An array consisting of N integers is given. There are several Right Circular Rotations of range[L..R] that we perform. After performing these rotations, we need to find an element at a given index.

Examples : 

Input : arr[] : {1, 2, 3, 4, 5}
        ranges[] = { {0, 2}, {0, 3} }
        index : 1
Output : 3
Explanation : After first given rotation {0, 2}
                arr[] = {3, 1, 2, 4, 5}
              After second rotation {0, 3} 
                arr[] = {4, 3, 1, 2, 5}
After all rotations we have element 3 at given
index 1. 

Method - 1: Brute-force Approach

The brute force approach is to actually rotate the array for all given ranges, finally return the element in at given index in the modified array.

Method - 2: Efficient Approach

We can do offline processing after saving all ranges. 

Suppose, our rotate ranges are : [0..2] and [0..3] 
We run through these ranges from reverse.
After range [0..3], index 0 will have the element which was on index 3. 
So, we can change 0 to 3, i.e. if index = left, index will be changed to right. 
After range [0..2], index 3 will remain unaffected.
So, we can make 3 cases : 
If index = left, index will be changed to right. 
If index is not bounds by the range, no effect of rotation. 
If index is in bounds, index will have the element at index-1.

For better explanation:

10 20 30 40 50
Index: 1
Rotations: {0,2} {1,4} {0,3}
Answer: 
Index 1 will have 30 after all the 3 rotations in the order {0,2} {1,4} {0,3}.
We performed {0,2} on A and now we have a new array A1.
We performed {1,4} on A1 and now we have a new array A2.
We performed {0,3} on A2 and now we have a new array A3.
Now we are looking for the value at index 1 in A3.
But A3 is {0,3} done on A2.
So index 1 in A3 is index 0 in A2.
But A2 is {1,4} done on A1.
So index 0 in A2 is also index 0 in A1 as it does not lie in the range {1,4}.
But A1 is {0,2} done on A.
So index 0 in A1 is index 2 in A.
On observing it, we are going deeper into the previous rotations 
starting from the latest rotation.
{0,3}
|
{1,4}
|
{0,2}
This is the reason we are processing the rotations in reverse order.

Please note that we are not rotating the elements in the reverse order, just processing the index from reverse.

Because if we actually rotate in reverse order, we might get a completely different answer as in case of rotations the order matters. 

<?php
// PHP code to rotate an array
// and answer the index query

// Function to compute the 
// element at given index
function findElement($arr, $ranges,
                     $rotations, $index)
{
    for ($i = $rotations - 1; 
         $i >= 0; $i--) 
    {

        // Range[left...right]
        $left = $ranges[$i][0];
        $right = $ranges[$i][1];

        // Rotation will not
        // have any effect
        if ($left <= $index && 
            $right >= $index)
        {
            if ($index == $left)
                $index = $right;
            else
                $index--;
        }
    }

    // Returning new element
    return $arr[$index];
}

// Driver Code
$arr = array(1, 2, 3, 4, 5);

// No. of rotations
$rotations = 2;

// Ranges according 
// to 0-based indexing
$ranges = array(array(0, 2),
                array(0, 3));

$index = 1;

echo findElement($arr, $ranges,
                 $rotations, $index);

// This code is contributed by ajit
?>

<?php

// PHP code to rotate an array

// and answer the index query

​

// Function to compute the 

// element at given index

function findElement($arr, $ranges,

                     $rotations, $index)

{

    for ($i = $rotations - 1; 

         $i >= 0; $i--) 

    {

​

        // Range[left...right]

        $left = $ranges[$i][0];

        $right = $ranges[$i][1];

​

        // Rotation will not

        // have any effect

        if ($left <= $index && 

            $right >= $index)

        {

            if ($index == $left)

                $index = $right;

            else

                $index--;

        }

    }

​

    // Returning new element

    return $arr[$index];

}

​

// Driver Code

$arr = array(1, 2, 3, 4, 5);

​

// No. of rotations

$rotations = 2;

​

// Ranges according 

// to 0-based indexing

$ranges = array(array(0, 2),

                array(0, 3));

​

$index = 1;

​

echo findElement($arr, $ranges,

                 $rotations, $index);

​

// This code is contributed by ajit

?>

3

Complexity Analysis: 

• Time Complexity: O(N), where N represents the given number of rotations.
• Auxiliary Space: O(1), no extra space is required, so it is a constant.

Please refer complete article on Find element at given index after a number of rotations for more details!

kartik

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Article Tags :

• Web Technologies
• PHP
• PHP Programs
• DSA
• Arrays
• array-range-queries
• rotation

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Practice Tags :

• Arrays