# Boyer–Moore–Horspool algorithm

In computer science, the Boyer–Moore–Horspool algorithm or Horspool's algorithm is an algorithm for finding substrings in strings. It was published by Nigel Horspool in 1980.[1]

It is a simplification of the Boyer–Moore string search algorithm which is related to the Knuth–Morris–Pratt algorithm. The algorithm trades space for time in order to obtain an average-case complexity of O(N) on random text, although it has O(MN) in the worst case, where the length of the pattern is M and the length of the search string is N.

## Example implementation (C)

```#include <string.h>
#include <limits.h>

/* Returns a pointer to the first occurrence of "needle"
* memmem().
*/

/* Note: In this example needle is a C string. The ending
* 0x00 will be cut off, so you could call this example with
* boyermoore_horspool_memmem(haystack, hlen, "abc", sizeof("abc")-1)
*/
const unsigned char *
boyermoore_horspool_memmem(const unsigned char* haystack, size_t hlen,
const unsigned char* needle,   size_t nlen)
{
size_t scan = 0;
size_t bad_char_skip[UCHAR_MAX + 1]; /* Officially called:

/* Sanity checks on the parameters */
if (nlen <= 0 || !haystack || !needle)
return NULL;

/* ---- Preprocess ---- */
/* Initialize the table to default value */
/* When a character is encountered that does not occur
* in the needle, we can safely skip ahead for the whole
* length of the needle.
*/
for (scan = 0; scan <= UCHAR_MAX; scan = scan + 1)

/* C arrays have the first byte at [0], therefore:
* [nlen - 1] is the last byte of the array. */
size_t last = nlen - 1;

/* Then populate it with the analysis of the needle */
for (scan = 0; scan < last; scan = scan + 1)

/* ---- Do the matching ---- */

/* Search the haystack, while the needle can still be within it. */
while (hlen >= nlen)
{
/* scan from the end of the needle */
for (scan = last; haystack[scan] == needle[scan]; scan = scan - 1)
if (scan == 0) /* If the first byte matches, we've found it. */
return haystack;

/* otherwise, we need to skip some bytes and start again.
Note that here we are getting the skip value based on the last byte
of needle, no matter where we didn't match. So if needle is: "abcd"
then we are skipping based on 'd' and that value will be 4, and
for "abcdd" we again skip on 'd' but the value will be only 1.
The alternative of pretending that the mismatched character was
the last character is slower in the normal case (E.g. finding
"abcd" in "...azcd..." gives 4 by using 'd' but only
4-2==2 using 'z'. */
}

return NULL;
}
```

## Example implementation (Python)

```#!/usr/bin/env python3
# Author: Milan Oberkirch <code_review nearby oberkirch dot org>
from collections import defaultdict

"""
Returns the index to the first character of the first occurrence of "needle"
in "heystack".
"""
def boyer_moore_horspool(heystack, needle):
# amount of characters to skip at start of heystack:
skip = 0
# skip can be increased by len(needle) if the current heystack-character is
# not in needle
skip_table = defaultdict(lambda x=None: len(needle))
# store the number of characters to skip for each letter in needle
for index in range(len(needle) - 1):
skip_table[needle[index]] = len(needle) - 1 - index
# do the actual matching
while len(heystack) - skip >= len(needle):
index = len(needle) - 1
while heystack[skip + index] == needle[index]:
if index == 0:
return skip
index = index - 1
skip += skip_table[heystack[skip + len(needle) - 1]]
return None
```

## Performance

The algorithm performs best with long needle strings, when it consistently hits a non-matching character at or near the final byte of the current position in the haystack and the final byte of the needle does not occur elsewhere within the needle. For instance a 32 byte needle ending in "z" searching through a 255 byte haystack which does not have a 'z' byte in it would take up to 224 byte comparisons.

The best case is the same as for the Boyer–Moore string search algorithm in big O notation, although the constant overhead of initialization and for each loop is less.

The worst case behavior happens when the bad character skip is consistently low (with the lower limit of 1 byte movement) and a large portion of the needle matches the haystack. The bad character skip is only low, on a partial match, when the final character of the needle also occurs elsewhere within the needle, with 1 byte movement happening when the same byte is in both of the last two positions.

The canonical degenerate case similar to the above "best" case is a needle of an 'a' byte followed by 31 'z' bytes in a haystack consisting of 255 'z' bytes. This will do 31 successful byte comparisons, a 1 byte comparison that fails and then move forward 1 byte. This process will repeat 223 more times (255 - 32), bringing the total byte comparisons to 7,168 (32 * 224).

The worst case is significantly higher than for the Boyer–Moore string search algorithm, although obviously this is hard to achieve in normal use cases. It is also worth noting that this worst case is also the worst case for the naive (but usual) memmem() algorithm, although the implementation of that tends to be significantly optimized (and is more cache friendly).