In this paper, we construct a data structure using a forest of redblack trees to efficiently compute the longest increasing subsequence of a dynamically updated sequence. Our data structure supports a query for the longest increasing subsequence in O(r + logn) worst case time and supports inserts anywhere in the sequence in O (r logn/r) worst case time, where r is the length of the longest incr...

Given a random permutation of the numbers 1, 2, . . . . n, let L, be the length of the longest descending subsequence of this permutation. Let F, be the minimal header (first element) of the descending subsequences having maximal length. It is known that EL,/&,,,, c and that c=2. However, the proofs that r=2 are far from elementary and involve limit processes. Several relationships between thes...

The paper presents a load balancing method for some CGM (Coarse-Grained Multicomputer) algorithms. This method can be applied on different dynamic programming problems such as: Longest Increasing Subsequence, Longest Common Subsequence, Longest Repeated Suffix Ending at each point in a word and Detection of Repetitions. We present also experimental results showing that our method is efficient.

Let Ln be the length of a longest increasing subsequence in a random permutation of {1, ..., n}. It is known that the expected value of Ln is asymptotically equal to 2 √ n as n gets large. This note derives upper bound on the probability that Ln − 2 √ n exceeds certain quantities. In particular, we prove that Ln − 2 √ n has order at most n1/6 with high probability. Our main result is an isoperi...

Standard tableaux. A Young tableau (also known as a standard tableau) is an arrangement of distinct integers in an array of left justiied rows, such that the entries of each row are in increasing order from left to right and the entries of each column are in increasing order from bottom to top 2. For instance, here is an array of size 15: 11 4 8 12 14 3 6 7 13 1 2 5 9 10 15 : In general a table...

The weak Bruhat order on Sn is the partial order ≺ so that σ ≺ τ whenever the set of inversions of σ is a subset of the set of inversions of τ . We investigate the time complexity of computing the size of intervals with respect to ≺. Using relationships between two-dimensional posets and the weak Bruhat order, we show that the size of the interval [σ1, σ2] can be computed in polynomial time whe...

Let S = s1, s2, . . . , sn be an integer sequence. The longest increasing subsequence problem is to find an increasing subsequence of S with the longest length. By regarding S as a weight sequence of the vertices in a path, we can redefine the longest increasing subsequence problem on graphs as follows. Let G = (V,E) be a graph in which every vertex v ∈ V has a weight w(v). A longest increasing...

where each ( ) ∈ , 1 ≤  ≤ , and  is arbitrary. For convenience, define (0 0) = (0 0) and (+1 +1) = (1 1). Define such a point sequence to be an up/right path if, for any  ≥ 1, we have −1 ≤  and −1 ≤ . Hence an up/right path joins points of  in a continuous, piecewise linear manner with line segments of slope , 0 ≤  ≤ ∞, attaching (−1 −1) and ( ) for al...