Find the Highest Altitude

Python solution:

class Solution:
    def largestAltitude(self, gain: List[int]) -> int:
        highest_altitude = 0  # Initialize the highest altitude
        current_altitude = 0  # Initialize the current altitude

        for change in gain:  # Iterate through each change in altitude
            current_altitude += change  # Update the current altitude
            highest_altitude = max(highest_altitude, current_altitude)  # Update the highest altitude if current is greater

        return highest_altitude  # Return the highest altitude

This works by keeping track of the current altitude and the highest altitude as we go through each change in altitude. The current altitude is updated by adding the change in altitude for each point, and the highest altitude is updated if the current altitude exceeds the current highest altitude. Finally, the highest altitude is returned.

Abstract Representation of the Problem

We have a sequence of integer values which represent changes in a cumulative quantity. We are required to find the maximum cumulative quantity achieved at any point in the sequence.

Here, the sequence starts at zero and each subsequent position is calculated by adding the change (which could be positive, negative, or zero) at that position to the cumulative quantity at the previous position.

The goal is to find the maximum cumulative quantity achieved throughout the entire sequence.

Coding Constructs

In addition to Running Maximum, another high-level construct or concept that is evident in the solution is Prefix Sum.

Prefix Sum: In the problem, we need to compute a sequence of altitudes from a sequence of changes in altitude (gain). This is essentially calculating the prefix sum of the changes, where each element in the prefix sum sequence is the sum of all the elements before it in the original sequence.

These two high-level constructs - Running Maximum and Prefix Sum - are commonly seen in problems involving sequence data and provide efficient ways to track certain properties (like maximums or cumulative sums) over the course of a sequence.

Identifying Problem Isomorphism

‘Find the Highest Altitude’ can be mapped to ‘Maximum Subarray Sum’.

Both problems involve working with an array of integers and finding a maximum cumulative value. In ‘Find the Highest Altitude’, we calculate the running sum of altitude changes (or a prefix sum) and find the maximum altitude achieved. In ‘Maximum Subarray Sum’, we calculate the maximum sum of any contiguous subarray. In both cases, we’re looking for a maximum cumulative or running value.

‘Maximum Subarray Sum’ is more complex because it involves finding maximums in a subset of array elements, whereas in ‘Find the Highest Altitude’ we consider the entire array.