String Matching
The KMP Algorithm

Terminologies

• Sub-String: Zero or more consecutive characters
• Prefix: A Sub-String that starts from the beginning of the string
• Suffix: A Sub-String that ends at the end of the string
• Proper-Prefix: A Prefix that's not the whole string
• Proper-Suffix: A Suffix that's not the whole string

Example

"aaab"

• Sub-Strings: {_, a, aa, aaa, aaab, a, aa, aab, a, ab, b}
• Prefixes: {_, a, aa, aaa, aaab}
• Suffixes: {_, b, ab, aab, aaab}
• Proper-Prefixes: {_, a, aa, aaa}
• Proper-Suffixes: {_, b, ab, aab}

Side Note:

• #Substrings =
• #Prefixes = #Suffixes = n + 1

Pattern Find Problem

Given 2 strings (text, pattern).

Print all the positions of the pattern's occurrences within the text.

Example:

  ababab ab

Output:

  3

     1 3 5

Optimizations?

Knuth
Morris
Pratt

Visualization

Failure (F) Function

The Longest Proper-Suffix that's also a Proper-Prefix