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

Embeddings in hypercubes

One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their inter-communication requirements can be modeled by a graph, and the assignment of subtasks to processors viewed as an embedding of the task graph into the graph of the hypercube network. We survey the known results concerning such embeddings, including expansion/dilation tradeoffs for general graphs, embeddings of meshes and trees, packings of multiple copies of a graph, the complexity of finding good embeddings, and critical graphs which are minimal with respect to some property. In addition, we describe several open problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27512/1/0000556.pd

The complex behavior of simple machines

This paper interprets work on understanding the actions of Turing machines operating on an initially blank tape. While this is impossible for arbitrary machines, complete characterizations of behavior are possible if the number of states is sufficiently constrained. The approach combines normalization to drastically reduce the number of machines considered, human-generated classification schemes, and computer-generated proofs of behavior. This approach can be applied to other computational systems, giving complete characterizations in sufficiently small domains. This is of interest in the area of emergent systems since the properties of such systems are often difficult to determine. By using computers to eliminate multitudes of machines with well understood behavior, some unanticipated exotic machines with complex behavior were discovered. These exotic machines show that it is quite difficult to estimate the number of states needed to produce a given behavior, and hence subjective estimates of complexity may be poor approximations of the true complexity.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28528/1/0000325.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