User:WhitePhosphorus/磷原子1号/Bitap算法
bitap算法（又称shift-or、shift-and或Baeza-Yates–Gonnet算法）是一种字符串近似匹配算法。此算法可以计算一段给定的字符串是否含有“约等于”给定模式串的子串，其中的“约等于”是用萊文斯坦距離定义的——如果子串和模式串的距离小于等于给定的k，算法就认为它们是匹配的。该算法先进行预处理计算一个掩码集，其中每个位代表一个字符是否在模式串中出现过。于是，几乎所有的操作都是位操作，速度非常快。
由Udi Manber、Sun Wu、Burra Gopal编写的UNIX实用程序agrep中，bitap算法是一种基础算法，这可能是它最为人知的应用。Manber和Wu最初发表的论文中还给出了该算法的扩展情形，可以用来处理正则表达式的模糊匹配。
Due to the data structures required by the algorithm, it performs best on patterns less than a constant length (typically the 字 of the machine in question), and also prefers inputs over a small alphabet. Once it has been implemented for a given alphabet and word length m, however, its 时间复杂度 is completely predictable — it runs in O(mn) operations, no matter the structure of the text or the pattern.
The bitap algorithm for exact string searching was invented by Bálint Dömölki in 1964 and extended by R. K. Shyamasundar in 1977, before being reinvented in the context of fuzzy string searching by Manber and Wu in 1991 based on work done by Ricardo Baeza-Yates and Gaston Gonnet. The algorithm was improved by Baeza-Yates and Navarro in 1996 and later by Gene Myers for long patterns in 1998.
精确匹配
The bitap algorithm for exact string searching, in full generality, looks like this in pseudocode:
algorithm bitap_search(text : string, pattern : string) returns string
m := length(pattern)
if m == 0
return text
/* Initialize the bit array R. */
R := new array[m+1] of bit, initially all 0
R[0] = 1
for i = 0; i < length(text); i += 1:
/* Update the bit array. */
for k = m; k >= 1; k -= 1:
R[k] = R[k-1] & (text[i] == pattern[k-1])
if R[m]:
return (text+i - m) + 1
return nil
Bitap distinguishes itself from other well-known string searching algorithms in its natural mapping onto simple bitwise operations, as in the following modification of the above program. Notice that in this implementation, counterintuitively, each bit with value zero indicates a match, and each bit with value 1 indicates a non-match. The same algorithm can be written with the intuitive semantics for 0 and 1, but in that case we must introduce another instruction into the inner loop to set R |= 1. In this implementation, we take advantage of the fact that left-shifting a value shifts in zeros on the right, which is precisely the behavior we need.
Notice also that we require CHAR_MAX additional bitmasks in order to convert the (text[i] == pattern[k-1]) condition in the general implementation into bitwise operations. Therefore, the bitap algorithm performs better when applied to inputs over smaller alphabets.
#include <string.h>
#include <limits.h>
const char *bitap_bitwise_search(const char *text, const char *pattern)
{
int m = strlen(pattern);
unsigned long R;
unsigned long pattern_mask[CHAR_MAX+1];
int i;
if (pattern[0] == '\0') return text;
if (m > 31) return "The pattern is too long!";
/* Initialize the bit array R */
R = ~1;
/* Initialize the pattern bitmasks */
for (i=0; i <= CHAR_MAX; ++i)
pattern_mask[i] = ~0;
for (i=0; i < m; ++i)
pattern_mask[pattern[i]] &= ~(1UL << i);
for (i=0; text[i] != '\0'; ++i) {
/* Update the bit array */
R |= pattern_mask[text[i]];
R <<= 1;
if (0 == (R & (1UL << m)))
return (text + i - m) + 1;
}
return NULL;
}
模糊匹配
To perform fuzzy string searching using the bitap algorithm, it is necessary to extend the bit array R into a second dimension. Instead of having a single array R that changes over the length of the text, we now have k distinct arrays R1..k. Array Ri holds a representation of the prefixes of pattern that match any suffix of the current string with i or fewer errors. In this context, an "error" may be an insertion, deletion, or substitution; see 萊文斯坦距離 for more information on these operations.
The implementation below performs fuzzy matching (returning the first match with up to k errors) using the fuzzy bitap algorithm. However, it only pays attention to substitutions, not to insertions or deletions — in other words, a 汉明距离 of k. As before, the semantics of 0 and 1 are reversed from their intuitive meanings.
#include <stdlib.h>
#include <string.h>
#include <limits.h>
const char *bitap_fuzzy_bitwise_search(const char *text, const char *pattern, int k)
{
const char *result = NULL;
int m = strlen(pattern);
unsigned long *R;
unsigned long pattern_mask[CHAR_MAX+1];
int i, d;
if (pattern[0] == '\0') return text;
if (m > 31) return "The pattern is too long!";
/* Initialize the bit array R */
R = malloc((k+1) * sizeof *R);
for (i=0; i <= k; ++i)
R[i] = ~1;
/* Initialize the pattern bitmasks */
for (i=0; i <= CHAR_MAX; ++i)
pattern_mask[i] = ~0;
for (i=0; i < m; ++i)
pattern_mask[pattern[i]] &= ~(1UL << i);
for (i=0; text[i] != '\0'; ++i) {
/* Update the bit arrays */
unsigned long old_Rd1 = R[0];
R[0] |= pattern_mask[text[i]];
R[0] <<= 1;
for (d=1; d <= k; ++d) {
unsigned long tmp = R[d];
/* Substitution is all we care about */
R[d] = (old_Rd1 & (R[d] | pattern_mask[text[i]])) << 1;
old_Rd1 = tmp;
}
if (0 == (R[k] & (1UL << m))) {
result = (text+i - m) + 1;
break;
}
}
free(R);
return result;
}
外部链接和参考文献
- ^ Bálint Dömölki, An algorithm for syntactical analysis, Computational Linguistics 3, Hungarian Academy of Science pp. 29–46, 1964.
- ^ Bálint Dömölki, A universal compiler system based on production rules, BIT Numerical Mathematics, 8(4), pp 262–275, 1968. doi:10.1007/BF01933436
- ^ R. K. Shyamasundar, Precedence parsing using Dömölki's algorithm, International Journal of Computer Mathematics, 6(2)pp 105–114, 1977
- ^ Udi Manber, Sun Wu. "Fast text searching with errors." Technical Report TR-91-11. Department of Computer Science, 亚利桑那大学, Tucson, June 1991. (gzipped PostScript)
- ^ Udi Manber, Sun Wu. "Fast text search allowing errors." ACM通讯, 35(10): pp. 83–91, October 1992, doi:10.1145/135239.135244.
- ^ Ricardo A. Baeza-Yates, Gastón H. Gonnet. "A New Approach to Text Searching." Communications of the ACM, 35(10): pp. 74–82, October 1992.
- ^ R. Baeza-Yates and G. Navarro. A faster algorithm for approximate string matching. In Dan Hirchsberg and Gene Myers, editors, Combinatorial Pattern Matching (CPM'96), LNCS 1075, pages 1–23, Irvine, CA, June 1996.
- ^ G. Myers. "A fast bit-vector algorithm for approximate string matching based on dynamic programming." Journal of the ACM 46 (3), May 1999, 395–415.
- libbitap, a free implementation that shows how the algorithm can easily be extended for most regular expressions. Unlike the code above, it places no limit on the pattern length.
- Baeza-Yates. Modern Information Retrieval. 1999. ISBN 0-201-39829-X.
- bitap.py - Python implementation of Bitap algorithm with Wu-Manber modifications.