By Jörg Arndt
This ebook offers algorithms and ideas for computationalists. matters handled contain low-level algorithms, bit wizardry, combinatorial iteration, quickly transforms just like the Fourier rework, and quick mathematics for either actual numbers and finite fields. numerous optimization thoughts are defined and the particular functionality of many given implementations is tested. the focal point is on fabric that doesn't often look in textbooks on algorithms. The implementations are performed in C++ and the GP language, written for POSIX-compliant structures akin to the Linux and BSD working systems.
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1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 diff ...... +1. +1.. 1-1. +1... + p . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 16-B: The Gray code equals the Gray code of doubled value shifted to the right once. Equivalently, we can separate the lowest bit which equals the parity of the other bits. The last column shows that the changes with each increment always happen one position left of the rightmost bit. Let g(k) be the Gray code of a number k. We are interested in efficiently generating g(k + 1). h]: 1 2 3 4 5 6 7 8 9 10 11 12 13 static inline ulong next_gray2(ulong x) // With input x==gray_code(2*k) the return is gray_code(2*k+2).
Return BITS_PER_LONG - bit_count_sparse( ~x ); } If the number of ones is guaranteed to be less than 16, then the following routine (suggested by Gunther Piez [priv. ]) can be used: 1 2 3 4 5 6 7 8 static inline ulong bit_count_15(ulong x) // Return number of set bits, must have at most 15 set bits. 2 // 0-2 in 2 bits Counting blocks Compute the number of bit-blocks in a binary word with the following function: 1 2 3 4 static inline ulong bit_block_count(ulong x) // Return number of bit blocks.
1. 1... 11... 1 g(2*k) ....... 11. 11.. 1. 11... 1111. 1.. 1. 11.... 11 g(k) ...... 1. 11. 1.. 11.. 111. 1. 1... 11... 1 p . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 diff ...... +1. +1.. 1-1. +1... + p . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 16-B: The Gray code equals the Gray code of doubled value shifted to the right once. Equivalently, we can separate the lowest bit which equals the parity of the other bits. The last column shows that the changes with each increment always happen one position left of the rightmost bit.