• Motivation:
def is_substr(source, query):
    """return if query is a substring of the source"""
    for i in range(len(source)):
        if source[i:i+len(query)] == query: return True
    return False

O(MN) time where N = len(source), M = len(query). Redaundant work is that when search fails, we go back to next i.

source = aabdcaaabdcaaabcdba
query  = aabdcaaabcdb
                  | match process started from i = 0 we fail here

Do we need to go back to i+1 and start over? No

source = aabdcaaabdcaaabcdba
query  =       aabdcaaabcdb
We should try to start from here. 

How do we know?

query = aabdca|aab|cdb
query =       |aab|dcaaabcdb

aab is the prefix but also the suffix for subtring of query aabdcaaab

If we know such information about the query string, we can move i back to where that suffix started.

  • Longest proper prefix that is also a suffix. LPS Generate an array that contains the length of the LPS of each part of target. ```python def lps(p): m = len(p) lps = [0, 0] j = 0 for i in range(1, m): while j and p[i] != p[j]: j = lps[j] if p[i] == p[j]: j += 1 lps.append(j) return lps

query = “aabdcaaabcdb” print(query) print(““.join(map(str, lps(query))))


lps[i] is: if match at query location i failed (does not match source[j]), we can go back to lps[i] of query and try to match query[lps[i]] with the source[j].

Matching prodedure.

def match(s, p):
    n, m = len(s), len(p)
    lps = lps(p)
    ans = []
    j = 0
    for i in range(n):
        while j and s[i] != p[j]: j = lps[j]  # don't match, use lps table to jump back
        if s[i] == p[j]: j += 1
        if j == m:
            ans.append(i - m + 1)
            j = lps[j]
    return ans

match('abcdeabcdabcbacad', 'abc')

partial use:

def kmp_table(s):
    m = len(s)
    table, j = [0, 0], 0
    for i in range(1, m):
        while j > 0 and s[i] != s[j]:  j = table[j]
        if s[i] == s[j]:               j += 1
    return table

class Solution:
    def shortestPalindrome(self, s: str) -> str:
        ss = s + "+" + s[::-1]
        tb = kmp_table(ss)
        k = tb[-1]  
        return s[k:][::-1] + s[:k] + s[k:]

Rabin-Karp Rolling Hash

  • Motivation:
# rolling hash a fixed size substring on DNA sequence. 
def update_code(code, c):
    code <<= 2
    code &= mask
    if c == 'A': code |= 0
    if c == 'C': code |= 1
    if c == 'G': code |= 2
    if c == 'T': code |= 3
    return code