#Permute a matrix python zip#

If a set of functions is given instead of the usual >, <, and = operators (or overridden in object-oriented languages), the array can be an arbitrary object. We can define these functions in any way appropriate for the data type. The key to establishing lexicographic order is the definition of a set of ordering functions (such as, , and ). Lexicographic order is a generalization of, for instance, alphabetic order. In each iteration, the algorithm will produce all the permutations that end with the current last element.

If is even, then swap the th element (in the loop).

If is odd, swap the first and last element.

While looping over the n-1 elements, there is a (mystical) step to the algorithm that depends on whether is odd or even. Then the (n-1)! permutations of the first n-1 elements are adjoined to this last element. The algorithm basically generates all the permutations that end with the last element. The principle of Heap’s algorithm is decrease and conquer. We see that the advantage of this algorithm, as opposed to the previous algorithm, is that we use less memory. This method is a systematic algorithm, which at each step chooses a pair of elements to switch in order to generate new permutations. It produces every possible permutation of these elements exactly once.

This algorithm is based on swapping elements to generate the permutations. One of the more traditional and effective algorithms used to generate permutations is the method developed by B. We have to rely on other methods of finding a password, such as guessing the owner’s dog’s name or “qwerty.” 2.3. It would take us several lifetimes of the universe to try to figure out the key. For instance, the standard 256-encryption key has 1.1 x 10 77 combinations of zeros and ones. For example, if we were to write a program to generate these permutations recursively (see below), we would quickly run out of memory space.Īlthough this is bad news for those of us who want to generate all the possible permutations, it is good news for encryption. Even if we could find a dealer in Las Vegas who could shuffle the cards once every nanosecond, he would still not even come close to all the possible combinations before the end of the universe.įurthermore, the amount of time it takes us to generate all permutations is not our only limitation. The age of the universe is approximately 10 13.813 years old. append ( t1 - t0 ) print ( 'time for sec.'.

#Permute a matrix python zip#

append ( t1 - t0 ) print ( 'matvec:' ) for fmt, name in zip ( fmts, names ): Afmt = fmt ( A ) fmt_ts = for _ in range ( niter_matvec ): t0 = time. empty_like ( x ) # first let's just do a little warmup for fmt, name in zip ( fmts, names ): Afmt = fmt ( A ) fmt_ts = for _ in range ( niter_matvec ): t0 = time. dok_matrix ) names = ( 'CSC', 'CSR', 'LIL', 'COO', 'DOK' ) niter_matvec = 100 # number of iterations of matvec niter_matmat = 20 # number of iterations of matmat x = np. This can make it much faster to apply CSC matrices.įmts = ( sparse. If you’re familiar with pointers in a language like C, ptr is an array of pointer offsets.īasically, the non-zero entries for each column are stored in contiguous blocks of memory. The non-zero row indices for column j are stored in row:ptr], and the non-zero values for those rows are stored in val:ptr].

If S is a CSC matrix with m rows, n columns, and nnz non-zeros, we specify S with three lists: ptr (length n+1), row (length nnz) and val (length nnz). We’ll describe CSC matrices - the transpose is a CSR matrix. You might think of these as the sparse equivalents of row-major and column-major dense matrices. One of the disadvantages of COO Matrices are that entries need not be ordered in any way, which can lead to inefficiencies in memory access in matrix-vector or matrix-matrix multiplication.Ĭommonly used formats which keeps entries in a sensible order (without additional structure assumed) are Compressed Sparse Row (CSR) and Compressed Sparse Column (CSC) matrices. Dimension Reduction and Data Visualization