What is a Jump (or Block) Search?

Jump Search, also known as Block Search, is a search algorithm used for finding a specific element in a sorted array. The primary concept behind Jump Search is to reduce the number of comparisons made during the search process, which is achieved by jumping ahead by a fixed number of elements instead of examining each element sequentially as in a linear search.

How Jump Search Works

1. Jumping Ahead: Instead of searching through each element one by one, Jump Search moves ahead in fixed increments or 'jumps'. Suppose the array has $$ n $$ elements, Jump Search will jump ahead by $$ \sqrt{n} $$ elements at a time, until it finds an element that is larger than the search element or it reaches the end of the array[1][2][3][4][5][6].

2. Linear Search: Once the block where the element could potentially exist is identified (i.e., the element following the jump is greater than the search element), a linear search is performed backwards from this point to find the element[1][2][3][4][5][6].

Optimal Block Size

The optimal size of the jump or block is $$ \sqrt{n} $$, where $$ n $$ is the total number of elements in the array. This block size minimizes the number of operations in the worst case[1][2][3][4][5][6].

Complexity

• Time Complexity: The worst-case time complexity of Jump Search is $$ O(\sqrt{n}) $$, which is more efficient than Linear Search ($$ O(n) $$) but less efficient than Binary Search ($$ O(\log n) $$) for large datasets[1][2][3][4][5][6].
• Space Complexity: The space complexity of Jump Search is $$ O(1) $$, as it requires no additional storage beyond the original array[5][6].

Implementation

Here is a basic implementation of Jump Search in Python:

import math

def jump_search(arr, x):
    n = len(arr)
    step = math.sqrt(n)
    
    # Finding the block where the element is present
   ...