Coping with Sparse Inputs on Enhanced Meshes – Semigroup Computation with COMMON CRCW Buses

Consider an p n � p n processor mesh where, in addition to the local links, each row and column is enhanced by a COM-MON CRCW bus. Assume that each processor stores an element of a commutative semigroup, and only k�nentries (in arbitrary positions) are nonzero. We wish to compute the sum of all entries. For this problem we easily obtain a lower time bound of ��k 1=4 � if k � n 2=3. Our main result is an O�k 1=4 log 2 k � time algorithm. It requires a composition of several data movement and compaction techniques which seem to be of general use for solving problems with sparse inputs scattered on the mesh, as it is typical e.g. for primal sketches in digital image processing. 1: Model, Motivation, and Problem The mesh-connected computer with row and column buses (other denotations in the literature are: mesh or processsor array with multiple broadcasting, enhanced mesh) has reached some attention as an architecture for parallel processing, especially suitable for digital geometry problems. It consists of a p n � p n grid of processors. Each processor is connected to its (at most four) neighbors by local links. Additionally, each row and column is equipped with a bus for long-distance communication. Each processor has local memory (of usually O�log n � bits) and applies in each time unit a global instruction to its own data. A processor can perform local computations and send and receive messages through the local links and buses. All processors in a row or column, respectively, can simultaneously read the message from the bus, but for technical reasons only one message per step can be broadcast by the bus. If only one processor per step is allowed to send a message, we speak of CREW buses, according to the terminology for PRAMs. Since the first treatment in [9], many complexity results have been obtained for this architecture. Usually one assumes that the input has length n and is pretiled onto the whole mesh, or it has length k � n and stands initially in k prescribed processors [2]. The mesh is often considered as a natural device for geometric problems on digital images ([9] [10] [3] and others) – every processor represents