#1824

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Sideway Jumps

Solve the Minimum Sideway Jumps problem using state transition dynamic programming to minimize side jumps while navigating obstacles.

array dynamic programming greedy

Problem Statement

You are given a 3-lane road with length n, where each point is labeled from 0 to n. A frog starts at point 0 on lane 2 and needs to reach point n while avoiding obstacles. You are provided with an array obstacles of length n + 1, where obstacles[i] represents the obstacle at point i. If obstacles[i] is 0, there's no obstacle at that point; otherwise, the frog must navigate around it. The frog can either move forward on the same lane if there's no obstacle or perform a side jump to another lane if it’s clear.

The goal is to determine the minimum number of side jumps the frog needs to make in order to reach point n. The frog starts at point 0 on lane 2 and the frog can switch to any lane at a given point, as long as there’s no obstacle on the lane it switches to. Use dynamic programming to track the frog's state at each point and optimize the number of side jumps.

Examples

Example 1

Input: obstacles = [0,1,2,3,0]

Output: 2

The optimal solution is shown by the arrows above. There are 2 side jumps (red arrows). Note that the frog can jump over obstacles only when making side jumps (as shown at point 2).

Example 2

Input: obstacles = [0,1,1,3,3,0]

Output: 0

There are no obstacles on lane 2. No side jumps are required.

Example 3

Input: obstacles = [0,2,1,0,3,0]

Output: 2

The optimal solution is shown by the arrows above. There are 2 side jumps.

Constraints

obstacles.length == n + 1
1 <= n <= 5 * 105
0 <= obstacles[i] <= 3
obstacles[0] == obstacles[n] == 0

Solution Approach

State Transition Dynamic Programming

The frog's movement through the road can be represented as a dynamic programming problem. We use a 2D array where each element represents the minimum number of side jumps needed to reach that point on a particular lane. At each point, the frog has three potential lane choices, and the transition to the next point depends on whether the lane has an obstacle or not.

Greedy Optimization

By optimizing the frog’s path dynamically based on the current lane, we can determine when to side jump. A greedy approach helps minimize the number of side jumps by selecting the least costly transitions at each point.

Final Path Calculation

At each step, consider all three lanes and calculate the side jumps needed to proceed. After processing all points, the value at the last point on any lane will give the minimum side jumps required to reach the end.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the final dynamic programming approach, typically O(n), where n is the length of the obstacle array. The space complexity also depends on the DP table used, which is O(3n) in the worst case due to tracking the three lanes at each step.

What Interviewers Usually Probe

Assess whether the candidate can identify the state transition dynamic programming pattern for pathfinding problems.
Evaluate the candidate’s understanding of how greedy choices fit within dynamic programming for optimization.
Check if the candidate is comfortable dealing with arrays representing multiple possible states at each step.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the problem by attempting unnecessary backtracking or multiple state recalculations.
Forgetting to account for all three lanes and the possibility of switching between them at each point.
Misunderstanding the role of side jumps and assuming that forward movement is always possible.

Follow-up variants

The problem can be extended to more than three lanes, increasing complexity.
Consider variations where side jumps are only allowed to adjacent lanes, limiting the flexibility of the solution.
Introduce time constraints or additional dynamic programming states based on different types of obstacles.

FAQ

What is the main pattern to solve the Minimum Sideway Jumps problem?

The main pattern is state transition dynamic programming, where the frog’s movement across lanes is optimized with minimum side jumps.

What’s the time complexity of the Minimum Sideway Jumps problem?

The time complexity is typically O(n), where n is the length of the obstacles array.

How do side jumps work in the Minimum Sideway Jumps problem?

Side jumps allow the frog to change lanes at the same point if there’s no obstacle on the new lane, minimizing the total number of jumps.

What’s the space complexity of the Minimum Sideway Jumps problem?

The space complexity is O(3n), as we need to track the frog's position on three possible lanes at each point.

Can the problem be solved without dynamic programming?

It can be solved without dynamic programming, but dynamic programming is the most efficient and optimal approach for minimizing side jumps.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j]$ as the minimum number of sidesteps for the frog to reach the $i$-th point and be on the $j$-th lane (index starts from $0$).

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class Solution:
    def minSideJumps(self, obstacles: List[int]) -> int:
        f = [1, 0, 1]
        for v in obstacles[1:]:
            for j in range(3):
                if v == j + 1:
                    f[j] = inf
                    break
            x = min(f) + 1
            for j in range(3):
                if v != j + 1:
                    f[j] = min(f[j], x)
        return min(f)

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