Efficient Range Updates Using Delta Encoding

In programming, we often encounter problems where we need to update a range of elements in an array multiple times and then retrieve the final state of the array efficiently. A straightforward approach would be to iterate through the range for each update, but this can be inefficient, especially for large arrays and numerous updates. Delta encoding provides an elegant solution to this problem.

What is Delta Encoding?

Delta encoding is a method of storing or transmitting data in the form of differences (deltas) between sequential data points rather than complete data points. This technique can significantly reduce the amount of data to be stored or transmitted, especially when the data points are similar or change incrementally.

Applying Delta Encoding to Range Updates

In the context of updating ranges in an array, delta encoding can be used to efficiently perform multiple range updates. The idea is to use a difference array (delta array) to record the changes and then compute the final values using prefix sums.

Step-by-Step Guide

Let's break down the process of using delta encoding for range updates with a practical example.

1. Initial Setup

Suppose we have an array arr of size n initialized to zeros:

int n = 5;

int arr[n] = {0, 0, 0, 0, 0};

We also create a delta array delta of size n+1 initialized to zeros:

int delta[n + 1] = {0, 0, 0, 0, 0, 0};

2. Range Updates

Let's say we want to perform the following updates:

• Increment the range [1, 3] by 5

• Increment the range [2, 4] by 10

For each update, we modify the delta array:

• First Update [1, 3] by 5:

  • Increment delta[1] by 5:

  delta[1] += 5;

  • Decrement delta[4] (i.e., r+1 where r=3) by 5:

  delta[4] -= 5;

• Second Update [2, 4] by 10:

  • Increment delta[2] by 10:

  delta[2] += 10;

  • Decrement delta[5] (i.e., r+1 where r=4) by 10:

  delta[5] -= 10;

3. Applying the Updates

To get the final values of the original array, compute the prefix sum of the delta array:

arr[0] = delta[0];
   for (int i = 1; i < n; ++i) {
       arr[i] = arr[i - 1] + delta[i];
   }

The final updated array is: arr = {0, 5, 15, 15, 10}

Benefits of Delta Encoding for Range Updates

1. Efficiency: Each range update operation is performed in O(1) time, and applying all updates is done in O(n) time.
2. Simplicity: The approach is easy to implement and understand, involving basic array manipulations and prefix sum calculations.
3. Scalability: This method is highly scalable and performs well even for large arrays and numerous update operations.

Try out Delta encoding in this problem : Tea Tasting