urn with colored balls but with an unknown composition i.e. you do not know what are the specific colors in the urn nor their relative proportions. The urn could be composed by infinitely many colors and to learn about its composition you have been sampling balls with replacement from it. In this extended abstract we will construct exact confidence intervals for the proportion in the urn of the so far unobserved colors when there is an upper-bound m for the additional number of samples permitted from the urn. The research is motivated by a variety of situations of practical interest. For instance, the different colors in the urn could represent different solutions to a particular binding site problem in a random RNA pool, or the number of different species of bacteria present in a sample of soil or the gut of a person with a digestive disorder.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117228596","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 : 2009-01-03DOI: 10.1137/1.9781611972993.4
P. Jacquet, L. Viennot
Motivated by the optimization of link state routing in ad hoc networks, and the concept of multipoint relays, we introduce the notion of remote-spanner. Given an unweighted graph G, a remote spanner is a set of links H such that for any pair of nodes (u, v) there exists a shortest path in G for which all links in the path that are not adjacent to u belong to H. The remote spanner is a kind of minimal topology information beyond its neighborhood that any node would need in order to compute its shortest paths in a distributed way. This can be extended to k-connected graphs by considering minimum length sum over k disjoint paths as distance. In this paper, we give distributed algorithms for computing remote-spanners in order to obtain sparse remote-spanners with various properties. We provide a polynomial distributed algorithm that computes a k-connecting unstretched remote-spanner whose number of edges is at a factor 2(1 + log Δ) from optimal where Δ is the maximum degree of a node. Interestingly, its expected compression ratio in number of edges is O(k/n log n) in Erdos-Renyi graph model and O((k/n)2/3) in the unit disk graph model with a uniform Poisson distribution of nodes.
{"title":"Average Size of Unstretched Remote-Spanners","authors":"P. Jacquet, L. Viennot","doi":"10.1137/1.9781611972993.4","DOIUrl":"https://doi.org/10.1137/1.9781611972993.4","url":null,"abstract":"Motivated by the optimization of link state routing in ad hoc networks, and the concept of multipoint relays, we introduce the notion of remote-spanner. Given an unweighted graph G, a remote spanner is a set of links H such that for any pair of nodes (u, v) there exists a shortest path in G for which all links in the path that are not adjacent to u belong to H. The remote spanner is a kind of minimal topology information beyond its neighborhood that any node would need in order to compute its shortest paths in a distributed way. This can be extended to k-connected graphs by considering minimum length sum over k disjoint paths as distance. \u0000 \u0000In this paper, we give distributed algorithms for computing remote-spanners in order to obtain sparse remote-spanners with various properties. We provide a polynomial distributed algorithm that computes a k-connecting unstretched remote-spanner whose number of edges is at a factor 2(1 + log Δ) from optimal where Δ is the maximum degree of a node. Interestingly, its expected compression ratio in number of edges is O(k/n log n) in Erdos-Renyi graph model and O((k/n)2/3) in the unit disk graph model with a uniform Poisson distribution of nodes.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115379435","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}