#1732

Easy

LeetCode Problem Workspace

Find the Highest Altitude

Find the highest altitude after a series of altitude changes during a road trip using prefix sum technique.

Problem Statement

A biker is going on a road trip that consists of n + 1 points at different altitudes. The biker starts at point 0 with an altitude of 0. You are given an integer array, gain, where each element gain[i] represents the net altitude change from point i to i + 1 for all 0 <= i < n. Your task is to determine the highest altitude the biker reaches during the trip.

For example, if the array gain is [-5, 1, 5, 0, -7], the biker's altitudes are calculated by starting from 0 and progressively adding each value in the gain array. The goal is to return the highest point reached during the trip.

Examples

Example 1

Input: gain = [-5,1,5,0,-7]

Output: 1

The altitudes are [0,-5,-4,1,1,-6]. The highest is 1.

Example 2

Input: gain = [-4,-3,-2,-1,4,3,2]

Output: 0

The altitudes are [0,-4,-7,-9,-10,-6,-3,-1]. The highest is 0.

Constraints

n == gain.length
1 <= n <= 100
-100 <= gain[i] <= 100

Solution Approach

Prefix Sum Calculation

You can calculate the altitude at each point by progressively adding the values from the gain array to a cumulative sum. Start at 0 and keep a running total of the altitude. As you go through the array, track the maximum altitude you encounter.

Iterate Once Over the Array

A single traversal through the gain array suffices to compute the result. This approach ensures O(N) time complexity, making it efficient even for larger arrays within the problem constraints.

Space Optimization

Since you only need to track the current altitude and the maximum altitude reached, the space complexity can be reduced to O(1). You don’t need any additional storage structures beyond a couple of variables.

Complexity Analysis

Metric	Value
Time	O(N)
Space	O(1)

The time complexity of this solution is O(N) because we only make one pass through the gain array. The space complexity is O(1) because we only maintain a few variables to track the current and maximum altitude, independent of the input size.

What Interviewers Usually Probe

Look for the candidate's ability to apply prefix sum in a simple problem setting.
Assess whether the candidate can explain the efficiency of the O(N) time complexity and O(1) space complexity.
Gauge how well the candidate can identify and optimize unnecessary space usage.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the problem by trying to store the altitudes at every point instead of just tracking the highest altitude.
Misunderstanding the prefix sum concept and attempting a more complex solution like nested loops.
Failing to recognize that only a few variables are needed to track the current and maximum altitude, leading to unnecessary space usage.

Follow-up variants

Extend the problem to handle negative altitude changes more effectively by considering additional edge cases.
Consider situations where the biker stays at the starting altitude or only increases or decreases in a simple pattern.
Explore the problem with varying input sizes to assess performance with maximum constraints.

FAQ

What is the prefix sum technique?

The prefix sum technique involves computing the cumulative sum of an array as you iterate through it, allowing you to efficiently calculate values like total altitude at each point in this problem.

How can I optimize the space complexity for this problem?

You can optimize space complexity by tracking only the current and maximum altitude reached, avoiding the need to store all altitudes.

Why does the time complexity of this solution remain O(N)?

The time complexity is O(N) because we traverse the gain array exactly once, updating the altitude and maximum altitude in each step.

What are some common mistakes in solving this problem?

Some common mistakes include unnecessary nested loops or storing all intermediate altitude values instead of just tracking the maximum altitude.

How does GhostInterview help with problems like this?

GhostInterview provides step-by-step guidance, helping you understand the core concepts like prefix sum and space optimization, so you can solve problems efficiently in interviews.

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Solution

Solution 1: Prefix Sum (Difference Array)

We assume the altitude of each point is $h_i$. Since $gain[i]$ represents the altitude difference between the $i$th point and the $(i + 1)$th point, we have $gain[i] = h_{i + 1} - h_i$. Therefore:

1

2

3

class Solution:
    def largestAltitude(self, gain: List[int]) -> int:
        return max(accumulate(gain, initial=0))

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