Unit SS, Part B: String Matching, Boyer-Moore and Variants

Standard B-M

• Shift maximum of 2 heuristics:
• 1) Match heuristic
  • Shift right so match all chars already matched; have a new char at match-check position
  • Example: abracacabra
    • ddhat[j] = min{s+m-j | ...} where s=skip amount
    • a: no match so shift 1 to try the "r"; 1=1+11-11
    • r: matched "a" so shift 3 to see if have "da"; 4=3+11-10
    • b: matched "ra" so shift fully; 12=10+11-9
• 2) Occurrence heuristic
  • Given text char X that didn't match pat, find rightmost place in pat where X occurs, to try again, and return distance from right to shift
  • Example: abracacabra
    • "a" is 0 from right, "b" is 2 in,
    • "c" is 6 in from right, . . .

Simplified Boyer-Moore

• Boyer-Moore with only occur. heur.
  • Why?
    • Patterns are not periodic
    • Space is less
  • Thus should be faster on average
• Actually, performs slightly less well than Boyer-Moore (see tests)

Boyer-Moore-Horspool

• Let X = last char in pattern; find T = char in text at X's position; use T to lookup skip in heuristic table
• Boyer-Moore-Horspool-Sunday:
  • go back to left to right checking
  • only use occurrence test but shift so char of text is at pat[m+1]
  • BMH gives best empirical results

Example of BM Searching for Pattern in Text

BOYER-MOORE
ababaxabracabracadabratcadabrax
abracadabra
          ^(no-match, k=11 +=max(d[c]=6,dd[a]=1) = 17)
            skip 6 to line up c in text with c in pattern
ababaxabracabracadabratcadabrax
      abracadabra
                ^(match)
ababaxabracabracadabratcadabrax
      abracadabra
               ^(no-match, k=16 +=max(d[c]=6,dd[r]=4) = 22)
                 skip 6 to line up c in text with c in pattern
ababaxabracabracadabratcadabrax
           abracadabra
                     ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                    ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                   ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                   ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                  ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                 ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
                ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
               ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
              ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
             ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
            ^(match)
ababaxabracabracadabratcadabrax
           abracadabra
           ^(match,full-match, skip so line up what matched before)
ababaxabracabracadabratcadabrax
                  abracadabra
                            ^(no-match, k+=max(d[r]=1,dd[a]=1)=1)
ababaxabracabracadabratcadabrax
                   abracadabra
                            ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                            ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                           ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                          ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                         ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                        ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                       ^(match)
ababaxabracabracadabratcadabrax
                   abracadabra
                      ^(no-match, k+=max(d[t]=11,dd[a]=14)=14)
ababaxabracabracadabratcadabrax
                          abracadabra
                           (past text-end so quit)