Minimum Time to Visit All Houses - Solution & Explanation

Problem Overview: You are given an array representing houses and the time required to move between them. The task is to compute the minimum total time needed to visit all houses while accounting for travel costs between indices. The key challenge is efficiently calculating cumulative travel time across ranges of houses.

Approach 1: Direct Range Summation (Brute Force) (Time: O(n^2), Space: O(1))

A straightforward method computes the travel time between houses by summing the required movement cost each time you move across a range. For every step or transition, iterate through the array segment and accumulate the cost manually. This approach repeatedly recalculates the same partial sums, which leads to quadratic behavior in the worst case. While simple to implement, it becomes inefficient when the number of houses grows large.

Approach 2: Prefix Sum Optimization (Time: O(n), Space: O(n))

The optimal strategy precomputes cumulative travel times using a prefix sum array. Each index stores the total time required to reach that point from the start. Once built, the travel time between any two houses can be computed in constant time using prefix[r] - prefix[l]. Instead of repeatedly iterating over ranges, you perform a single pass to build the prefix array and then use fast lookups during the traversal logic.

This technique is widely used when working with cumulative ranges in arrays. The key insight is that many travel calculations overlap. Storing cumulative totals eliminates redundant work. After computing prefix sums, you iterate through the houses and accumulate the required movement time based on prefix differences.

Recommended for interviews: Interviewers typically expect the prefix sum optimization. A brute force implementation shows you understand the problem mechanics, but recognizing repeated range calculations and replacing them with a prefix array demonstrates stronger algorithmic intuition. The optimal solution runs in O(n) time with O(n) space and scales efficiently for large inputs.