Non-square Matrices: Following The Cycles
For non-square matrices, the algorithms are more complicated. Many of the algorithms prior to 1980 could be described as "follow-the-cycles" algorithms. That is, they loop over the cycles, moving the data from one location to the next in the cycle. In pseudocode form:for each length>1 cycle C of the permutation pick a starting address s in C let D = data at s let x = predecessor of s in the cycle while x ≠ s move data from x to successor of x let x = predecessor of x move data from D to successor of s
The differences between the algorithms lie mainly in how they locate the cycles, how they find the starting addresses in each cycle, and how they ensure that each cycle is moved exactly once. Typically, as discussed above, the cycles are moved in pairs, since s and MN−1−s are in cycles of the same length (possibly the same cycle). Sometimes, a small scratch array, typically of length M+N (e.g. Brenner, 1973; Cate & Twigg, 1977) is used to keep track of a subset of locations in the array that have been visited, to accelerate the algorithm.
In order to determine whether a given cycle has been moved already, the simplest scheme would be to use O(MN) auxiliary storage, one bit per element, to indicate whether a given element has been moved. To use only O(M+N) or even O(log MN) auxiliary storage, more complicated algorithms are required, and the known algorithms have a worst-case linearithmic computational cost of O(MN log MN) at best, as first proved by Knuth (Fich et al., 1995; Gustavson & Swirszcz, 2007).
Such algorithms are designed to move each data element exactly once. However, they also involve a considerable amount of arithmetic to compute the cycles, and require heavily non-consecutive memory accesses since the adjacent elements of the cycles differ by multiplicative factors of N, as discussed above.
Famous quotes containing the word cycles:
“The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
—Linda Goodman (b. 1929)