Problem Description

In this problem, we are dealing with an engine comprised of multiple pistons, and our goal is to determine the maximum possible area beneath these pistons. The pistons are capable of moving up and down within a defined range.

You are given:

• An integer height, indicating the maximum height a piston can reach.
• An integer array positions, where each element positions[i] represents the current position of piston i, which is equal to the current area beneath it.
• A string directions, where each character directions[i] indicates the current moving direction of piston i, either 'U' for up or 'D' for down.

Each second, every piston moves by 1 unit in its given direction. If a piston reaches one of the boundaries (either position 0 or height), its moving direction will switch.

The task is to return the maximum possible area that can be achieved beneath all pistons.

Intuition

The key to solving this problem is to track the movement of the pistons and determine how their changing positions impact the total area beneath them. Initially, we calculate the sum of the current positions of the pistons to determine the starting area.

For each piston, depending on its direction, we adjust an incremental value (diff) that reflects how the movement of pistons will alter the total area. When pistons shift directions upon reaching the boundaries, we use a dictionary to track events that occur at specific time intervals (delta). This involves adjusting the effect on diff at specific positions in time, effectively simulating the pistons' movement over time.

The solution iterates through these changes and updates the running total area (res). We keep track of the maximum area observed (ans) as the pistons continue moving up and down.

This approach efficiently calculates the maximum potential area by leveraging the predictive modeling of each piston's movement pattern over time.

Learn more about Prefix Sum patterns.

Solution Approach

To solve the problem of calculating the maximum possible area under the pistons, we utilize a combination of calculations and simulations.

Here's a breakdown of the approach:

1. Initialize Variables:

   • Use a defaultdict named delta to keep track of changes in direction at certain time intervals.
   • Initialize diff to track the net change in area over time.
   • Calculate the initial res, which is the total area under the pistons at the starting configuration by summing up all positions.

2. Loop through Pistons:

   • For each piston, based on its direction (U for up or D for down), update the diff and prepare the delta mapping:
     • If the direction is U, increase diff, prepare an event to decrease it when reaching the top boundary (height), and to revert at double-height.
     • If the direction is D, decrease diff, prepare an event to increase it when reaching zero, and to revert at the reach of height.

3. Simulate Movements:

   • Iterate over the sorted delta dictionary to simulate the changes:
     • Update res (the current area under the pistons) by adding (cur - pre) * diff, where cur is the current event's position, and pre is the previous event's position.
     • Adjust diff based on the current event.
     • Determine if the current res is greater than the recorded ans (maximum area so far), and update ans accordingly.

4. Return Result:

   • After processing all the events, return ans, which contains the maximum possible area achieved.

This approach effectively leverages the predictive modeling of piston movement to efficiently calculate the maximum possible area beneath them.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach for the given problem.

Consider the following inputs:

• height = 5
• positions = [2, 3]
• directions = "UD"

The pistons start at positions [2, 3], with the first piston moving up ('U') and the second piston moving down ('D').

Step-by-Step Simulation:

1. Initial Setup:

   • The initial area (res) is the sum of the starting positions: 2 + 3 = 5.
   • We initialize diff as 0 and a defaultdict delta to manage direction changes.

2. Loop Through Pistons:

   • Piston 0 (Moving Up):

     • Current position: 2, starts moving up.
     • Increment diff by 1 because moving up increases area.
     • In delta, set an event to decrease diff at time 3 (when it hits the top boundary, position 5).

   • Piston 1 (Moving Down):

     • Current position: 3, starts moving down.
     • Decrement diff by 1 because moving down decreases area.
     • In delta, set an event to increase diff at time 4 (when it hits the bottom boundary, position 0).

3. Simulate Movements:

   • Start by iterating over the sorted keys of delta (events at time 3 and 4).

   • Time 3 Event:

     • diff changes from 1 to 0 (Piston 0 hits the top).
     • Before adjusting diff, compute the area addition: (3 - 0) * 1 = 3, update res to 8.
     • ans is updated to 8 as it's greater than initial res.

   • Time 4 Event:

     • Adjust diff for Piston 1 hitting the bottom boundary: from 0 to 1.
     • Compute area addition from 3 to 4 with diff as 0: 0. Area remains 8.
     • ans remains 8 as it's the maximum so far.

4. Return Result:

   • After processing all events, the maximum possible area (ans) is 8.

This walkthrough demonstrates how the solution tracks pistons' movements with events to compute the maximum area efficiently over time.

Solution Implementation

Time and Space Complexity

• Time Complexity: The time complexity of the code is primarily determined by the loop iterating through the positions and directions arrays and the sorting operation on the delta dictionary. The loop has a time complexity of O(n), where n is the length of the positions. The sorting operation has a time complexity of O(m log m), where m is the number of unique keys in the delta dictionary. Therefore, the overall time complexity is O(n + m log m).

• Space Complexity: The space complexity is determined by the space used for the delta dictionary, which can contain up to n keys in the worst-case scenario. Thus, the overall space complexity is O(m), where m is the number of distinct keys in the delta dictionary.

Learn more about how to find time and space complexity quickly.