Boyer-Moore 字符串搜索算法
Boyer-Moore 字符串搜索算法是一种高效的字符串搜索算法，是实用字符串搜索文献的标准基准。
Boyer-Moore 算法通过在不同的对齐方式上执行显式字符比较来搜索 $T$ 中出现的 $P$。 Boyer–Moore 使用通过预处理 P 获得的信息来跳过尽可能多的对齐，而不是对所有对齐进行暴力搜索。
该算法的关键在于，如果将模式的结尾与文本进行比较，则可以沿着文本跳转，而不是检查文本的每个字符。这样做的原因是在将模式与文本对齐时，模式的最后一个字符与文本中的字符进行比较。如果字符不匹配，则无需继续沿文本向后搜索。如果文本中的字符与模式中的任何字符都不匹配，则文本中要检查的下一个字符位于文本中较远的 n 个字符处，其中 n 是模式的长度。如果文本中的字符在模式中，则沿文本进行模式的部分移动以沿匹配字符排列并重复该过程。沿着文本跳转进行比较，而不是检查文本中的每个字符，减少了必须进行的比较次数，这是算法效率的关键。
更正式地说，算法从对齐 $k=n$ 开始，因此 P 的开头与 T 的开头对齐。然后从 P 中的索引 n 和 T 中的 k 开始比较 P 和 T 中的字符，移动落后。字符串从 P 的结尾匹配到 P 的开头。比较继续进行，直到到达 P 的开头（这意味着存在匹配）或发生不匹配时对齐向前移动（向右）根据许多规则允许的最大值。在新的比对处再次进行比较，并重复该过程，直到比对移动到 T 的末尾，这意味着将找不到进一步的匹配项。
使用在 P 的预处理期间生成的表，将移位规则实现为恒定时间表查找。
Python 实现：
from typing import *
# This version is sensitive to the English alphabet in ASCII for case-insensitive matching.
# To remove this feature, define alphabet_index as ord(c), and replace instances of "26"
# with "256" or any maximum code-point you want. For Unicode you may want to match in UTF-8
# bytes instead of creating a 0x10FFFF-sized table.
ALPHABET_SIZE = 26
def alphabet_index(c: str) -> int:
"""Return the index of the given character in the English alphabet, counting from 0."""
val = ord(c.lower()) - ord("a")
assert val >= 0 and val < ALPHABET_SIZE
return val
def match_length(S: str, idx1: int, idx2: int) -> int:
"""Return the length of the match of the substrings of S beginning at idx1 and idx2."""
if idx1 == idx2:
return len(S) - idx1
match_count = 0
while idx1 < len(S) and idx2 < len(S) and S[idx1] == S[idx2]:
match_count += 1
idx1 += 1
idx2 += 1
return match_count
def fundamental_preprocess(S: str) -> List[int]:
"""Return Z, the Fundamental Preprocessing of S.
Z[i] is the length of the substring beginning at i which is also a prefix of S.
This pre-processing is done in O(n) time, where n is the length of S.
"""
if len(S) == 0: # Handles case of empty string
return []
if len(S) == 1: # Handles case of single-character string
return [1]
z = [0 for x in S]
z[0] = len(S)
z[1] = match_length(S, 0, 1)
for i in range(2, 1 + z[1]): # Optimization from exercise 1-5
z[i] = z[1] - i + 1
# Defines lower and upper limits of z-box
l = 0
r = 0
for i in range(2 + z[1], len(S)):
if i <= r: # i falls within existing z-box
k = i - l
b = z[k]
a = r - i + 1
if b < a: # b ends within existing z-box
z[i] = b
else: # b ends at or after the end of the z-box, we need to do an explicit match to the right of the z-box
z[i] = a + match_length(S, a, r + 1)
l = i
r = i + z[i] - 1
else: # i does not reside within existing z-box
z[i] = match_length(S, 0, i)
if z[i] > 0:
l = i
r = i + z[i] - 1
return z
def bad_character_table(S: str) -> List[List[int]]:
"""
Generates R for S, which is an array indexed by the position of some character c in the
English alphabet. At that index in R is an array of length |S|+1, specifying for each
index i in S (plus the index after S) the next location of character c encountered when
traversing S from right to left starting at i. This is used for a constant-time lookup
for the bad character rule in the Boyer-Moore string search algorithm, although it has
a much larger size than non-constant-time solutions.
"""
if len(S) == 0:
return [[] for a in range(ALPHABET_SIZE)]
R = [[-1] for a in range(ALPHABET_SIZE)]
alpha = [-1 for a in range(ALPHABET_SIZE)]
for i, c in enumerate(S):
alpha[alphabet_index(c)] = i
for j, a in enumerate(alpha):
R[j].append(a)
return R
def good_suffix_table(S: str) -> List[int]:
"""
Generates L for S, an array used in the implementation of the strong good suffix rule.
L[i] = k, the largest position in S such that S[i:] (the suffix of S starting at i) matches
a suffix of S[:k] (a substring in S ending at k). Used in Boyer-Moore, L gives an amount to
shift P relative to T such that no instances of P in T are skipped and a suffix of P[:L[i]]
matches the substring of T matched by a suffix of P in the previous match attempt.
Specifically, if the mismatch took place at position i-1 in P, the shift magnitude is given
by the equation len(P) - L[i]. In the case that L[i] = -1, the full shift table is used.
Since only proper suffixes matter, L[0] = -1.
"""
L = [-1 for c in S]
N = fundamental_preprocess(S[::-1]) # S[::-1] reverses S
N.reverse()
for j in range(0, len(S) - 1):
i = len(S) - N[j]
if i != len(S):
L[i] = j
return L
def full_shift_table(S: str) -> List[int]:
"""
Generates F for S, an array used in a special case of the good suffix rule in the Boyer-Moore
string search algorithm. F[i] is the length of the longest suffix of S[i:] that is also a
prefix of S. In the cases it is used, the shift magnitude of the pattern P relative to the
text T is len(P) - F[i] for a mismatch occurring at i-1.
"""
F = [0 for c in S]
Z = fundamental_preprocess(S)
longest = 0
for i, zv in enumerate(reversed(Z)):
longest = max(zv, longest) if zv == i + 1 else longest
F[-i - 1] = longest
return F
def string_search(P, T) -> List[int]:
"""
Implementation of the Boyer-Moore string search algorithm. This finds all occurrences of P
in T, and incorporates numerous ways of pre-processing the pattern to determine the optimal
amount to shift the string and skip comparisons. In practice it runs in O(m) (and even
sublinear) time, where m is the length of T. This implementation performs a case-insensitive
search on ASCII alphabetic characters, spaces not included.
"""
if len(P) == 0 or len(T) == 0 or len(T) < len(P):
return []
matches = []
# Preprocessing
R = bad_character_table(P)
L = good_suffix_table(P)
F = full_shift_table(P)
k = len(P) - 1 # Represents alignment of end of P relative to T
previous_k = -1 # Represents alignment in previous phase (Galil's rule)
while k < len(T):
i = len(P) - 1 # Character to compare in P
h = k # Character to compare in T
while i >= 0 and h > previous_k and P[i] == T[h]: # Matches starting from end of P
i -= 1
h -= 1
if i == -1 or h == previous_k: # Match has been found (Galil's rule)
matches.append(k - len(P) + 1)
k += len(P) - F[1] if len(P) > 1 else 1
else: # No match, shift by max of bad character and good suffix rules
char_shift = i - R[alphabet_index(T[h])][i]
if i + 1 == len(P): # Mismatch happened on first attempt
suffix_shift = 1
elif L[i + 1] == -1: # Matched suffix does not appear anywhere in P
suffix_shift = len(P) - F[i + 1]
else: # Matched suffix appears in P
suffix_shift = len(P) - 1 - L[i + 1]
shift = max(char_shift, suffix_shift)
previous_k = k if shift >= i + 1 else previous_k # Galil's rule
k += shift
return matches
更多信息：
https://en.m.wikipedia.org/wiki/Boyer%E2%80%93Moore_string-search_algorithm