#1654

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Jumps to Reach Home

Find the minimum number of jumps needed to reach a target position while avoiding forbidden positions.

array dynamic programming breadth first search

Problem Statement

A bug starts at position 0 on the x-axis and must reach its home at position x. The bug can make jumps forward and backward, where the forward jump length is a and the backward jump length is b.

There are forbidden positions where the bug cannot land. The goal is to determine the minimum number of jumps needed to reach home or return -1 if it's impossible to do so. The bug can jump forward beyond its home but cannot land on negative positions.

Examples

Example 1

Input: forbidden = [14,4,18,1,15], a = 3, b = 15, x = 9

Output: 3

3 jumps forward (0 -> 3 -> 6 -> 9) will get the bug home.

Example 2

Input: forbidden = [8,3,16,6,12,20], a = 15, b = 13, x = 11

Output: -1

Example details omitted.

Example 3

Input: forbidden = [1,6,2,14,5,17,4], a = 16, b = 9, x = 7

Output: 2

One jump forward (0 -> 16) then one jump backward (16 -> 7) will get the bug home.

Constraints

1 <= forbidden.length <= 1000
1 <= a, b, forbidden[i] <= 2000
0 <= x <= 2000
All the elements in forbidden are distinct.
Position x is not forbidden.

Solution Approach

State Transition Dynamic Programming

The problem can be approached using dynamic programming by treating each position on the x-axis as a state. From each state, the bug can transition to a new state by either jumping forward or backward. Use a breadth-first search (BFS) approach to find the minimum path while avoiding forbidden positions.

Breadth-First Search (BFS)

Implement BFS to explore all possible positions the bug can reach. Each position is added to the BFS queue, and the bug either moves forward or backward, ensuring forbidden positions are avoided. Track the number of jumps taken to reach each position.

Handling Forbidden Positions

Forbidden positions must be marked and never revisited during the BFS traversal. This ensures that the bug does not get stuck in a loop or land on an invalid position, helping find the optimal number of jumps.

Complexity Analysis

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

The time and space complexity depend on the number of positions the bug may traverse. Given that the bug can move up to 2000 units in either direction, the BFS approach runs in O(N) time, where N is the number of unique positions being explored.

What Interviewers Usually Probe

Candidates should demonstrate an understanding of BFS and dynamic programming.
Look for the ability to efficiently handle state transitions without revisiting forbidden positions.
Candidates should be comfortable thinking of the problem as a graph traversal challenge.

Common Pitfalls or Variants

Common pitfalls

Failing to properly mark forbidden positions, leading to unnecessary loops in the BFS.
Misunderstanding the jump constraints, especially the ability to jump past the target.
Not considering the backward jumps in the BFS search, which could lead to missing a solution.

Follow-up variants

Change the jumping lengths a and b to test scalability.
Introduce more complex forbidden position patterns.
Increase the target position x for larger-scale problems.

FAQ

What is the best approach for solving the 'Minimum Jumps to Reach Home' problem?

The optimal approach is to use dynamic programming with a breadth-first search (BFS) to explore all reachable positions while avoiding forbidden ones.

How does dynamic programming help in this problem?

Dynamic programming helps by treating each position as a state and exploring transitions between states efficiently using BFS, ensuring minimal jumps to the target position.

What are the forbidden positions in this problem?

Forbidden positions are specific positions on the x-axis that the bug cannot land on. These positions must be avoided during the BFS traversal.

Can the bug jump beyond its home?

Yes, the bug can jump beyond its home but cannot land on negative positions or forbidden positions.

What happens if the bug cannot reach the target position?

If the bug cannot reach the target position, the function should return -1.

terminal

Solution

Solution 1: BFS

We can use the position and jumping direction of the flea as the state, and use BFS to search for the shortest path. The key point of this problem is to determine the right boundary, that is, how far the flea can jump.

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class Solution:
    def minimumJumps(self, forbidden: List[int], a: int, b: int, x: int) -> int:
        s = set(forbidden)
        q = deque([(0, 1)])
        vis = {(0, 1)}
        ans = 0
        while q:
            for _ in range(len(q)):
                i, k = q.popleft()
                if i == x:
                    return ans
                nxt = [(i + a, 1)]
                if k & 1:
                    nxt.append((i - b, 0))
                for j, k in nxt:
                    if 0 <= j < 6000 and j not in s and (j, k) not in vis:
                        q.append((j, k))
                        vis.add((j, k))
            ans += 1
        return -1

Continue Practicing

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