Abstract
In a generalized shuffle permutation an address ( a q−1 a q−2 … a 0) receives its content from an address obtained through a cyclic shift on a subset of the q dimensions used for the encoding of the addresses. Bit-complementation may be combined with the shift. We give an algorithm that requires K 2 + 2 exchanges for K elements per processor, when storage dimensions are part of the permutation, and concurrent communication on all ports of every processor is possible. The number of element exchanges in sequence is independent of the number of processor dimensions σ r in the permutation. With no storage dimensions in the permutation our best algorithm requires ( σ 4 r + 1)[ K 2σ r ] element exchanges. We also give an algorithm for σ r = 2, or the real shuffle consists of a number of cycles of length two, that requires K 2 + 1 element exchanges in sequence when there is no bit complement. The lower bound is K 2 for both real and mixed shuffles with no bit-complementation. The minimum number of communication start-ups is σ r for both cases, which is also the lower bound. The data transfer time for communication restricted to one port per processor is σ r ( K 2 ) , and the minimum number of start-ups is σ r. The analysis is verified by experimental results on the Intel iPSC/1, and for one case also on the Connection Machine model CM-2.
Highlights
The main contributions of this paper are optimal algorithms for dimension permutations on Boolean cube con gured distributed memory multi-processors, and lower bounds for such permutations with concurrent communication on all channels
An extended-cube permutation (ECP) is an algorithm for dimension permutation in which the routing is extended to an ne-cube in which the n-cube holding data is embedded
Lemma 1 The time complexity of an Stable Dimension Permutation (SDP) of real order r < n cannot be improved by communication in the n r processor dimensions not included in the index set, if a dimension 4 permutation is required within all r-cubes, and the SDP algorithm uses full bandwidth within each r-cube
Summary
The main contributions of this paper are optimal algorithms for dimension permutations on Boolean cube con gured distributed memory multi-processors, and lower bounds for such permutations with concurrent communication on all channels. Virtual dimension permutations that only include local storage addresses require no communication, and are not considered here. In a Connection Machine model CM-2 each processor has a single 32-bit wide data path to memory, while inter-processor communication channels are 1-bit wide. Examples of dimension permutations are k-shu e/unshu e permutations, matrix transposition, bit-reversal, vector-reversal, and conversion between various data allocation schemes, such as consecutive and cyclic storage 3, 4], reshaping of arrays 7], and multi-sectioning. We consider concurrent communication on all channels of all processors Such communication is possible on the Connection Machine.
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