We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1991 Mathematics Subject Classifications: Primary 60C05, 05E10, 15A52, 60F05. Research supported by N.S.F. Grant MCS 96-22859 1 Introduction This survey paper treats two themes in parallel. One theme is a purely mathematical question: describe the asymptotic law (probability distribution) of the length of the longest increasing subsequence of a random permutation. This question has been studied by a variety of increasingly technically sophisticated methods over the last 30 years. We outline three, apparently quite unrelated, methods in sections 2 - 4. The other theme is a card game, patience sorting. This gam...

In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

In this paper we present three algorithms that solve three combinatorial optimization problems related to each other. One of them is the patience sorting game, invented as a practical method of sorting real decks of cards. The second problem is computing the longest monotone increasing subsequence of the given sequence of n positive integers in the range 1; : : : ; n. The third problem is to enumerate all the longest monotone increasing subsequences of the given permutation.

Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and improve fixed-parameter tractability results for these problems. We show that Feedback Vertex Set in tournaments (FVST) is amenable to the novel iterative compression technique, and we provide a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization. Moreover, we apply the iterative compression technique to d-Hitting Set, which generalizes Feedback Vertex Set in tournaments, and obtain improved upper bounds for the time needed to solve 4-Hitting Set and 5-Hitting Set. Using our parameterized algorithm for Feedback Vertex Set in tournaments, we also give an exact (not parameterized) algorithm for it running in O(1.709 n) time, where n is the number of input graph vertices, answering a question of Woeginger [Discrete Appl. Math. 156(3):397–405, 2008].

We study a scheduling problem in which jobs have locations. For example, consider a repairman that is supposed to visit customers at their homes. Each customer is given a time window during which the repairman is allowed to arrive. The goal is to find a schedule that visits as many homes as possible. We refer to this problem as the Prize-Collecting Traveling Salesman Problem with time windows (TW-TSP).

We explore various techniques to compress a permutation π over n integers, taking advantage of ordered subsequences in π, while supporting its application π(i) and the application of its inverse π −1 (i) in small time. Our compression schemes yield several interesting byproducts, in many cases matching, improving or extending the best existing results on applications such as the encoding of a permutation in order to support iterated applications π k (i) of it, of integer functions, and of inverted lists and suffix arrays.

In this paper, we present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data streaming model.

A brief survey of the theory and practice of sequence comparison is made focusing on diff, the UNIX 1 file difference utility. 1 Sequence comparison Sequence comparison is a deep and fascinating subject in Computer Science, both theoretical and practical. However, in our opinion, neither the theoretical nor the practical aspects of the problem are well understood and we feel that their mastery is a true challenge for Computer Science. The central problem can be stated very easily: find an algorithm, as efficient and practical as possible, to compute a longest common subsequence (lcs for short) of two given sequences 2 . As usual, a subsequence of a sequence is another sequence obtained from it by deleting some (not necessarily contiguous) terms. Thus, both en/pri and en/pai are longest common subsequences of sequence/comparison and theory/and/practice. Part of this work was done while the author was visiting the Universit'e de Rouen, in 1987. That visit was partially supported...

Let A = 〈a1, a2,..., am 〉 and B = 〈b1, b2,..., bn 〉 be two sequences, where each pair of elements in the sequences is comparable. A common increasing subsequence of A and B is a subsequence 〈ai1 = bj1, ai2 = bj2,..., ail = bjl 〉, where i1 < i2 < · · · < il and j1 < j2 < · · · < jl, such that for all 1 ≤ k < l, we have aik < aik+1. A longest common increasing subsequence of A and B is a common increasing subsequence of the maximum length. This paper presents an algorithm for delivering a longest common increasing subsequence in O(mn) time and O(mn) space.

Given a sequence π1π2... πn, a longest increasing subsequence (LIS) in a window π〈l, r〉 = πlπl+1... πr is a longest subsequence σ = πi1πi2... πiT such that l ≤ i1 < i2 < · · · < iT ≤ r and πi1 < πi2 < · · · < πiT. We consider the Lisw problem, which is to find the longest increasing subsequences in a sliding window of fixed-size w over a sequence. Formally, it is to find a LIS for every window in a set SFIX = � π〈i + 1, i + w 〉 � � 0 ≤ i ≤ n − w � ∪ � π〈1, i〉, π〈n − i, n 〉 � � i < w �. By maintaining a canonical antichain partition in windows, we present an optimal output-sensitive algorithm to solve this problem in O(output) time, where output is the sum of the lengths of the n+w −1 LISs in those windows of SFIX. In addition, we propose a more generalized problem called Lisset problem, which is to find a LIS for every window in a set SVAR containing variable-size windows. By applying our algorithm, we provide an efficient solution for the Lisset problem to output a LIS (or all the LISs) in every window which is better than the straightforward generalization of classical LIS algorithms. An upper bound of our algorithm on the Lisset problem is discussed.