Problem Description

In this problem, we have n plants arranged in a row, numbered from 0 to n - 1. Each plant requires a specific amount of water. We have a watering can with a finite capacity and a river located at x = -1 where we can refill the can. The goal is to find out the minimum number of steps needed to water all the plants by following these rules:

1. We must water the plants in order from left to right.
2. If the watering can does not have enough water to fully water the next plant, we must go back to the river to refill the can to its full capacity before watering that plant.
3. We are not allowed to refill the can before it is completely empty.
4. We start at the river (at x = -1), and each step equates to moving one unit on the x-axis.

The problem asks us to calculate the total number of steps we must take to water all the plants when we are given an array plants, where plants[i] represents the amount of water needed by the ith plant, and an integer capacity, which is the total capacity of the watering can.

Intuition

The intuitive approach to solve this problem is by a simple simulation of the watering process, keeping track of the current position, and the amount of water left in the can.

1. Start from the river with the can filled to capacity.
2. Move towards the plants and water them in sequence until the can is depleted.
3. When the can doesn't have enough water for the next plant, calculate the number of steps to go back to the river, refill the can, and return to the current plant.
4. Each watering step and each walking step is counted to calculate the number of total steps.

The crux of the solution involves calculating additional steps each time a refill is required, which is equal to double the current index (because we need to go back to the river and then return to the plant at index i). After refilling, we subtract the amount of water needed by the current plant from the can's capacity, move to the next plant, and continue this process until all plants are watered. The total number of steps taken throughout this process is the answer.

Solution Approach

The solution is implemented as a function wateringPlants within the Solution class. It takes two arguments: plants, which is a list of integers where each integer represents the water requirement of a plant, and capacity, which is an integer representing the full capacity of the watering can. The function returns an integer that is the total number of steps needed to water all the plants.

Here's a step-by-step explanation of the solution's implementation:

1. We initialize a variable ans to store the total number of steps needed, and set it to 0. We also create a variable cap to keep track of the current water level in the can and initialize it to capacity.

2. We use a for loop that goes through each plant (and its index i) by enumerating over plants. The enumerate function is useful here as it provides both the index and the value from the list.

3. For each plant, we check if the current water level (cap) is sufficient to water the plant (x):

   • If cap >= x, it means we have enough water for the current plant. We water the plant by subtracting x from cap and increment ans by 1 to account for the step taken to water the plant.
   • If cap < x, we do not have enough water and need to refill the can. Before refilling, we calculate the number of steps needed to go back to the river and return to the current plant. This is i * 2 (double the distance from the river to the plant) plus 1 more step to water the plant. We update ans with these additional steps. We then reset cap to capacity - x since we refill the can and use x amount of water to water the current plant.

4. When the loop is completed, all plants have been watered, and ans contains the total number of steps required. We return ans as the final answer.

The algorithm's time complexity is O(n), where n is the number of plants since every plant is visited at most twice (once while moving forward and once while moving backward if a refill is needed). The space complexity is O(1) as we only use a fixed amount of additional memory (variables ans and cap).

Here is the core implementation encapsulated by the for loop:

for i, x in enumerate(plants):
    if cap >= x:
        cap -= x
        ans += 1
    else:
        cap = capacity - x
        ans += i * 2 + 1

This approach straightforwardly solves the problem efficiently and effectively without the need for complex data structures or patterns.

Example Walkthrough

Let's assume we have 5 plants with the following water requirements given in the array plants = [2, 4, 5, 1, 2] and a watering can with a capacity of 6 units of water. Using the solution approach, let's walk through the process to determine the minimum number of steps needed to water all the plants.

1. Start with the can at full capacity (6 units of water) at the river location which is at x=-1.
2. Move to plant 0 (1 step). The plant requires 2 units of water, and we have enough water. Water the plant (cap = 6 - 2 = 4, ans = 1).
3. Move to plant 1 (1 step). The plant requires 4 units of water, and we have enough water. Water the plant (cap = 4 - 4 = 0, ans = 2).
4. Since the watering can is now empty, move back to the river to refill the can (2 steps back, +2 steps forward to return to plant 1, total 4 steps). Refill the can to full capacity and water plant 2 which requires 5 units of water (cap = 6 - 5 = 1, ans = 6 after refilling and watering).
5. Move to plant 3 (1 step). The plant requires 1 unit of water, and we have enough water. Water the plant (cap = 1 - 1 = 0, ans = 7).
6. Again, the watering can is empty, so move back to the river and refill the can (3 steps back, +3 steps forward to return to plant 3, total 6 steps). Water plant 4 which requires 2 units of water (cap = 6 - 2 = 4, ans = 13 after refilling and watering).

Now all plants have been watered, and the total number of steps taken is 13. Therefore, the wateringPlants function would return 13 for this example.

The steps taken for each action are summarized in the following sequence:

• Start [can=6]
• Water plant 0 [can=4, steps=1]
• Water plant 1 [can=0, steps=2]
• Refill at river, water plant 2 [can=1, steps=6]
• Water plant 3 [can=0, steps=7]
• Refill at river, water plant 4 [can=4, steps=13]
• Done

The answer to how many steps are needed to water all plants is 13 steps.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n), where n is the number of plants. This is because the code iterates through each plant exactly once.

The space complexity of the code is O(1) since a fixed amount of extra space is used regardless of the input size. Additional variables ans and cap are used, but their use does not scale with the number of plants.

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