Min Cost Climbing Stairs

Problem

https://leetcode.com/problems/min-cost-climbing-stairs/

You are given an integer array cost where cost[i] is the cost of i:sup:`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

Pattern

Array, Dynamic Programming

Approaches

Explanation

Note that the minimum cost to reach step i is the the minimum of 2 values: 1. the minimum cost to reach step i-1 plus the cost of step i-1 2. the minimum cost to reach step i-2 plus the cost of step i-2 The base cases are the minimum cost to reach step 0 and step 1 are cost[0] and cost[1].

Thus, we can use dynamic programming to solve this problem. We create an array min_cost_to_climb where min_cost_to_climb[i] is the minimum cost to reach step i. We then calculate the minimum cost to reach each step from 2 to n+1 (n+1 because the top is 1 step past cost[-1]) using the recurrence relation above. The value min_cost_to_climb[n + 1] is the final answer.

Code

def minCostClimbingStairs(cost: list[int]) -> int:
    """Return the minimum cost to reach the top of the staircase."""
    n = len(cost)
    min_cost_to_climb = [0] * (n + 1)
    for i in range(2, n + 1):
        min_cost_to_climb[i] = min(
            min_cost_to_climb[i - 1] + cost[i - 1],
            min_cost_to_climb[i - 2] + cost[i - 2],
        )
    return min_cost_to_climb[n]

Test

>>> from min_cost_climbing_stairs__dynamic_programming import minCostClimbingStairs
>>> minCostClimbingStairs([10, 15, 20])
15
>>> minCostClimbingStairs([1, 100, 1, 1, 1, 100, 1, 1, 100, 1])
6
min_cost_climbing_stairs__dynamic_programming.minCostClimbingStairs(cost: list[int]) → int

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