Longest Increasing Subsequence Problem Research Articles

The Longest Wave Subsequence Problem: Generalizations of the Longest Increasing Subsequence Problem

The longest increasing subsequence (LIS) problem aims to find the subsequence exhibiting an increasing trend in a numeric sequence with the maximum length. In this paper, we generalize the LIS problem to the longest wave subsequence (LWS) problem, which encompasses two versions: LWSt and LWSr. Given a numeric sequence [Formula: see text] of distinct values and a target trend sequence [Formula: see text], the LWSt problem aims to identify the longest subsequence of [Formula: see text] that preserves the trend of the prefix of [Formula: see text]. And, the LWSr problem aims to find the longest subsequence of [Formula: see text] within [Formula: see text] segments, alternating increasing and decreasing subsequences. We propose two efficient algorithms for solving the two versions of the LWS problem. For the LWSt problem, the time complexity of our algorithm is O[Formula: see text], where [Formula: see text] represents the length of the given numeric sequence [Formula: see text]. Additionally, we propose an O[Formula: see text]-time algorithm for solving the LWSr problem. In both algorithms, we utilize the priority queues for the insertion, deletion, and successor operations.

EDC-LISP: An Efficient Divide-and-Conquer Solution to The Longest Increasing Subsequence Problem

EDC-LISP: An Efficient Divide-and-Conquer Solution to The Longest Increasing Subsequence Problem

Improvised divide and conquer approach for the LIS problem

There exist many optimal (using single and multiple processors) and approximate solutions to the longest increasing subsequence (LIS) problem. Through this paper, we present the enhancement to the divide-and-conquer approach presented in paper [1]. An improved D&C algorithmic solution is proposed which outputs optimal solution in all cases. The proposed algorithm takes O(nlog⁡n) time in best and average cases and o(nlog2⁡n) time in worst case. The portion of the proposed solution can run in parallel using multiprocessors.

Asymptotics of the Jordan Normal Form of a Random Nilpotent Matrix

We study the Jordan normal form of an upper triangular matrix constructed from a random acyclic graph or a random poset. Some limit theorems and concentration results for the number and sizes of Jordan blocks are obtained. In particular, we study a linear algebraic analog of Ulam’s longest increasing subsequence problem.

The surprising mathematics of longest increasing subsequences

In a surprising sequence of developments, the longest increasing subsequence problem, originally mentioned as merely a curious example in a 1961 paper, has proven to have deep connections to many seemingly unrelated branches of mathematics, such as random permutations, random matrices, Young tableaux, and the corner growth model. The detailed and playful study of these connections makes this book suitable as a starting point for a wider exploration of elegant mathematical ideas that are of interest to every mathematician and to many computer scientists, physicists and statisticians. The specific topics covered are the Vershik-Kerov–Logan-Shepp limit shape theorem, the Baik–Deift–Johansson theorem, the Tracy–Widom distribution, and the corner growth process. This exciting body of work, encompassing important advances in probability and combinatorics over the last forty years, is made accessible to a general graduate-level audience for the first time in a highly polished presentation.

Longest Increasing Subsequences of Randomly Chosen Multi-Row Arrays

A two-row array of integers \[ \alpha_{n}= \begin{pmatrix}a_1 & a_2 & \cdots & a_n\\ b_1 & b_2 & \cdots & b_n \end{pmatrix} \] is said to be in lexicographic order if its columns are in lexicographic order (where character significance decreases from top to bottom, i.e., either ak < ak+1, or bk ≤ bk+1 when ak = ak+1). A length ℓ (strictly) increasing subsequence of αn is a set of indices i1 < i2 < ⋅⋅⋅ < iℓ such that ai1 < ai2 < ⋅⋅⋅ < aiℓ and bi1 < bi2 < ⋅⋅⋅ < biℓ. We are interested in the statistics of the length of a longest increasing subsequence of αn chosen according to ${\cal D}$n, for different families of distributions ${\cal D} = ({\cal D}_{n})_{n\in\NN}$, and when n goes to infinity. This general framework encompasses well-studied problems such as the so-called longest increasing subsequence problem, the longest common subsequence problem, and problems concerning directed bond percolation models, among others. We define several natural families of different distributions and characterize the asymptotic behaviour of the length of a longest increasing subsequence chosen according to them. In particular, we consider generalizations to d-row arrays as well as symmetry-restricted two-row arrays.

A linear time approximation algorithm for permutation flow shop scheduling

In the last 40 years, the permutation flow shop scheduling (PFS) problem with makespan minimization has been a central problem, known for its intractability, that has been well studied from both theoretical and practical aspects. The currently best performance ratio of a deterministic approximation algorithm for the PFS was recently presented by Nagarajan and Sviridenko, using a connection between the PFS and the longest increasing subsequence problem. In a different and independent way, this paper employs monotone subsequences in the approximation analysis techniques. To do this, an extension of the Erdös–Szekeres theorem to weighted monotone subsequences is presented. The result is a simple deterministic algorithm for the PFS with a similar approximation guarantee, but a much lower time complexity.

Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm

We define and study the Plancherel–Hecke probability measure on Young diagrams; the Hecke algorithm of Buch–Kresch–Shimozono–Tamvakis–Yong is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of Thomas–Yong on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of Schensted and of Knuth, respectively, on the Schensted and Robinson–Schensted–Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of Vershik–Kerov, Logan–Shepp and others on the “longest increasing subsequence problem” for permutations. We also include a related extension of Aldous–Diaconis on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K-theoretic Schubert calculus.

Tight Bounds for Permutation Flow Shop Scheduling

In flow shop scheduling there are m machines and n jobs, such that every job has to be processed on the machines in the fixed order 1,…,m. In the permutation flow shop problem, it is also required that each machine process the set of all jobs in the same order. Formally, given n jobs along with their processing times on each machine, the goal is to compute a single permutation of the jobs σ: [n] → [n] that minimizes the maximum job completion time (makespan) of the schedule resulting from σ. The previously best known approximation guarantee for this problem was O((m log m)1/2) [Sviridenko, M. 2004. A note on permutation flow shop problem. Ann. Oper. Res. 129 247–252]. In this paper, we obtain an improved O(min{m1/2,n1/2}) approximation algorithm for the permutation flow shop scheduling problem, by finding a connection between the scheduling problem and the longest increasing subsequence problem. Our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load, and is the best possible such result. This also resolves an open question from Potts et al. [Potts, C., D. Shmoys, D. Williamson. 1991. Permutation vs. nonpermutation flow shop schedules. Oper. Res. Lett. 10 281–284], by algorithmically matching the gap between permutation and nonpermutation schedules. We also consider the weighted completion time objective for the permutation flow shop scheduling problem. Using a natural linear programming relaxation and our algorithm for the makespan objective, we obtain an O(min{m1/2,n1/2}) approximation algorithm for minimizing the total weighted completion time, improving on the previously best known guarantee of εm for any constant ε > 0 [Smutnicki, C. 1998. Some results of the worst-case analysis for flow shop scheduling. Eur. J. Oper. Res. 109 66–87]. We give a matching lower bound on the integrality gap of our linear programming relaxation.

Largest planar matching in random bipartite graphs

For a distribution g over labeled bipartite (multi) graphs G = (W, M, E), |W| = |M| = n, let L(n) denote the size of the largest planar matching of G (here W and M are posets drawn on the plane as two ordered rows of nodes and edges are drawn as straight lines). We study the asymptotic (in n) behavior of L(n) for different distributions g. Two interesting instances of this problem are Ulam's longest increasing subsequence problem and the longest common subsequence problem. We focus on the case where g is the uniform distribution over the k-regular bipartite graphs on W and M. For k = o(n1/4), we establish that L(n)/√kn tends to 2 in probability when n → ∞. Convergence in mean is also studied. Furthermore, we show that if each of the n2 possible edges between W and M are chosen independently with probability 0 < p < 1, then L(n)/n tends to a constant γp in probability and in mean when n → ∞.

EFFICIENT PARALLEL ALGORITHMS FOR THE LIS AND LCS PROBLEMS ON BSR MODEL USING CONSTANT NUMBER OF SELECTIONS

Recently Akl et al. introduced a new model of parallel computation, called BSR (broadcasting with selective reduction) and showed that it is more powerful than any CRCW PRAM and yet requires no more resources for implementation than even EREW PRAM. The model allows constant time solutions to sorting, parallel prefix and other problems. In this paper, we describe constant time solution to the longest common subsequence (LCS) and longest increasing subsequence (LIS) problems using the BSR model. The number of processors used to solve the LCS problem is O(N × M) (where N and M are the length of the two input sequences). Our BSR solution of the LIS problem needs O(N) processors (where N is the length of the input sequence). These two solutions use BSR instructions with a constant number of selections. It is an improvement of our former BSR algorithm for the LCS in (Journal of Parallel and Distributed Computing, to appear) which uses 3N + 3 selections.

Tight Ω(n lg n) lower bound for finding a longest increasing subsequence

The longest increasing subsequence problem is as follows: Given a sequence of n real numbers, find a longest increasing subsequence of . There is a well-known O(n lg n)-time comparison tree algorithm for solving this problem. Also, a tight Ω(n lg n) lower bound in the comparison tree model is known. We prove a tight Ω(n lg n) lower bound in the more powerful algebraic decision tree model. The above lower bounds also apply to the apparently simpler problem of finding the length of a longest increasing subsequence.