Best LpL_p Isotonic Regressions, p∈{0,1,∞}p \in \{0, 1, \infty\}

Given a real-valued weighted function $f$ on a finite dag, the $L_p$ isotonic regression of $f$, $p \in [0,\infty]$, is unique except when $p \in [0,1] \cup \{\infty\}$. We are interested in determining a ``best'' isotonic regression for $p \in \{0, 1, \infty\}$, where by best we mean a regression satisfying stronger properties than merely having minimal norm. One approach is to use strict $L_p$ regression, which is the limit of the best $L_q$ approximation as $q$ approaches $p$, and another is lex regression, which is based on lexical ordering of regression errors. For $L_\infty$ the strict and lex regressions are unique and the same. For $L_1$, strict $q \scriptstyle\searrow 1$ is unique, but we show that $q \scriptstyle\nearrow 1$ may not be, and even when it is unique the two limits may not be the same. For $L_0$, in general neither of the strict and lex regressions are unique, nor do they always have the same set of optimal regressions, but by expanding the objectives of $L_p$ optimization to $p < 0$ we show $p{ \scriptstyle \nearrow} 0$ is the same as lex regression. We also give algorithms for computing the best $L_p$ isotonic regression in certain situations

Supporting divide-and-conquer algorithms for image processing

Divide-and-conquer is an important algorithm strategy, but it is not widely used in image processing. For higher-level, symbolic operations it should often be the strategy of choice for parallel computers. It is natural for a machine with a regular interconnection scheme such as a mesh, mesh with broadcasting, tree, pyramid, mesh-of-trees, PRAM, or hypercube, and can be used either on a machine with a pixel per processor or on one with many pixels per processor. However, divide-and-conquer algorithms use parallel computers in a different manner than, say, local edge detection, so machines optimized for local neighborhood algorithms may be poor for divide-and-conquer algorithms. Some characteristics of divide-and-conquer algorithms are examined, along with some of their implications for the design of machines and languages which can support the efficient programming and execution of divide-and-conquer algorithms.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26821/1/0000380.pd

Optimal Parallel Construction of Hamiltonian Cycles and Spanning Trees in Random Graphs

We give tight bounds on the parallel complexity of some problems involving random graphs. Specifically, we show that a Hamiltonian cycle, a breadth first spanning tree, and a maximal matching can all be constructed in \Theta(log n) expected time using n= log n processors on the CRCW PRAM. This is a substantial improvement over the best previous algorithms, which required \Theta((log log n) 2 ) time and n log 2 n processors. We then introduce a technique which allows us to prove that constructing an edge cover of a random graph from its adjacency matrix requires \Omega\Gammaequ n) expected time on a CRCW PRAM with O(n) processors. Constructing an edge cover is implicit in constructing a spanning tree, a Hamiltonian cycle, and a maximal matching, so this lower bound holds for all these problems, showing that our algorithms are optimal. This new lower bound technique is one of the very few lower bound techniques known which apply to randomized CRCW PRAM algorithms, and it pro..

Special issue on algorithms for hypercube computers : Guest editor's introduction

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28649/1/0000465.pd

Linear Time Distance Transforms for Quadtrees

Linear time algorithms are given for computing the chessboard distance transform for both pointer-based and linear quadtree representations. Comparisons between algorithmic styles for the two representations are made. Both versions of the algorithm consist of a pair of tree traversals

A parallel solution-adaptive scheme for ideal magnetohydrodynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77232/1/AIAA-1999-3273-200.pd