Problem Description

Alice and Bob are watering plants numbered from 0 to n - 1 in a row. Each plant requires a specific amount of water. Alice starts watering from the left (plant 0) moving to the right, while Bob begins from the right (plant n - 1) moving to the left. Both Alice and Bob are equipped with a watering can which is initially full, and they start watering simultaneously.

The key rules for watering the plants are as follows:

1. The time to water each plant is the same, regardless of the water needed.
2. They can only water a plant if their can has enough water for the whole plant. If not, they refill their can instantly and then water the plant.
3. If Alice and Bob meet at the same plant, the one with more water will water it. If they have the same amount, Alice does it.

The goal is to find out the total number of times Alice and Bob have to refill their watering cans in order to water all the plants.

The input to the problem is an array plants containing the amount of water each plant needs, and two integers, capacityA and capacityB, indicating the capacity of Alice's and Bob's watering cans, respectively.

Intuition

To solve this problem, we can simulate the process of watering the plants. Alice and Bob move towards each other from opposite ends of the row. They each water the plants according to their capacities. We track the remaining water in their cans after watering each plant. If a can doesn't have enough water to fully water the current plant:

• We increment a counter representing the number of refills.
• We refill the can, subtracting the water needed for the current plant from the full capacity.

We continue this process until Alice and Bob meet at the same plant or pass by each other, meaning all plants have been watered. At the moment they reach the same plant, we compare their remaining water and make a decision based on the rule.

The intuition behind this approach is to mimic the real-world actions Alice and Bob would take, updating values and counters as necessary. By simulating the watering process step by step, we avoid missing any cases where a refill might be necessary and ensure we account for every plant.

Learn more about Two Pointers patterns.

Solution Approach

The solution adopts a two-pointer approach, with one pointer (i) starting from the beginning of the array (Alice's side), and another pointer (j) starting from the end of the array (Bob's side). These pointers represent the current position of Alice and Bob, respectively. As they move toward each other, we calculate the number of refills needed based on the rules given.

To implement the solution, the following steps are followed:

1. Initialize two variables a and b to represent the full capacities of Alice's and Bob's watering cans (capacityA and capacityB) as they'll be refilled to these values.

2. Initialize the pointer i to 0 and j to len(plants) - 1.

3. Initialize a counter ans to 0 for counting the total number of refills made by Alice and Bob.

4. Use a while loop to iterate as long as i <= j (meaning Alice and Bob are not past each other):

   • Check if i == j, which means Alice and Bob are at the same plant.

     • If so, check if their can capacity (max(capacityA, capacityB)) is less than the water needed for this plant (plants[i]). If it is, increment the ans counter as this will require a refill.
     • Then break the loop as all plants have been watered.

   • For Alice:

     • If the current water (capacityA) is less than what the current plant needs (plants[i]), refill Alice's can and increment the ans counter.
     • Otherwise, reduce Alice's current water by the amount needed for the plant.
     • Move to the next plant by incrementing i.

   • For Bob (similar to Alice):

     • If Bob's current water (capacityB) is less than what the current plant needs (plants[j]), refill Bob's can and increment the ans counter.
     • Otherwise, reduce Bob's current water by the amount needed for the plant.
     • Move to the previous plant by decrementing j.

5. After the while loop, return the ans counter which now holds the total number of refills required.

The two-pointer technique allows us to effectively simulate the action of both individuals as they meet in the middle, considering both directions simultaneously. This approach ensures that both capacities and refill requirements are checked at every step, thus finding the minimum number of refills needed to water all plants.

Example Walkthrough

Let's illustrate the solution approach using a small example:

Suppose we have an array plants = [1, 2, 4, 2, 3], and capacityA (Alice's can capacity) is 5, and capacityB (Bob's can capacity) is 6.

Here is a step-by-step walkthrough:

• Initialize Alice's current water a = capacityA = 5 and Bob's current water b = capacityB = 6.
• Initialize the pointers for Alice and Bob i = 0 and j = 4 respectively, pointing at the start and end of the plants array.
• Initialize the number of refills counter ans = 0.

Start the while loop:

• Step 1: Alice at i = 0, Bob at j = 4

  • Alice has enough water for plant 0 (needs 1 water), so capacityA = 5 - 1 = 4. No refills needed.
  • Bob has enough water for plant 4 (needs 3 water), so capacityB = 6 - 3 = 3. No refills needed.
  • Move Alice to i = 1 and Bob to j = 3.

• Step 2: Alice at i = 1, Bob at j = 3

  • Alice has enough water for plant 1 (needs 2 water), so capacityA = 4 - 2 = 2. No refills needed.
  • Bob has enough water for plant 3 (needs 2 water), so capacityB = 3 - 2 = 1. No refills needed.
  • Move Alice to i = 2 and Bob to j = 2.

• Step 3: Alice at i = 2, Bob at j = 2 (they meet at the same plant)

  • The current plant 2 requires 4 water.
  • Alice only has capacityA = 2, which is not enough. She refills her can and capacityA is now 5 - 4 = 1, and ans is incremented to 1.
  • Bob doesn't water since Alice has already watered the plant.
  • Since Alice and Bob meet at the same plant and have gone through all plants, the while loop ends.

The total number of refills ans is 1. Therefore, Alice and Bob needed to refill their cans only once combined to water all the plants.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is O(n), where n is the length of the plants list. This is because the code uses a single while loop to traverse the list from both ends towards the center, performing a constant number of calculations at each step. In the worst case, the loop runs for the entire length of the list if there's only one refill station (when i starts at 0 and j at n-1, incrementing and decrementing respectively until they meet), thus the time complexity is linear with respect to the number of plants.

Space Complexity

The space complexity of the code is O(1). This is because the code uses a fixed amount of extra space (variables to keep track of the positions i, j, the remaining capacities capacityA and capacityB, and the counter ans for the number of refills). This space requirement does not grow with the size of the input (plants list), hence the constant space complexity.

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