mirror of https://github.com/python/cpython
bpo-41972: Use the two-way algorithm for string searching (GH-22904)
Implement an enhanced variant of Crochemore and Perrin's Two-Way string searching algorithm, which reduces worst-case time from quadratic (the product of the string and pattern lengths) to linear. This applies to forward searches (like``find``, ``index``, ``replace``); the algorithm for reverse searches (like ``rfind``) is not changed. Co-authored-by: Tim Peters <tim.peters@gmail.com>
This commit is contained in:
Dennis Sweeney
GitHub
parent
2183d06bc8
commit
73a85c4e1d
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@ -5,6 +5,7 @@ Common tests shared by test_unicode, test_userstring and test_bytes.
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import unittest, string, sys, struct
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from test import support
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from collections import UserList
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import random
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class Sequence:
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def __init__(self, seq='wxyz'): self.seq = seq
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@ -317,6 +318,44 @@ class BaseTest:
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else:
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self.checkraises(TypeError, 'hello', 'rindex', 42)
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def test_find_periodic_pattern(self):
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"""Cover the special path for periodic patterns."""
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def reference_find(p, s):
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for i in range(len(s)):
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if s.startswith(p, i):
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return i
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return -1
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rr = random.randrange
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choices = random.choices
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for _ in range(1000):
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p0 = ''.join(choices('abcde', k=rr(10))) * rr(10, 20)
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p = p0[:len(p0) - rr(10)] # pop off some characters
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left = ''.join(choices('abcdef', k=rr(2000)))
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right = ''.join(choices('abcdef', k=rr(2000)))
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text = left + p + right
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with self.subTest(p=p, text=text):
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self.checkequal(reference_find(p, text),
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text, 'find', p)
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def test_find_shift_table_overflow(self):
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"""When the table of 8-bit shifts overflows."""
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N = 2**8 + 100
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# first check the periodic case
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# here, the shift for 'b' is N + 1.
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pattern1 = 'a' * N + 'b' + 'a' * N
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text1 = 'babbaa' * N + pattern1
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self.checkequal(len(text1)-len(pattern1),
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text1, 'find', pattern1)
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# now check the non-periodic case
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# here, the shift for 'd' is 3*(N+1)+1
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pattern2 = 'ddd' + 'abc' * N + "eee"
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text2 = pattern2[:-1] + "ddeede" * 2 * N + pattern2 + "de" * N
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self.checkequal(len(text2) - N*len("de") - len(pattern2),
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text2, 'find', pattern2)
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def test_lower(self):
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self.checkequal('hello', 'HeLLo', 'lower')
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self.checkequal('hello', 'hello', 'lower')
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@ -0,0 +1 @@
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Substring search functions such as ``str1 in str2`` and ``str2.find(str1)`` now sometimes use the "Two-Way" string comparison algorithm to avoid quadratic behavior on long strings.
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@ -9,10 +9,16 @@
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/* note: fastsearch may access s[n], which isn't a problem when using
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Python's ordinary string types, but may cause problems if you're
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using this code in other contexts. also, the count mode returns -1
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if there cannot possible be a match in the target string, and 0 if
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if there cannot possibly be a match in the target string, and 0 if
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it has actually checked for matches, but didn't find any. callers
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beware! */
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/* If the strings are long enough, use Crochemore and Perrin's Two-Way
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algorithm, which has worst-case O(n) runtime and best-case O(n/k).
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Also compute a table of shifts to achieve O(n/k) in more cases,
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and often (data dependent) deduce larger shifts than pure C&P can
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deduce. */
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#define FAST_COUNT 0
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#define FAST_SEARCH 1
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#define FAST_RSEARCH 2
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@ -160,6 +166,353 @@ STRINGLIB(rfind_char)(const STRINGLIB_CHAR* s, Py_ssize_t n, STRINGLIB_CHAR ch)
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#undef MEMCHR_CUT_OFF
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/* Change to a 1 to see logging comments walk through the algorithm. */
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#if 0 && STRINGLIB_SIZEOF_CHAR == 1
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# define LOG(...) printf(__VA_ARGS__)
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# define LOG_STRING(s, n) printf("\"%.*s\"", n, s)
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#else
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# define LOG(...)
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# define LOG_STRING(s, n)
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#endif
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Py_LOCAL_INLINE(Py_ssize_t)
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STRINGLIB(_lex_search)(const STRINGLIB_CHAR *needle, Py_ssize_t len_needle,
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Py_ssize_t *return_period, int invert_alphabet)
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{
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/* Do a lexicographic search. Essentially this:
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>>> max(needle[i:] for i in range(len(needle)+1))
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Also find the period of the right half. */
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Py_ssize_t max_suffix = 0;
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Py_ssize_t candidate = 1;
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Py_ssize_t k = 0;
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// The period of the right half.
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Py_ssize_t period = 1;
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while (candidate + k < len_needle) {
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// each loop increases candidate + k + max_suffix
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STRINGLIB_CHAR a = needle[candidate + k];
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STRINGLIB_CHAR b = needle[max_suffix + k];
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// check if the suffix at candidate is better than max_suffix
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if (invert_alphabet ? (b < a) : (a < b)) {
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// Fell short of max_suffix.
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// The next k + 1 characters are non-increasing
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// from candidate, so they won't start a maximal suffix.
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candidate += k + 1;
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k = 0;
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// We've ruled out any period smaller than what's
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// been scanned since max_suffix.
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period = candidate - max_suffix;
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}
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else if (a == b) {
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if (k + 1 != period) {
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// Keep scanning the equal strings
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k++;
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}
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else {
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// Matched a whole period.
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// Start matching the next period.
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candidate += period;
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k = 0;
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}
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}
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else {
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// Did better than max_suffix, so replace it.
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max_suffix = candidate;
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candidate++;
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k = 0;
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period = 1;
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}
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}
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*return_period = period;
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return max_suffix;
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}
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Py_LOCAL_INLINE(Py_ssize_t)
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STRINGLIB(_factorize)(const STRINGLIB_CHAR *needle,
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Py_ssize_t len_needle,
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Py_ssize_t *return_period)
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{
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/* Do a "critical factorization", making it so that:
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>>> needle = (left := needle[:cut]) + (right := needle[cut:])
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where the "local period" of the cut is maximal.
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The local period of the cut is the minimal length of a string w
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such that (left endswith w or w endswith left)
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and (right startswith w or w startswith left).
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The Critical Factorization Theorem says that this maximal local
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period is the global period of the string.
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Crochemore and Perrin (1991) show that this cut can be computed
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as the later of two cuts: one that gives a lexicographically
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maximal right half, and one that gives the same with the
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with respect to a reversed alphabet-ordering.
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This is what we want to happen:
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>>> x = "GCAGAGAG"
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>>> cut, period = factorize(x)
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>>> x[:cut], (right := x[cut:])
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('GC', 'AGAGAG')
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>>> period # right half period
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2
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>>> right[period:] == right[:-period]
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True
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This is how the local period lines up in the above example:
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GC | AGAGAG
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AGAGAGC = AGAGAGC
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The length of this minimal repetition is 7, which is indeed the
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period of the original string. */
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Py_ssize_t cut1, period1, cut2, period2, cut, period;
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cut1 = STRINGLIB(_lex_search)(needle, len_needle, &period1, 0);
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cut2 = STRINGLIB(_lex_search)(needle, len_needle, &period2, 1);
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// Take the later cut.
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if (cut1 > cut2) {
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period = period1;
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cut = cut1;
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}
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else {
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period = period2;
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cut = cut2;
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}
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LOG("split: "); LOG_STRING(needle, cut);
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LOG(" + "); LOG_STRING(needle + cut, len_needle - cut);
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LOG("\n");
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*return_period = period;
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return cut;
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}
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#define SHIFT_TYPE uint8_t
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#define NOT_FOUND ((1U<<(8*sizeof(SHIFT_TYPE))) - 1U)
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#define SHIFT_OVERFLOW (NOT_FOUND - 1U)
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#define TABLE_SIZE_BITS 6
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#define TABLE_SIZE (1U << TABLE_SIZE_BITS)
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#define TABLE_MASK (TABLE_SIZE - 1U)
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typedef struct STRINGLIB(_pre) {
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const STRINGLIB_CHAR *needle;
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Py_ssize_t len_needle;
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Py_ssize_t cut;
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Py_ssize_t period;
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int is_periodic;
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SHIFT_TYPE table[TABLE_SIZE];
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} STRINGLIB(prework);
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Py_LOCAL_INLINE(void)
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STRINGLIB(_preprocess)(const STRINGLIB_CHAR *needle, Py_ssize_t len_needle,
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STRINGLIB(prework) *p)
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{
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p->needle = needle;
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p->len_needle = len_needle;
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p->cut = STRINGLIB(_factorize)(needle, len_needle, &(p->period));
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assert(p->period + p->cut <= len_needle);
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p->is_periodic = (0 == memcmp(needle,
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needle + p->period,
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p->cut * STRINGLIB_SIZEOF_CHAR));
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if (p->is_periodic) {
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assert(p->cut <= len_needle/2);
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assert(p->cut < p->period);
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}
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else {
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// A lower bound on the period
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p->period = Py_MAX(p->cut, len_needle - p->cut) + 1;
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}
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// Now fill up a table
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memset(&(p->table[0]), 0xff, TABLE_SIZE*sizeof(SHIFT_TYPE));
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assert(p->table[0] == NOT_FOUND);
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assert(p->table[TABLE_MASK] == NOT_FOUND);
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for (Py_ssize_t i = 0; i < len_needle; i++) {
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Py_ssize_t shift = len_needle - i;
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if (shift > SHIFT_OVERFLOW) {
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shift = SHIFT_OVERFLOW;
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}
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p->table[needle[i] & TABLE_MASK] = Py_SAFE_DOWNCAST(shift,
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Py_ssize_t,
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SHIFT_TYPE);
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}
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}
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Py_LOCAL_INLINE(Py_ssize_t)
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STRINGLIB(_two_way)(const STRINGLIB_CHAR *haystack, Py_ssize_t len_haystack,
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STRINGLIB(prework) *p)
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{
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// Crochemore and Perrin's (1991) Two-Way algorithm.
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// See http://www-igm.univ-mlv.fr/~lecroq/string/node26.html#SECTION00260
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Py_ssize_t len_needle = p->len_needle;
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Py_ssize_t cut = p->cut;
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Py_ssize_t period = p->period;
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const STRINGLIB_CHAR *needle = p->needle;
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const STRINGLIB_CHAR *window = haystack;
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const STRINGLIB_CHAR *last_window = haystack + len_haystack - len_needle;
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SHIFT_TYPE *table = p->table;
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LOG("===== Two-way: \"%s\" in \"%s\". =====\n", needle, haystack);
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if (p->is_periodic) {
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LOG("Needle is periodic.\n");
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Py_ssize_t memory = 0;
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periodicwindowloop:
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while (window <= last_window) {
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Py_ssize_t i = Py_MAX(cut, memory);
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// Visualize the line-up:
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LOG("> "); LOG_STRING(haystack, len_haystack);
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LOG("\n> "); LOG("%*s", window - haystack, "");
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LOG_STRING(needle, len_needle);
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LOG("\n> "); LOG("%*s", window - haystack + i, "");
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LOG(" ^ <-- cut\n");
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if (window[i] != needle[i]) {
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// Sunday's trick: if we're going to jump, we might
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// as well jump to line up the character *after* the
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// current window.
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STRINGLIB_CHAR first_outside = window[len_needle];
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SHIFT_TYPE shift = table[first_outside & TABLE_MASK];
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if (shift == NOT_FOUND) {
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LOG("\"%c\" not found. Skipping entirely.\n",
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first_outside);
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window += len_needle + 1;
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}
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else {
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LOG("Shifting to line up \"%c\".\n", first_outside);
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Py_ssize_t memory_shift = i - cut + 1;
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window += Py_MAX(shift, memory_shift);
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}
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memory = 0;
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goto periodicwindowloop;
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}
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for (i = i + 1; i < len_needle; i++) {
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if (needle[i] != window[i]) {
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LOG("Right half does not match. Jump ahead by %d.\n",
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i - cut + 1);
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window += i - cut + 1;
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memory = 0;
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goto periodicwindowloop;
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}
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}
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for (i = memory; i < cut; i++) {
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if (needle[i] != window[i]) {
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LOG("Left half does not match. Jump ahead by period %d.\n",
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period);
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window += period;
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memory = len_needle - period;
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goto periodicwindowloop;
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}
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}
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LOG("Left half matches. Returning %d.\n",
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window - haystack);
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return window - haystack;
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}
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}
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else {
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LOG("Needle is not periodic.\n");
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assert(cut < len_needle);
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STRINGLIB_CHAR needle_cut = needle[cut];
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windowloop:
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while (window <= last_window) {
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// Visualize the line-up:
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LOG("> "); LOG_STRING(haystack, len_haystack);
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LOG("\n> "); LOG("%*s", window - haystack, "");
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LOG_STRING(needle, len_needle);
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LOG("\n> "); LOG("%*s", window - haystack + cut, "");
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LOG(" ^ <-- cut\n");
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if (window[cut] != needle_cut) {
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// Sunday's trick: if we're going to jump, we might
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// as well jump to line up the character *after* the
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// current window.
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STRINGLIB_CHAR first_outside = window[len_needle];
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SHIFT_TYPE shift = table[first_outside & TABLE_MASK];
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if (shift == NOT_FOUND) {
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LOG("\"%c\" not found. Skipping entirely.\n",
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first_outside);
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window += len_needle + 1;
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}
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else {
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LOG("Shifting to line up \"%c\".\n", first_outside);
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window += shift;
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}
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goto windowloop;
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}
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for (Py_ssize_t i = cut + 1; i < len_needle; i++) {
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if (needle[i] != window[i]) {
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LOG("Right half does not match. Advance by %d.\n",
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i - cut + 1);
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window += i - cut + 1;
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goto windowloop;
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}
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}
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for (Py_ssize_t i = 0; i < cut; i++) {
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if (needle[i] != window[i]) {
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LOG("Left half does not match. Advance by period %d.\n",
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period);
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window += period;
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goto windowloop;
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}
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}
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LOG("Left half matches. Returning %d.\n", window - haystack);
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return window - haystack;
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}
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}
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LOG("Not found. Returning -1.\n");
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return -1;
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}
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Py_LOCAL_INLINE(Py_ssize_t)
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STRINGLIB(_two_way_find)(const STRINGLIB_CHAR *haystack,
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Py_ssize_t len_haystack,
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const STRINGLIB_CHAR *needle,
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Py_ssize_t len_needle)
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{
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LOG("###### Finding \"%s\" in \"%s\".\n", needle, haystack);
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STRINGLIB(prework) p;
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STRINGLIB(_preprocess)(needle, len_needle, &p);
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return STRINGLIB(_two_way)(haystack, len_haystack, &p);
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}
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Py_LOCAL_INLINE(Py_ssize_t)
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STRINGLIB(_two_way_count)(const STRINGLIB_CHAR *haystack,
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Py_ssize_t len_haystack,
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const STRINGLIB_CHAR *needle,
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Py_ssize_t len_needle,
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Py_ssize_t maxcount)
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{
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LOG("###### Counting \"%s\" in \"%s\".\n", needle, haystack);
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STRINGLIB(prework) p;
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STRINGLIB(_preprocess)(needle, len_needle, &p);
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Py_ssize_t index = 0, count = 0;
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while (1) {
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Py_ssize_t result;
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result = STRINGLIB(_two_way)(haystack + index,
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len_haystack - index, &p);
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if (result == -1) {
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return count;
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}
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count++;
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if (count == maxcount) {
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return maxcount;
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}
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index += result + len_needle;
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}
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return count;
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}
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#undef SHIFT_TYPE
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#undef NOT_FOUND
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#undef SHIFT_OVERFLOW
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#undef TABLE_SIZE_BITS
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#undef TABLE_SIZE
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#undef TABLE_MASK
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#undef LOG
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#undef LOG_STRING
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Py_LOCAL_INLINE(Py_ssize_t)
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FASTSEARCH(const STRINGLIB_CHAR* s, Py_ssize_t n,
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const STRINGLIB_CHAR* p, Py_ssize_t m,
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@ -195,10 +548,22 @@ FASTSEARCH(const STRINGLIB_CHAR* s, Py_ssize_t n,
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}
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mlast = m - 1;
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skip = mlast - 1;
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skip = mlast;
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mask = 0;
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if (mode != FAST_RSEARCH) {
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if (m >= 100 && w >= 2000 && w / m >= 5) {
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/* For larger problems where the needle isn't a huge
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percentage of the size of the haystack, the relatively
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expensive O(m) startup cost of the two-way algorithm
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will surely pay off. */
|
||||
if (mode == FAST_SEARCH) {
|
||||
return STRINGLIB(_two_way_find)(s, n, p, m);
|
||||
}
|
||||
else {
|
||||
return STRINGLIB(_two_way_count)(s, n, p, m, maxcount);
|
||||
}
|
||||
}
|
||||
const STRINGLIB_CHAR *ss = s + m - 1;
|
||||
const STRINGLIB_CHAR *pp = p + m - 1;
|
||||
|
||||
|
|
@ -207,41 +572,118 @@ FASTSEARCH(const STRINGLIB_CHAR* s, Py_ssize_t n,
|
|||
/* process pattern[:-1] */
|
||||
for (i = 0; i < mlast; i++) {
|
||||
STRINGLIB_BLOOM_ADD(mask, p[i]);
|
||||
if (p[i] == p[mlast])
|
||||
if (p[i] == p[mlast]) {
|
||||
skip = mlast - i - 1;
|
||||
}
|
||||
}
|
||||
/* process pattern[-1] outside the loop */
|
||||
STRINGLIB_BLOOM_ADD(mask, p[mlast]);
|
||||
|
||||
if (m >= 100 && w >= 8000) {
|
||||
/* To ensure that we have good worst-case behavior,
|
||||
here's an adaptive version of the algorithm, where if
|
||||
we match O(m) characters without any matches of the
|
||||
entire needle, then we predict that the startup cost of
|
||||
the two-way algorithm will probably be worth it. */
|
||||
Py_ssize_t hits = 0;
|
||||
for (i = 0; i <= w; i++) {
|
||||
if (ss[i] == pp[0]) {
|
||||
/* candidate match */
|
||||
for (j = 0; j < mlast; j++) {
|
||||
if (s[i+j] != p[j]) {
|
||||
break;
|
||||
}
|
||||
}
|
||||
if (j == mlast) {
|
||||
/* got a match! */
|
||||
if (mode != FAST_COUNT) {
|
||||
return i;
|
||||
}
|
||||
count++;
|
||||
if (count == maxcount) {
|
||||
return maxcount;
|
||||
}
|
||||
i = i + mlast;
|
||||
continue;
|
||||
}
|
||||
/* miss: check if next character is part of pattern */
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1])) {
|
||||
i = i + m;
|
||||
}
|
||||
else {
|
||||
i = i + skip;
|
||||
}
|
||||
hits += j + 1;
|
||||
if (hits >= m / 4 && i < w - 1000) {
|
||||
/* We've done O(m) fruitless comparisons
|
||||
anyway, so spend the O(m) cost on the
|
||||
setup for the two-way algorithm. */
|
||||
Py_ssize_t res;
|
||||
if (mode == FAST_COUNT) {
|
||||
res = STRINGLIB(_two_way_count)(
|
||||
s+i, n-i, p, m, maxcount-count);
|
||||
return count + res;
|
||||
}
|
||||
else {
|
||||
res = STRINGLIB(_two_way_find)(s+i, n-i, p, m);
|
||||
if (res == -1) {
|
||||
return -1;
|
||||
}
|
||||
return i + res;
|
||||
}
|
||||
}
|
||||
}
|
||||
else {
|
||||
/* skip: check if next character is part of pattern */
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1])) {
|
||||
i = i + m;
|
||||
}
|
||||
}
|
||||
}
|
||||
if (mode != FAST_COUNT) {
|
||||
return -1;
|
||||
}
|
||||
return count;
|
||||
}
|
||||
/* The standard, non-adaptive version of the algorithm. */
|
||||
for (i = 0; i <= w; i++) {
|
||||
/* note: using mlast in the skip path slows things down on x86 */
|
||||
if (ss[i] == pp[0]) {
|
||||
/* candidate match */
|
||||
for (j = 0; j < mlast; j++)
|
||||
if (s[i+j] != p[j])
|
||||
for (j = 0; j < mlast; j++) {
|
||||
if (s[i+j] != p[j]) {
|
||||
break;
|
||||
}
|
||||
}
|
||||
if (j == mlast) {
|
||||
/* got a match! */
|
||||
if (mode != FAST_COUNT)
|
||||
if (mode != FAST_COUNT) {
|
||||
return i;
|
||||
}
|
||||
count++;
|
||||
if (count == maxcount)
|
||||
if (count == maxcount) {
|
||||
return maxcount;
|
||||
}
|
||||
i = i + mlast;
|
||||
continue;
|
||||
}
|
||||
/* miss: check if next character is part of pattern */
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1]))
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1])) {
|
||||
i = i + m;
|
||||
else
|
||||
}
|
||||
else {
|
||||
i = i + skip;
|
||||
} else {
|
||||
}
|
||||
}
|
||||
else {
|
||||
/* skip: check if next character is part of pattern */
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1]))
|
||||
if (!STRINGLIB_BLOOM(mask, ss[i+1])) {
|
||||
i = i + m;
|
||||
}
|
||||
}
|
||||
}
|
||||
} else { /* FAST_RSEARCH */
|
||||
}
|
||||
else { /* FAST_RSEARCH */
|
||||
|
||||
/* create compressed boyer-moore delta 1 table */
|
||||
|
||||
|
|
@ -250,28 +692,36 @@ FASTSEARCH(const STRINGLIB_CHAR* s, Py_ssize_t n,
|
|||
/* process pattern[:0:-1] */
|
||||
for (i = mlast; i > 0; i--) {
|
||||
STRINGLIB_BLOOM_ADD(mask, p[i]);
|
||||
if (p[i] == p[0])
|
||||
if (p[i] == p[0]) {
|
||||
skip = i - 1;
|
||||
}
|
||||
}
|
||||
|
||||
for (i = w; i >= 0; i--) {
|
||||
if (s[i] == p[0]) {
|
||||
/* candidate match */
|
||||
for (j = mlast; j > 0; j--)
|
||||
if (s[i+j] != p[j])
|
||||
for (j = mlast; j > 0; j--) {
|
||||
if (s[i+j] != p[j]) {
|
||||
break;
|
||||
if (j == 0)
|
||||
}
|
||||
}
|
||||
if (j == 0) {
|
||||
/* got a match! */
|
||||
return i;
|
||||
}
|
||||
/* miss: check if previous character is part of pattern */
|
||||
if (i > 0 && !STRINGLIB_BLOOM(mask, s[i-1]))
|
||||
if (i > 0 && !STRINGLIB_BLOOM(mask, s[i-1])) {
|
||||
i = i - m;
|
||||
else
|
||||
}
|
||||
else {
|
||||
i = i - skip;
|
||||
} else {
|
||||
}
|
||||
}
|
||||
else {
|
||||
/* skip: check if previous character is part of pattern */
|
||||
if (i > 0 && !STRINGLIB_BLOOM(mask, s[i-1]))
|
||||
if (i > 0 && !STRINGLIB_BLOOM(mask, s[i-1])) {
|
||||
i = i - m;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,431 @@
|
|||
This document explains Crochemore and Perrin's Two-Way string matching
|
||||
algorithm, in which a smaller string (the "pattern" or "needle")
|
||||
is searched for in a longer string (the "text" or "haystack"),
|
||||
determining whether the needle is a substring of the haystack, and if
|
||||
so, at what index(es). It is to be used by Python's string
|
||||
(and bytes-like) objects when calling `find`, `index`, `__contains__`,
|
||||
or implicitly in methods like `replace` or `partition`.
|
||||
|
||||
This is essentially a re-telling of the paper
|
||||
|
||||
Crochemore M., Perrin D., 1991, Two-way string-matching,
|
||||
Journal of the ACM 38(3):651-675.
|
||||
|
||||
focused more on understanding and examples than on rigor. See also
|
||||
the code sample here:
|
||||
|
||||
http://www-igm.univ-mlv.fr/~lecroq/string/node26.html#SECTION00260
|
||||
|
||||
The algorithm runs in O(len(needle) + len(haystack)) time and with
|
||||
O(1) space. However, since there is a larger preprocessing cost than
|
||||
simpler algorithms, this Two-Way algorithm is to be used only when the
|
||||
needle and haystack lengths meet certain thresholds.
|
||||
|
||||
|
||||
These are the basic steps of the algorithm:
|
||||
|
||||
* "Very carefully" cut the needle in two.
|
||||
* For each alignment attempted:
|
||||
1. match the right part
|
||||
* On failure, jump by the amount matched + 1
|
||||
2. then match the left part.
|
||||
* On failure jump by max(len(left), len(right)) + 1
|
||||
* If the needle is periodic, don't re-do comparisons; maintain
|
||||
a "memory" of how many characters you already know match.
|
||||
|
||||
|
||||
-------- Matching the right part --------
|
||||
|
||||
We first scan the right part of the needle to check if it matches the
|
||||
the aligned characters in the haystack. We scan left-to-right,
|
||||
and if a mismatch occurs, we jump ahead by the amount matched plus 1.
|
||||
|
||||
Example:
|
||||
|
||||
text: ........EFGX...................
|
||||
pattern: ....abcdEFGH....
|
||||
cut: <<<<>>>>
|
||||
|
||||
Matched 3, so jump ahead by 4:
|
||||
|
||||
text: ........EFGX...................
|
||||
pattern: ....abcdEFGH....
|
||||
cut: <<<<>>>>
|
||||
|
||||
Why are we allowed to do this? Because we cut the needle very
|
||||
carefully, in such a way that if the cut is ...abcd + EFGH... then
|
||||
we have
|
||||
|
||||
d != E
|
||||
cd != EF
|
||||
bcd != EFG
|
||||
abcd != EFGH
|
||||
... and so on.
|
||||
|
||||
If this is true for every pair of equal-length substrings around the
|
||||
cut, then the following alignments do not work, so we can skip them:
|
||||
|
||||
text: ........EFG....................
|
||||
pattern: ....abcdEFGH....
|
||||
^ (Bad because d != E)
|
||||
text: ........EFG....................
|
||||
pattern: ....abcdEFGH....
|
||||
^^ (Bad because cd != EF)
|
||||
text: ........EFG....................
|
||||
pattern: ....abcdEFGH....
|
||||
^^^ (Bad because bcd != EFG)
|
||||
|
||||
Skip 3 alignments => increment alignment by 4.
|
||||
|
||||
|
||||
-------- If len(left_part) < len(right_part) --------
|
||||
|
||||
Above is the core idea, and it begins to suggest how the algorithm can
|
||||
be linear-time. There is one bit of subtlety involving what to do
|
||||
around the end of the needle: if the left half is shorter than the
|
||||
right, then we could run into something like this:
|
||||
|
||||
text: .....EFG......
|
||||
pattern: cdEFGH
|
||||
|
||||
The same argument holds that we can skip ahead by 4, so long as
|
||||
|
||||
d != E
|
||||
cd != EF
|
||||
?cd != EFG
|
||||
??cd != EFGH
|
||||
etc.
|
||||
|
||||
The question marks represent "wildcards" that always match; they're
|
||||
outside the limits of the needle, so there's no way for them to
|
||||
invalidate a match. To ensure that the inequalities above are always
|
||||
true, we need them to be true for all possible '?' values. We thus
|
||||
need cd != FG and cd != GH, etc.
|
||||
|
||||
|
||||
-------- Matching the left part --------
|
||||
|
||||
Once we have ensured the right part matches, we scan the left part
|
||||
(order doesn't matter, but traditionally right-to-left), and if we
|
||||
find a mismatch, we jump ahead by
|
||||
max(len(left_part), len(right_part)) + 1. That we can jump by
|
||||
at least len(right_part) + 1 we have already seen:
|
||||
|
||||
text: .....EFG.....
|
||||
pattern: abcdEFG
|
||||
Matched 3, so jump by 4,
|
||||
using the fact that d != E, cd != EF, and bcd != EFG.
|
||||
|
||||
But we can also jump by at least len(left_part) + 1:
|
||||
|
||||
text: ....cdEF.....
|
||||
pattern: abcdEF
|
||||
Jump by len('abcd') + 1 = 5.
|
||||
|
||||
Skip the alignments:
|
||||
text: ....cdEF.....
|
||||
pattern: abcdEF
|
||||
text: ....cdEF.....
|
||||
pattern: abcdEF
|
||||
text: ....cdEF.....
|
||||
pattern: abcdEF
|
||||
text: ....cdEF.....
|
||||
pattern: abcdEF
|
||||
|
||||
This requires the following facts:
|
||||
d != E
|
||||
cd != EF
|
||||
bcd != EF?
|
||||
abcd != EF??
|
||||
etc., for all values of ?s, as above.
|
||||
|
||||
If we have both sets of inequalities, then we can indeed jump by
|
||||
max(len(left_part), len(right_part)) + 1. Under the assumption of such
|
||||
a nice splitting of the needle, we now have enough to prove linear
|
||||
time for the search: consider the forward-progress/comparisons ratio
|
||||
at each alignment position. If a mismatch occurs in the right part,
|
||||
the ratio is 1 position forward per comparison. On the other hand,
|
||||
if a mismatch occurs in the left half, we advance by more than
|
||||
len(needle)//2 positions for at most len(needle) comparisons,
|
||||
so this ratio is more than 1/2. This average "movement speed" is
|
||||
bounded below by the constant "1 position per 2 comparisons", so we
|
||||
have linear time.
|
||||
|
||||
|
||||
-------- The periodic case --------
|
||||
|
||||
The sets of inequalities listed so far seem too good to be true in
|
||||
the general case. Indeed, they fail when a needle is periodic:
|
||||
there's no way to split 'AAbAAbAAbA' in two such that
|
||||
|
||||
(the stuff n characters to the left of the split)
|
||||
cannot equal
|
||||
(the stuff n characters to the right of the split)
|
||||
for all n.
|
||||
|
||||
This is because no matter how you cut it, you'll get
|
||||
s[cut-3:cut] == s[cut:cut+3]. So what do we do? We still cut the
|
||||
needle in two so that n can be as big as possible. If we were to
|
||||
split it as
|
||||
|
||||
AAbA + AbAAbA
|
||||
|
||||
then A == A at the split, so this is bad (we failed at length 1), but
|
||||
if we split it as
|
||||
|
||||
AA + bAAbAAbA
|
||||
|
||||
we at least have A != b and AA != bA, and we fail at length 3
|
||||
since ?AA == bAA. We already knew that a cut to make length-3
|
||||
mismatch was impossible due to the period, but we now see that the
|
||||
bound is sharp; we can get length-1 and length-2 to mismatch.
|
||||
|
||||
This is exactly the content of the *critical factorization theorem*:
|
||||
that no matter the period of the original needle, you can cut it in
|
||||
such a way that (with the appropriate question marks),
|
||||
needle[cut-k:cut] mismatches needle[cut:cut+k] for all k < the period.
|
||||
|
||||
Even "non-periodic" strings are periodic with a period equal to
|
||||
their length, so for such needles, the CFT already guarantees that
|
||||
the algorithm described so far will work, since we can cut the needle
|
||||
so that the length-k chunks on either side of the cut mismatch for all
|
||||
k < len(needle). Looking closer at the algorithm, we only actually
|
||||
require that k go up to max(len(left_part), len(right_part)).
|
||||
So long as the period exceeds that, we're good.
|
||||
|
||||
The more general shorter-period case is a bit harder. The essentials
|
||||
are the same, except we use the periodicity to our advantage by
|
||||
"remembering" periods that we've already compared. In our running
|
||||
example, say we're computing
|
||||
|
||||
"AAbAAbAAbA" in "bbbAbbAAbAAbAAbbbAAbAAbAAbAA".
|
||||
|
||||
We cut as AA + bAAbAAbA, and then the algorithm runs as follows:
|
||||
|
||||
First alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^^X
|
||||
- Mismatch at third position, so jump by 3.
|
||||
- This requires that A!=b and AA != bA.
|
||||
|
||||
Second alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^^^^^^^^
|
||||
X
|
||||
- Matched entire right part
|
||||
- Mismatch at left part.
|
||||
- Jump forward a period, remembering the existing comparisons
|
||||
|
||||
Third alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
mmmmmmm^^X
|
||||
- There's "memory": a bunch of characters were already matched.
|
||||
- Two more characters match beyond that.
|
||||
- The 8th character of the right part mismatched, so jump by 8
|
||||
- The above rule is more complicated than usual: we don't have
|
||||
the right inequalities for lengths 1 through 7, but we do have
|
||||
shifted copies of the length-1 and length-2 inequalities,
|
||||
along with knowledge of the mismatch. We can skip all of these
|
||||
alignments at once:
|
||||
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
~ A != b at the cut
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
~~ AA != bA at the cut
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^^^^X 7-3=4 match, and the 5th misses.
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
~ A != b at the cut
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
~~ AA != bA at the cut
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^X 7-3-3=1 match and the 2nd misses.
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
~ A != b at the cut
|
||||
|
||||
Fourth alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^X
|
||||
- Second character mismatches, so jump by 2.
|
||||
|
||||
Fifth alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
^^^^^^^^
|
||||
X
|
||||
- Right half matches, so use memory and skip ahead by period=3
|
||||
|
||||
Sixth alignment:
|
||||
bbbAbbAAbAAbAAbbbAAbAAbAAbAA
|
||||
AAbAAbAAbA
|
||||
mmmmmmmm^^
|
||||
- Right part matches, left part is remembered, found a match!
|
||||
|
||||
The one tricky skip by 8 here generalizes: if we have a period of p,
|
||||
then the CFT says we can ensure the cut has the inequality property
|
||||
for lengths 1 through p-1, and jumping by p would line up the
|
||||
matching characters and mismatched character one period earlier.
|
||||
Inductively, this proves that we can skip by the number of characters
|
||||
matched in the right half, plus 1, just as in the original algorithm.
|
||||
|
||||
To make it explicit, the memory is set whenever the entire right part
|
||||
is matched and is then used as a starting point in the next alignment.
|
||||
In such a case, the alignment jumps forward one period, and the right
|
||||
half matches all except possibly the last period. Additionally,
|
||||
if we cut so that the left part has a length strictly less than the
|
||||
period (we always can!), then we can know that the left part already
|
||||
matches. The memory is reset to 0 whenever there is a mismatch in the
|
||||
right part.
|
||||
|
||||
To prove linearity for the periodic case, note that if a right-part
|
||||
character mismatches, then we advance forward 1 unit per comparison.
|
||||
On the other hand, if the entire right part matches, then the skipping
|
||||
forward by one period "defers" some of the comparisons to the next
|
||||
alignment, where they will then be spent at the usual rate of
|
||||
one comparison per step forward. Even if left-half comparisons
|
||||
are always "wasted", they constitute less than half of all
|
||||
comparisons, so the average rate is certainly at least 1 move forward
|
||||
per 2 comparisons.
|
||||
|
||||
|
||||
-------- When to choose the periodic algorithm ---------
|
||||
|
||||
The periodic algorithm is always valid but has an overhead of one
|
||||
more "memory" register and some memory computation steps, so the
|
||||
here-described-first non-periodic/long-period algorithm -- skipping by
|
||||
max(len(left_part), len(right_part)) + 1 rather than the period --
|
||||
should be preferred when possible.
|
||||
|
||||
Interestingly, the long-period algorithm does not require an exact
|
||||
computation of the period; it works even with some long-period, but
|
||||
undeniably "periodic" needles:
|
||||
|
||||
Cut: AbcdefAbc == Abcde + fAbc
|
||||
|
||||
This cut gives these inequalities:
|
||||
|
||||
e != f
|
||||
de != fA
|
||||
cde != fAb
|
||||
bcde != fAbc
|
||||
Abcde != fAbc?
|
||||
The first failure is a period long, per the CFT:
|
||||
?Abcde == fAbc??
|
||||
|
||||
A sufficient condition for using the long-period algorithm is having
|
||||
the period of the needle be greater than
|
||||
max(len(left_part), len(right_part)). This way, after choosing a good
|
||||
split, we get all of the max(len(left_part), len(right_part))
|
||||
inequalities around the cut that were required in the long-period
|
||||
version of the algorithm.
|
||||
|
||||
With all of this in mind, here's how we choose:
|
||||
|
||||
(1) Choose a "critical factorization" of the needle -- a cut
|
||||
where we have period minus 1 inequalities in a row.
|
||||
More specifically, choose a cut so that the left_part
|
||||
is less than one period long.
|
||||
(2) Determine the period P_r of the right_part.
|
||||
(3) Check if the left part is just an extension of the pattern of
|
||||
the right part, so that the whole needle has period P_r.
|
||||
Explicitly, check if
|
||||
needle[0:cut] == needle[0+P_r:cut+P_r]
|
||||
If so, we use the periodic algorithm. If not equal, we use the
|
||||
long-period algorithm.
|
||||
|
||||
Note that if equality holds in (3), then the period of the whole
|
||||
string is P_r. On the other hand, suppose equality does not hold.
|
||||
The period of the needle is then strictly greater than P_r. Here's
|
||||
a general fact:
|
||||
|
||||
If p is a substring of s and p has period r, then the period
|
||||
of s is either equal to r or greater than len(p).
|
||||
|
||||
We know that needle_period != P_r,
|
||||
and therefore needle_period > len(right_part).
|
||||
Additionally, we'll choose the cut (see below)
|
||||
so that len(left_part) < needle_period.
|
||||
|
||||
Thus, in the case where equality does not hold, we have that
|
||||
needle_period >= max(len(left_part), len(right_part)) + 1,
|
||||
so the long-period algorithm works, but otherwise, we know the period
|
||||
of the needle.
|
||||
|
||||
Note that this decision process doesn't always require an exact
|
||||
computation of the period -- we can get away with only computing P_r!
|
||||
|
||||
|
||||
-------- Computing the cut --------
|
||||
|
||||
Our remaining tasks are now to compute a cut of the needle with as
|
||||
many inequalities as possible, ensuring that cut < needle_period.
|
||||
Meanwhile, we must also compute the period P_r of the right_part.
|
||||
|
||||
The computation is relatively simple, essentially doing this:
|
||||
|
||||
suffix1 = max(needle[i:] for i in range(len(needle)))
|
||||
suffix2 = ... # the same as above, but invert the alphabet
|
||||
cut1 = len(needle) - len(suffix1)
|
||||
cut2 = len(needle) - len(suffix2)
|
||||
cut = max(cut1, cut2) # the later cut
|
||||
|
||||
For cut2, "invert the alphabet" is different than saying min(...),
|
||||
since in lexicographic order, we still put "py" < "python", even
|
||||
if the alphabet is inverted. Computing these, along with the method
|
||||
of computing the period of the right half, is easiest to read directly
|
||||
from the source code in fastsearch.h, in which these are computed
|
||||
in linear time.
|
||||
|
||||
Crochemore & Perrin's Theorem 3.1 give that "cut" above is a
|
||||
critical factorization less than the period, but a very brief sketch
|
||||
of their proof goes something like this (this is far from complete):
|
||||
|
||||
* If this cut splits the needle as some
|
||||
needle == (a + w) + (w + b), meaning there's a bad equality
|
||||
w == w, it's impossible for w + b to be bigger than both
|
||||
b and w + w + b, so this can't happen. We thus have all of
|
||||
the ineuqalities with no question marks.
|
||||
* By maximality, the right part is not a substring of the left
|
||||
part. Thus, we have all of the inequalities involving no
|
||||
left-side question marks.
|
||||
* If you have all of the inequalities without right-side question
|
||||
marks, we have a critical factorization.
|
||||
* If one such inequality fails, then there's a smaller period,
|
||||
but the factorization is nonetheless critical. Here's where
|
||||
you need the redundancy coming from computing both cuts and
|
||||
choosing the later one.
|
||||
|
||||
|
||||
-------- Some more Bells and Whistles --------
|
||||
|
||||
Beyond Crochemore & Perrin's original algorithm, we can use a couple
|
||||
more tricks for speed in fastsearch.h:
|
||||
|
||||
1. Even though C&P has a best-case O(n/m) time, this doesn't occur
|
||||
very often, so we add a Boyer-Moore bad character table to
|
||||
achieve sublinear time in more cases.
|
||||
|
||||
2. The prework of computing the cut/period is expensive per
|
||||
needle character, so we shouldn't do it if it won't pay off.
|
||||
For this reason, if the needle and haystack are long enough,
|
||||
only automatically start with two-way if the needle's length
|
||||
is a small percentage of the length of the haystack.
|
||||
|
||||
3. In cases where the needle and haystack are large but the needle
|
||||
makes up a significant percentage of the length of the
|
||||
haystack, don't pay the expensive two-way preprocessing cost
|
||||
if you don't need to. Instead, keep track of how many
|
||||
character comparisons are equal, and if that exceeds
|
||||
O(len(needle)), then pay that cost, since the simpler algorithm
|
||||
isn't doing very well.
|
||||
Loading…
Reference in New Issue