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.

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++

// 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

//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

# 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#

//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
// 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
?>

Javascript

<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>

Output:

1

Time Complexity : O(log n) ,where n is size of given array

Space Complexity : O(1) ,as we are not using any extra space

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