#1871

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Jump Game VII

The problem asks to determine if we can reach the last index of a binary string, with a set range of jumps between indices.

string dynamic programming sliding window prefix sum

Problem Statement

You are given a binary string s and two integers minJump and maxJump. Starting from index 0, which is always '0', you can move from index i to index j if the following conditions are fulfilled: 1) i < j <= i + maxJump, 2) s[j] == '0', 3) j >= i + minJump. Determine whether you can reach index s.length - 1.

Return true if you can reach the last index in the string s, otherwise return false. For example, given s = "011010", minJump = 2, and maxJump = 3, the correct answer is true, as there is a valid path to the end.

Examples

Example 1

Input: s = "011010", minJump = 2, maxJump = 3

Output: true

In the first step, move from index 0 to index 3. In the second step, move from index 3 to index 5.

Example 2

Input: s = "01101110", minJump = 2, maxJump = 3

Output: false

Example details omitted.

Constraints

2 <= s.length <= 105
s[i] is either '0' or '1'.
s[0] == '0'
1 <= minJump <= maxJump < s.length

Solution Approach

State Transition Dynamic Programming

Track each reachable index starting from index 0 using dynamic programming. For each index i, maintain the interval of possible jumps [i + minJump, i + maxJump], and check if any valid position leads to the last index.

Sliding Window for Optimization

Use a sliding window to track valid jumps within the range [i + minJump, i + maxJump]. This reduces unnecessary recalculations and optimizes the process of finding the next reachable index.

Prefix Sum or Cumulative Reach

Leverage a prefix sum or cumulative reach approach to store the status of whether an index can be reached, updating the possible jumps efficiently as you iterate over the string.

Complexity Analysis

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

The time complexity depends on the final approach but will generally range from O(n) to O(n^2) based on how efficiently you implement the jump range evaluation. Space complexity can be reduced with sliding window techniques or kept linear with a dynamic programming approach.

What Interviewers Usually Probe

Look for a candidate's understanding of state transition DP.
Evaluate their ability to optimize using sliding windows.
See how they manage time and space complexity when implementing dynamic programming in a real-world scenario.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the solution by checking all possible combinations instead of limiting the range of valid moves.
Failure to optimize the solution by using a sliding window, leading to inefficient time complexity.
Not managing memory effectively when using dynamic programming to track reachable indices, potentially causing excessive space usage.

Follow-up variants

Consider cases where the string starts with a larger number of '1's, blocking early jumps.
What if there were more than two jump ranges instead of just minJump and maxJump?
What if the string had a large number of consecutive '1's, requiring careful interval handling?

FAQ

How do I handle the state transition in Jump Game VII?

In Jump Game VII, handle state transitions by tracking the valid indices you can jump to within the range [i + minJump, i + maxJump], ensuring that all positions fall within this interval.

What is the time complexity of the Jump Game VII problem?

The time complexity of Jump Game VII depends on the approach, but a well-optimized solution using dynamic programming or sliding window would typically run in O(n) time.

How does dynamic programming apply to Jump Game VII?

Dynamic programming helps track reachable indices and manage overlapping subproblems efficiently. Each state transition from index to index is stored, reducing redundant checks.

What are the challenges when solving Jump Game VII?

Challenges include managing the range of jumps efficiently, optimizing the process to avoid recalculating all potential jumps, and handling large input sizes without exceeding time and space limits.

Can I use greedy algorithms to solve Jump Game VII?

Greedy algorithms are not ideal for Jump Game VII, as they may overlook optimal paths. The problem requires dynamic programming to ensure that all possible jumps are considered and evaluated.

terminal

Solution

Solution 1: Prefix Sum + Dynamic Programming

We define a prefix sum array $pre$ of length $n+1$, where $pre[i]$ represents the number of reachable positions in the first $i$ positions of $s$. We define a boolean array $f$ of length $n$, where $f[i]$ indicates whether $s[i]$ is reachable. Initially, $pre[1] = 1$ and $f[0] = true$.

1

2

3

4

5

6

7

8

9

10

11

12

class Solution:
    def canReach(self, s: str, minJump: int, maxJump: int) -> bool:
        n = len(s)
        pre = [0] * (n + 1)
        pre[1] = 1
        f = [True] + [False] * (n - 1)
        for i in range(1, n):
            if s[i] == "0":
                l, r = max(0, i - maxJump), i - minJump
                f[i] = l <= r and pre[r + 1] - pre[l] > 0
            pre[i + 1] = pre[i] + f[i]
        return f[-1]

Continue Practicing

#1888 minimum number of flips to make the binary string alternating #2024 maximize the confusion of an exam #2209 minimum white tiles after covering with carpets