Pipeline Vector Evaluator 49 — Problem Statement & Solution Guide
Problem Description
Given a sequence of integers representing pipeline and vector metrics, construct an optimal algorithm to compute the sum of the elements under given operational constraints.
DSA Pattern Breakdown
DSA Pattern Breakdown
"Pipeline Vector Evaluator 49"
WHY DOES IT MATTER?
DP transforms combinatorial explosion into tractable linear work.
OPTIMIZATION CHALLENGE
The key is collapsing the state space to constant size while preserving correctness.
REAL-WORLD CONNECTION
It mirrors pipeline scheduling where each stage's optimal throughput depends on a few prior stages.
Always identify the minimal set of previous results needed; extra dimensions waste memory and time.
COMPLEXITY AT A GLANCE
O(n)O(1)Core Theory — Why This Approach?
Dynamic programming solves this problem by recognizing that the optimal sum up to any index depends only on a fixed number of previous states, turning an exponential search into a linear recurrence. A naive recursive enumeration explores all 2^n subsets, quickly exhausting time limits for large n, whereas DP caches intermediate results and leverages optimal substructure to compute the answer in O(n) time.
Interview Questions on This Problem
Q1How does the DP recurrence for this problem avoid recomputation of overlapping subproblems?
It stores the best sum for each prefix in an array and reuses those values when extending the prefix, turning exponential calls into linear updates.
Q2What is the impact of allowing or disallowing adjacent selections on the DP state definition?
If adjacent selections are forbidden, the recurrence must reference the state two positions back; otherwise it can reference the immediate previous state.
Q3Can this DP be reduced to O(1) space, and if so, how?
Yes, because each step only needs the last two computed values, we can keep two scalar variables instead of the full array.
Examples
Input
[1, 2, 3, 4, 5]
Output
15
Explanation: Step-by-step: with input [1, 2, 3, 4, 5], we calculate the sum of the elements, giving output 15
Input
[10, 20, 30, 40, 50]
Output
150
Explanation: Step-by-step: with input [10, 20, 30, 40, 50], we calculate the sum of the elements, giving output 150
Constraints
- 1 <= N <= 10^5
- -10^4 <= metrics[i] <= 10^4
- 1 <= K <= N
Optimal Approach & Strategy
Iterate once, updating a DP array (or two variables) with the recurrence that captures the constraint, achieving O(n) time.
Brute Force Approach
Enumerate every subset of indices and compute its sum, checking constraints for each, which is O(2^n) time.
Verified Code Solutions
function solution(nums) { let sum = 0; for (let i = 0; i < nums.length; i++) { sum += nums[i]; } return sum; }class Solution { public: int solution(vector<int>& nums) { int sum = 0; for (int num : nums) { sum += num; } return sum; } }class Solution { public int solution(int[] nums) { int sum = 0; for (int num : nums) { sum += num; } return sum; } }def solution(nums): return sum(nums)function solution(nums) { let sum = 0; for (let i = 0; i < nums.length; i++) { sum += nums[i]; } return sum; }Asked in Top Tech Interviews
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