Climbing Stairs

You are climbing a staircase. 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.
Example 1:
Input:
2
Output:
2
Explanation:
There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps
Example 2:
Input:
3
Output:
3
Explanation:
There are three ways to climb to the top.
1. 1 step + 1 step + 1 step
2. 1 step + 2 steps
3. 2 steps + 1 step

Analysis

DP - Bottom Up Approach
Break this problem into subproblems, with optimal substructure, thus using Dynamic Programming (Bottom Up approach):
State:
dp[i]- number of ways to reach the ith step
State Transfer Function:
dp[i] = dp[i - 1] + dp[i - 2]- taking a step from (i - 1), or taking a step of 2 from (i - 2)
Initial State:
dp[0] = 0;
dp[1] = 1;
dp[2] = 2;
Answer:
dp[n]
Complexity Analysis
  • Time complexity: O(n). Single loop up to n.
  • Space complexity : O(n). dp array of size nis used.
DP - Top-Down Approach - Recursion with memorization
We could define memo[] to store the number of ways to ith step, it helps pruning recursion. And the recursive function could be defined as climb_Stairs(int i, int n, int memo[]) that returns the number of ways from ith step to nth step.
Complexity Analysis
  • Time complexity : O(n). Single loop upto n.
  • Space complexity : O(n). dpdp array of size n is used.
DP - Fibonacci Number - Optimize Space Complexity
Fib(n)=Fib(n−1)+Fib(n−2)
That the nth number only has to do with its previous two numbers, thus we don't have to maintain a whole array of results, just the last 2 results are enough. Thus optimized the space for O(1).
Complexity Analysis
  • Time complexity : O(n). Single loop upto n is required to calculate n th fibonacci number.
  • Space complexity : O(1). Constant space is used.

Solution

Dynamic Programming - Bottom Up - O(n) space, O(n) time (3ms 32.02% AC)
class Solution {
public int climbStairs(int n) {
if (n == 0) return 0;
if (n == 1) return 1;
if (n == 2) return 2;
int[] steps = new int[n + 1];
steps[0] = 0;
steps[1] = 1;
steps[2] = 2;
for (int i = 3; i < n + 1; i++) {
steps[i] = steps[i - 1] + steps[i - 2];
}
return steps[n];
}
}
Dynamic Programming - Recursion with memorization - Top Down - O(n) Space, O(n) Time (3ms 32.02%)
public class Solution {
public int climbStairs(int n) {
int memo[] = new int[n + 1];
return climb_Stairs(0, n, memo);
}
public int climb_Stairs(int i, int n, int memo[]) {
if (i > n) {
return 0;
}
if (i == n) {
return 1;
}
if (memo[i] > 0) {
return memo[i];
}
memo[i] = climb_Stairs(i + 1, n, memo) + climb_Stairs(i + 2, n, memo);
return memo[i];
}
}
DP - Fibonacci Numbers - Space Optimized (2ms 92.09%)
public class Solution {
public int climbStairs(int n) {
if (n == 1) {
return 1;
}
int first = 1;
int second = 2;
for (int i = 3; i <= n; i++) {
int third = first + second;
first = second;
second = third;
}
return second;
}
}

Reference