Improved Selection on Totally Monotone Arrays

TL;DR: An O(n lg m)-time algorithm for computing an approximate median in each row of an m×n totally monotone array; this approximate median is an entry whose rank in its row lies between [ n/4] and [3n/4].

Abstract: This paper's main result is an O((√m lg m)(n lg n)+m lg n)-time algorithm for computing the kth smallest entry in each row of an m×n totally monotone array. (A two-dimensional array A = {a[i,j]} is totally monotone if for all i1

TL;DR: An O(n3/2lg1/2 n)-time algorithm for computing the kth nearest neighbor of each vertex of a convex n-gon and an approximate median in each row of an m×n totally monotone array.

Abstract: This paper's main result is an -time algorithm for computing the kth smallest entry in each row of an m×n totally monotone array. (A two-dimensional array A={a[i, j]} is totally monotone if for all i1

TL;DR: The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm-PICK.

Abstract: The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm-PICK. Specifically, no more than 5.4305 n comparisons are ever required. This bound is improved for extreme values of i, and a new lower bound on the requisite number of comparisons is also proved.

TL;DR: The Θ(m) bound on finding the maxima of wide totally monotone matrices is used to speed up several geometric algorithms by a factor of logn.

Abstract: LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi 1 >i 2 implies thatj(i 1) ≥J(i 2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm

TL;DR: Efficient PRAM parallel algorithms for the string editing problem for input strings x and y and the CREW bound is $O(\log m \log n)$ time with $O({{mn} / {\log m}})$ processors.

Abstract: The string editing problem for input strings x and y consists of transforming x into y by performing a series of weighted edit operations on x of overall minimum cost. An edit operation on x can be the deletion of a symbol from x, the insertion of a symbol in x or the substitution of a symbol x with another symbol. This problem has a well known O((absolute value of x)(absolute value of y)) time sequential solution (25). The efficient Program Requirements Analysis Methods (PRAM) parallel algorithms for the string editing problem are given. If m = ((absolute value of x),(absolute value of y)) and n = max((absolute value of x),(absolute value of y)), then the CREW bound is O (log m log n) time with O (mn/log m) processors. In all algorithms, space is O (mn).

TL;DR: The notion of two-dimensional totally monotone arrays is generalized to multidimensional arrays, and a wide variety of problems are exhibited involving computational geometry, dynamic programming, VLSI river routing, and finding certain kinds of shortest paths that can be solved efficiently by finding maxima in totally monOTone arrays.

Abstract: A two-dimensional array A=(a/sub i,j/) is called monotone if the maximum entry in its ith row lies below or to the right of the maximum entry in its (i- 1)-st row. An array A is called totally monotone if every 2*2 subarray (i.e., every 2*2 minor) is monotone. The notion of two-dimensional totally monotone arrays is generalized to multidimensional arrays, and a wide variety of problems are exhibited involving computational geometry, dynamic programming, VLSI river routing, and finding certain kinds of shortest paths that can be solved efficiently by finding maxima in totally monotone arrays. >

TL;DR: It is shown that the concave least weight subsequence problem can always be solved in O(n) time, without any extra conditions.

Abstract: We are given an integer n and a real-valued function w(i, j) defined for integers 0 ≤ i < j ≤ n and with the property that w(i0, j0) + w(i1, j1) ≤ w(i0, j1) + w(i1, j0) for 0 ≤ i0 < i1 < j0 < j1 ≤ n. The concave least-weight subsequence problem is to find an integer k ≥ 1 and a sequence of integers 0 = l0 < l1 < … < lk−1 < lk = n such that ∑i=0k−w(li, li+1) is minimized. One application of this problem is determining optimal line breaks in a text formatting system. D. S. Hirschberg and L. L. Larmore (SIAM J. Comput.16 (1987), 628–638) showed that the concave least-weight subsequence problem can be solved in O(n log n) time and that if a certain extra condition is imposed it can be solved in O(n) time. Here we show that the concave least weight subsequence problem can always be solved in O(n) time, without any extra conditions.