Description
leetcode 70
You are climbing a stair case. It takes n steps to reach to the top.

Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?

Note: Given n will be a positive integer.

Java DP solution; Time O(n) Space O(n)

Induction could be visualized as picture below (n=3 example),
imgn=3 example
Iterating i from 3 to n. For every i, it could be from reached from two ways – one step from position “i-1”, or two steps jump from position “i-2”.

Look at code below, and think about it.

class Solution {
    public int climbStairs(int n) {
        /**
        * Java Dynamic Programming solution
        */
        if (n==1)   return 1;
        int[] dp = new int[n+1];
        dp[1] = 1; dp[2] = 2;
        for (int i=3; i<=n; i++) {
            dp[i] = dp[i-1] + dp[i-2];
        }
        
        return dp[n];
    }
}

imgRuntime: 0ms

Python fabonacci number solution Time O(n), O(1)

class Solution(object):
    def climbStairs(self, n):
        """
        :type n: int
        :rtype: int
        :algorithm: fibonacci number
        """
        if n==1:
            return 1
        if n==2:
            return 2
        ret = 0
        fib_i_minus_2 = 1
        fib_i_minus_1 = 2
        for i in xrange(3, n+1):
            ret = fib_i_minus_2 + fib_i_minus_1
            fib_i_minus_2 = fib_i_minus_1
            fib_i_minus_1 = ret
        return ret

imgRuntime: 40ms