Chapter 2: Using Objects

Design and Analysis of Algorithms – Chapter 7

Space-Time Tradeoffs:

String Matching Algorithms*

Dr. Ying Lu

ylu@cse.unl.edu

RAIK 283: Data Structures & AlgorithmsRAIK 283: Data Structures & Algorithms

*slides referred to

http://www.aw-bc.com/info/levitinhttp://www.aw-bc.com/info/levitin

Motivation exampleMotivation example

bHow to sort array a[1], a[2], …., a[n], whose valuesare known to come from a set, e.g., {a, b, c, d, e, f,…. x, y, z}?bHow to sort array a[1], a[2], …., a[n], whose valuesare known to come from a set, e.g., {a, b, c, d, e, f,…. x, y, z}?

Design and Analysis of Algorithms – Chapter 7

Design and Analysis of Algorithms – Chapter 7

Space-time tradeoffsSpace-time tradeoffs

For many problems some extra space really pays off:For many problems some extra space really pays off:

bExtra space in tablesbExtra space in tables

•hashing•hashing

•non comparison-based sorting (e.g., sorting by countingthe array a[1], a[2], …., a[n], whose values are known tocome from a set, e.g., {a, b, c, d, e, f, …. x, y, z})•non comparison-based sorting (e.g., sorting by countingthe array a[1], a[2], …., a[n], whose values are known tocome from a set, e.g., {a, b, c, d, e, f, …. x, y, z})

bInput enhancementbInput enhancement

•auxiliary tables (shift tables for pattern matching)•auxiliary tables (shift tables for pattern matching)

•indexing schemes (e.g., B-trees)•indexing schemes (e.g., B-trees)

bTables of solutions for overlapping subproblemsbTables of solutions for overlapping subproblems

•dynamic programming•dynamic programming

Design and Analysis of Algorithms – Chapter 7

String matchingString matching

bpattern: a string of m characters to search forbpattern: a string of m characters to search for

btext: a (long) string of n characters to search inbtext: a (long) string of n characters to search in

bBrute force algorithm?bBrute force algorithm?

Design and Analysis of Algorithms – Chapter 7

String matchingString matching

bpattern: a string of m characters to search forbpattern: a string of m characters to search for

btext: a (long) string of n characters to search inbtext: a (long) string of n characters to search in

bBrute force algorithm:bBrute force algorithm:

1.Align pattern at beginning of text1.Align pattern at beginning of text

2.moving from left to right, compare each character of pattern to thecorresponding character in text until2.moving from left to right, compare each character of pattern to thecorresponding character in text until

–all characters are found to match (successful search); or–all characters are found to match (successful search); or

–a mismatch is detected–a mismatch is detected

3.while pattern is not found and the text is not yet exhausted, realignpattern one position to the right and repeat step 2.3.while pattern is not found and the text is not yet exhausted, realignpattern one position to the right and repeat step 2.

bTime efficiency:bTime efficiency:

Design and Analysis of Algorithms – Chapter 7

String matchingString matching

bpattern: a string of m characters to search forbpattern: a string of m characters to search for

btext: a (long) string of n characters to search inbtext: a (long) string of n characters to search in

bBrute force algorithm:bBrute force algorithm:

1.Align pattern at beginning of text1.Align pattern at beginning of text

–all characters are found to match (successful search); or–all characters are found to match (successful search); or

–a mismatch is detected–a mismatch is detected

bTime efficiency: (nm)bTime efficiency: (nm)

Design and Analysis of Algorithms – Chapter 7

Horspool’s AlgorithmHorspool’s Algorithm

bA simplified version of Boyer-Moore algorithm that retainskey insights:bA simplified version of Boyer-Moore algorithm that retainskey insights:

•compare pattern characters to text from right to left•compare pattern characters to text from right to left

•given a pattern, create a shift table that determines howmuch to shift the pattern when a mismatch occurs (inputenhancement)•given a pattern, create a shift table that determines howmuch to shift the pattern when a mismatch occurs (inputenhancement)

Design and Analysis of Algorithms – Chapter 7

How far to shift when a mismatch occurs ?How far to shift when a mismatch occurs ?

Look at first (rightmost) character in text that was compared. Three cases:Look at first (rightmost) character in text that was compared. Three cases:

•The character is not in the pattern•The character is not in the pattern

.....c...................... (c not in pattern) .....c...................... (c not in pattern)

BAOBAB BAOBAB

•The character is in the pattern (but not at rightmost position)•The character is in the pattern (but not at rightmost position)

.....O...................... (O occurs once in pattern) .....O...................... (O occurs once in pattern)

BAOBAB BAOBAB

.....A...................... (A occurs twice in pattern) .....A...................... (A occurs twice in pattern)

BAOBAB BAOBAB

•The rightmost characters produced a match•The rightmost characters produced a match

.....B...................... .....B......................

BAOBAB BAOBAB

.....B...................... .....B......................

AAOAAB AAOAAB

bShift Table: Stores number of characters to shift depending on firstcharacter comparedbShift Table: Stores number of characters to shift depending on firstcharacter compared

Shift tableShift table

bIndexed by text and pattern alphabetbIndexed by text and pattern alphabet

bIf a character c is not among the first m-1 characters of thepattern, then t(c) = the pattern’s length mbIf a character c is not among the first m-1 characters of thepattern, then t(c) = the pattern’s length m

bOtherwise, t(c) = the distance from the rightmost c amongthe first m-1 characters of the pattern to its last characterbOtherwise, t(c) = the distance from the rightmost c amongthe first m-1 characters of the pattern to its last character

Design and Analysis of Algorithms – Chapter 7

Design and Analysis of Algorithms – Chapter 7

Shift tableShift table

bConstructed by scanning pattern before search beginsbConstructed by scanning pattern before search begins

bIndexed by text and pattern alphabetbIndexed by text and pattern alphabet

All entries are initialized to length of pattern. E.g.,BAOBAB:All entries are initialized to length of pattern. E.g.,BAOBAB:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

For c occurring in the first m-1 characters of the pattern,update table entry to distance of rightmost c among thefirst m-1 characters of the pattern to its last characterFor c occurring in the first m-1 characters of the pattern,update table entry to distance of rightmost c among thefirst m-1 characters of the pattern to its last character

Design and Analysis of Algorithms – Chapter 7

Shift tableShift table

bConstructed by scanning pattern before search beginsbConstructed by scanning pattern before search begins

bIndexed by text and pattern alphabetbIndexed by text and pattern alphabet

All entries are initialized to length of pattern. E.g.,BAOBAB:All entries are initialized to length of pattern. E.g.,BAOBAB:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

Shift tableShift table

bWe can create the shift table by processing pattern fromL→R:bWe can create the shift table by processing pattern fromL→R:

Algorithm ShiftTable(P[0…m-1]) {Algorithm ShiftTable(P[0…m-1]) {

initialize all the elements of Table with minitialize all the elements of Table with m

for j = 0 to m-2 do Table[P[j]] = m-1-jfor j = 0 to m-2 do Table[P[j]] = m-1-j

return Tablereturn Table

}}

Design and Analysis of Algorithms – Chapter 7

Design and Analysis of Algorithms – Chapter 7

ExampleExample

BARD LOVED BANANASBARD LOVED BANANAS

BAOBABBAOBAB

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Design and Analysis of Algorithms – Chapter 7

In-class exercisesIn-class exercises

bP267: 7.2.1--- Apply Horspool’s algorithm to search for thepattern BAOBAB in the textBESS_KNEW_ABOUT_BAOBABSbP267: 7.2.1--- Apply Horspool’s algorithm to search for thepattern BAOBAB in the textBESS_KNEW_ABOUT_BAOBABS

bP268: 7.2.3--- How many character comparisons will bemade by Horspool’s algorithm in searching for each of thefollowing patterns in the binary text of 1000 zeros?bP268: 7.2.3--- How many character comparisons will bemade by Horspool’s algorithm in searching for each of thefollowing patterns in the binary text of 1000 zeros?

–00001–00001

–10000–10000

Design and Analysis of Algorithms – Chapter 7

In-class exercisesIn-class exercises

bP264: 7.2.1--- Apply Horspool’s algorithm to search for thepattern BAOBAB in the textBESS_KNEW_ABOUT_BAOBABSbP264: 7.2.1--- Apply Horspool’s algorithm to search for thepattern BAOBAB in the textBESS_KNEW_ABOUT_BAOBABS

bP264: 7.2.3--- How many character comparisons will bemade by Horspool’s algorithm in searching for each of thefollowing patterns in the binary text of 1000 zeros?bP264: 7.2.3--- How many character comparisons will bemade by Horspool’s algorithm in searching for each of thefollowing patterns in the binary text of 1000 zeros?

–00001 (1*996 = 996 comparisons)–00001 (1*996 = 996 comparisons)

–10000 (5*996 = 4980 comparisons)–10000 (5*996 = 4980 comparisons)

Time EfficiencyTime Efficiency

bHorspool’s algorithmbHorspool’s algorithm

•(nm) in the worst-case.•(nm) in the worst-case.

•(n) for random texts•(n) for random texts

Design and Analysis of Algorithms – Chapter 7

In-Class ExercisesIn-Class Exercises

bYou have an array of positive integers likear[]={1,3,2,4,5,4,2}. You need to create another arrayar_low[] such that ar_low[i] = number of elements lowerthan or equal to ar[i]. So the output of above should be{1,4,3,6,7,6,3}.bYou have an array of positive integers likear[]={1,3,2,4,5,4,2}. You need to create another arrayar_low[] such that ar_low[i] = number of elements lowerthan or equal to ar[i]. So the output of above should be{1,4,3,6,7,6,3}.

bAlgorithm time complexity should be O(n) and use of extraspace is allowed.bAlgorithm time complexity should be O(n) and use of extraspace is allowed.

Design and Analysis of Algorithms – Chapter 7

Boyer-Moore algorithmBoyer-Moore algorithm

bBased on same two ideas:bBased on same two ideas:

•compare pattern characters to text from right toleft•compare pattern characters to text from right toleft

•given a pattern, create a shift table thatdetermines how much to shift the pattern whena mismatch occurs (input enhancement)•given a pattern, create a shift table thatdetermines how much to shift the pattern whena mismatch occurs (input enhancement)

bUses additional shift table with same idea appliedto the number of matched charactersbUses additional shift table with same idea appliedto the number of matched characters

Design and Analysis of Algorithms – Chapter 7

In-class exercisesIn-class exercises

bP264: 7.2.7 How many character comparisons will Boyer-Moore algorithm make in searching for each of thefollowing patterns in the binary text of 1000 zeros?bP264: 7.2.7 How many character comparisons will Boyer-Moore algorithm make in searching for each of thefollowing patterns in the binary text of 1000 zeros?

–00001–00001

–10000–10000

Design and Analysis of Algorithms – Chapter 7

In-class exercisesIn-class exercises

–00001 (1*996 = 996 comparisons)–00001 (1*996 = 996 comparisons)

–10000 (5*200 = 1000 comparisons)–10000 (5*200 = 1000 comparisons)

In-class exercisesIn-class exercises

–01010–01010

Design and Analysis of Algorithms – Chapter 7