Prefix Sum Array in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a Prefix Sum Array?

A Prefix Sum Array is a new array where each element at index i is the sum of all elements from the start up to index i in the original array.

beginner

How do you calculate the prefix sum for an array element at index i?

PrefixSum[i] = PrefixSum[i-1] + OriginalArray[i], with PrefixSum[0] = OriginalArray[0].

intermediate

Why is a Prefix Sum Array useful?

It helps quickly find the sum of any subarray by subtracting prefix sums, reducing repeated addition and improving efficiency.

beginner

What is the time complexity to build a Prefix Sum Array for an array of size n?

O(n), because you calculate each prefix sum once by adding the current element to the previous prefix sum.

intermediate

How do you find the sum of elements between indices i and j using a Prefix Sum Array?

Sum(i to j) = PrefixSum[j] - PrefixSum[i-1] (if i > 0), else PrefixSum[j] if i = 0.

What does the element at index 3 in a prefix sum array represent?

A Value of the original array at index 3

B Difference between elements at index 3 and 2

C Sum of elements from index 3 to end of the array

D Sum of elements from index 0 to 3 in the original array

The prefix sum at index 3 is the sum of all elements from the start (index 0) up to index 3.

What is the time complexity to find the sum of any subarray using a prefix sum array?

A O(n)

B O(1)

C O(log n)

D O(n^2)

Using prefix sums, subarray sums can be found in constant time by subtracting two prefix sums.

If the original array is [2, 4, 6, 8], what is the prefix sum array?

A [2, 6, 12, 20]

B [2, 4, 6, 8]

C [2, 8, 14, 22]

D [2, 6, 10, 18]

Prefix sums: 2, 2+4=6, 6+6=12, 12+8=20.

How do you initialize the first element of a prefix sum array?

A Double the first element

B Zero

C Same as the first element of the original array

D Sum of all elements

The first prefix sum is just the first element of the original array.

Which of these is NOT a benefit of using a prefix sum array?

A Less memory usage than original array

B Avoid repeated addition for sums

C Faster subarray sum queries

D Simplifies range sum calculations

Prefix sum arrays use extra memory equal to the original array size, so memory usage is not less.

Explain how to build a prefix sum array from a given array.

Think about adding elements one by one to get running totals.

Describe how to use a prefix sum array to find the sum of elements between two indices i and j.

Use subtraction of prefix sums to get subarray sums quickly.

Practice

1. What is the main purpose of a Prefix Sum Array in algorithms?

easy

A. To quickly find the sum of elements in any subarray

B. To sort the array elements in ascending order

C. To find the maximum element in the array

D. To reverse the array elements

Solution

1. Step 1: Understand what prefix sums represent

   A prefix sum array stores cumulative sums of elements from the start up to each index.

2. Step 2: Identify the use of prefix sums

   Using prefix sums, we can find the sum of any subarray quickly by subtracting two prefix sums.

3. Final Answer:

   To quickly find the sum of elements in any subarray -> Option A

4. Quick Check:

   Prefix Sum Array = Fast subarray sums [OK]

Hint: Prefix sums help sum subarrays fast by subtraction [OK]

Common Mistakes:

• Thinking prefix sums sort the array
• Confusing prefix sums with maximum element search
• Assuming prefix sums reverse the array

2. Which of the following C# code snippets correctly initializes a prefix sum array prefix from an integer array arr?

easy

A. int[] prefix = new int[arr.Length+1]; for(int i=1; i<=arr.Length; i++) prefix[i] = prefix[i-1] + arr[i];

B. int[] prefix = new int[arr.Length]; for(int i=0; i

C. int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

D. int[] prefix = new int[arr.Length]; prefix[0] = 0; for(int i=1; i

Solution

1. Step 1: Check initialization of prefix array

   int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

2. Step 2: Verify loop for cumulative sum

   The loop starts from 1 and adds prefix[i-1] + arr[i], correctly building prefix sums.

3. Final Answer:

   int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i -> Option C

4. Quick Check:

   Correct prefix sum initialization = int[] prefix = new int[arr.Length]; prefix[0] = arr[0]; for(int i=1; i

Hint: Start prefix[0] = arr[0], then add previous prefix + current element [OK]

Common Mistakes:

• Using wrong array length for prefix
• Starting loop from 0 instead of 1
• Accessing arr out of bounds (like arr[i] when i == arr.Length)

3. Given the array arr = new int[] {2, 4, 6, 8} and its prefix sum array prefix computed as usual, what is the output of prefix[3] - prefix[1]?

medium

A. 18

B. 10

C. 12

D. 14

Solution

1. Step 1: Compute prefix sum array

   prefix[0]=2, prefix[1]=2+4=6, prefix[2]=6+6=12, prefix[3]=12+8=20

2. Step 2: Calculate prefix[3] - prefix[1]

   20 - 6 = 14

3. Final Answer:

   14 -> Option D

4. Quick Check:

   Sum from index 2 to 3 = 6+8 = 14 [OK]

Hint: Subtract prefix sums to get subarray sum [OK]

Common Mistakes:

• Confusing indices and subtracting wrong prefix values
• Adding prefix sums instead of subtracting
• Forgetting prefix sums start at index 0

4. Consider the following code snippet to build a prefix sum array. Identify the error:

int[] prefix = new int[arr.Length];
prefix[0] = 0;
for(int i = 1; i < arr.Length; i++) {
    prefix[i] = prefix[i-1] + arr[i];
}

medium

A. Loop should start from i = 0, not 1

B. prefix[0] should be arr[0], not 0

C. Array prefix should have length arr.Length + 1

D. No error, code is correct

Solution

1. Step 1: Check initialization of prefix[0]

   Setting prefix[0] = 0 is incorrect; it should be arr[0] to start cumulative sum correctly.

2. Step 2: Verify loop and array length

   The loop starting at 1 and array length are fine; only prefix[0] initialization is wrong.

3. Final Answer:

   prefix[0] should be arr[0], not 0 -> Option B

4. Quick Check:

   Correct prefix start = arr[0] [OK]

Hint: Initialize prefix[0] with arr[0], not zero [OK]

Common Mistakes:

• Starting loop at 0 causing index errors
• Using wrong prefix array size
• Assuming zero initialization is always correct

5. You have an integer array arr of length 6: {3, 1, 4, 1, 5, 9}. You want to find the sum of elements from index 2 to 5 inclusive using a prefix sum array prefix where prefix[i] stores sum of elements from index 0 to i. Which expression correctly computes this sum?

hard

A. prefix[5] - prefix[1]

B. prefix[5] - prefix[2]

C. prefix[4] - prefix[1]

D. prefix[5] + prefix[1]

Solution

1. Step 1: Understand prefix sum indexing

   prefix[i] = sum of arr[0] to arr[i]. To get sum from index 2 to 5, subtract prefix[1] (sum up to index 1) from prefix[5].

2. Step 2: Write the correct expression

   Sum = prefix[5] - prefix[1]

3. Final Answer:

   prefix[5] - prefix[1] -> Option A

4. Quick Check:

   Sum(2 to 5) = prefix[5] - prefix[1] [OK]

Hint: Sum(i to j) = prefix[j] - prefix[i-1] [OK]

Common Mistakes:

• Subtracting prefix[i] instead of prefix[i-1]
• Adding prefix sums instead of subtracting
• Using wrong indices for prefix array