#838

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Push Dominoes

In the "Push Dominoes" problem, you simulate the falling dominoes based on their initial states and determine their final positions.

two pointers string dynamic programming

Problem Statement

You are given a string representing a line of dominoes. Each domino is either upright ('.'), falling to the left ('L'), or falling to the right ('R'). Initially, some dominoes are pushed to the left or right simultaneously. In each subsequent second, any domino that is falling to the left pushes the adjacent domino on the left, and similarly, any domino that is falling to the right pushes its neighboring domino to the right. A domino that has forces acting on it from both sides remains upright.

Your task is to determine the final state of the dominoes after all movements have occurred. The simulation continues until all dominoes either fall or remain standing. The final state should be represented by the dominoes' positions at the end of the process.

Examples

Example 1

Input: dominoes = "RR.L"

Output: "RR.L"

The first domino expends no additional force on the second domino.

Example 2

Input: dominoes = ".L.R...LR..L.."

Output: "LL.RR.LLRRLL.."

Example details omitted.

Constraints

n == dominoes.length
1 <= n <= 105
dominoes[i] is either 'L', 'R', or '.'.

Solution Approach

Simulate the Dominoes Falling

First, iterate through the dominoes from left to right and then right to left, updating the dominoes' states based on whether they are pushed by others. This state transition simulation mimics the forces acting on the dominoes as they fall.

Dynamic Programming to Track the States

Use dynamic programming to track the forces acting on each domino. Keep a running state of the left and right pushes and update each domino's state based on its neighbors. The DP approach allows for efficient state transitions over time.

Two Pointers for Optimization

Two pointers can be employed to track the effects of falling dominoes in both directions. By processing from left to right and then right to left, the problem can be broken down efficiently, ensuring a time complexity of O(N).

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	O(N)

The time complexity depends on the final approach, but the solution typically runs in O(N) due to the single pass required to simulate the dominoes' movements in both directions. Space complexity is O(N) as we store the states of each domino during the simulation process.

What Interviewers Usually Probe

Look for understanding of state transitions in dynamic systems.
Check how well the candidate can optimize a problem involving multiple iterations over a large dataset.
Evaluate the candidate's ability to apply dynamic programming or two pointers in a string manipulation context.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the force balance when both directions act on the same domino, leading to incorrect final states.
Failing to optimize the algorithm, leading to unnecessary time complexity when simulating the falling process.
Incorrect handling of edge cases like dominoes with no initial movement (all '.' characters).

Follow-up variants

Consider variations where dominoes can be pushed in a random sequence instead of initially falling left or right.
Simulate the scenario with additional constraints such as multiple groups of falling dominoes.
Implement the solution with a different space complexity requirement, optimizing the algorithm further.

FAQ

What is the key approach to solving the Push Dominoes problem?

The key approach is to simulate the falling dominoes using dynamic programming, where each domino's state is updated based on its neighbors, and two pointers help optimize the solution.

How can dynamic programming be applied in this problem?

Dynamic programming is used to track the forces acting on each domino, allowing for efficient state transitions as dominoes fall over time.

What is the time complexity of the Push Dominoes problem?

The time complexity is typically O(N), where N is the number of dominoes, as the solution requires a single pass through the dominoes to simulate the falling process.

Can you explain the two-pointer approach for Push Dominoes?

The two-pointer approach processes the dominoes from both directions, updating their states efficiently and ensuring an optimal solution in linear time.

What are some common pitfalls in solving the Push Dominoes problem?

Common pitfalls include incorrect handling of the balance between forces acting on a domino and failing to optimize the solution for time complexity.

terminal

Solution

Solution 1: Multi-Source BFS

Treat all initially pushed dominoes (`L` or `R`) as **sources**, which simultaneously propagate their forces outward. Use a queue to perform BFS layer by layer (0, 1, 2, ...):

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class Solution:
    def pushDominoes(self, dominoes: str) -> str:
        n = len(dominoes)
        q = deque()
        time = [-1] * n
        force = defaultdict(list)
        for i, f in enumerate(dominoes):
            if f != '.':
                q.append(i)
                time[i] = 0
                force[i].append(f)
        ans = ['.'] * n
        while q:
            i = q.popleft()
            if len(force[i]) == 1:
                ans[i] = f = force[i][0]
                j = i - 1 if f == 'L' else i + 1
                if 0 <= j < n:
                    t = time[i]
                    if time[j] == -1:
                        q.append(j)
                        time[j] = t + 1
                        force[j].append(f)
                    elif time[j] == t + 1:
                        force[j].append(f)
        return ''.join(ans)

Continue Practicing

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