A Parallel Algorithm for Approximate Regularity, by Laurence Boxer and Russ Miller, A presentation for the Niagara University Research Council, Nov., 2000 Or: Recognizing Patterns Amidst Seeming Chaos Information Processing Letters, to appear

Areas of investigation Mathematical theory of computing Algorithms – solutions of problems; study resources (chiefly time; also memory, accuracy of approximations) required Computational geometry – algorithms to compute solutions to questions of geometric nature Parallel algorithms – algorithms for parallel computers

Growth of relevant functions n: size of sample set Regard these functions as measures of time Time units not given. They rely on factors like speed of computer - important, but irrelevant to analysis of algorithms.

Example – analysis of 2 algorithms Problem: given n unsorted real numbers, find their span = maximum - minimum Solution 1: Solution 2:

The Problem: Given a finite set S of points in the Euclidean plane, identify all maximal subsets of S that consist of at least 3 (approximately) collinear points that are (approximately) equally spaced. Applications (aerial photos): Military: recognizing land mines from infrared “hot spots” photos Surveying: recognizing property boundaries (fence posts) Urban: recognizing street lights

PRAM - Parallel Random Access Machine Shared memory yields fast communications Source processor writes data to memory Destination processor reads data from memory Fast communications make this model theoretical ideal for fastest possible parallel algorithms for given # of processors Impractical - too many wires if lots of processors

Mesh architecture Square grid of processors Each processor connected by communication link to N, S, E, W neighbors Next slide illustrates how the communication diameter is an important limitation on the speed of a mesh (or any parallel computer based on networked processors).

Semigroup operation (e.g., total) in mesh 1. “Roll up” columns in parallel, totaling each column in last row by sending data downward. 2. Roll up last row to get total in a corner. 3. Broadcast total from corner to all processors. This takes time.

Versions of our problem – Exact version Exploit fact that no pair of input points can appear as consecutive points in more than 1 maximal equally-spaced collinear subset of input set

The exact version is discussed in the following book (available in fine brick&mortar & online bookstores):

Versions of our problem – Approximate version More useful (practical) version An error tolerance parameter specifies the margin of approximation. A segment determined by input points may appear in multiple maximal approximately equally-spaced collinear subsets. Therefore, there may be more output than in the exact version of the problem.

Our Solution We developed an architecture-independent algorithm, then considered its implementation on various parallel architectures. Ours was NOT a straightforward adaptation of the Robins, et al., algorithm, which seems inherently sequential.