Prefix Sum Array in DSA Python - 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 i in the original array. It helps to quickly find the sum of any subarray.

beginner

How do you calculate the prefix sum array for [2, 4, 6, 8]?

Start with the first element: 2. Then add each next element to the sum so far: [2, 2+4=6, 6+6=12, 12+8=20]. So the prefix sum array is [2, 6, 12, 20].

beginner

Why is a prefix sum array useful?

It allows you to find the sum of any part of the array quickly by subtracting two prefix sums instead of adding many elements each time.

intermediate

What is the time complexity to build a prefix sum array of size n?

The time complexity is O(n) because you calculate each prefix sum by adding one element to the previous sum in a single pass.

intermediate

How do you find the sum of elements from index i to j using a prefix sum array?

Sum from i to j = prefix_sum[j] - prefix_sum[i-1] if i > 0, else prefix_sum[j] if i == 0.

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

A Difference between elements at index 3 and 2

B Value of the original array at index 3

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

Each prefix sum element at index i is the sum of all original elements from 0 to i.

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

A O(1)

B O(log n)

C O(n)

D O(n^2)

Using prefix sums, subarray sums can be found in constant time O(1) by subtraction.

If prefix_sum = [3, 7, 12, 18], what is the sum of elements from index 1 to 3?

A 18

B 15

C 12

D 9

Sum = prefix_sum[3] - prefix_sum[0] = 18 - 3 = 15.

Which of the following is NOT true about prefix sum arrays?

A They store the product of elements up to index i

B They require O(n) time to build

C They help in quick range sum queries

D They can be used to optimize sum calculations

Prefix sum arrays store sums, not products.

What is the prefix sum array of [1, 1, 1, 1]?

A [4, 3, 2, 1]

B [1, 1, 1, 1]

C [1, 2, 3, 4]

D [0, 1, 2, 3]

Each element is the sum of all previous elements: 1, 1+1=2, 2+1=3, 3+1=4.

Explain how to build a prefix sum array from a given array and why it is useful.

Think about adding elements one by one and how it helps in fast queries.

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

Use subtraction of prefix sums to get the subarray sum.

Practice

1. What does a prefix sum array represent for a given array arr?

easy

A. An array where each element is the product of all elements from the start up to that index in arr.

B. An array where each element is the maximum value found in arr up to that index.

C. An array where each element is the difference between consecutive elements in arr.

D. An array where each element is the sum of all elements from the start up to that index in arr.

Solution

1. Step 1: Understand prefix sum definition

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

2. Step 2: Compare options with definition

   Only An array where each element is the sum of all elements from the start up to that index in arr. correctly describes this sum accumulation; others describe different operations.

3. Final Answer:

   An array where each element is the sum of all elements from the start up to that index in arr. -> Option D

4. Quick Check:

   Prefix sum = cumulative sum [OK]

Hint: Prefix sum means adding elements from start to current index [OK]

Common Mistakes:

• Confusing prefix sum with product or max operations
• Thinking prefix sum stores differences
• Assuming prefix sum stores original array values

2. Which of the following Python code snippets correctly initializes a prefix sum array prefix for an input list arr?

easy

A. prefix = [arr[0]]
for i in range(1, len(arr)):
prefix.append(arr[i] - prefix[i-1])

B. prefix = []
for i in range(len(arr)):
prefix.append(arr[i] + arr[i-1])

C. prefix = [arr[0]]
for i in range(1, len(arr)):
prefix.append(prefix[i-1] + arr[i])

D. prefix = [0] * len(arr)
for i in range(len(arr)):
prefix[i] = arr[i] * prefix[i-1]

Solution

1. Step 1: Analyze code for prefix sum logic

   prefix = [arr[0]] for i in range(1, len(arr)): prefix.append(prefix[i-1] + arr[i]) starts prefix with first element and adds current element to previous prefix sum correctly.

2. Step 2: Check other options for errors

   prefix = [] for i in range(len(arr)): prefix.append(arr[i] + arr[i-1]) adds current and previous array elements incorrectly; C multiplies instead of sums; A subtracts instead of sums.

3. Final Answer:

   prefix = [arr[0]]\nfor i in range(1, len(arr)):\n prefix.append(prefix[i-1] + arr[i]) -> Option C

4. Quick Check:

   Prefix sum uses cumulative addition [OK]

Hint: Start prefix with first element, then add previous prefix sum [OK]

Common Mistakes:

• Using multiplication instead of addition
• Adding wrong elements or indices
• Initializing prefix array incorrectly

3. What is the output of the following code?

arr = [3, 1, 4, 1, 5]
prefix = [arr[0]]
for i in range(1, len(arr)):
    prefix.append(prefix[i-1] + arr[i])
print(prefix)

medium

A. [3, 4, 8, 9, 14]

B. [3, 1, 4, 1, 5]

C. [3, 3, 7, 8, 13]

D. [0, 3, 4, 8, 9]

Solution

1. Step 1: Calculate prefix sums step-by-step

   Start with prefix = [3]. Then add 1: 3+1=4, add 4: 4+4=8, add 1: 8+1=9, add 5: 9+5=14.

2. Step 2: Form the final prefix array

   prefix = [3, 4, 8, 9, 14]

3. Final Answer:

   [3, 4, 8, 9, 14] -> Option A

4. Quick Check:

   Sum accumulates correctly [OK]

Hint: Add current element to last prefix sum [OK]

Common Mistakes:

• Confusing prefix array with original array
• Off-by-one errors in indexing
• Incorrect initial prefix value

4. Find the error in this prefix sum code and choose the corrected version:

arr = [2, 4, 6]
prefix = [0]
for i in range(len(arr)):
    prefix.append(prefix[i] + arr[i])
print(prefix)

medium

A. Change prefix = [0] to prefix = [arr[0]] and loop from 1 to len(arr)-1.

B. Change prefix.append(prefix[i] + arr[i]) to prefix.append(prefix[i-1] + arr[i]).

C. Remove the initial zero from prefix and start loop from 0 as is.

D. No error; code works correctly and prints prefix sums.

Solution

1. Step 1: Identify initialization and loop issues

   Initializing prefix with [0] and looping from 0 adds an extra leading zero. Tracing: i=0 append(0+2)=2 -> [0,2]; i=1 append(2+4)=6 -> [0,2,6]; i=2 append(6+6)=12 -> [0,2,6,12]. This produces prefix sums of length n+1 with prefix[0]=0, whereas standard is length n starting from arr[0].

2. Step 2: Correct initialization and loop range

   Start prefix with first element arr[0], then loop from index 1 to end to add sums correctly.

3. Final Answer:

   Change prefix = [0] to prefix = [arr[0]] and loop from 1 to len(arr)-1. -> Option A

4. Quick Check:

   Proper start and loop fix prefix sums [OK]

Hint: Initialize prefix with first element, loop from second element [OK]

Common Mistakes:

• Starting prefix with zero instead of first element
• Looping from zero causing off-by-one errors
• Using wrong prefix index inside loop

5. Given an array arr of length n, how can you use a prefix sum array to find the sum of elements between indices l and r (inclusive) efficiently?

hard

A. Calculate sum as prefix[r] * prefix[l].

B. Calculate sum as prefix[r] - prefix[l-1] if l > 0, else prefix[r].

C. Calculate sum as prefix[r] + prefix[l] always.

D. Calculate sum by looping from l to r and adding elements.

Solution

1. Step 1: Understand prefix sum usage for range sums

   Prefix sum at index r stores sum from start to r. To get sum from l to r, subtract sum from start to l-1.

2. Step 2: Handle edge case when l is zero

   If l is zero, sum is prefix[r] directly since no subtraction needed.

3. Final Answer:

   Calculate sum as prefix[r] - prefix[l-1] if l > 0, else prefix[r]. -> Option B

4. Quick Check:

   Range sum = prefix[r] - prefix[l-1] [OK]

Hint: Subtract prefix before l from prefix at r for range sum [OK]

Common Mistakes:

• Adding prefix sums instead of subtracting
• Ignoring edge case when l = 0
• Using multiplication or looping unnecessarily