Min Cost Climbing Stairs

LeetCode 747 | Difficulty: Easy

Easy

Problem Description

You are given an integer array cost where cost[i] is the cost of i^th step on a staircase. Once you pay the cost, you can either climb one or two steps.

You can either start from the step with index 0, or the step with index 1.

Return the minimum cost to reach the top of the floor.

Example 1:

Input: cost = [10,15,20]
Output: 15
Explanation: You will start at index 1.
- Pay 15 and climb two steps to reach the top.
The total cost is 15.

Example 2:

Input: cost = [1,100,1,1,1,100,1,1,100,1]
Output: 6
Explanation: You will start at index 0.
- Pay 1 and climb two steps to reach index 2.
- Pay 1 and climb two steps to reach index 4.
- Pay 1 and climb two steps to reach index 6.
- Pay 1 and climb one step to reach index 7.
- Pay 1 and climb two steps to reach index 9.
- Pay 1 and climb one step to reach the top.
The total cost is 6.

Constraints:

- `2 <= cost.length <= 1000`

- `0 <= cost[i] <= 999`

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: 146 ms)

Metric	Value
Runtime	146 ms
Memory	36.8 MB
Date	2022-01-24

Solution

public class Solution {
    public int MinCostClimbingStairs(int[] cost) {
        int n = cost.Length;
    int[] dp = new int[n];
    dp[0] = cost[0];
    dp[1] = cost[1];
    for (int i = 2; i < n; i++)
    {
        dp[i] = cost[i]+Math.Min(dp[i-1], dp[i-2]);
    }
    return Math.Min(dp[n-1], dp[n-2]);
    }
}

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Complexity Analysis

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

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Interview Tips

Key Points

• Start by clarifying edge cases: empty input, single element, all duplicates.
• 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.
• LeetCode provides 3 hint(s) for this problem — try solving without them first.

💡 Hints

Hint 1: Build an array dp where dp[i] is the minimum cost to climb to the top starting from the ith staircase.

Hint 2: Assuming we have n staircase labeled from 0 to n - 1 and assuming the top is n, then dp[n] = 0, marking that if you are at the top, the cost is 0.

Hint 3: Now, looping from n - 1 to 0, the dp[i] = cost[i] + min(dp[i + 1], dp[i + 2]). The answer will be the minimum of dp[0] and dp[1]