Workshop on Analytic Algorithmics and Combinatorics最新文献

{"title":"On the Number of Hamilton Cycles in Bounded Degree Graphs","authors":"Heidi Gebauer","doi":"10.1137/1.9781611972986.8","DOIUrl":"https://doi.org/10.1137/1.9781611972986.8","url":null,"abstract":"The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs -- deriving thereby extremal bounds. \u0000 \u0000We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276n), improving on Eppstein's previous bound. The resulting new upper bound of O(1.276n) for the maximum number of Hamilton cycles in 3-regular n-vertex graphs gets close to the best known lower bound of Ω(1.259n). Our method differs from Eppstein's in that he considers in each step a new graph and modifies it, while we fix (at the very beginning) one Hamilton cycle C and then proceed around C, succesively producing partial Hamilton cycles. \u0000 \u0000Our approach can also be used to show that the number of Hamilton cycles of a 4-regular n-vertex graph is at most O(18n/5) ≤ O(1.783n), which improves a previous bound by Sharir and Welzl. This result is complemented by a lower bound of 48n/8 ≥ 1.622n. \u0000 \u0000Then we present an algorithm which finds the minimum weight Hamilton cycle of a given 4-regular graph in time √3n · poly(n) = O(1.733n), improving a previous result of Eppstein. This algorithm can be modified to compute the number of Hamilton cycles in the same time bound and to enumerate all Hamilton cycles in time (√3n +hc(G))·poly(n) with hc(G) denoting the number of Hamilton cycles of the given graph G. So our upper bound of O(1.783n) for the number of Hamilton cycles serves also as a time bound for enumeration. \u0000 \u0000Using similar techniques as in the 3-regular case we establish upper bounds for the number of Hamilton cycles in 5-regular graphs and in graphs of average degree 3, 4, and 5.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126146191","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}

{"title":"Generating Random Derangements","authors":"C. Martínez, A. Panholzer, H. Prodinger","doi":"10.1137/1.9781611972986.7","DOIUrl":"https://doi.org/10.1137/1.9781611972986.7","url":null,"abstract":"In this short note, we propose a simple and efficient algorithm to generaterandom derangements, that is, permutations without fixed points. We discuss the algorithm correctness and its performance and compare it to other alternatives. We find that the algorithm has expected linear complexity, works in-place with little additional auxiliary memory and qualitatively behaves like the well-known Fisher-Yates shuffle for random permutations or Sattolo's algorithm for random cyclic permutations.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127404371","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}

{"title":"Augmented Graph Models for Small-World Analysis with Geographic Factors","authors":"V. Nguyen, C. Martel","doi":"10.1137/1.9781611972986.5","DOIUrl":"https://doi.org/10.1137/1.9781611972986.5","url":null,"abstract":"Small-world properties, such as small-diameter and clustering, and the power-law property are widely recognized as common features of large-scale real-world networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distance-bias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for real-world complex networks usually don't fully consider these geographical factors. \u0000 \u0000We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz's [33], and Kleinberg's [20] models) and present important initial results on a refined model where we focus on the small-diameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial node-weighted graph H is augmented with additional random links specified by a generic 'distribution rule' τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growth-bounded metric, and τ has a distance-bias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2 > 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β1 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 ≤ Nu(2r)/Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α > 0 such that the probability that a link from u goes to v ≠ u is ∝ 1/dα(u,v). We show which parameters produce a small-diameter graph and how the diameter changes depending on the relationship between the distance-bias parameter α and the two bounded growth parameters β1, β2 > 0. In particular, for most connected base graphs, the diameter of our augmented graph is logarithmic if α ≤ β1, and poly-log if β2 ≤ α 2β2. These results also suggest promising implications for applications in designing routing algorithms.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133057068","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}

{"title":"Exact Analysis of the Recurrence Relations Generalized from the Tower of Hanoi","authors":"A. Matsuura","doi":"10.1137/1.9781611972986.6","DOIUrl":"https://doi.org/10.1137/1.9781611972986.6","url":null,"abstract":"In this paper, we analyze the recurrence relations generalized from the Tower of Hanoi problem of the form T(n, α, β) = min1≤t≤n{α T(n − t, α, β)+β S(t, 3)}, where S(t, 3) = 2t − 1 is the optimal solution for the 3-peg Tower of Hanoi problem. It is shown that when α and β are natural numbers and α ≥ 2, the sequence of differences of T(n, α, β)'s, i.e., T(n, α, β) − T(n − 1, α, β), consists of numbers of the form β2iαj (i, j ≥ 0) lined in the increasing order.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122177848","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}

{"title":"Bloom Maps","authors":"David Talbot, J. Talbot","doi":"10.1137/1.9781611972986.4","DOIUrl":"https://doi.org/10.1137/1.9781611972986.4","url":null,"abstract":"We consider the problem of succinctly encoding a static map to support approximate queries. We derive upper and lower bounds on the space requirements in terms of the error rate and the entropy of the distribution of values over keys: our bounds differ by a small constant factor. For the upper bound we introduce a novel data structure, the Bloom map, generalising the Bloom filter to this problem. The lower bound follows from an information theoretic argument.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128893184","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}

{"title":"Analysis of Insertion Costs in Priority Trees","authors":"Markus Kuba, A. Panholzer","doi":"10.1137/1.9781611972979.2","DOIUrl":"https://doi.org/10.1137/1.9781611972979.2","url":null,"abstract":"Priority trees are a data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . ., n are equally likely to construct a priority tree of size n we give a detailed average-case analysis of insertion cost measures: we study the recursion depth and the number of key comparisons when inserting an element into a random size-n priority tree. For inserting a random element we obtain exact and asymptotic results for the expectation and the variance and can further show a central limit law of the parameters studied and for inserting an element with specified priority we can show exact and asymptotic results for the expectation of these quantities.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"41 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131865507","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}

{"title":"The Average Profile of Suffix Trees","authors":"Mark Daniel Ward","doi":"10.1137/1.9781611972979.3","DOIUrl":"https://doi.org/10.1137/1.9781611972979.3","url":null,"abstract":"The internal profile of a tree structure denotes the number of internal nodes found at a specific level of the tree. Similarly, the external profile denotes the number of leaves on a level. The profile is of great interest because of its intimate connection to many other parameters of trees. For instance, the depth, fill-up level, height, path length, shortest path, and size of trees can each be interpreted in terms of the profile. \u0000 \u0000The current study is motivated by the work of Park et al. [22], which was a comprehensive study of the profile of tries constructed from independent strings (also, each string generated by a memoryless source). In the present paper, however, we consider suffix trees, which are constructed from suffixes of a common string. The dependency between suffixes demands a careful, intricate treatment of overlaps in words. \u0000 \u0000We precisely analyze the average internal and external profiles of suffix trees generated by a memoryless source. We utilize combinatorics on words (in particular, autocorrelation, i.e., the degree to which a word overlaps with itself) generating functions, singularity analysis, and the Mellin transform. We make comparisons of the average profile of suffix trees to the average profile of tries constructed from independent strings. We emphasize that our methods are extensible to higher moments. The present report describes the first moment of both the internal and external profiles of suffix trees.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130817424","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}

{"title":"Fast Sorting and Pattern-avoiding Permutations","authors":"David Arthur","doi":"10.1137/1.9781611972979.1","DOIUrl":"https://doi.org/10.1137/1.9781611972979.1","url":null,"abstract":"We say a permutation π \"avoids\" a pattern σ if no length |σ| subsequence of π is ordered in precisely the same way as σ. For example, π avoids (1, 2, 3) if it contains no increasing subsequence of length three. It was recently shown by Marcus and Tardos that the number of permutations of length n avoiding any fixed pattern is at most exponential in n. This suggests the possibility that if π is known a priori to avoid a fixed pattern, it may be possible to sort π in as little as linear time. Fully resolving this possibility seems very challenging, but in this paper, we demonstrate a large class of patterns σ for which σ-avoiding permutations can be sorted in O(n log log log n) time.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124503632","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}

{"title":"Estimating the Number of Active Flows in a Data Stream over a Sliding Window","authors":"Éric Fusy, F. Giroire","doi":"10.1137/1.9781611972979.9","DOIUrl":"https://doi.org/10.1137/1.9781611972979.9","url":null,"abstract":"A new algorithm is introduced to estimate the number of distinct flows (or connections) in a data stream. The algorithm maintains an accurate estimate of the number of distinct flows over a sliding window. It is simple to implement, parallelizes optimally, and has a very good trade-off between auxiliary memory and accuracy of the estimate: a relative accuracy of order 1/√m requires essentially a memory of order mln(n/m) words, where n is an upper bound on the number of flows to be seen over the sliding window. For instance, a memory of only 64kB is sufficient to maintain an estimate with accuracy of order 4 percents for a stream with several million flows. The algorithm has been validated both by simulations and experimentations on real traffic. It proves very efficient to monitor traffic and detect attacks.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125339633","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}

{"title":"On the Average Cost of Insertions on Random Relaxed K-d Trees","authors":"Amalia Duch Brown, C. Martínez","doi":"10.1137/1.9781611972979.4","DOIUrl":"https://doi.org/10.1137/1.9781611972979.4","url":null,"abstract":"In this work we refine the average case analysis of randomized insertions and deletions in random relaxed K-d trees, first given by Broutin et al. in [3]. The analysis is based in the analysis of the split and join algorithms, which recursively call one another and are the basis of the randomized update operations under consideration. \u0000 \u0000For K = 2 the average cost of insertions and deletions is Θ(log n). For K > 2, this average cost is Θ(np(K)-1), for some p(K) > 1. This immediately follows from the analysis of the expected cost sn of splitting a tree of size n, which is the same as the expected cost mn of joining a pair of trees with total size n. These costs are, for K = 2, sn = mn = Θ(n) and, for K > 2, sn = mn = Ω(np(K)). In this abstract we find a closed form for the value of the exponent p(K), as well as the constant factor multiplying the main order term in sn.","PeriodicalId":340112,"journal":{"name":"Workshop on Analytic Algorithmics and Combinatorics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121682160","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}