2007 Proceedings of the Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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.

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.

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.

The 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.

We 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.

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.

For K = 2 the average cost of insertions and deletions is Θ(log n). For K > 2, this average cost is Θ(n^φ(K)−1), for some φ(K) > 1. This immediately follows from the analysis of the expected cost s_n of splitting a tree of size n, which is the same as the expected cost m_n of joining a pair of trees with total size n. These costs are, for K = 2, s_n = m_n = Θ(n) and, for K > 2, s_n = m_n = Ω(n^φ(K)). In this abstract we find a closed form for the value of the exponent φ(K), as well as the constant factor multiplying the main order term in s_n.

Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they do so by means of real-arithmetic computations. We present a collection of construction rules for such samplers, which applies to a wide variety of combinatorial classes, including integer partitions, necklaces, unlabelled functional graphs, dictionaries, series-parallel circuits, term trees and acyclic molecules obeying a variety of constraints, and so on. Under an abstract real-arithmetic computation model, the algorithms are, for many classical structures, of linear complexity provided a small tolerance is allowed on the size of the object drawn. As opposed to many of their discrete competitors, the resulting programs routinely make it possible to generate random objects of sizes in the range 10⁴−10⁶.

Majority of practical caching algorithms, in particular those used in the World Wide Web applications, are based on the so-called Least-Recently-Used (LRU) cache replacement heuristic whose desirable attributes include low complexity, quick adaptability and high cache hit (low fault) probability. Recent studies have developed asymptotic characterization of the LRU fault probability for the generalized Zipf's (power) law request distributions. In this paper, we extend these results to include the distributions that decay faster than power laws but slower than exponential, hence named moderately heavy distributions. Informally, for these types of distributions and the independent reference model, the main result of this paper shows that the ratio between the cache fault probabilities of the LRU heuristic and the optimal static algorithm is, for large caches, equal to e^γ ≈ 1.78, where γ is Euler's constant. Interestingly enough, this limiting ratio is constant, i.e., it is invariant to the underlying characteristics of the request distributions.

The height of a tree is the maximal path distance between a leaf and the root. It is convenient to attach weights to edges and consider the weighted height in order to obtain a more universal model of tree. We introduce a random weighted model that captures most random search trees and random recursive trees that we know of. For most nodes in such trees, subtree sizes and weights of edges emanating from it are allowed to be dependent in a rather general way. Using the multivariate theory of large deviations, we obtain the first asymptotic term of the height. In the talk, we will survey many applications of our result.

Majority of practical caching algorithms, in particular those used in the World Wide Web applications, are based on the so-called Least-Recently-Used (LRU) cache replacement heuristic whose desirable attributes include low complexity, quick adaptability and high cache hit (low fault) probability. Recent studies have developed asymptotic characterization of the LRU fault probability for the generalized Zipf's (power) law request distributions. In this paper, we extend these results to include the distributions that decay faster than power laws but slower than exponential, hence named moderately heavy distributions. Informally, for these types of distributions and the independent reference model, the main result of this paper shows that the ratio between the cache fault probabilities of the LRU heuristic and the optimal static algorithm is, for large caches, equal to e^γ ≈ 1.78, where γ is Euler's constant. Interestingly enough, this limiting ratio is constant, i.e., it is invariant to the underlying characteristics of the request distributions.

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 tradeoff between auxiliary memory and accuracy of the estimate: a relative accuracy of order requires essentially a memory of order m ln(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.

A graph G is a 2-tree if G = K₃, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v is a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees.

Let T(G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T(G) for circulant graphs.

The circulant graph is the 2k regular graph with n vertices labelled 0, 1, 2, …, n − 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s₁, i + s₂, …, i + s_k mod n. In this note we give a closed formula for the asymptotic limit as a function of s₁, s₂, …, s_n. We then extend this by permitting the s_i to be linear functions of n, i.e., we give a closed formula for

where the d_i and e_i are arbitrary integers. One consequence of our derivation is that if we let the s_i go to infinity then

Interestingly, this value is same as the asymptotic number of spanning trees in the k-dimensional square lattice obtained in Garcia, Noy and Tejel in [7].

The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input's combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples required to produce an accurate probabilistic roadmap, and thus on the running time of the algorithm, in an environment of simplices. This sheds light on its widespread empirical success.

For g < n, let b_l, …, b_n−g be n−g independent vectors in ℝⁿ with a same distribution invariant by rotation and without mass in 0. These vectors form a.s. a basis for the Euclidean lattice they generate. The aim of this paper is to provide asymptotic results when n → +∞ concerning the property that such a random basis is reduced in the sense of Lenstra, Lenstra and Lovász (LLL). If is the new basis obtained by the Gram–Schmidt orthogonalization, the quality of reduction depends on the ratios of squared lengths of consecutive vectors . We show that, as n → +∞, the process (r_j⁽ⁿ⁾ − 1)_j tends in distribution in some sense to an explicit process ; some properties of this latter are provided.

There exist fast variants of the gcd algorithm which are all based on principles due to Knuth and Schönhage. On inputs of size n, these algorithms use a Divide and Conquer approach, perform FFT multiplications and stop the recursion at a depth slightly smaller than lg n. A rough estimate of the worst–case complexity of these fast versions provides the bound O(n(log n)² log log n). However, this estimate is based on some heuristics and is not actually proven. Here, we provide a precise probabilistic analysis of some of these fast variants, and we prove that their average bit–complexity on random inputs of size n is Θ(n(log n)² log log n), with a precise remainder term. We view such a fast algorithm as a sequence of what we call interrupted algorithms, and we obtain three results about the (plain) Euclid Algorithm which may be of independent interest. We precisely describe the evolution of the distribution during the execution of the (plain) Euclid Algorithm; we obtain a sharp estimate for the probability that all the quotients produced by the (plain) Euclid Algorithm are small enough; we also exhibit a strong regularity phenomenon, which proves that these interrupted algorithms are locally “similar” to the total algorithm. This finally leads to the precise evaluation of the average bit–complexity of these fast algorithms. This work uses various tools, and is based on a precise study of generalised transfer operators related to the dynamical system underlying the Euclid Algorithm.