How do you solve Find Number of Ways to Reach the K-th Stair on LeetCode?

Find Number of Ways to Reach the K-th Stair (LeetCode #3154) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n) time complexity and O(1) space complexity. Key insight: Understanding the relationship between jumps and downs is crucial.

What is the brute force solution for Find Number of Ways to Reach the K-th Stair?

The brute force approach for Find Number of Ways to Reach the K-th Stair is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach explores all possible sequences of moves Alice can make to reach stair k. It involves recursively trying every combination of going down and jumping up, counting valid paths. The optimal Optimal Solution approach improves this to O(n).

What are the interview tips for Find Number of Ways to Reach the K-th Stair?

Always clarify the problem statement and constraints with the interviewer. Think out loud to demonstrate your thought process. Consider edge cases, such as k = 0 or k = 1, early in your solution.

What are common mistakes when solving Find Number of Ways to Reach the K-th Stair?

Not considering the constraints on consecutive down moves. Failing to account for all possible jump values.

#3154

Find Number of Ways to Reach the K-th Stair

Hard

MathDynamic ProgrammingBit ManipulationMemoizationCombinatoricsDynamic ProgrammingCombinatorics

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n)
Space	O(1)	O(1)

💡

Intuition

Time O(n)Space O(1)

The optimal solution uses combinatorial mathematics to calculate the number of ways to reach stair k based on the number of jumps and downward moves. It leverages the fact that the position can be expressed as a function of jumps and downs.

⚙️

Algorithm

3 steps

1Step 1: For each valid number of jumps x, calculate the maximum number of downs y that can be performed.
2Step 2: Use the formula 2^x to determine the number of ways to arrange x jumps and y downs.
3Step 3: Sum the results for all possible x values until the stair k is reached.

solution.py12 lines

1def countWays(k):
2    if k == 0:
3        return 2
4    total_ways = 0
5    for x in range(k + 1):
6        y = k - (2 ** x)
7        if y >= 0 and y % 2 == 0:
8            total_ways += 2 ** x
9    return total_ways
10
11# Example usage
12print(countWays(1))  # Output: 4

ℹ

Complexity note: The time complexity is O(n) because we iterate through possible jump values up to k, and the space complexity is O(1) since we only use a few variables.

1Understanding the relationship between jumps and downs is crucial.
2Recognizing that the problem can be framed in terms of combinatorial arrangements helps simplify the solution.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.