Cumulative Matrix Traversal — Problem Statement & Solution Guide
Problem Description
Given an array or sequence of length N representing numerical values or system metrics, compute the cumulative matrix traversal according to the target algorithm rules. The cumulative matrix traversal is the sum of all elements in the array.
DSA Pattern Breakdown
DSA Pattern Breakdown
"Cumulative Matrix Traversal"
WHY DOES IT MATTER?
Cumulative sum (prefix‑sum) is a foundational DP pattern that appears in range‑query problems, sliding‑window calculations, and probability distributions. Mastery of this pattern enables you to convert quadratic‑time aggregations into linear‑time solutions.
OPTIMIZATION CHALLENGE
The key insight is to recognize that each element's contribution to the final answer is additive and can be accumulated incrementally, removing the need for nested loops or repeated summations.
REAL-WORLD CONNECTION
Think of a bank ledger where each transaction updates the account balance. Instead of recomputing the balance from scratch for every statement, the system maintains a running total, mirroring the prefix‑sum technique.
During an interview, write the recurrence first (sum[i] = sum[i‑1] + arr[i]) and immediately point out that you only need the previous sum, so you can collapse the DP array into a single variable.
COMPLEXITY AT A GLANCE
O(N)O(1)Core Theory — Why This Approach?
The cumulative matrix traversal problem is a classic illustration of prefix‑sum dynamic programming. The naive interpretation treats each element independently and recomputes the total for every query, leading to O(N^2) time when multiple cumulative queries are required. By recognizing that the sum of the first i elements can be expressed recursively as sum(i) = sum(i‑1) + a[i], we transform the problem into a linear‑time DP where each state depends only on its immediate predecessor. This eliminates redundant work, because each array element contributes exactly once to the final answer, and the recurrence can be evaluated in a single pass. The optimal paradigm therefore leverages the overlapping sub‑structure of cumulative sums and stores only the most recent aggregate, achieving O(N) time with O(1) auxiliary space.
Interview Questions on This Problem
Q1How would you compute the sum of all elements in a massive stream of integers where you cannot store the entire array in memory?
Maintain a running total variable that you update with each incoming integer: total += value. This yields the cumulative sum in O(1) extra space and O(N) time, regardless of stream size.
Q2Given an array, how can you answer multiple range‑sum queries efficiently after a single preprocessing step?
Build a prefix‑sum array pref where pref[i] = sum of first i elements. Then any range sum [l, r] is pref[r] - pref[l‑1] in O(1) time, with O(N) preprocessing.
Q3Why might a recursive solution that sums the array by splitting it in half be less optimal than an iterative prefix‑sum approach?
The recursive divide‑and‑conquer method incurs O(log N) call‑stack depth and additional overhead for function calls, while an iterative prefix‑sum traverses the array once with minimal constant factors and O(1) extra space.
Examples
Input
[1, 2, 3, 4, 5]
Output
15
Explanation: Step-by-step: Given the input array [1, 2, 3, 4, 5], we need to compute the cumulative sum. We start with the first element 1, then add the second element 2 to get 3, then add the third element 3 to get 6, then add the fourth element 4 to get 10, and finally add the fifth element 5 to get 15.
Input
[5, 4, 3, 2, 1]
Output
15
Explanation: Step-by-step: Given the input array [5, 4, 3, 2, 1], we need to compute the cumulative sum. We start with the first element 5, then add the second element 4 to get 9, then add the third element 3 to get 12, then add the fourth element 2 to get 14, and finally add the fifth element 1 to get 15.
Constraints
- 1 <= N <= 10^5
- -10^9 <= arr[i] <= 10^9
- Time Complexity expected: O(N) or O(N log N)
- Space Complexity expected: O(1) or O(N)
Optimal Approach & Strategy
Compute a running total in one pass (or build a prefix‑sum array) so that the overall sum is obtained in O(N) time with O(1) extra space.
Brute Force Approach
Iterate over the array for each query and sum all elements each time, resulting in O(N^2) time for multiple queries.
Verified Code Solutions
function solution(nums) {
let cumulativeSum = 0;
for (let num of nums) {
cumulativeSum += num;
}
return cumulativeSum;
}class Solution {
public:
int solution(vector<int> nums) {
int cumulativeSum = 0;
for (int num : nums) {
cumulativeSum += num;
}
return cumulativeSum;
}
};class Solution {
public int solution(int[] nums) {
int cumulativeSum = 0;
for (int num : nums) {
cumulativeSum += num;
}
return cumulativeSum;
}
}def solution(nums):
cumulative_sum = 0
for num in nums:
cumulative_sum += num
return cumulative_sumfunction solution(nums) {
let cumulativeSum = 0;
for (let num of nums) {
cumulativeSum += num;
}
return cumulativeSum;
}Asked in Top Tech Interviews
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