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Search element in a Spirally sorted Matrix

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  • Difficulty Level : Hard
  • Last Updated : 11 Aug, 2021

Given a spirally sorted matrix with N × N elements and an integer X, the task is to find the position of this given integer in the matrix if it exists, else print -1. Note that all the matrix elements are distinct.

Examples:  

Input: arr[] = { 
{1, 2, 3, 4}, 
{12, 13, 14, 5}, 
{11, 16, 15, 6}, 
{10, 9, 8, 7}}, X = 9 
Output: 3 1 
9 appears in row number 3 and column number 1 (0-based indexing) 
Thus, output is (3, 1).

Input: arr[] = { 
{1, 2, 3}, 
{8, 9, 4}, 
{7, 6, 5}}, X = 9 
Output: 1 1  

A simple solution is to search through all the elements in the array. The worst-case time complexity of this approach will be O(n2).

A better solution is to use binary search. We apply binary search in two phases. 
But before jumping to that, let’s define what a ring means here. A ring is defined as a set of all the cells in the array such that their minimum of the distances from all four sides is equal. 
First, we try to determine the ring the number ‘X’ will belong to. We will do this using binary search. For that, observe the diagonal elements of the matrix. The first ceil(N/2) of the diagonal matrix is guaranteed to be sorted in increasing order. So, each one of the ceil(N/2) diagonal elements can represent a ring. By, applying binary on the first ceil(N/2) diagonal elements, we determine the ring the number ‘X’ belongs to in O(log(n)) time. 
After that, we apply a binary search on the elements of the ring. Before that we determine the side of the ring, the number ‘X’ will belong to. Then, we apply the binary search correspondingly.
So, the total time complexity becomes O(log(n)).

Below is the implementation of the above approach: 

C++




// C++ implementation of the above approach
 
#include <iostream>
#define n 4
using namespace std;
 
// Function to return the ring, the number x
// belongs to.
int findRing(int arr[][n], int x)
{
    // Returns -1 if number x is smaller than
    // least element of arr
    if (arr[0][0] > x)
        return -1;
 
    // l and r represent the diagonal
    // elements to search in
    int l = 0, r = (n + 1) / 2 - 1;
 
    // Returns -1 if number x is greater
    // than the largest element of arr
    if (n % 2 == 1 && arr[r][r] < x)
        return -1;
    if (n % 2 == 0 && arr[r + 1][r] < x)
        return -1;
 
    while (l < r) {
        int mid = (l + r) / 2;
        if (arr[mid][mid] <= x)
            if (mid == (n + 1) / 2 - 1
                || arr[mid + 1][mid + 1] > x)
                return mid;
            else
                l = mid + 1;
        else
            r = mid - 1;
    }
     
    return r;
}
 
// Function to perform binary search
// on an array sorted in increasing order
// l and r represent the left and right
// index of the row to be searched
int binarySearchRowInc(int arr[][n], int row,
                    int l, int r, int x)
{
    while (l <= r) {
        int mid = (l + r) / 2;
         
        if (arr[row][mid] == x)
            return mid;
             
        if (arr[row][mid] < x)
            l = mid + 1;
        else
            r = mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// a particular column of the 2D array
// t and b represent top and
// bottom rows
int binarySearchColumnInc(int arr[][n], int col,
                        int t, int b, int x)
{
    while (t <= b) {
         
        int mid = (t + b) / 2;
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            t = mid + 1;
        else
            b = mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// an array sorted in decreasing order
int binarySearchRowDec(int arr[][n], int row,
                    int l, int r, int x)
{
    while (l <= r) {
         
        int mid = (l + r) / 2;
         
        if (arr[row][mid] == x)
            return mid;
         
        if (arr[row][mid] < x)
            r = mid - 1;
        else
            l = mid + 1;
    }
     
    return -1;
}
 
// Function to perform binary search on a
// particular column of the 2D array
int binarySearchColumnDec(int arr[][n], int col,
                        int t, int b, int x)
{
    while (t <= b) {
        int mid = (t + b) / 2;
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            b = mid - 1;
        else
            t = mid + 1;
    }
     
    return -1;
}
 
// Function to find the position of the number x
void spiralBinary(int arr[][n], int x)
{
 
    // Finding the ring
    int f1 = findRing(arr, x);
 
    // To store row and column
    int r, c;
 
    if (f1 == -1) {
        cout << "-1";
        return;
    }
 
    // Edge case if n is odd
    if (n % 2 == 1 && f1 == (n + 1) / 2 - 1) {
        cout << f1 << " " << f1 << endl;
        return;
    }
 
    // Check which of the 4 sides, the number x
    // lies in
    if (x < arr[f1][n - f1 - 1]) {
        c = binarySearchRowInc(arr, f1, f1,
                            n - f1 - 2, x);
        r = f1;
    }
    else if (x < arr[n - f1 - 1][n - f1 - 1]) {
        c = n - f1 - 1;
         
        r = binarySearchColumnInc(arr, n - f1 - 1, f1,
                                n - f1 - 2, x);
    }
    else if (x < arr[n - f1 - 1][f1]) {
         
        c = binarySearchRowDec(arr, n - f1 - 1, f1 + 1,
                            n - f1 - 1, x);
        r = n - f1 - 1;
    }
    else {
         
        r = binarySearchColumnDec(arr, f1, f1 + 1,
                                n - f1 - 1, x);
        c = f1;
    }
 
    // Printing the position
    if (c == -1 || r == -1)
        cout << "-1";
    else
        cout << r << " " << c;
 
    return;
}
 
// Driver code
int main()
{
    int arr[][n] = { { 1, 2, 3, 4 },
                    { 12, 13, 14, 5 },
                    { 11, 16, 15, 6 },
                    { 10, 9, 8, 7 } };
 
    spiralBinary(arr, 7);
 
    return 0;
}

Java




// Java implementation of the above approach
class GFG
{
 
final static int n =4;
 
// Function to return the ring,
// the number x belongs to.
static int findRing(int arr[][], int x)
{
    // Returns -1 if number x is 
    // smaller than least element of arr
    if (arr[0][0] > x)
        return -1;
 
    // l and r represent the diagonal
    // elements to search in
    int l = 0, r = (n + 1) / 2 - 1;
 
    // Returns -1 if number x is greater
    // than the largest element of arr
    if (n % 2 == 1 && arr[r][r] < x)
        return -1;
    if (n % 2 == 0 && arr[r + 1][r] < x)
        return -1;
 
    while (l < r)
    {
        int mid = (l + r) / 2;
        if (arr[mid][mid] <= x)
            if (mid == (n + 1) / 2 - 1
                || arr[mid + 1][mid + 1] > x)
                return mid;
            else
                l = mid + 1;
        else
            r = mid - 1;
    }
    return r;
}
 
// Function to perform binary search
// on an array sorted in increasing order
// l and r represent the left and right
// index of the row to be searched
static int binarySearchRowInc(int arr[][], int row,
                    int l, int r, int x)
{
    while (l <= r)
    {
        int mid = (l + r) / 2;
         
        if (arr[row][mid] == x)
            return mid;
             
        if (arr[row][mid] < x)
            l = mid + 1;
        else
            r = mid - 1;
    }
    return -1;
}
 
// Function to perform binary search on
// a particular column of the 2D array
// t and b represent top and
// bottom rows
static int binarySearchColumnInc(int arr[][], int col,
                        int t, int b, int x)
{
    while (t <= b)
    {
        int mid = (t + b) / 2;
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            t = mid + 1;
        else
            b = mid - 1;
    }
    return -1;
}
 
// Function to perform binary search on
// an array sorted in decreasing order
static int binarySearchRowDec(int arr[][], int row,
                    int l, int r, int x)
{
    while (l <= r) {
         
        int mid = (l + r) / 2;
         
        if (arr[row][mid] == x)
            return mid;
         
        if (arr[row][mid] < x)
            r = mid - 1;
        else
            l = mid + 1;
    }
    return -1;
}
 
// Function to perform binary search on a
// particular column of the 2D array
static int binarySearchColumnDec(int arr[][], int col,
                        int t, int b, int x)
{
    while (t <= b)
    {
        int mid = (t + b) / 2;
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            b = mid - 1;
        else
            t = mid + 1;
    }
    return -1;
}
 
// Function to find the position of the number x
static void spiralBinary(int arr[][], int x)
{
 
    // Finding the ring
    int f1 = findRing(arr, x);
 
    // To store row and column
    int r, c;
 
    if (f1 == -1)
    {
            System.out.print("-1");
        return;
    }
 
    // Edge case if n is odd
    if (n % 2 == 1 && f1 == (n + 1) / 2 - 1)
    {
            System.out.println(f1+" "+f1);
        return;
    }
 
    // Check which of the 4 sides, the number x
    // lies in
    if (x < arr[f1][n - f1 - 1])
    {
        c = binarySearchRowInc(arr, f1, f1,
                            n - f1 - 2, x);
        r = f1;
    }
    else if (x < arr[n - f1 - 1][n - f1 - 1])
    {
        c = n - f1 - 1;
         
        r = binarySearchColumnInc(arr, n - f1 - 1, f1,
                                n - f1 - 2, x);
    }
    else if (x < arr[n - f1 - 1][f1])
    {
        c = binarySearchRowDec(arr, n - f1 - 1, f1 + 1,
                            n - f1 - 1, x);
        r = n - f1 - 1;
    }
    else
    {
        r = binarySearchColumnDec(arr, f1, f1 + 1,
                                n - f1 - 1, x);
        c = f1;
    }
 
    // Printing the position
    if (c == -1 || r == -1)
        System.out.print("-1");
    else
            System.out.print(r+" "+c);
 
    return;
}
 
// Driver code
public static void main(String[] args)
{
        int arr[][] = { { 1, 2, 3, 4 },
                    { 12, 13, 14, 5 },
                    { 11, 16, 15, 6 },
                    { 10, 9, 8, 7 } };
 
    spiralBinary(arr, 7);
}
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 implementation of the above approach
 
# Function to return the ring,
# the number x belongs to.
def findRing(arr, x):
 
    # Returns -1 if number x is smaller
    # than least element of arr
    if arr[0][0] > x:
        return -1
 
    # l and r represent the diagonal
    # elements to search in
    l, r = 0, (n + 1) // 2 - 1
 
    # Returns -1 if number x is greater
    # than the largest element of arr
    if n % 2 == 1 and arr[r][r] < x:
        return -1
    if n % 2 == 0 and arr[r + 1][r] < x:
        return -1
 
    while l < r:
        mid = (l + r) // 2
        if arr[mid][mid] <= x:
             
            if (mid == (n + 1) // 2 - 1 or
                arr[mid + 1][mid + 1] > x):
                return mid
            else:
                l = mid + 1
         
        else:
            r = mid - 1
     
    return r
 
# Function to perform binary search
# on an array sorted in increasing order
# l and r represent the left and right
# index of the row to be searched
def binarySearchRowInc(arr, row, l, r, x):
 
    while l <= r:
        mid = (l + r) // 2
         
        if arr[row][mid] == x:
            return mid
        elif arr[row][mid] < x:
            l = mid + 1
        else:
            r = mid - 1
     
    return -1
 
# Function to perform binary search on
# a particular column of the 2D array
# t and b represent top and
# bottom rows
def binarySearchColumnInc(arr, col, t, b, x):
 
    while t <= b:
         
        mid = (t + b) // 2
         
        if arr[mid][col] == x:
            return mid
        elif arr[mid][col] < x:
            t = mid + 1
        else:
            b = mid - 1
     
    return -1
 
# Function to perform binary search on
# an array sorted in decreasing order
def binarySearchRowDec(arr, row, l, r, x):
 
    while l <= r:
         
        mid = (l + r) // 2
         
        if arr[row][mid] == x:
            return mid
        elif arr[row][mid] < x:
            r = mid - 1
        else:
            l = mid + 1
     
    return -1
 
# Function to perform binary search on a
# particular column of the 2D array
def binarySearchColumnDec(arr, col, t, b, x):
 
    while t <= b:
        mid = (t + b) // 2
         
        if arr[mid][col] == x:
            return mid
        elif arr[mid][col] < x:
            b = mid - 1
        else:
            t = mid + 1
     
    return -1
 
# Function to find the position of the number x
def spiralBinary(arr, x):
 
    # Finding the ring
    f1 = findRing(arr, x)
 
    # To store row and column
    r, c = None, None
 
    if f1 == -1:
        print("-1")
        return
 
    # Edge case if n is odd
    if n % 2 == 1 and f1 == (n + 1) // 2 - 1:
        print(f1, f1)
        return
 
    # Check which of the 4 sides,
    # the number x lies in
    if x < arr[f1][n - f1 - 1]:
        c = binarySearchRowInc(arr, f1, f1,
                            n - f1 - 2, x)
        r = f1
     
    elif x < arr[n - f1 - 1][n - f1 - 1]:
        c = n - f1 - 1
         
        r = binarySearchColumnInc(arr, n - f1 - 1, f1,
                                n - f1 - 2, x)
     
    elif x < arr[n - f1 - 1][f1]:
         
        c = binarySearchRowDec(arr, n - f1 - 1, f1 + 1,
                            n - f1 - 1, x)
        r = n - f1 - 1
     
    else:
         
        r = binarySearchColumnDec(arr, f1, f1 + 1,
                                n - f1 - 1, x)
        c = f1
     
    # Printing the position
    if c == -1 or r == -1:
        print("-1")
    else:
        print("{0} {1}".format(r, c))
 
# Driver code
if __name__ == "__main__":
     
    n = 4
    arr = [[1, 2, 3, 4],
        [12, 13, 14, 5],
        [11, 16, 15, 6],
        [10, 9, 8, 7]]
 
    spiralBinary(arr, 7)
 
# This code is contributed by Rituraj Jain

C#




// C# implementation of the above approach
using System;
class GFG
{
 
 static int n =4;
 
// Function to return the ring,
// the number x belongs to.
static int findRing(int [,]arr, int x)
{
    // Returns -1 if number x is
    // smaller than least element of arr
    if (arr[0,0] > x)
        return -1;
 
    // l and r represent the diagonal
    // elements to search in
    int l = 0, r = (n + 1) / 2 - 1;
 
    // Returns -1 if number x is greater
    // than the largest element of arr
    if (n % 2 == 1 && arr[r,r] < x)
        return -1;
    if (n % 2 == 0 && arr[r + 1,r] < x)
        return -1;
 
    while (l < r)
    {
        int mid = (l + r) / 2;
        if (arr[mid,mid] <= x)
            if (mid == (n + 1) / 2 - 1
                || arr[mid + 1,mid + 1] > x)
                return mid;
            else
                l = mid + 1;
        else
            r = mid - 1;
    }
    return r;
}
 
// Function to perform binary search
// on an array sorted in increasing order
// l and r represent the left and right
// index of the row to be searched
static int binarySearchRowInc(int [,]arr, int row,
                    int l, int r, int x)
{
    while (l <= r)
    {
        int mid = (l + r) / 2;
         
        if (arr[row,mid] == x)
            return mid;
             
        if (arr[row,mid] < x)
            l = mid + 1;
        else
            r = mid - 1;
    }
    return -1;
}
 
// Function to perform binary search on
// a particular column of the 2D array
// t and b represent top and
// bottom rows
static int binarySearchColumnInc(int [,]arr, int col,
                        int t, int b, int x)
{
    while (t <= b)
    {
        int mid = (t + b) / 2;
         
        if (arr[mid,col] == x)
            return mid;
         
        if (arr[mid,col] < x)
            t = mid + 1;
        else
            b = mid - 1;
    }
    return -1;
}
 
// Function to perform binary search on
// an array sorted in decreasing order
static int binarySearchRowDec(int [,]arr, int row,
                    int l, int r, int x)
{
    while (l <= r) {
         
        int mid = (l + r) / 2;
         
        if (arr[row,mid] == x)
            return mid;
         
        if (arr[row,mid] < x)
            r = mid - 1;
        else
            l = mid + 1;
    }
    return -1;
}
 
// Function to perform binary search on a
// particular column of the 2D array
static int binarySearchColumnDec(int [,]arr, int col,
                        int t, int b, int x)
{
    while (t <= b)
    {
        int mid = (t + b) / 2;
         
        if (arr[mid,col] == x)
            return mid;
         
        if (arr[mid,col] < x)
            b = mid - 1;
        else
            t = mid + 1;
    }
    return -1;
}
 
// Function to find the position of the number x
static void spiralBinary(int [,]arr, int x)
{
 
    // Finding the ring
    int f1 = findRing(arr, x);
 
    // To store row and column
    int r, c;
 
    if (f1 == -1)
    {
            Console.Write("-1");
        return;
    }
 
    // Edge case if n is odd
    if (n % 2 == 1 && f1 == (n + 1) / 2 - 1)
    {
            Console.WriteLine(f1+" "+f1);
        return;
    }
 
    // Check which of the 4 sides, the number x
    // lies in
    if (x < arr[f1,n - f1 - 1])
    {
        c = binarySearchRowInc(arr, f1, f1,
                            n - f1 - 2, x);
        r = f1;
    }
    else if (x < arr[n - f1 - 1,n - f1 - 1])
    {
        c = n - f1 - 1;
         
        r = binarySearchColumnInc(arr, n - f1 - 1, f1,
                                n - f1 - 2, x);
    }
    else if (x < arr[n - f1 - 1,f1])
    {
        c = binarySearchRowDec(arr, n - f1 - 1, f1 + 1,
                            n - f1 - 1, x);
        r = n - f1 - 1;
    }
    else
    {
        r = binarySearchColumnDec(arr, f1, f1 + 1,
                                n - f1 - 1, x);
        c = f1;
    }
 
    // Printing the position
    if (c == -1 || r == -1)
        Console.Write("-1");
    else
            Console.Write(r+" "+c);
 
    return;
}
 
// Driver code
public static void Main(String []args)
{
        int [,]arr = { { 1, 2, 3, 4 },
                    { 12, 13, 14, 5 },
                    { 11, 16, 15, 6 },
                    { 10, 9, 8, 7 } };
 
    spiralBinary(arr, 7);
}
}
 
// This code is contributed by Arnab Kundu

PHP




<?php
// PHP implementation of the above approach
$n = 4;
 
// Function to return the ring, the number x
// belongs to.
function findRing($arr, $x)
{
    global $n;
     
    // Returns -1 if number x is smaller than
    // least element of arr
    if ($arr[0][0] > $x)
        return -1;
 
    // l and r represent the diagonal
    // elements to search in
    $l = 0;
    $r = (int)(($n + 1) / 2 - 1);
 
    // Returns -1 if number x is greater
    // than the largest element of arr
    if ($n % 2 == 1 && $arr[$r][$r] < $x)
        return -1;
    if ($n % 2 == 0 && $arr[$r + 1][$r] < $x)
        return -1;
 
    while ($l < $r)
    {
        $mid = (int)(($l + $r) / 2);
        if ($arr[$mid][$mid] <= $x)
            if ($mid == (int)(($n + 1) / 2 - 1) ||
                $arr[$mid + 1][$mid + 1] > $x)
                return $mid;
            else
                $l = $mid + 1;
        else
            $r = $mid - 1;
    }
     
    return $r;
}
 
// Function to perform binary search
// on an array sorted in increasing order
// l and r represent the left and right
// index of the row to be searched
function binarySearchRowInc($arr, $row,
                            $l, $r, $x)
{
    while ($l <= $r)
    {
        $mid = (int)(($l + $r) / 2);
         
        if ($arr[$row][$mid] == $x)
            return $mid;
             
        if ($arr[$row][$mid] < $x)
            $l = $mid + 1;
        else
            $r = $mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// a particular column of the 2D array
// t and b represent top and
// bottom rows
function binarySearchColumnInc($arr, $col,
                               $t, $b, $x)
{
    while ($t <= $b)
    {
        $mid = (int)(($t + b) / 2);
         
        if ($arr[$mid][$col] == $x)
            return $mid;
         
        if ($arr[$mid][$col] < $x)
            $t = $mid + 1;
        else
            $b = $mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// an array sorted in decreasing order
function binarySearchRowDec($arr, $row, $l, $r, $x)
{
    while ($l <= $r)
    {
         
        $mid = (int)(($l + $r) / 2);
         
        if ($arr[$row][$mid] == $x)
            return $mid;
         
        if ($arr[$row][$mid] < $x)
            $r = $mid - 1;
        else
            $l = $mid + 1;
    }
     
    return -1;
}
 
// Function to perform binary search on a
// particular column of the 2D array
function binarySearchColumnDec($arr, $col,
                               $t, $b, $x)
{
    while ($t <= $b)
    {
        $mid = (int)(($t + $b) / 2);
         
        if ($arr[$mid][$col] == $x)
            return $mid;
         
        if ($arr[$mid][$col] < $x)
            $b = $mid - 1;
        else
            $t = $mid + 1;
    }
     
    return -1;
}
 
// Function to find the position of the number x
function spiralBinary($arr, $x)
{
    global $n;
 
    // Finding the ring
    $f1 = findRing($arr, $x);
 
    // To store row and column
    $r = -1;
    $c = -1;
 
    if ($f1 == -1)
    {
        echo "-1";
        return;
    }
 
    // Edge case if n is odd
    if ($n % 2 == 1 &&
        $f1 == (int)(($n + 1) / 2 - 1))
    {
        echo $f1 . " " . $f1 . "\n";
        return;
    }
 
    // Check which of the 4 sides, the number x
    // lies in
    if ($x < $arr[$f1][$n - $f1 - 1])
    {
        $c = binarySearchRowInc($arr, $f1, $f1,
                                   $n - $f1 - 2, $x);
        $r = $f1;
    }
    else if ($x < $arr[$n - $f1 - 1][$n - $f1 - 1])
    {
        $c = $n - $f1 - 1;
         
        $r = binarySearchColumnInc($arr, $n - $f1 - 1,
                                    $f1, $n - $f1 - 2, $x);
    }
    else if ($x < $arr[$n - $f1 - 1][$f1])
    {
         
        $c = binarySearchRowDec($arr, $n - $f1 - 1,
                                $f1 + 1, $n - $f1 - 1, $x);
        $r = $n - $f1 - 1;
    }
    else
    {
        $r = binarySearchColumnDec($arr, $f1, $f1 + 1,
                                   $n - $f1 - 1, $x);
        $c = $f1;
    }
 
    // Printing the position
    if ($c == -1 || $r == -1)
        echo "-1";
    else
        echo $r . " " . $c;
 
    return;
}
 
// Driver code
$arr = array(array( 1, 2, 3, 4 ),
             array( 12, 13, 14, 5 ),
             array( 11, 16, 15, 6 ),
             array( 10, 9, 8, 7 ));
 
spiralBinary($arr, 7);
 
// This code is contributed by mits
?>

Javascript




<script>
 
// Javascript implementation of the above approach
var n = 4;
 
// Function to return the ring, the number x
// belongs to.
function findRing(arr, x)
{
    // Returns -1 if number x is smaller than
    // least element of arr
    if (arr[0][0] > x)
        return -1;
 
    // l and r represent the diagonal
    // elements to search in
    var l = 0, r = parseInt((n + 1) / 2) - 1;
 
    // Returns -1 if number x is greater
    // than the largest element of arr
    if (n % 2 == 1 && arr[r][r] < x)
        return -1;
    if (n % 2 == 0 && arr[r + 1][r] < x)
        return -1;
 
    while (l < r) {
        var mid = parseInt((l + r) / 2);
        if (arr[mid][mid] <= x)
            if (mid == (n + 1) / 2 - 1
                || arr[mid + 1][mid + 1] > x)
                return mid;
            else
                l = mid + 1;
        else
            r = mid - 1;
    }
     
    return r;
}
 
// Function to perform binary search
// on an array sorted in increasing order
// l and r represent the left and right
// index of the row to be searched
function binarySearchRowInc(arr, row, l, r, x)
{
    while (l <= r) {
        var mid = parseInt((l + r) / 2);
         
        if (arr[row][mid] == x)
            return mid;
             
        if (arr[row][mid] < x)
            l = mid + 1;
        else
            r = mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// a particular column of the 2D array
// t and b represent top and
// bottom rows
function binarySearchColumnInc(arr, col, t, b, x)
{
    while (t <= b) {
         
        var mid = parseInt((t + b) / 2);
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            t = mid + 1;
        else
            b = mid - 1;
    }
     
    return -1;
}
 
// Function to perform binary search on
// an array sorted in decreasing order
function binarySearchRowDec(arr, row, l, r, x)
{
    while (l <= r) {
         
        var mid = parseInt((l + r) / 2);
         
        if (arr[row][mid] == x)
            return mid;
         
        if (arr[row][mid] < x)
            r = mid - 1;
        else
            l = mid + 1;
    }
     
    return -1;
}
 
// Function to perform binary search on a
// particular column of the 2D array
function binarySearchColumnDec(arr, col, t, b, x)
{
    while (t <= b) {
        var mid = parseInt((t + b) / 2);
         
        if (arr[mid][col] == x)
            return mid;
         
        if (arr[mid][col] < x)
            b = mid - 1;
        else
            t = mid + 1;
    }
     
    return -1;
}
 
// Function to find the position of the number x
function spiralBinary(arr, x)
{
 
    // Finding the ring
    var f1 = findRing(arr, x);
 
    // To store row and column
    var r, c;
 
    if (f1 == -1) {
        document.write( "-1");
        return;
    }
 
    // Edge case if n is odd
    if (n % 2 == 1 && f1 == (n + 1) / 2 - 1) {
        document.write( f1 + " " + f1 + "<br>");
        return;
    }
 
    // Check which of the 4 sides, the number x
    // lies in
    if (x < arr[f1][n - f1 - 1]) {
        c = binarySearchRowInc(arr, f1, f1,
                            n - f1 - 2, x);
        r = f1;
    }
    else if (x < arr[n - f1 - 1][n - f1 - 1]) {
        c = n - f1 - 1;
         
        r = binarySearchColumnInc(arr, n - f1 - 1, f1,
                                n - f1 - 2, x);
    }
    else if (x < arr[n - f1 - 1][f1]) {
         
        c = binarySearchRowDec(arr, n - f1 - 1, f1 + 1,
                            n - f1 - 1, x);
        r = n - f1 - 1;
    }
    else {
         
        r = binarySearchColumnDec(arr, f1, f1 + 1,
                                n - f1 - 1, x);
        c = f1;
    }
 
    // Printing the position
    if (c == -1 || r == -1)
        document.write( "-1");
    else
        document.write( r + " " + c);
 
    return;
}
 
// Driver code
var arr = [ [ 1, 2, 3, 4 ],
                [ 12, 13, 14, 5 ],
                [ 11, 16, 15, 6 ],
                [ 10, 9, 8, 7 ] ];
spiralBinary(arr, 7);
 
</script>
Output: 
3 3

 

Time Complexity: O(logN)
Auxiliary Space: O(logN)


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