Searching Strings

1
Searching Strings

• Brandon Ochs

2
The Basics

• Pattern Smaller string that we are looking for
• String Larger than the pattern the part being
  searched

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Simple Method

• Traverse the string from left to right
• Compare pattern to left most part of the string
• If a match is found, compare with the patten
• If no match is found shift the pattern right one
  unit
• If we run out of space, there is no match

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Simple Method Example

• We want to find the word base in this sentence

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Simple Method Example

• ALL YOUR BASE ARE BELONG TO US
• BASE
• Left Most Character B

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• ALL YOUR BASE ARE BELONG TO US
• BASE
• Left Most Character L

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• ALL YOUR BASE ARE BELONG TO US
• BASE
• Left Most Character L

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Eventually

• ALL YOUR BASE ARE BELONG TO US
• BASE
• O(mn) where m is the length of the pattern and n
  is the length of the string

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Worst Case Scenario

• String AAAAAAAAAAAAAB
• Pattern AAAB
• Total of 44 comparisons

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Boyer-Moore Algorithm

• Compare to pattern from right to left
• If the strings character being matched is not in
  the pattern, shift by the pattern length

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Boyer-Moore Example

• ALL YOUR BASE ARE BELONG TO US
• BASE
• Right Most Character
• Shift 4

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• ALL YOUR BASE ARE BELONG TO US
• BASE
• Right Most Character R
• Shift 4

13

• ALL YOUR BASE ARE BELONG TO US
• BASE
• Right Most Character S
• Shift 1

14

• ALL YOUR BASE ARE BELONG TO US
• BASE
• Right Most Character E
• Match Found

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Worst Case

• String AAAAAAAAAAAAAB
• Pattern AAAB
• Total of 24 comparisons

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Improvements

• Create a table which contains the distance from
  each unique character in the pattern, starting
  from the right.
• Create a second table, which will indicate the
  rightmost recurrence of each possible terminal
  portion of the pattern.

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Calculating the First Table

• Start at the last character of the pattern and
  move towards the first character.
• If the current character is not in the table
  already, add it.
• The Shift value is its distance from the
  rightmost character.
• All other characters receive a count equal to
  the length of the pattern.

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Table 1 Example

• Pattern Banana
• Character    Shift        A        0        N
         1        B        5 Other        6

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Calculating the Second Table

• Traverse the pattern and create sub-strings for
  each value of i less than the total length.
• Calculate the sub-string consisting of the last i
  characters preceded by a mis-match for the
  character before it (meaning anything but the
  current character).

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Table 2 Example

• Pattern Banana
• i Character Shift
• 0 A 1
• 1 NA 4
• 2 ANA 6
• 3 NANA 2
• 4 ANANA 6
• 5 BANANA 6

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More On Table 2

• Remember, we are moving the Pattern, not the
  substring.
• Example
• ABA
• --ABABABABA
• Match is found after 2 shifts

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Pseudo Code

• ilt-width
• while string remains
• j lt- width
• if j 0 then print Match at i 1
• if string(i) pattern(j)
• then j lt- j 1
• i lt- j 1
• //if
• i lt- i max (table1(string(i)), table2(j))
• //if
• //while

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How About That Worst Case?

• String AAAAAAAAAAAAAB
• Pattern AAAB
• Table1 Table2
• B 0 0 B 1
• A 1 1 AB 4
• Other 4 2 AAB 4
• 3 AAAB 4
• Total of 14 comparisons

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Summary of Boyer Moore

• Preprocesses the pattern
• Time Complexity is Sub-lilnear because it
  doesnt need to check every character of the
  string to be searched.
• Requires fewer than (i width) instructions
• The algorithm improves in efficiency as the
  pattern becomes longer

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Table 1 vs. Table 2

• Table 1 is more useful with large alphabets and
  small patterns
• Table 2 is more useful for small alphabets and
  large patterns

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Applications to Computer Science

• Really only useful for one thing
• Searching text!!
• Text based searching is very important in many
  applications and web pages

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References

• Dewdney, A.K. The (New) Turing Omnibus. New York
  Henry Holt and Company, 1993.
• R.S. Boyer, J.S. Moore A Fast String Searching
  Algorithm. Communications of the ACM, 20, 10,
  762-772 (1977)

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Homework Questions

• 1) Construct one of the two types of shift tables
  for the pattern PICNIC
• 2) List the number of comparisons required for
  the simple algorithm to find the pattern AAB in
  the string AAAAAAB