# Search element in a Spirally sorted Matrix

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

Given a spirally sorted matrix with N Ã— 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 ``#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

 ` ``\$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

 ``
Output:
`3 3`

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

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