Meta Binary Search | One-Sided Binary Search

Meta binary search (also called one-sided binary search by Steven Skiena in The Algorithm Design Manual on page 134) is a modified form of binary search that incrementally constructs the index of the target value in the array. Like normal binary search, meta binary search takes O(log n) time.

Meta Binary Search, also known as One-Sided Binary Search, is a variation of the binary search algorithm that is used to search an ordered list or array of elements. This algorithm is designed to reduce the number of comparisons needed to search the list for a given element.

The basic idea behind Meta Binary Search is to start with an initial interval of size n that includes the entire array. The algorithm then computes a middle element, as in binary search, and compares it to the target element. If the target element is found, the search terminates. If the middle element is greater than the target element, the algorithm sets the new interval to the left half of the previous interval, and if the middle element is less than the target element, the new interval is set to the right half of the previous interval. However, unlike binary search, Meta Binary Search does not perform a comparison for each iteration of the loop.

Instead, the algorithm uses a heuristic to determine the size of the next interval. It computes the difference between the value of the middle element and the value of the target element, and divides the difference by a predetermined constant, usually 2. This result is then used as the size of the new interval. The algorithm continues until it finds the target element or determines that it is not in the list.

The advantage of Meta Binary Search over binary search is that it can perform fewer comparisons in some cases, particularly when the target element is close to the beginning of the list. The disadvantage is that the algorithm may perform more comparisons than binary search in other cases, particularly when the target element is close to the end of the list. Therefore, Meta Binary Search is most effective when the list is ordered in a way that is consistent with the distribution of the target elements.

Here is the pseudocode for Meta Binary Search:

function meta_binary_search(A, target):
    n = length(A)
    interval_size = n
    while interval_size > 0:
        index = min(n - 1, interval_size / 2)
        mid = A[index]
        if mid == target:
            return index
        elif mid < target:
            interval_size = (n - index) / 2
        else:
            interval_size = index / 2
    return -1

Examples:  

Input: [-10, -5, 4, 6, 8, 10, 11], key_to_search = 10
Output: 5

Input: [-2, 10, 100, 250, 32315], key_to_search = -2
Output: 0

The exact implementation varies, but the basic algorithm has two parts:  

1. Figure out how many bits are necessary to store the largest array index.
2. Incrementally construct the index of the target value in the array by determining whether each bit in the index should be set to 1 or 0.

Approach:

1. Store number of bits to represent the largest array index in variable lg.
2. Use lg to start off the search in a for loop.
3. If the element is found return pos.
4. Otherwise, incrementally construct an index to reach the target value in the for loop.
5. If element found return pos otherwise -1.

Below is the implementation of the above approach: 
 

// C++ implementation of above approach

#include <iostream>
#include <cmath>
#include <vector>
using namespace std;

// Function to show the working of Meta binary search
int bsearch(vector<int> A, int key_to_search)
{
    int n = (int)A.size();
    // Set number of bits to represent largest array index
    int lg = log2(n-1)+1; 

    //while ((1 << lg) < n - 1)
        //lg += 1;

    int pos = 0;
    for (int i = lg ; i >= 0; i--) {
        if (A[pos] == key_to_search)
            return pos;

        // Incrementally construct the
        // index of the target value
        int new_pos = pos | (1 << i);

        // find the element in one
        // direction and update position
        if ((new_pos < n) && (A[new_pos] <= key_to_search))
            pos = new_pos;
    }

    // if element found return pos otherwise -1
    return ((A[pos] == key_to_search) ? pos : -1);
}

// Driver code
int main(void)
{

    vector<int> A = { -2, 10, 100, 250, 32315 };
    cout << bsearch(A, 10) << endl;

    return 0;
}

// This implementation was improved by Tanin

//Java implementation of above approach 
import java.util.Vector;
import com.google.common.math.BigIntegerMath;
import java.math.*;

class GFG {

// Function to show the working of Meta binary search 
    static int bsearch(Vector<Integer> A, int key_to_search) {
        int n = (int) A.size();
        // Set number of bits to represent largest array index
        int lg = BigIntegerMath.log2(BigInteger.valueOf(n-1),RoundingMode.UNNECESSARY) + 1;
 
        //while ((1 << lg) < n - 1) {
        //    lg += 1;
        //}

        int pos = 0;
        for (int i = lg - 1; i >= 0; i--) {
            if (A.get(pos) == key_to_search) {
                return pos;
            }

            // Incrementally construct the 
            // index of the target value 
            int new_pos = pos | (1 << i);

            // find the element in one 
            // direction and update position 
            if ((new_pos < n) && (A.get(new_pos) <= key_to_search)) {
                pos = new_pos;
            }
        }

        // if element found return pos otherwise -1 
        return ((A.get(pos) == key_to_search) ? pos : -1);
    }

// Driver code 
    static public void main(String[] args) {
        Vector<Integer> A = new Vector<Integer>();
        int[] arr = {-2, 10, 100, 250, 32315};
        for (int i = 0; i < arr.length; i++) {
            A.add(arr[i]);
        }
        System.out.println(bsearch(A, 10));
    }
}

// This code is contributed by 29AjayKumar
// This implementation was improved by Tanin

# Python 3 implementation of 
# above approach

# Function to show the working
# of Meta binary search
import math
def bsearch(A, key_to_search):

    n = len(A)
    # Set number of bits to represent
    lg = int(math.log2(n-1)) + 1;

    # largest array index
    #while ((1 << lg) < n - 1):
        #lg += 1

    pos = 0
    for i in range(lg - 1, -1, -1) :
        if (A[pos] == key_to_search):
            return pos

        # Incrementally construct the
        # index of the target value
        new_pos = pos | (1 << i)

        # find the element in one
        # direction and update position
        if ((new_pos < n) and 
            (A[new_pos] <= key_to_search)):
            pos = new_pos

    # if element found return
    # pos otherwise -1
    return (pos if(A[pos] == key_to_search) else -1)

# Driver code
if __name__ == "__main__":

    A = [ -2, 10, 100, 250, 32315 ]
    print( bsearch(A, 10))

# This implementation was improved by Tanin

# This code is contributed
# by ChitraNayal

//C# implementation of above approach 
using System;
using System.Collections.Generic;

class GFG 
{

    // Function to show the working of Meta binary search 
    static int bsearch(List<int> A, int key_to_search)
    {
        int n = (int) A.Count;
        //int lg = 0;
        // Set number of bits to represent largest array index 
        int lg = (int)Math.Log(n-1, 2.0) + 1;
         
        // This is redundant and will cause error
        //while ((1 << lg) < n - 1)
        //{
        //    lg += 1;
        //}

        int pos = 0;
        for (int i = lg - 1; i >= 0; i--)
        {
            if (A[pos] == key_to_search)
            {
                return pos;
            }

            // Incrementally construct the 
            // index of the target value 
            int new_pos = pos | (1 << i);

            // find the element in one 
            // direction and update position 
            if ((new_pos < n) && (A[new_pos] <= key_to_search))
            {
                pos = new_pos;
            }
        }

        // if element found return pos otherwise -1 
        return ((A[pos] == key_to_search) ? pos : -1);
    }

    // Driver code 
    static public void Main()
    {
        List<int> A = new List<int>();
        int[] arr = {-2, 10, 100, 250, 32315};
        for (int i = 0; i < arr.Length; i++) 
        {
            A.Add(arr[i]);
        }
        Console.WriteLine(bsearch(A, 10));
    }
}

// This code is contributed by Rajput-Ji
// This implementation was improved by Tanin

<?php
// PHP implementation of above approach

// Function to show the working of 
// Meta binary search
function bsearch($A, $key_to_search, $n)
{
    // Set number of bits to represent
    $lg = log($n-1, 2) + 1;
 
    // largest array index
     // This is redundant and will cause error for some case
    //while ((1 << $lg) < $n - 1)
        //$lg += 1;

    $pos = 0;
    for ($i = $lg - 1; $i >= 0; $i--) 
    {
        if ($A[$pos] == $key_to_search)
            return $pos;

        // Incrementally construct the
        // index of the target value
        $new_pos = $pos | (1 << $i);

        // find the element in one
        // direction and update $position
        if (($new_pos < $n) && 
            ($A[$new_pos] <= $key_to_search))
            $pos = $new_pos;
    }

    // if element found return $pos 
    // otherwise -1
    return (($A[$pos] == $key_to_search) ? 
                              $pos : -1);
}

// Driver code
$A = [ -2, 10, 100, 250, 32315 ];
$ans = bsearch($A, 10, 5);
echo $ans;

// This code is contributed by AdeshSingh1
// This implementation was improved by Tanin
?>

<script>
// Javascript implementation of above approach

// Function to show the working of Meta binary search
function bsearch(A, key_to_search)
{
    let n = A.length;
    // Set number of bits to represent largest array index
    let lg = parseInt(Math.log(n-1) / Math.log(2)) + 1; 

    //while ((1 << lg) < n - 1)
        //lg += 1;

    let pos = 0;
    for (let i = lg ; i >= 0; i--) {
        if (A[pos] == key_to_search)
            return pos;

        // Incrementally construct the
        // index of the target value
        let new_pos = pos | (1 << i);

        // find the element in one
        // direction and update position
        if ((new_pos < n) && (A[new_pos] <= key_to_search))
            pos = new_pos;
    }

    // if element found return pos otherwise -1
    return ((A[pos] == key_to_search) ? pos : -1);
}

// Driver code

    let A = [ -2, 10, 100, 250, 32315 ];
    document.write(bsearch(A, 10));

</script>

1

Time Complexity: O(log n), where n is the size of the given array
Auxiliary Space: O(1) , as we are not using any extra space

Reference: https://www.quora.com/What-is-meta-binary-search