Minimum Jumps to Reach Home
1. What is this problem about?
In "Minimum Jumps to Reach Home," you control a bug that starts at position 0 and wants to reach a target position x. The bug has two jump rules:
- It can jump
a steps forward.
- It can jump
b steps backward.
However, there are restrictions:
- The bug cannot jump to any "forbidden" positions.
- The bug cannot perform two backward jumps in a row.
- The bug cannot jump to a negative position.
- There is usually a practical upper limit to how far the bug can jump (often around 2000 + a + b).
The goal is to find the minimum number of jumps to reach x. If it's impossible, return -1.
2. Why is this asked in interviews?
This is a classic Breadth-First Search (BFS) problem used by companies like Microsoft and Amazon. It tests:
- Shortest Path Intuition: BFS is the standard for finding the minimum number of steps in an unweighted graph.
- State Space Definition: Recognizing that the "state" isn't just the position, but also how you got there (was the last jump backward?).
- Boundary Conditions: Handling the forbidden list and the upper limit effectively.
3. Algorithmic pattern used
The pattern is BFS on a Graph.
- State:
(current_position, can_jump_backward).
- Queue: Store the state and the current jump count.
- Visited Set: Use a set to keep track of visited states
(position, direction) to avoid infinite loops.
- Logic:
- If
current_position == x, return the count.
- Forward jump is always an option if the new position is not forbidden and within the upper bound.
- Backward jump is only an option if the previous jump was not backward and the new position is non-negative and not forbidden.
4. Example explanation
a = 3, b = 2, target = 9, forbidden = [1, 6]
- Start at 0. Queue:
[(0, True, 0)]
- From 0, jump forward to 3. Queue:
[(3, True, 1)] (Backward from 0 is negative).
- From 3, jump forward to 6 (Forbidden) or backward to 1 (Forbidden).
- Wait, the bug must reach 9.
If we go forward from 0 to 3, then forward to 6 (No), we must find another path.
If we jump forward to 3, then 6 (No), maybe jump to 3, then back to 1 (No)...
Eventually, the BFS will explore all valid combinations of
+3 and -2 to find the shortest path to 9.
5. Common mistakes candidates make
- Incomplete State: Only tracking the position in the
visited set. Since you can visit the same position twice (once from a forward jump and once from a backward jump), only tracking position will cause you to miss valid paths.
- No Upper Bound: Without a limit on how far forward the bug can go, the BFS might run forever if the target is unreachable.
- Forgetting the "No Double Backward" Rule: This is the most common logic error.
6. Interview preparation tip
When solving BFS problems, always identify what defines a unique "state." If the rules say "you can't do X twice in a row" or "you can only do Y if Z happened," that information MUST be part of your state and your visited set.
