Data Structure and Algorithm (CS 102)

Ashok K Turuk

Searching

Finding the location of an given item in a collection of item

Linear Search with Array

2 7 9 12

1 2 3 4

Algorithm[1] i = 1[2] If K = A[i] , Print “Search is

Successful” and Stop [3] i = i + 1[4] If (i <= n) then Go To Step [2][5] Else Print “Search is

Unsuccessful” and Stop [6] Exit

Complexity Of Linear Search Array

Case 1: Key matches with the first element

T(n) = Number of Comparison T(n) = 1, Best Case = O(1)

Case 2: Key does not existT(n) = n, Worst Case = O(n)

Case 3: Key is present at any locationT(n) = (n+1)/2, Average Case =

O(n)

Linear Search with Linked List

Head

A B C

Best Case Worst CaseAverage Case

Linear Search with Ordered List

5 7 9 10 13 56 76 82 110 112

1 2 3 4 5 6 7 8 9 10

Algorithm[1] i = 1[2] If K = A[i] , Print “Search is

Successful” and Stop [3] i = i + 1[4] If (i <= n) and (A[i] <= K) then Go

To Step [2][5] Else Print “Search is

Unsuccessful” and Stop [6] Exit

Complexity

Case 1: Key matches with the first element

T(n) = Number of Comparison T(n) = 1, Best Case = O(1)

Case 2: Key does not existT(n) = (n + 1)/2, Worst Case =

O(n)Case 3: Key is present at any location

T(n) = (n+1)/2, Average Case = O(n)

Binary Search

l u

mid = (l + u) /2

K = A[mid] then done

l u mid

mid

l u

If K < A[mid]

If K > A[mid]

u = mid -1

mid

l = mid +1

Algorithm[1] l =1, u =n[2] while (l < u) repeat steps 3 to 7[3] mid = (l + u) / 2[4] if K = A[mid] then print Successful

and Stop [5] if K < A[mid] then [6] u = mid -1[7] else l = mid + 1[8] Print Unsuccessful and Exit

Example

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95

K = 75l = 1u =8

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95l = 1u =8mid = 4K = 75 > A[4]

l u

l = 4 + 1

u

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95

l = 4 + 1

u

l = 5u =8mid = 6K = 75 = A[6]

Example

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95

K = 55l = 1u =8

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95l = 1u =8mid = 4K = 55 > A[4]

l u

l = 4 + 1

u

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95

l = 4 + 1u= 6-1

l = 5u =8mid = 6K = 55 < A[6]

1 2 3 4 5 6 7 8

15 25 35 45 65 75 85 95

l = 5u =8mid = 5K = 55 < A[5]

l = 5u= 5-1

Fibonacci Search

Fn = Fn – 1 + Fn – 2 with F0 = 0 and F1 = 1

If we expand the nth Fibonacci number into the form of a recurrence tree, its result into a binary tree and we can term this as Fibonacci tree of order n.

In a Fibonacci tree, a node correspond to Fi the ith Fibonacci number

Fi-1 is the left subtree

Fi-2 is the right subtree

Fi-1 and Fi-2 are further expanded until F0 and F1.

Fibonacci Tree

c

c

c

c

c

c

cc

c

c

cc c

c c

cc

c

cc

c

c

c

c

F6

F5 F4

F4 F3

F3

F2F2 F1

F1 F0

F2

F1

F1 F0

F1

F0

F3

F2

F0

F1F2F2

F1 F0

A Fibonacci Tree of Order 6

Apply the following two rules to obtain the Fibonacci search tree of order n from a Fibonacci tree of order n

Rule 1: For all leaf nodes corresponding to F1 and F0, we denote them as external nodes and redraw them as squares, and the value of each external node is set to zero. Value of each node is replaced by its corresponding Fibonacci number.

Rule 2: For all nodes corresponding to Fi (i>=2), each of them as a Fibonacci search tree of order i such that

(a) the root is Fi

(b) the left-subtree is a Fibonacci search tree of order i -1

(c) the right-subtree is a Fibonacci search tree of order i -2 with all numbers increased by Fi

Fibonacci Tree

c

c

c

c

c

cc

c

c

c c

F6

F5 F4

F4 F3

F3

F2F2

F2

F3

F2

F2

Rule 1 applied to Fibonacci Tree of Order 6

0 0

0

0 0

0 0

0

0 0

00

0

Fibonacci Tree

c

c

c

c

c

cc

c

c

c c

8

5 3

3 2

2 1

1

12

1

1

Rule 1 applied to Fibonacci Tree of Order 6

0 0

0

0 0

0 0

0

0 0

00

0

Fibonacci Tree

c

c

c

c

c

cc

c

c

c c

8

5 11

3 7

2 4

1

610

12

9

Rule 2 applied to Fibonacci Tree of Order 6

0 1

2

3 4

5 6

7

8 9

11

10

12

Algorithm

Elements are in sorted order. Number of elements n is related to a perfect Fibonacci number Fk+1 such that Fk+1 = n+1

[1] i = Fk

[2] p = Fk-1 , q = Fk-2

[3] If ( K < Ki ) then

[4] If (q==0) then Print “unsuccessful and exit”

[5] Else i = i – q, pold = p, p =q, q =pold – q

[6] Goto step 3

[7] If (K > Ki ) then

[8] If (p ==1) then Print “Unsuccessful and Exit”

[9] Else i = i + q, p = p+q, q=q-p[10] Goto step 3[11] If ( k == Ki ) then Print “Successful

at ith location”[12] Stop

Example 1 2 3 4 5 6 7 8 9 10 11 12

15 20 25 30 35 40 45 50 65 75 85 95

Initialization: i = Fk = 8, p = Fk-1 = 5, q = Fk-2 = 3Iteration 1;K8 = A[8] = 50, K < K8 , q == 3 i = i –q = 8 -3 = 5, pold = p =5p=q= 3, q = pold – q = 5-3 = 2

Iteration 2;K5 = A[5] = 35, K < K5 , q == 2 i = i –q = 5 -2 = 3, pold = p =3p=q= 2, q = pold – q = 3-2 = 1

K = 25

Iteration 2;K3 = A[3] = 25, K = K3

Search is successful

1 2 3 4 5 6 7 8 9 10 11 12

15 20 25 30 35 40 45 50 65 75 85 95