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Chapter 2.8

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 record with a given key String Search A text is represented by an array of characters

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Chapter 2.8

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  1. Chapter 2.8 Search Algorithms

  2. 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

  3. 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

  4. 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

  5. TYPE ArrayOfItems = 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; Array Search

  6. Search ( PROCEDURE VAR A: ARRAYOF Item, Key: KeyType) : CARDINAL; VAR i : CARDINAL; BEGIN i := HIGH(A); DEC(i) WHILE A[i].Key # Key AND i # 0 DO END; RETURN i END Search; Straight Search

  7. Key: KeyType) : CARDINAL; VAR i : CARDINAL; WHILE A[i].Key # Key AND i # 0 DO RETURN i END Search; Straight Search Search ( PROCEDURE VAR A: ARRAYOF Item, BEGIN i := HIGH(A); DEC(i) END;

  8. 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; Sentinel Search

  9. Binary Search # elements > 1 Yes No Key < Keymiddle Key = Keyelement Yes Yes No No Binary Search right half Binary Search left half Not Found Found # elements > 1 Binary Search

  10. 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 END END Search;

  11. 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; END END Src;

  12. Iterative Binary Search PROCEDURE 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 THENRETURN m ELSERETURN 0 END END Search;

  13. Unordered Array Straight search : 2n Sentinel search : n Ordered Array Binary search : log 2 n Array Search Performance(Number of comparisons)

  14. 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

  15. 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

  16. WHILE current char.in Text # String[1] Move to next character in Text End of Text reached ? No WHILE char. in Text = char. in String Move to next character pair String matched OR Text exhausted String Searchby Brute Force this algorithm tries to find a string string string string

  17. 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 - 1 END Search;

  18. String Search (1)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in first position of string, Move string 1 position in text

  19. String Search(2)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in fourth position of string, Move string 4 positions in text

  20. String Search(3)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in fifth position of string, Move string 4 positions in text

  21. String Search(4)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in first position of string, Move string 1 position in text

  22. String Search(5)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in first position of string, Move string 1 position in text

  23. String Search(6)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in second position of string, Move string 1 position in text

  24. String Search(7)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT Mismatch in eight position of string, Move string 3 positions in text

  25. String Search(8)by the KMP algorithm GATCGATCAGCAATCATCATCACATC ATCATCACAT String found !

  26. Step: 1 ? ? ? ? ? ? ? ? ? The KMP algorithmThe Next function 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 A T C A T C A C A T

  27. Step: 1 1 ? ? ? ? ? ? ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  28. Step: 1 1 2 ? ? ? ? ? ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  29. Step: 1 1 2 4 ? ? ? ? ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  30. Step: 1 1 2 4 4 ? ? ? ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  31. Step: 1 1 2 4 4 5 ? ? ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  32. Step: 1 1 2 4 4 5 7 ? ? ? The KMP algorithmThe Next function / x x x A T C A T C A x x x x x x x A T C A T C A C A T A T C A T C A

  33. Step: 1 1 2 4 4 5 7 3 ? ? The KMP algorithmThe Next function / 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 A T C A T C A C A T

  34. Step: 1 1 2 4 4 5 7 3 9 ? The KMP algorithmThe Next function / x x x A T C A T C A C A x x x x x A T C A T C A C A T A T C A T

  35. Step: 1 1 2 4 4 5 7 3 9 9 The KMP algorithmThe Next function / x x x A T C A T C A C A T x x x x A T C A T C A C A T A T C A T

  36. String: A T C A T C A C A T Step: 1 1 2 4 4 5 7 3 9 9 Next: 0 1 1 0 1 1 0 5 0 1 The KMP algorithmThe Next function i = 1 2 3 4 5 6 7 8 9 10 Next[i] = i – Step[i]

  37. 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 *)

  38. The KMP algorithmBuilding the Next function 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 *) i : 1 String A T C A T C A C A T Next 0 k : 0

  39. The KMP algorithmBuilding the Next function 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 *) i : 1 String A T C A T C A C A T Next 0 k : 0

  40. The KMP algorithmBuilding the Next function 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 *) i : 1 > 2 String A T C A T C A C A T Next 0 1 k : 0 > 1

  41. The KMP algorithmBuilding the Next function 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 *) i : 2 String A T C A T C A C A T Next 0 1 k : 1 > 0

  42. The KMP algorithmBuilding the Next function 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 *) i : 2 > 3 String A T C A T C A C A T Next 0 1 1 k : 0 > 1

  43. The KMP algorithmBuilding the Next function 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 *) i : 3 String A T C A T C A C A T Next 0 1 1 k : 1 > 0

  44. The KMP algorithmBuilding the Next function 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 *) i : 3 > 4 String A T C A T C A C A T Next 0 1 1 0 k : 0 > 1

  45. The KMP algorithmBuilding the Next function 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 *) i : 4 String A T C A T C A C A T Next 0 1 1 0 k : 1

  46. The KMP algorithmBuilding the Next function 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 *) i : 4 > 5 String A T C A T C A C A T Next 0 1 1 0 k : 1 > 2

  47. The KMP algorithmBuilding the Next function 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 *) i : 5 String A T C A T C A C A T Next 0 1 1 0 1 k : 2

  48. The KMP algorithmBuilding the Next function 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 *) i : 5 > 6 String A T C A T C A C A T Next 0 1 1 0 1 1 k : 2 > 3

  49. The KMP algorithmBuilding the Next function 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 *) i : 6 String A T C A T C A C A T Next 0 1 1 0 1 1 k : 3

  50. The KMP algorithmBuilding the Next function 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 *) i : 6 > 7 String A T C A T C A C A T Next 0 1 1 0 1 1 0 k : 3 > 4

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