Suppose you have a sorted array of infinite numbers, how would you search an element in the array? Since array is sorted, the first thing clicks into mind is binary search, but the problem here is that we don’t know size of array.

If the array is infinite, that means we don’t have proper bounds to apply binary search. So in order to find position of key, first we find bounds and then apply binary search algorithm.

Let low be pointing to 1st element and high pointing to 2nd element of array, Now compare key with high index element,

->if it is greater than high index element then copy high index in low index and double the high index.

->if it is smaller, then apply binary search on high and low indices found.

Below are implementations of above algorithm

`<?php`

`// PHP program to demonstrate working`

`// of an algorithm that finds an`

`// element in an array of infinite size`

`// Simple binary search algorithm`

`function`

`binarySearch(`

`$arr`

`, `

`$l`

`,`

` `

`$r`

`, `

`$x`

`)`

`{`

` `

`if`

`(`

`$r`

`>= `

`$l`

`)`

` `

`{`

` `

`$mid`

`= `

`$l`

`+ (`

`$r`

`- `

`$l`

`)/2;`

` `

`if`

`(`

`$arr`

`[`

`$mid`

`] == `

`$x`

`)`

` `

`return`

`$mid`

`;`

` `

`if`

`(`

`$arr`

`[`

`$mid`

`] > `

`$x`

`)`

` `

`return`

`binarySearch(`

`$arr`

`, `

`$l`

`,`

` `

`$mid`

`- 1, `

`$x`

`);`

` `

`return`

`binarySearch(`

`$arr`

`, `

`$mid`

`+ 1,`

` `

`$r`

`, `

`$x`

`);`

` `

`}`

` `

`return`

`-1;`

`}`

`// function takes an infinite`

`// size array and a key to be`

`// searched and returns its`

`// position if found else -1.`

`// We don't know size of arr[]`

`// and we can assume size to be`

`// infinite in this function.`

`// NOTE THAT THIS FUNCTION ASSUMES`

`// arr[] TO BE OF INFINITE SIZE`

`// THEREFORE, THERE IS NO INDEX`

`// OUT OF BOUND CHECKING`

`function`

`findPos( `

`$arr`

`, `

`$key`

`)`

`{`

` `

`$l`

`= 0; `

`$h`

`= 1;`

` `

`$val`

`= `

`$arr`

`[0];`

` `

`// Find h to do binary search`

` `

`while`

`(`

`$val`

`< `

`$key`

`)`

` `

`{`

` `

` `

`// store previous high`

` `

`$l`

`= `

`$h`

`; `

` `

` `

`// double high index`

` `

`$h`

`= 2 * `

`$h`

`;`

` `

` `

`// update new val`

` `

`$val`

`= `

`$arr`

`[`

`$h`

`];`

` `

`}`

` `

`// at this point we have`

` `

`// updated low and high`

` `

`// indices, Thus use binary`

` `

`// search between them`

` `

`return`

`binarySearch(`

`$arr`

`, `

`$l`

`,`

` `

`$h`

`, `

`$key`

`);`

`}`

` `

`// Driver Code`

` `

`$arr`

`= `

`array`

`(3, 5, 7, 9, 10, 90, 100,`

` `

`130, 140, 160, 170);`

` `

`$ans`

`= findPos(`

`$arr`

`, 10);`

` `

`if`

`(`

`$ans`

`==-1)`

` `

`echo`

`"Element not found"`

`;`

` `

`else`

` `

`echo`

`"Element found at index "`

`, `

`$ans`

`;`

`?>`

Output:

Element found at index 4

Let p be the position of element to be searched. Number of steps for finding high index ‘h’ is O(Log p). The value of ‘h’ must be less than 2*p. The number of elements between h/2 and h must be O(p). Therefore, time complexity of Binary Search step is also O(Log p) and overall time complexity is 2*O(Log p) which is O(Log p).