# Prefix Sum 1D and 2D Array Algorithm

TLDRThis video script introduces the concept and implementation of the prefix sum algorithm for both one-dimensional and two-dimensional arrays. It explains how to calculate the prefix sum by accumulating values from the given array and presents an efficient solution using formulas based on the index positions. The script provides examples and a step-by-step explanation of the algorithm, emphasizing its time complexity and practical application through code snippets for one-dimensional and two-dimensional arrays.

###### Takeaways

- π The prefix sum algorithm is used to find the cumulative sum of elements in an array, both 1D and 2D.
- π’ A 1D prefix sum array adds each element to the sum of all preceding elements, including itself.
- π In a 2D array (matrix), the prefix sum includes sums from elements above and to the left, including itself.
- π‘ The prefix sum array's size matches the original array, maintaining the same number of elements.
- π― The concept is to efficiently calculate the sum of a subarray without repetitive computations.
- π The general formula for a 1D element at position 'i' depends on the index 'i', with two cases.
- π For 2D elements at position '(i, j)', the formula considers four cases based on the indices 'i' and 'j'.
- π The algorithm's efficiency comes from using these formulas instead of a brute force approach.
- π The time complexity for a 1D array prefix sum is O(n), and for a 2D array, it's O(r * c) where r and c are the number of rows and columns.
- π The script provides detailed examples of how to calculate prefix sums for both 1D and 2D arrays.
- π οΈ The implementation of the algorithm involves initializing a prefix sum array and populating it according to the defined formulas.
- π€ Understanding the logic behind the prefix sum algorithm is crucial for its effective application in various computational problems.

###### Q & A

### What is the prefix sum algorithm?

-The prefix sum algorithm is a technique used to calculate the sum of elements in a given array or matrix, where each element of the resulting array (prefix sum array) represents the sum of all elements in the original array up to and including that position.

### How does the prefix sum algorithm work for a 1D array?

-For a 1D array, the prefix sum algorithm starts with the first element and iteratively adds the current element's value to the sum of all previous elements to calculate the prefix sum for each position in the array.

### How does the prefix sum algorithm work for a 2D array or matrix?

-For a 2D array, the algorithm calculates the prefix sum by considering the sum of all elements above and to the left of the current element, in addition to the value of the current element itself, for each position in the matrix.

### What is the time complexity of the prefix sum algorithm for a 1D array?

-The time complexity of the prefix sum algorithm for a 1D array is O(n), where n is the number of elements in the array, as the algorithm iterates through the array once.

### What is the time complexity of the prefix sum algorithm for a 2D array?

-The time complexity of the prefix sum algorithm for a 2D array is O(r * c), where r is the number of rows and c is the number of columns in the matrix, as the algorithm iterates through the matrix once.

### What is the general formula for calculating the prefix sum of a 1D element at position i?

-The general formula for a 1D element at position i depends on the index i. There are two cases: when i is equal to 0, the prefix sum is the value of the given array at index i; when i is greater than 0, the prefix sum is the previous value in the prefix sum array plus the value of the given array at index i.

### What is the general formula for calculating the prefix sum of a 2D element at position (i, j)?

-The general formula for a 2D element at position (i, j) depends on the indices i and j. There are four cases: when both i and j are 0, when i is 0 and j is greater than 0, when i is greater than 0 and j is 0, and when both i and j are greater than 0. The prefix sum is calculated by summing the values of elements above and to the left of the current position, plus the value of the current element in the given array.

### How is the size of the prefix sum array determined?

-The size of the prefix sum array is always the same as the original array. Whether it's a 1D array or a 2D matrix, the prefix sum array maintains the same dimensions as the input array.

### What is the initial value of the prefix sum array?

-The initial value of the prefix sum array is zero. Each element in the prefix sum array is initialized to zero before the algorithm starts calculating the prefix sums.

### How can the prefix sum array be represented for a 1D array?

-For a 1D array, the prefix sum array can be represented as an array where each element at index i represents the sum of all elements from the start of the array up to index i in the original array.

### How can the prefix sum array be represented for a 2D array?

-For a 2D array, the prefix sum array (or prefix sum matrix) can be represented as a matrix where each element at position (i, j) represents the sum of all elements above and to the left of (i, j) in the original matrix, including the value at (i, j) itself.

### What is the purpose of the prefix sum algorithm?

-The prefix sum algorithm is useful for efficiently querying the sum of elements within a range in an array or matrix. It allows for quick calculation of the sum of elements from any starting point to the current index without needing to iterate through all the elements each time.

###### Outlines

##### π Introduction to Prefix Sum Algorithm

This paragraph introduces the concept of the prefix sum algorithm, applicable to both 1D and 2D arrays of integers. It explains the algorithm's purpose, which is to calculate the prefix sum array/matrices by summing elements from the original array/matrix. The explanation includes examples of how the prefix sum would look for given 1D and 2D arrays, highlighting the initial zero value and the accumulation of sums along the way. The paragraph also touches on the concept of prefix sum arrays in the context of sets and matrices, emphasizing the relationship between the size of the prefix sum array and the original array.

##### π Understanding the Prefix Sum Array

The paragraph delves deeper into the concept of prefix sum arrays, using a set of numbers to illustrate the calculation process. It outlines the initial zero value of the set and how each subsequent number in the prefix sum set is the sum of all previous numbers. The explanation extends to 2D arrays, detailing how the first number of the prefix sum matrix is determined and how subsequent numbers are calculated by summing elements above and to the left. The paragraph also compares the brute force method of finding prefix sums to the efficient formula-based approach, setting the stage for the explanation of the algorithm's logic and implementation.

##### π Explanation of the Prefix Sum Algorithm

This section provides a step-by-step explanation of the prefix sum algorithm for 1D arrays, including the general formula for calculating the prefix sum at any index. It describes the logic behind the algorithm, which involves adding the value of the current element of the given array to the previous value in the prefix sum array. The paragraph then walks through the algorithm's application to a specific array, demonstrating how each element of the prefix sum array is updated based on the formula and previous values. The explanation is clear and methodical, ensuring understanding of the algorithm's process.

##### π’ Time Complexity and 2D Array Prefix Sum

The paragraph discusses the time complexity of the prefix sum algorithm for 2D arrays, emphasizing the efficiency of iterating through the array only once. It then explains the application of the prefix sum algorithm to 2D arrays, outlining the logic behind calculating the element values in the prefix sum matrix. The explanation includes the four cases based on the indices i and j, which determine whether an element has elements above or to the left of it. The paragraph continues with a practical example of applying the algorithm to a given matrix, illustrating the process of updating the prefix sum array with detailed steps. The conclusion highlights the removal of duplicate additions and the finalization of the prefix sum array, reinforcing the algorithm's effectiveness.

##### π¨βπ» Implementation of Prefix Sum Algorithm

The final paragraph focuses on the implementation of the prefix sum algorithm for both one-dimensional and two-dimensional arrays. It provides a clear and concise explanation of the code structure and logic used in the actual implementation of the algorithm. The paragraph describes the initialization of the prefix sum arrays, the iteration process, and the printing of the results. It concludes with a summary of the algorithm's purpose and a thank you note to the viewer for their attention, encapsulating the entire explanation of the prefix sum algorithm in a neat and accessible manner.

###### Mindmap

###### Keywords

##### π‘Prefix Sum Algorithm

##### π‘1D Array

##### π‘2D Array

##### π‘Cumulative Sum

##### π‘Time Complexity

##### π‘Efficient Solution

##### π‘Brute Force Approach

##### π‘Formula

##### π‘Index

##### π‘Element

##### π‘Implementation

###### Highlights

Introduction to the prefix sum algorithm and its application on 1D and 2D arrays.

Explanation of the prefix sum array concept with an example of a 1D array of size 6.

Illustration of the prefix sum for a 2D array of size 2x3 with specific numerical values.

Detailed explanation of how to calculate the prefix sum for a set of numbers.

Clarification that the size of the prefix sum array matches the original array's size.

Description of the inefficient brute force approach for finding the prefix sum.

Introduction of a more efficient solution using formulas for 1D and 2D arrays.

General formula for calculating the 1D prefix sum element at position 'i'.

General formula for calculating the 2D prefix sum element at position 'i, j'.

Step-by-step example of the prefix sum algorithm for a 1D array of integers.

Explanation of the time complexity for the 1D prefix sum algorithm being O(n).

Detailed example of the prefix sum algorithm applied to a 2D array.

Explanation of the time complexity for the 2D prefix sum algorithm being O(r*c).

Implementation of the one-dimensional prefix sum algorithm with code explanation.

Implementation of the two-dimensional prefix sum algorithm with code explanation.

Summary of the algorithm's logic and its efficient calculation of prefix sums.

Appreciation for the viewer's attention and understanding of the prefix sum algorithm.

###### Transcripts

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