Prerequisite
Problem Statement
Given a sorted array of n elements, write a function to search for the index of a given element (target)
Approach
- Search for the range within which the target is included increasing index by powers of 2
- If this range exists in array apply the Binary Search algorithm over it
- Else return -1
Example
arr = [1, 2, 3, 4, 5, 6, 7, ... 998, 999, 1_000]
target = 998
index = 0
1. SEARCHING FOR THE RANGE
index = 1, 2, 4, 8, 16, 32, 64, ..., 512, ..., 1_024
after 10 iteration we have the index at 1_024 and outside of the array
2. BINARY SEARCH
Now we can apply the binary search on the subarray from 512 and 1_000.
Note : Note: we apply the Binary Search from 512
to 1_000
because at i = 2^10 = 1_024
the array is finished and the target number is less than the latest index of the array ( 1_000
).
Time Complexity
worst case : O(log i)
where i = index
(position) of the target
best case : O(1)
Complexity Explanation
- The complexity of the first part of the algorithm is
O( log I )
because if i is the position of the target in the array, after doubling the search index⌈log(i)⌉
times, the algorithm will be at a search index that is greater than or equal to i. We can write2^⌈log(i)⌉ >= I
- The complexity of the second part of the algorithm also is
O ( log I )
because that is a simple Binary Search. The Binary Search complexity ( as explained here ) isO( n )
where n is the length of the array. In the Exponential Search, the length of the array on which the algorithm is applied is2^i - 2^(i-1)
, put into words it means '( the length of the array from start to i ) - ( the part of array skipped until the previous iteration )'. Is simple verify that2^i - 2^(i-1) = 2^(i-1)
After this detailed explanation we can say that the the complexity of the Exponential Search is:
O(log i) + O(log i) = 2O(log i) = O(log i)
Binary Search vs Exponential Search
Let's take a look at this comparison with a less theoretical example. Imagine we have an array with 1_000_000
elements and we want to search an element that is in the 4th
position. It's easy to see that:
- The Binary Search start from the middle of the array and arrive to the
4th
position after many iterations - The Exponential Search arrive at the
4th
index after only 2 iterations
Code Implementation Links
My implementation : C
Others : C++, JavaScript