Jump Game VII

Jump Game VII

1. What is this problem about?

The Jump Game VII interview question is a reachability problem on a binary string. You start at index 0, which is always '0'. You can jump from index i to index j if:

1. s[j] == '0'
2. i + minJump <= j <= min(i + maxJump, n - 1) You need to determine if you can reach the last index of the string.

2. Why is this asked in interviews?

Companies like Apple and Amazon use the Jump Game VII coding problem to see if a candidate can optimize a range-based reachability search. A standard BFS or DFS would be $O(N \cdot ( ext{maxJump} - ext{minJump}))$, which is too slow. It tests your ability to use Prefix Sums or a Sliding Window to track "how many reachable points exist in the current jump range."

3. Algorithmic pattern used

This problem is solved using Dynamic Programming with a Sliding Window or Prefix Sum.

1. DP State: dp[i] is true if index i is reachable.
2. The "Reachable Count" trick: To determine if dp[j] is true, you need to know if any index i in the range [j - maxJump, j - minJump] has dp[i] == true.
3. Window Logic: Maintain a running count reachable_count of indices within the current jump window that are marked as reachable.
4. Transition: As you move to index j:
   • Add dp[j - minJump] to the count (if valid).
   • Subtract dp[j - maxJump - 1] from the count (if valid).
   • If s[j] == '0' and reachable_count > 0, then dp[j] = true.

4. Example explanation

s = "011010", minJump = 2, maxJump = 3.

• dp[0] = true.
• $j=2$: Window [2-3, 2-2] = [-1, 0]. dp[0] is in window. count=1. But s[2] is '1'. dp[2]=false.
• $j=3$: Window [0, 1]. dp[0] is in window. count=1. s[3] is '0'. dp[3]=true.
• $j=5$: Window [2, 3]. dp[3] is in window. count=1. s[5] is '0'. dp[5]=true. Result: True.

5. Common mistakes candidates make

• Naive BFS: Visiting every potential jump from every reachable point, which leads to $O(N^2)$ in the worst case.
• Incorrect window boundaries: Miscalculating the indices for the reachable count.
• Ignoring '1's: Forgetting that you can only land on '0's.

6. Interview preparation tip

Range-based reachability can often be optimized by asking: "Is there at least one valid source in my range?" This turns a search problem into a Prefix Sum interview pattern or a sliding window count problem.