Frog Jump

LeetCode 403 | Difficulty: Hard

Hard

Problem Description

A frog is crossing a river. The river is divided into some number of units, and at each unit, there may or may not exist a stone. The frog can jump on a stone, but it must not jump into the water.

Given a list of stones positions (in units) in sorted ascending order, determine if the frog can cross the river by landing on the last stone. Initially, the frog is on the first stone and assumes the first jump must be 1 unit.

If the frog's last jump was k units, its next jump must be either k - 1, k, or k + 1 units. The frog can only jump in the forward direction.

Example 1:

Input: stones = [0,1,3,5,6,8,12,17]
Output: true
Explanation: The frog can jump to the last stone by jumping 1 unit to the 2nd stone, then 2 units to the 3rd stone, then 2 units to the 4th stone, then 3 units to the 6th stone, 4 units to the 7th stone, and 5 units to the 8th stone.

Example 2:

Input: stones = [0,1,2,3,4,8,9,11]
Output: false
Explanation: There is no way to jump to the last stone as the gap between the 5th and 6th stone is too large.

Constraints:

- `2 <= stones.length <= 2000`

- `0 <= stones[i] <= 2^31 - 1`

- `stones[0] == 0`

- `stones` is sorted in a strictly increasing order.

Topics: Array, Dynamic Programming

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Approach

Dynamic Programming

Break the problem into overlapping subproblems. Define a state (what information do you need?), a recurrence (how does state[i] depend on smaller states?), and a base case. Consider both top-down (memoization) and bottom-up (tabulation) approaches.

When to use

Optimal substructure + overlapping subproblems (counting ways, min/max cost, feasibility).

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Solutions

Solution 1: C# (Best: 239 ms)

Metric	Value
Runtime	239 ms
Memory	45.7 MB
Date	2022-02-05

Solution

public class Solution {
    public bool CanCross(int[] stones) {
        Dictionary<int, HashSet<int>> d = new Dictionary<int, HashSet<int>>();
    d.Add(0, new HashSet<int>() {1});
    for (int i = 1; i < stones.Length; i++)
    {
        d.Add(stones[i], new HashSet<int>() );
    }
    int target = stones[stones.Length-1];
    
    for (int i = 0; i < stones.Length-1; i++)
    {
        int stone = stones[i];
        foreach (var step in d[stone])
        {
            int reach = stone+step;
            if(reach == target)
            {
                return true;
            }
            if(d.ContainsKey(reach))
            {
                var set = d[reach];
                set.Add(step);
                if(step-1 > 0) set.Add(step-1);
                set.Add(step+1);
            }
        }
    }

    return false;
}
    }

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Complexity Analysis

Approach	Time	Space
Dynamic Programming	$O(n)$	$O(n)$

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Interview Tips

Key Points

• Break the problem into smaller subproblems. Communicate your approach before coding.
• Define the DP state clearly. Ask: "What is the minimum information I need to make a decision at each step?"
• Consider if you can reduce space by only keeping the last row/few values.