Chapter 2.8 Search Algorithms. Array Search –An array contains a certain number of records –Each record is identified by a certain key –One searches the.

Chapter 2.8

Search Algorithms

Search Algorithms

• Array Search– An array contains a certain number of records– Each record is identified by a certain key– One searches the record with a given key

• String Search– A text is represented by an array of characters– One searches one or all occurrences of a certain

string

Search Algorithms



string

Array Search

PROCEDURE Search– Two parameters:• The Array to be searched–Contains n Items–Index in the range 0..n–Item with index 0 is not used

• The Key of the Item to be found– Search returns a Cardinal value• the index of the Item where the key has been

found• if the key has not been found , 0.

Array Search

TYPEArrayOfItems = ARRAY[0..n] OF Item;(* By convention, the element with index 0 is not used *)

Item = RECORD Key : KeyType (* any ordinal type *); Other record fields ; END;

Straight Search

VAR

PROCEDURE Search( A: ARRAY OF Item,Key: KeyType): CARDINAL;

VAR i : CARDINAL;

BEGINi := HIGH(A);

WHILE A[i].Key # Key AND i # 0 DO

DEC(i)

END;

RETURN iEND Search;

Straight Search

VAR

PROCEDURE Search( A: ARRAY OF Item,

VAR i : CARDINAL;

BEGINi := HIGH(A);

WHILE A[i].Key # Key AND i # 0 DO

DEC(i)

END;

RETURN iEND Search;

Key: KeyType): CARDINAL;

Sentinel Search

PROCEDURE Search( VAR A: ARRAY OF Item,Key: KeyType): CARDINAL;

VAR i : CARDINAL;

BEGIN

i := HIGH(A);

A[0].Key := Key;

WHILE A[i].Key # Key DO DEC(i) END;RETURN i

END Search;

Binary Search

# elements > 1

Binary Search

No Yes# elements > 1

No YesKey < Keymiddle

No YesKey = Keyelement

BinarySearchrighthalf

BinarySearch

lefthalf

NotFound Found

Binary Search (1)

PROCEDURE Search(VAR a: ARRAY OF Item, Key:KeyType):CARDINAL;

VAR Min,Max,m: CARDINAL; PROCEDURE src(Min,Max: CARDINAL); … END src;BEGIN Min := 1; Max := HIGH(a); src(Min,Max); IF a[m].Key = Key THEN RETURN m ELSE RETURN 0 ENDEND Search;

Binary Search (2)

PROCEDURE Src(Min,Max : CARDINAL);BEGIN m := (Min+Max) DIV 2; IF Min # Max THEN IF a[m].Key >= Key THEN src(Min,m) ELSE src(m+1,Max) END; ENDEND Src;

Iterative Binary SearchPROCEDURE Search(VAR a: ARRAY OF Item,

Key:KeyType):CARDINAL;VAR Min,Max,m: CARDINAL;

BEGIN Min := 1; Max := HIGH(a); WHILE Min < Max DO m := (Min+Max) DIV 2; IF a[m].Key >= Key THEN Max := m ELSE Min := m+1 END; (* IF *) END; (* WHILE *) IF a[m].Key = Key THEN RETURN m ELSE RETURN 0 ENDEND Search;

Array Search Performance

(Number of comparisons)

• Unordered Array–Straight search : 2n–Sentinel search : n

• Ordered Array– Binary search : log 2 n

Search Algorithms



string

String Search• The problem: Find a given string in a text.

• Data structures:

Text : ARRAY[1..TextSize] OF CHAR;

String : ARRAY[1..StringSize] OF CHAR;• The Algorithms:– Brute Force– Knuth, Morris & Pratt (KPM - 1974)– Boyer & Moore (BM - 1977)

String Search

by Brute Forcethis algorithm tries to find a string

string stringstring

String matched OR Text exhausted

Move to next character in Text

End of Text reached ?No

WHILE current char.in Text # String[1]

WHILE char. in Text = char. in String

Move to next character pair

Brute Force String Search

(Straightforward coding)PROCEDURE Search (VAR Text: TextType; TLength:CARDINAL; VAR String: StringType; SLength:CARDINAL):CARDINAL; VAR j, jmax : CARDINAL;BEGIN j := 0; jmax := TLength - SLength; REPEAT WHILE (Text[j] # String[1]) AND (j <= jmax) DO j := j+1 END; IF j <= jmax THEN i := 2; WHILE (Text[j+i] = String[i]) AND (i < SLength) DO i := i+1 END; END; (* IF *) j := j + 1; UNTIL (i = SLength) OR (j > jmax); RETURN j - 1END Search;

String Search (1)

by the KMP algorithmGATCGATCAGCAATCATCATCACATC

ATCATCACAT

Mismatch in first position of string,Move string 1 position in text

String Search(2)by the KMP algorithm

GATCGATCAGCAATCATCATCACATC

ATCATCACAT

Mismatch in fourth position of string,Move string 4 positions in text



ATCATCACAT

Mismatch in fifth position of string,Move string 4 positions in text



ATCATCACAT




ATCATCACAT




ATCATCACAT

Mismatch in second position of string,Move string 1 position in text



ATCATCACAT

Mismatch in eight position of string,Move string 3 positions in text



ATCATCACAT

String found !

The KMP algorithm

The Next function

A T C A T C A C A T

1 ? ? ? ? ? ? ? ? ?Step:

x x x A x x x x x x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 ? ? ? ? ? ? ? ?Step:

x x x A T x x x x x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 ? ? ? ? ? ? ?Step:

x x x A T C x x x x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 ? ? ? ? ? ?Step:

x x x A T C A x x x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 ? ? ? ? ?Step:

x x x A T C A T x x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 5 ? ? ? ?Step:

x x x A T C A T C x x x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 5 7 ? ? ?Step:

x x x A T C A T C A x x x x x x x /

A T C A T C A

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 5 7 3 ? ?Step:

x x x A T C A T C A C x x x x x x /

A T C A T C A C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 5 7 3 9 ?Step:

x x x A T C A T C A C A x x x x x /

A T C A T

The KMP algorithm

The Next function

A T C A T C A C A T

1 1 2 4 4 5 7 3 9 9Step:

x x x A T C A T C A C A T x x x x /

A T C A T

The KMP algorithm

The Next functionA T C A T C A C A T

String:

1 1 2 4 4 5 7 3 9 9Step:

0 1 1 0 1 1 0 5 0 1Next:

i = 1 2 3 4 5 6 7 8 9 10

Next[i] = i – Step[i]

Computation of the Next table

i := 1; k := 0; Next[1] := 0; WHILE i < SLength DO WHILE (k > 0) AND (String[i]#String[k]) DO k := Next[k] END; (* WHILE *) k := k + 1; i := i + 1; IF String[i] = String[k] THEN Next[i] := Next[k] ELSE Next[i] := k END; (* IF *) END; (* WHILE *)

The KMP algorithm

Building the Next function

A T C A T C A C A T String0 Next


i : 1

k : 0

The KMP algorithm


A T C A T C A C A T String0 Next


i : 1

k : 0

The KMP algorithm


A T C A T C A C A T String0 1 Next


i : 1 > 2

k : 0 > 1

The KMP algorithm


A T C A T C A C A T String0 1 Next


i : 2

k : 1 > 0

The KMP algorithm


A T C A T C A C A T String0 1 1 Next


i : 2 > 3

k : 0 > 1

The KMP algorithm


A T C A T C A C A T String0 1 1 Next


i : 3

k : 1 > 0

The KMP algorithm


A T C A T C A C A T String0 1 1 0 Next


i : 3 > 4

k : 0 > 1

The KMP algorithm




i : 4

k : 1

The KMP algorithm




i : 4 > 5

k : 1 > 2

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 Next


i : 5

k : 2

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 Next


i : 5 > 6

k : 2 > 3

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 Next


i : 6

k : 3

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 Next


i : 6 > 7

k : 3 > 4

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 Next


i : 7

k : 4

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 5 Next


i : 7 > 8

k : 4 > 5

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 5 Next


i : 8

k : 5 > 1 > 0

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 5 0 Next


i : 8 > 9

k : 0 > 1

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 5 0 Next


i : 9

k : 1

The KMP algorithm


A T C A T C A C A T String0 1 1 0 1 1 0 5 0 1 Next


i : 9 > 10

k : 1 > 2

Actual KMP search

j := 1; i := 1;WHILE (i <= SLength) AND (j <= TLength) DO WHILE (i > 0) AND (Text[j] # String[i]) DO i := Next[i] END; (* WHILE *) i := i + 1; j := j + 1;END; (* WHILE *)IF i > SLength THEN RETURN j - SLength ELSE RETURN TLengthEND; (* IF *)

String Search Performance• Text length = n characters

• String length = m characters• The Algorithms:– Brute Force• Worst case search time : O(n*m)

– Knuth, Morris & Pratt (KMP - 1974)• Worst case search time : O(n+m)

– Boyer & Moore (BM - 1977)• Similar to but slightly better than KMP.

Chapter 2.8 Search Algorithms. Array Search –An array contains a certain number of records –Each record is identified by a certain key –One searches the.

Documents