The document explains the prefix sum technique, which allows for efficient calculations of the sum of elements in specified ranges of an array. It provides examples and formulas for calculating sums, demonstrating the time complexity of operations such as preprocessing and querying. Overall, it emphasizes the efficiency gained through the use of prefix sums to facilitate quick range sum queries.

2. Prefix sum
It is a simple yet powerful technique that allows to perform fast calculation on the sum
of elements in a given range (called contiguous segments of array).

3. Prefix sum
It is a simple yet powerful technique that allows to perform fast calculation on the sum
of elements in a given range (called contiguous segments of array).
6 3 -2 4 -1 0 -5
Example -
A[ ] =

4. Prefix sum
It is a simple yet powerful technique that allows to perform fast calculation on the sum
of elements in a given range (called contiguous segments of array).
6 3 -2 4 -1 0 -5
Example - i = 0 1 2 3 4 5 6
A[ ] =

5. Prefix sum
It is a simple yet powerful technique that allows to perform fast calculation on the sum
of elements in a given range (called contiguous segments of array).
6 3 -2 4 -1 0 -5
Example - i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5Prefix Sum Array -
A[ ] =
A[ ] =

6. Prefix sum
6 3 -2 4 -1 0 -5
0 1 2 3 4 5 6
6 9 -2 4 -1 0 -5
6 9 7 4 -1 0 -5
6 9 7 11 -1 0 -5
6 9 7 11 10 0 -5
6 9 7 11 10 10 5
6 9 7 11 10 10 -5
Example -
i =
A[ ] =

7. Prefix sum
6 3 -2 4 -1 0 -5
0 1 2 3 4 5 6
6 9 -2 4 -1 0 -5
6 9 7 4 -1 0 -5
6 9 7 11 -1 0 -5
6 9 7 11 10 0 -5
6 9 7 11 10 10 5
6 9 7 11 10 10 -5
for( int i=1 ; i<7 ; i++ ){
A[i] = A[i]+A[i-1];
}
n
Example -
i =
A[ ] =

8. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6

9. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6

10. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6

11. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
i = 0 1 2 3 4 5 6

12. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
i = 0 1 2 3 4 5 6

13. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
i = 0 1 2 3 4 5 6

14. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
i = 0 1 2 3 4 5 6

15. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
i = 0 1 2 3 4 5 6

16. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
sum between range [0, 6]
i = 0 1 2 3 4 5 6

17. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
sum between range [0, 6]
sum between range [0, 6] =
sum between range [2, 6]
sum between range [0, 1] +
i = 0 1 2 3 4 5 6

18. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
sum between range [0, 6]
sum between range [0, 6] =
sum between range [2, 6]
sum between range [0, 1] +
A[6] = A[1] + sum between range [2, 6]
A[6] - A[1] = sum between range [2, 6]
i = 0 1 2 3 4 5 6

19. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
sum between range [0, 6]
sum between range [0, 6] =
sum between range [2, 6]
sum between range [0, 1] +
A[6] = A[1] + sum between range [2, 6]
A[6] - A[1] = sum between range [2, 6]
sum between range [2, 6] = A[6] - A[1]
i = 0 1 2 3 4 5 6

20. Prefix sum
6 3 -2 4 -1 0 -5
Example -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
Calculate the sum between range [0, 4] ?
A[ ] =
A[ ] =
Ans : 10 (=A[4]) O(1)
Calculate the sum between range [0, 6] ?
Ans : 5 (=A[6]) O(1)
Calculate the sum between range [2, 6] ?
sum between range [0, 6]
sum between range [0, 6] =
sum between range [2, 6]
sum between range [0, 1] +
A[6] = A[1] + sum between range [2, 6]
A[6] - A[1] = sum between range [2, 6]
sum between range [2, 6] = A[6] - A[1]
sum between range [2, 6] = 5 - 9
sum between range [2, 6] = - 4
i = 0 1 2 3 4 5 6

21. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6

22. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
A[2, 6]

23. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
A[6]
A[2, 6]

24. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
A[1]
A[6]
A[1]
A[2, 6]

25. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
A[2, 6] = A[6] - A[1]
A[6]
A[1]
A[2, 6]

26. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
To calculate the sum between range [I, j]
A[2, 6] = A[6] - A[1]
A[6]
A[1]
A[2, 6]

27. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
Formula -
To calculate the sum between range [I, j]
A[2, 6] = A[6] - A[1]
A[i, j] = A[j] - A[i - 1]
A[6]
A[1]
A[2, 6]

28. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
Formula -
To calculate the sum between range [I, j]
A[2, 6] = A[6] - A[1]
Calculate the sum between range [3, 5] ?
A[i, j] = A[j] - A[i - 1]
A[6]
A[1]
A[2, 6]

29. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
Formula -
To calculate the sum between range [I, j]
A[2, 6] = A[6] - A[1]
Calculate the sum between range [3, 5] ?
A[3, 5] = A[5] - A[3 - 1]
A[3, 5] = A[5] - A[2]
A[3, 5] = 10 - 7
A[3, 5] = 3
A[i, j] = A[j] - A[i - 1]
A[6]
A[1]
A[2, 6]

30. Prefix sum
6 3 -2 4 -1 0 -5
Generalization -
i = 0 1 2 3 4 5 6
6 9 7 11 10 10 5
Prefix Sum Array -
A[ ] =
A[ ] =
i = 0 1 2 3 4 5 6
A[6]
A[1]
A[2, 6]
Formula -
To calculate the sum between range [I, j]
A[2, 6] = A[6] - A[1]
Calculate the sum between range [3, 5] ?
A[3, 5] = A[5] - A[3 - 1]
A[3, 5] = A[5] - A[2]
A[3, 5] = 10 - 7
A[3, 5] = 3
O(1)
A[i, j] = A[j] - A[i - 1]

31. • To calculate prefix sum array of n size array
Time complexity - O(n)
Analysis of Algorithm -
• Time taken to perform range sum query is -
Time complexity - O(1)

32. Analysis of Algorithm -
• Total time taken to pre process the n size array and to perform range query is -
Time complexity - + O(1)O(n)
~ O(n)
• To calculate prefix sum array of n size array
Time complexity - O(n)
• Time taken to perform range sum query is -
Time complexity - O(1)

33. • It takes O(n) time to calculate prefix sum array of n size array.
Key takeaway from this lesson -
• It takes O(1) time to perform range sum query on n size array.

34. Here is the link to video tutorial :-
https://youtu.be/pVS3yhlzrlQ