Observations Concerning Multidimensional Ultracomputers
Observations Concerning Multidimensional Ultracomputers
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Thus, in general, the N-D UC must perform N independent shuffles to affect all the bit reverses necessary. However, in the number constructed above, one bit v/ill be wrong m each dimension even after these preliminary operations, except for the bit in d^ when N is odd. Thus, when N is even N additional data communication steps are needed, and when N is odd N-1 steps are needed. Q. E. D.
N-D UCs can be viewed not only as 1-D UCs, but also as intermediate dimensional UCs of a given total size. Th...is use is realized by splitting the N dimensions on an N-D UC into subgroups, and using each group to simulate its 1-D UC counterpjart.
1 ^"^ Proposition For each N, a size P N-D UC can simulate -j— - 2 (-1)' (M-i)'^ size P M-D UCs, M ^ N, with no asymptotic loss of efficiency.
M!i=Q Proof Partition the N dimensions into M nonempty groups. Let each group independently simulate a 1-D UC. Since there are M independent dimensions of 1-D UCs, by definition the entire connection scheme is an M-D UC.
What to read after Observations Concerning Multidimensional Ultracomputers?
You can find similar books in the "Read Also" column, or choose other free books by Clyde Kruskal to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Thus, in general, the N-D UC must perform N independent shuffles to affect all the bit reverses necessary. However, in the number constructed above, one bit v/ill be wrong m each dimension even after these preliminary operations, except for the bit in d^ when N is odd. Thus, when N is even N additional data communication steps are needed, and when N is odd N-1 steps are needed. Q. E. D.
N-D UCs can be viewed not only as 1-D UCs, but also as intermediate dimensional UCs of a given total size. Th...is use is realized by splitting the N dimensions on an N-D UC into subgroups, and using each group to simulate its 1-D UC counterpjart.
1 ^"^ Proposition For each N, a size P N-D UC can simulate -j— - 2 (-1)' (M-i)'^ size P M-D UCs, M ^ N, with no asymptotic loss of efficiency.
M!i=Q Proof Partition the N dimensions into M nonempty groups. Let each group independently simulate a 1-D UC. Since there are M independent dimensions of 1-D UCs, by definition the entire connection scheme is an M-D UC.
What to read after Observations Concerning Multidimensional Ultracomputers?
You can find similar books in the "Read Also" column, or choose other free books by Clyde Kruskal to read onlineMoreLess
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