Find an element in Bitonic array
Given a bitonic sequence of n distinct elements, and an integer x, the task is to write a program to find given element x in the bitonic sequence in O(log n) time.
A Bitonic Sequence is a sequence of numbers that is first strictly increasing then after a point decreasing.
Examples:
Input : arr[] = {-3, 9, 18, 20, 17, 5, 1}, key = 20
Output : Found at index 3Input : arr[] = {5, 6, 7, 8, 9, 10, 3, 2, 1}, key = 30
Output : Not Found
Naive Approach: A simple solution is to do a linear search. The time complexity of this solution would be O(n).
Efficient Approach: An efficient solution is based on Binary Search.
- The idea is to find the bitonic point ‘k’ which is the index of the maximum element of a given sequence.
- If the element to be searched is greater than the maximum element return -1,
- else search the element in both halves.
Below is the step by step algorithm on how to do this.
- Find the bitonic point in the given array, i.e the maximum element in the given bitonic array. This can be done in log(n) time by modifying the binary search algorithm. You can refer to this post on how to do this.
- If the element to be searched is equal to the element at the bitonic point then print the index of the bitonic point.
- If the element to be searched is greater than the element at a bitonic point then the element does not exist in the array.
- If the element to be searched is less than the element at a bitonic point then search for the element in both halves of the array using binary search.
Below is the implementation of the above idea:
C++
// C++ code to search key in bitonic array #include <iostream> using namespace std; // Function for binary search in ascending part int ascendingBinarySearch( int arr[], int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2; if (arr[mid] == key) return mid; if (arr[mid] > key) high = mid - 1; else low = mid + 1; } return -1; } // Function for binary search in // descending part of array int descendingBinarySearch( int arr[], int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2; if (arr[mid] == key) return mid; if (arr[mid] < key) high = mid - 1; else low = mid + 1; } return -1; } // finding bitonic point int findBitonicPoint( int arr[], int n, int l, int r) { int mid; int bitonicPoint = 0; mid = (r + l) / 2; if (arr[mid] > arr[mid - 1] && arr[mid] > arr[mid + 1]) { return mid; } else if (arr[mid] > arr[mid - 1] && arr[mid] < arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, mid, r); } else if (arr[mid] < arr[mid - 1] && arr[mid] > arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, l, mid); } return bitonicPoint; } // Function to search key in // bitonic array int searchBitonic( int arr[], int n, int key, int index) { if (key > arr[index]) return -1; else if (key == arr[index]) return index; else { int temp = ascendingBinarySearch(arr, 0, index - 1, key); if (temp != -1) { return temp; } // Search in right of k return descendingBinarySearch(arr, index + 1, n - 1, key); } } // Driver code int main() { int arr[] = { -8, 1, 2, 3, 4, 5, -2, -3 }; int key = 1; int n, l, r; n = sizeof (arr) / sizeof (arr[0]); l = 0; r = n - 1; int index; // Function call index = findBitonicPoint(arr, n, l, r); int x = searchBitonic(arr, n, key, index); if (x == -1) cout << "Element Not Found" << endl; else cout << "Element Found at index " << x << endl; return 0; } |
Java
// Java code to search key in bitonic array public class GFG { // Function for binary search // in ascending part static int ascendingBinarySearch( int arr[], int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2 ; if (arr[mid] == key) { return mid; } if (arr[mid] > key) { high = mid - 1 ; } else { low = mid + 1 ; } } return - 1 ; } // Function for binary search in // descending part of array static int descendingBinarySearch( int arr[], int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2 ; if (arr[mid] == key) { return mid; } if (arr[mid] < key) { high = mid - 1 ; } else { low = mid + 1 ; } } return - 1 ; } // finding bitonic point static int findBitonicPoint( int arr[], int n, int l, int r) { int mid; int bitonicPoint = 0 ; mid = (r + l) / 2 ; if (arr[mid] > arr[mid - 1 ] && arr[mid] > arr[mid + 1 ]) { return mid; } else { if (arr[mid] > arr[mid - 1 ] && arr[mid] < arr[mid + 1 ]) { bitonicPoint = findBitonicPoint(arr, n, mid, r); } else { if (arr[mid] < arr[mid - 1 ] && arr[mid] > arr[mid + 1 ]) { bitonicPoint = findBitonicPoint(arr, n, l, mid); } } } return bitonicPoint; } // Function to search key in bitonic array static int searchBitonic( int arr[], int n, int key, int index) { if (key > arr[index]) { return - 1 ; } else if (key == arr[index]) { return index; } else { int temp = ascendingBinarySearch( arr, 0 , index - 1 , key); if (temp != - 1 ) { return temp; } // Search in right of k return descendingBinarySearch(arr, index + 1 , n - 1 , key); } } // Driver code public static void main(String args[]) { int arr[] = { - 8 , 1 , 2 , 3 , 4 , 5 , - 2 , - 3 }; int key = 5 ; int n, l, r; n = arr.length; l = 0 ; r = n - 1 ; int index; index = findBitonicPoint(arr, n, l, r); int x = searchBitonic(arr, n, key, index); if (x == - 1 ) { System.out.println( "Element Not Found" ); } else { System.out.println( "Element Found at index " + x); } } } /*This code is contributed by 29AjayKumar*/ |
Python3
# Python code to search key in bitonic array # Function for binary search in ascending part def ascendingBinarySearch(arr, low, high, key): while low < = high: mid = low + (high - low) / / 2 if arr[mid] = = key: return mid if arr[mid] > key: high = mid - 1 else : low = mid + 1 return - 1 # Function for binary search in descending part of array def descendingBinarySearch(arr, low, high, key): while low < = high: mid = low + (high - low) / / 2 if arr[mid] = = key: return mid if arr[mid] < key: high = mid - 1 else : low = mid + 1 return - 1 # Find bitonic point def findBitonicPoint(arr, n, l, r): bitonicPoint = 0 mid = (r + l) / / 2 if arr[mid] > arr[mid - 1 ] and arr[mid] > arr[mid + 1 ]: return mid elif arr[mid] > arr[mid - 1 ] and arr[mid] < arr[mid + 1 ]: bitonicPoint = findBitonicPoint(arr, n, mid, r) else : bitonicPoint = finsBitonicPoint(arr, n, l, mid) return bitonicPoint # Function to search key in bitonic array def searchBitonic(arr, n, key, index): if key > arr[index]: return - 1 elif key = = arr[index]: return index else : temp = ascendingBinarySearch(arr, 0 , index - 1 , key) if temp ! = - 1 : return temp # search in right of k return descendingBinarySearch(arr, index + 1 , n - 1 , key) # Driver code def main(): arr = [ - 8 , 1 , 2 , 3 , 4 , 5 , - 2 , - 3 ] key = 1 n = len (arr) l = 0 r = n - 1 # Function call index = findBitonicPoint(arr, n, l, r) x = searchBitonic(arr, n, key, index) if x = = - 1 : print ( "Element Not Found" ) else : print ( "Element Found at index" , x) main() # This code is contributed by stutipathak31jan |
C#
// C# code to search key in bitonic array using System; class GFG { // Function for binary search in ascending part static int ascendingBinarySearch( int [] arr, int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2; if (arr[mid] == key) { return mid; } if (arr[mid] > key) { high = mid - 1; } else { low = mid + 1; } } return -1; } // Function for binary search in descending part of // array static int descendingBinarySearch( int [] arr, int low, int high, int key) { while (low <= high) { int mid = low + (high - low) / 2; if (arr[mid] == key) { return mid; } if (arr[mid] < key) { high = mid - 1; } else { low = mid + 1; } } return -1; } // finding bitonic point static int findBitonicPoint( int [] arr, int n, int l, int r) { int mid; int bitonicPoint = 0; mid = (r + l) / 2; if (arr[mid] > arr[mid - 1] && arr[mid] > arr[mid + 1]) { return mid; } else { if (arr[mid] > arr[mid - 1] && arr[mid] < arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, mid, r); } else { if (arr[mid] < arr[mid - 1] && arr[mid] > arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, l, mid); } } } return bitonicPoint; } // Function to search key in bitonic array static int searchBitonic( int [] arr, int n, int key, int index) { if (key > arr[index]) { return -1; } else if (key == arr[index]) { return index; } else { int temp = ascendingBinarySearch( arr, 0, index - 1, key); if (temp != -1) { return temp; } // Search in right of k return descendingBinarySearch(arr, index + 1, n - 1, key); } } // Driver Code static public void Main() { int [] arr = { -8, 1, 2, 3, 4, 5, -2, -3 }; int key = 1; int n, l, r; n = arr.Length; l = 0; r = n - 1; int index; index = findBitonicPoint(arr, n, l, r); int x = searchBitonic(arr, n, key, index); if (x == -1) { Console.WriteLine( "Element Not Found" ); } else { Console.WriteLine( "Element Found at index " + x); } } } // This code is contributed by ajit |
Javascript
<script> // JavaScript code to search key in bitonic array // Function for binary search in ascending part function ascendingBinarySearch(arr, low, high, key) { while (low <= high) { let mid = Math.floor(low + (high - low) / 2); if (arr[mid] == key) return mid; if (arr[mid] > key) high = mid - 1; else low = mid + 1; } return -1; } // Function for binary search in // descending part of array function descendingBinarySearch(arr, low, high, key) { while (low <= high) { let mid = Math.floor(low + (high - low) / 2); if (arr[mid] == key) return mid; if (arr[mid] < key) high = mid - 1; else low = mid + 1; } return -1; } // finding bitonic point function findBitonicPoint(arr, n, l, r) { let mid; let bitonicPoint = 0; mid = Math.floor((r + l) / 2); if (arr[mid] > arr[mid - 1] && arr[mid] > arr[mid + 1]) { return mid; } else if (arr[mid] > arr[mid - 1] && arr[mid] < arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, mid, r); } else if (arr[mid] < arr[mid - 1] && arr[mid] > arr[mid + 1]) { bitonicPoint = findBitonicPoint(arr, n, l, mid); } return bitonicPoint; } // Function to search key in // bitonic array function searchBitonic(arr, n, key, index) { if (key > arr[index]) return -1; else if (key == arr[index]) return index; else { let temp = ascendingBinarySearch(arr, 0, index - 1, key); if (temp != -1) { return temp; } // Search in right of k return descendingBinarySearch(arr, index + 1, n - 1, key); } } // Driver code let arr = [-8, 1, 2, 3, 4, 5, -2, -3]; let key = 1; let n, l, r; n = arr.length; l = 0; r = n - 1; let index; // Function call index = findBitonicPoint(arr, n, l, r); let x = searchBitonic(arr, n, key, index); if (x == -1) document.write( "Element Not Found" + "<br>" ); else document.write( "Element Found at index " + x + "<br>" ); </script> |
Element Found at index 1
Time complexity: O(log n)
Auxiliary Space: O(1)
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