#45

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Jump Game II

Jump Game II requires finding the minimum jumps to reach the end of an array using dynamic programming and greedy techniques.

array dynamic programming greedy

Problem Statement

In Jump Game II, you are given an array nums where each element represents the maximum jump length from that index. Starting at index 0, you need to determine the minimum number of jumps to reach the last index of the array. Each jump can take you forward, with the goal being to minimize the number of jumps.

The problem requires navigating through the array by leveraging the maximum jumps available at each index. You must determine the fewest jumps required to reach the last index, using the transition states of reachable indices while ensuring you can always reach the end. The test cases guarantee a valid solution.

Examples

Example 1

Input: nums = [2,3,1,1,4]

Output: 2

The minimum number of jumps to reach the last index is 2. Jump 1 step from index 0 to 1, then 3 steps to the last index.

Example 2

Input: nums = [2,3,0,1,4]

Output: 2

Example details omitted.

Constraints

1 <= nums.length <= 104
0 <= nums[i] <= 1000
It's guaranteed that you can reach nums[n - 1].

Solution Approach

Greedy Approach

A greedy approach can be used where we iteratively select the farthest reachable index from the current position. By keeping track of the current range and the next farthest index we can reach, we can optimize the number of jumps needed. This method ensures that we minimize jumps by taking the largest possible steps within each jump range.

State Transition Dynamic Programming

State transition dynamic programming is the main pattern for this problem. By maintaining the minimum jumps needed to reach each index, we can compute the jumps required from each state. The transition function updates based on the farthest reachable index, making the state space dynamic, and thus efficiently minimizing the total jumps.

Efficient Complexity Management

The optimal solution can be obtained using an approach where we update the number of jumps required based on the reachable indices from the current range. Managing the state transitions between indices ensures that we achieve both time and space efficiency, avoiding the need for exhaustive search or recursive calls, which would result in higher time complexities.

Complexity Analysis

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

The time complexity for the optimal approach is O(n), where n is the length of the input array. Each index is processed once, with updates occurring based on reachable ranges. The space complexity can vary based on the method used but generally remains O(1) for the greedy approach, as no additional space is required beyond tracking the current range and number of jumps.

What Interviewers Usually Probe

Can you describe how the greedy approach minimizes the number of jumps?
Do you understand how state transitions optimize the solution in Jump Game II?
Are you able to efficiently implement a solution with a time complexity of O(n)?

Common Pitfalls or Variants

Common pitfalls

A common pitfall is not updating the farthest reachable index correctly, which can result in missed opportunities to minimize jumps.
Failing to handle edge cases, such as when the array has only one element, can cause incorrect jump calculations.
Using a brute force solution instead of greedy or dynamic programming can lead to inefficient performance, especially for large inputs.

Follow-up variants

Jump Game I - Minimum Jumps with no backtracking.
Jump Game III - Adding obstacles and varying jump limits.
Jump Game IV - Multi-dimensional arrays with jumps.

FAQ

How does the greedy approach solve Jump Game II?

The greedy approach in Jump Game II works by choosing the farthest reachable index at each step, ensuring that we minimize the number of jumps needed to reach the last index.

What is the time complexity of the optimal solution for Jump Game II?

The time complexity of the optimal solution is O(n), where n is the length of the array. This is achieved by processing each index once.

Why is dynamic programming used in Jump Game II?

Dynamic programming is used in Jump Game II to track the minimum jumps required to reach each index and efficiently calculate the optimal path to the end.

What are the edge cases in Jump Game II?

Edge cases include small arrays, such as those with only one element, and arrays with large jump values. Handling these ensures the solution works for all inputs.

How does state transition help in Jump Game II?

State transition in Jump Game II helps by tracking reachable indices and dynamically updating the minimum jumps, optimizing the process of reaching the last index.

terminal

Solution

Solution 1: Greedy Algorithm

We can use a variable $mx$ to record the farthest position that can be reached from the current position, a variable $last$ to record the position of the last jump, and a variable $ans$ to record the number of jumps.

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class Solution:
    def jump(self, nums: List[int]) -> int:
        ans = mx = last = 0
        for i, x in enumerate(nums[:-1]):
            mx = max(mx, i + x)
            if last == i:
                ans += 1
                last = mx
        return ans

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