Solution Explanation: Find the Highest Altitude

This problem involves finding the highest altitude reached during a bike trip, given an array gain where gain[i] represents the net gain in altitude between points i and i+1. The biker starts at altitude 0.

Approach:

The most efficient approach leverages the concept of prefix sums (also known as cumulative sums). The altitude at each point can be calculated by accumulating the gains. The highest altitude will be the maximum of these accumulated altitudes.

Algorithm:

1. Initialization: Start with a variable highestAltitude set to 0 (the initial altitude). Also, initialize a variable currentAltitude to 0.

2. Iteration: Iterate through the gain array:

   • Add the current gain[i] to currentAltitude.
   • Update highestAltitude if currentAltitude is greater.

3. Return: After iterating through the entire gain array, return highestAltitude.

Time Complexity Analysis:

The algorithm iterates through the gain array once. Therefore, the time complexity is O(n), where n is the length of the gain array.

Space Complexity Analysis:

The algorithm uses a few variables to store the current altitude and the highest altitude. The space used is constant regardless of the input size. Therefore, the space complexity is O(1).

Code Implementation (Python):

class Solution:
    def largestAltitude(self, gain: List[int]) -> int:
        highest_altitude = 0
        current_altitude = 0
        for gain_value in gain:
            current_altitude += gain_value
            highest_altitude = max(highest_altitude, current_altitude)
        return highest_altitude

Code Implementation (Java):

class Solution {
    public int largestAltitude(int[] gain) {
        int highestAltitude = 0;
        int currentAltitude = 0;
        for (int gainValue : gain) {
            currentAltitude += gainValue;
            highestAltitude = Math.max(highestAltitude, currentAltitude);
        }
        return highestAltitude;
    }
}

Code Implementation (C++):

class Solution {
public:
    int largestAltitude(vector<int>& gain) {
        int highestAltitude = 0;
        int currentAltitude = 0;
        for (int gainValue : gain) {
            currentAltitude += gainValue;
            highestAltitude = max(highestAltitude, currentAltitude);
        }
        return highestAltitude;
    }
};

The code in other languages (JavaScript, Go, Rust, PHP, C) follows a very similar structure, adapting the syntax to each language's specifics while maintaining the same algorithmic logic. The core idea remains consistent across all implementations.