# Searching

Question 1 |

What is the output of following program?

#include <stdio.h> void print(int n, int j) { if (j >= n) return; if (n-j > 0 && n-j >= j) printf("%d %dn", j, n-j); print(n, j+1); } int main() { int n = 8; print(n, 1); }

1 7 2 6 3 5 4 4 4 4 | |

1 7 2 6 3 5 4 4 | |

1 7 2 6 3 5 | |

1 2 3 4 5 6 7 8 |

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Question 1 Explanation:

For a given number n, the program prints all distinct pairs of positive integers with sum equal to n.

Question 2 |

Which of the following is correct recurrence for worst case of Binary Search?

T(n) = 2T(n/2) + O(1) and T(1) = T(0) = O(1) | |

T(n) = T(n-1) + O(1) and T(1) = T(0) = O(1) | |

T(n) = T(n/2) + O(1) and T(1) = T(0) = O(1) | |

T(n) = T(n-2) + O(1) and T(1) = T(0) = O(1) |

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Question 2 Explanation:

Following is typical implementation of Binary Search.
In Binary Search, we first compare the given element x with middle of the array. If x matches with middle element, then we return middle index. Otherwise, we either recur for left half of array or right half of array.
So recurrence is T(n) = T(n/2) + O(1)

Question 3 |

Given a sorted array of integers, what can be the minimum worst case time complexity to find ceiling of a number x in given array? Ceiling of an element x is the smallest element present in array which is greater than or equal to x. Ceiling is not present if x is greater than the maximum element present in array. For example, if the given array is {12, 67, 90, 100, 300, 399} and x = 95, then output should be 100.

O(LogLogn) | |

O(n) | |

O(Logn) | |

O(Logn * Logn) |

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Question 3 Explanation:

We modify standard binary search to find ceiling. The time complexity T(n) can be written as
T(n) <= T(n/2) + O(1)
Solution of above recurrence can be obtained by Master Method. It falls in case 2 of Master Method. Solution is O(Logn).
[sourcecode language="C"]
#include
/* Function to get index of ceiling of x in arr[low..high]*/
int ceilSearch(int arr[], int low, int high, int x)
{
int mid;
/* If x is smaller than or equal to the first element,
then return the first element */
if(x <= arr[low])
return low;
/* If x is greater than the last element, then return -1 */
if(x > arr[high])
return -1;
/* get the index of middle element of arr[low..high]*/
mid = (low + high)/2; /* low + (high - low)/2 */
/* If x is same as middle element, then return mid */
if(arr[mid] == x)
return mid;
/* If x is greater than arr[mid], then either arr[mid + 1]
is ceiling of x or ceiling lies in arr[mid+1...high] */
else if(arr[mid] < x)
{
if(mid + 1 <= high && x <= arr[mid+1])
return mid + 1;
else
return ceilSearch(arr, mid+1, high, x);
}
/* If x is smaller than arr[mid], then either arr[mid]
is ceiling of x or ceiling lies in arr[mid-1...high] */
else
{
if(mid - 1 >= low && x > arr[mid-1])
return mid;
else
return ceilSearch(arr, low, mid - 1, x);
}
}
/* Driver program to check above functions */
int main()
{
int arr[] = {1, 2, 8, 10, 10, 12, 19};
int n = sizeof(arr)/sizeof(arr[0]);
int x = 20;
int index = ceilSearch(arr, 0, n-1, x);
if(index == -1)
printf("Ceiling of %d doesn't exist in array ", x);
else
printf("ceiling of %d is %d", x, arr[index]);
getchar();
return 0;
}
[/sourcecode]

Question 4 |

Consider the following C program that attempts to locate an element x in an array Y[] using binary search. The program is erroneous. (GATE CS 2008)

1. f(int Y[10], int x) { 2. int i, j, k; 3. i = 0; j = 9; 4. do { 5. k = (i + j) /2; 6. if( Y[k] < x) i = k; else j = k; 7. } while(Y[k] != x && i < j); 8. if(Y[k] == x) printf ("x is in the array ") ; 9. else printf (" x is not in the array ") ; 10. }On which of the following contents of Y and x does the program fail?

Y is [1 2 3 4 5 6 7 8 9 10] and x < 10 | |

Y is [1 3 5 7 9 11 13 15 17 19] and x < 1 | |

Y is [2 2 2 2 2 2 2 2 2 2] and x > 2 | |

Y is [2 4 6 8 10 12 14 16 18 20] and 2 < x < 20 and x is even |

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Question 4 Explanation:

The above program doesn’t work for the cases where element to be searched is the last element of Y[] or greater than the last element (or maximum element) in Y[]. For such cases, program goes in an infinite loop because i is assigned value as k in all iterations, and i never becomes equal to or greater than j. So while condition never becomes false.

Question 5 |

In the above question, the correction needed in the program to make it work properly is (GATE CS 2008)

Change line 6 to: if (Y[k] < x) i = k + 1; else j = k-1; | |

Change line 6 to: if (Y[k] < x) i = k - 1; else j = k+1; | |

Change line 6 to: if (Y[k] <= x) i = k; else j = k; | |

Change line 7 to: } while ((Y[k] == x) && (i < j)); |

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Question 5 Explanation:

Below is the corrected function
f(int Y[10], int x) {
int i, j, k;
i = 0; j = 9;
do {
k = (i + j) /2;
if( Y[k] < x) i = k + 1; else j = k - 1;
} while(Y[k] != x && i < j);
if(Y[k] == x) printf ("x is in the array ") ;
else printf (" x is not in the array ") ;
}[/sourcecode]
Reference: http://en.wikipedia.org/wiki/Binary_search_algorithm#Implementations

Question 6 |

You are given a list of 5 integers and these integers are in the range from 1 to 6. There are no duplicates in list. One of the integers is missing in the list. Which of the following expression would give the missing number.

**^**is bitwise XOR operator.**~**is bitwise NOT operator. Let elements of list can be accessed as list[0], list[1], list[2], list[3], list[4]list[0] ^ list[1] ^ list[2] ^ list[3] ^ list[4] | |

list[0] ^ list[1] ^ list[2] ^ list[3] ^ list[4] ^ 1 ^ 2 ^ 3 ^ 4 ^ 5 ^ 6 | |

list[0] ^ list[1] ^ list[2] ^ list[3] ^ list[4] ^ 1 ^ 2 ^ 3 ^ 4 ^ 5 | |

~(list[0] ^ list[1] ^ list[2] ^ list[3] ^ list[4]) |

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Question 6 Explanation:

XOR of all list elements and numbers from 1 to 6 gives the missing number. See this for details

Question 7 |

Consider the C function given below. Assume that the array listA contains n (> 0) elements, sorted in ascending order.

int ProcessArray(int *listA, int x, int n) { int i, j, k; i = 0; j = n-1; do { k = (i+j)/2; if (x <= listA[k]) j = k-1; if (listA[k] <= x) i = k+1; } while (i <= j); if (listA[k] == x) return(k); else return -1; }Which one of the following statements about the function ProcessArray is CORRECT?

It will run into an infinite loop when x is not in listA. | |

It is an implementation of binary search. | |

It will always find the maximum element in listA. | |

It will return −1 even when x is present in listA. |

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**GATE-CS-2014-(Set-3)**

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Question 7 Explanation:

The function is iterative implementation of Binary Search. k keeps track of current middle element. i and j keep track of left and right corners of current subarray

Question 8 |

Consider a sorted array of n numbers. What would be the time complexity of the best known algorithm to find a pair 'a' and 'b' such that |a-b| = k , k being a positive integer.

O(n) | |

O(n log n) | |

O(n ^ 2) | |

O(log n) |

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Question 8 Explanation:

Just maintain two pointers at the start and accordingly increment one of them depending upon whether difference is less than or greater than k. Just a single pass is required so the answer is O(n).

Question 9 |

Consider a sorted array of n numbers. What would be the time complexity of the best known algorithm to find a pair 'a' and 'b' such that |a-b| = k , k being a positive integer.

O(n) | |

O(n log n) | |

O(n ^ 2) | |

O(log n) |

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Question 9 Explanation:

Just maintain two pointers at the start and accordingly increment one of them depending upon whether difference is less than or greater than k. Just a single pass is required so the answer is O(n).

Question 10 |

The average number of key comparisons done in a successful sequential search in a list of length it is

log n | |

(n-1)/2 | |

n/2 | |

(n+1)/2 |

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**GATE CS 1996**

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Question 10 Explanation:

If element is at 1 position then it requires 1 comparison.
If element is at 2 position then it requires 2 comparison.
If element is at 3 position then it requires 3 comparison.
Similarly , If element is at n position then it requires n comparison.

Total comparison = n(n+1)/2 For average comparison = (n(n+1)/2) / n = (n+1)/2Option (D) is correct.

There are 21 questions to complete.