For many problems some extra space really pays off
(extra space in tables - breathing room)
 input enhancement
• non comparison-based sorting
• auxiliary tables (shift tables for pattern matching)
 prestructuring
• hashing
• indexing schemes (eg, B-trees)
 tables of information that do all the work
• dynamic programming
Design and Analysis of Algorithms - Chapter 7 1

Algorithm ComparisonCountingSort(A[0..n-1]) for i

0 to n-1 do Count[i]

0 for i

0 to n-2 do for j
 i+1 to n-1 do if A[i] <A[j] then Count[j]

Count[j]+1 else Count[i]

Count[i]+1 for i

0 to n-1 do S[Count[i]]

A[i]
 Example: 62 31 84 96 19 47
 Efficiency
Design and Analysis of Algorithms - Chapter 7 2

Algorithm DistributionCountingSort(A[0..n-1]) for j

0 to u-l do D[j]

0 for i

0 to n-1 do D[A[i]-l]

D[A[i]-l] + 1 for j

1 to u-l do D[j]

D[j-1]+D[j] for i
 n-1 down to 0 do j

A[i]-l; S[D[j]-1]

A[i]; D[j]

D[j]-1
 Example: 13 11 12 13 12 12
 Efficiency
Design and Analysis of Algorithms - Chapter 7 3



pattern: a string of m characters to search for
text: a (long) string of n characters to search in
 Brute force algorithm:
1.
Align pattern at beginning of text
2.
moving from left to right, compare each character of pattern to the corresponding character in text until
– all characters are found to match (successful search); or
– a mismatch is detected
3.
while pattern is not found and the text is not yet exhausted, realign pattern one position to the right and
4



1970: Cook shows (using finite-state machines) that problem can be solved in time proportional to n+m
1976 Knuth and Pratt find algorithm based on Cook’s idea;
Morris independently discovers same algorithm in attempt to avoid “backing up” over text
 At about the same time Boyer and Moore find an algorithm that examines only a fraction of the text in most cases (by comparing characters in pattern and text from right to left, instead of left to right)
 1980 Another algorithm proposed by Rabin and Karp virtually always runs in time proportional to n+m and has the advantage of extending easily to two-dimensional pattern matching and being almost as simple as the bruteforce method.
Design and Analysis of Algorithms - Chapter 7 5

 A simplified version of Boyer-Moore algorithm that retains key insights:
• compare pattern characters to text from right to left
• given a pattern, create a shift table that determines how much to shift the pattern when a mismatch occurs (input
enhancement)
Design and Analysis of Algorithms - Chapter 7 6

Look at first (rightmost) character in text that was compared. Three cases:
•
The character is not in the pattern
.....c...................... ( c not in pattern)
BAOBAB
• The character is in the pattern (but not at rightmost position)
.....O...................... ( O occurs once in pattern)
BAOBAB

.....A...................... ( A occurs twice in pattern)
BAOBAB
• The rightmost characters produced a match
.....B......................
BAOBAB
Shift Table: Stores number of characters to shift by depending on first character compared
Design and Analysis of Algorithms - Chapter 7 7

 Constructed by scanning pattern before search begins


All entries are initialized to length of pattern.
For c occurring in pattern, update table entry to distance of rightmost occurrence of c from end of pattern
 Algorithm ShiftTable(P[0..m-1]) for i

0 to size-1 do Table[i]
 m for j

0 to m-2 do Table[P[j]]
 m-1-j return Table
Design and Analysis of Algorithms - Chapter 7 8

 Example for pattern 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
 Then:
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 9

 Horspool Matching(P[0..m-1,T[0..n-1]])
ShiftTable(P[0..m-1]) i
 m-1 while i<=n-1 do k

0 while k<=m-1 and P[m-1-k]=T[i-k] do k
 k+1 if k=m return i-m+1 else i
 i+Table[T[i]] return -1
Design and Analysis of Algorithms - Chapter 7 10

 Based on same two ideas:
• compare pattern characters to text from right to left
• given a pattern, create a shift table that determines how much to shift the pattern when a mismatch occurs (input enhancement)
 Uses additional shift table with same idea applied to the number of matched characters
Design and Analysis of Algorithms - Chapter 7 11

 Based on the Horspool idea of using the extra table
 However, this table is computed differently
•
If c, the text character corresponding to the last pattern character, is not in the pattern, then shift in the same way by m characters (actually c is a bad symbol)
• If the mismatching character (the bad symbol) does not appear in the pattern, then shift to overpass it
• If the mismathcing character (the bad symbol) appears in the pattern, then shift to align the bad symbol to the same text character (lying to the left of the mismatching position).
Design and Analysis of Algorithms - Chapter 7 12

The bad symbol IS NOT in the pattern
...
S ER......................
BAR B ER
BARBER shift 4 positions
The bad symbol IS in the pattern
...
A ER......................
B A R B ER
BARBER shift 2 positions,
This shift is given by: d=max[t1(c)-k,1], where t1 is the Horspool table, k the distance between the bad symbol from the end of the pattern
Design and Analysis of Algorithms - Chapter 7 13

 What happens if a matched suffix appears again in the pattern (eg. AB RA CADAB RA )
 Important to find another suffix with a different previous character. Calculate the shift as the distance between two occurrences of the suffix.
 Also, important to find the longest prefix of size l<k that matches the suffix of size l.
Calculate the shift as the distance between the suffix and the prefix.
Design and Analysis of Algorithms - Chapter 7 14

K pattern d2
1 ABCBA B 2
K pattern d2
1 BAOBA B 2
2 ABCB AB 4 2 BAOB AB 5
3 ABC BAB 4
4 AB CBAB 4
5 A BCBAB 4
3 BAO BAB 5
4 BA OBAB 5
5 B AOBAB 5
Design and Analysis of Algorithms - Chapter 7 15

 Calculate shift as if k=0 d =
{ d1 max(d1,d2) if k>0 where d1=max(t1(c)-k)
 Example
BESS_KNEW_ABOUT_BAOBABS
BAOBAB
Design and Analysis of Algorithms - Chapter 7 16

 A very efficient method for implementing a
dictionary, i.e., a set with the operations:
– insert
– find
– delete
 Applications:
– databases
– symbol tables
Design and Analysis of Algorithms - Chapter 7 17

 Hash table: an array with indices that correspond to buckets
 Hash function: determines the bucket for each record
 Example: student records, key=SSN. Hash function:
h(k) = k mod m
(k is a key and m is the number of buckets)
• if m=1000, where is record with SSN= 315-17-4251 stored?
 Hash function must:
• be easy to compute
• distribute keys evenly throughout the table
Design and Analysis of Algorithms - Chapter 7 18

 If h(k1) = h(k2) then there is a collision.
 Good hash functions result in fewer collisions.
 Collisions can never be completely eliminated.
 Two types handle collisions differently:
•
Open hashing bucket points to linked list of all keys hashing to it.
•
Closed hashing – one key per bucket, in case of collision, find another bucket for one of the keys
– linear probing: use next bucket
– double hashing: use second hash function to compute increment
Design and Analysis of Algorithms - Chapter 7 19

 If hash function distributes keys uniformly, average length of linked list will be n/m
 Average number of probes S = 1+α/2, U = α
 Worst-case is still linear!
 Open hashing still works if n>m.
Design and Analysis of Algorithms - Chapter 7 20

 Does not work if n>m.
 Avoids pointers.
 Deletions are not straightforward.
 Number of probes to insert/find/delete a key depends on load factor α = n/m (hash table density)
– successful search: (½) (1+ 1/(1- α))
– unsuccessful search: (½) (1+ 1/(1- α)²)
 As the table gets filled (α approaches 1), number of probes increases dramatically:
Design and Analysis of Algorithms - Chapter 7 21