70. Climbing Stairs

September 27, 2024 · View on GitHub

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Description

You are climbing a staircase. It takes n steps to reach the top.

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

 

Example 1:

Input: n = 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps

Example 2:

Input: n = 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

 

Constraints:

  • 1 <= n <= 45

Solutions

Solution 1: Recursion

We define f[i]f[i] to represent the number of ways to climb to the ii-th step, then f[i]f[i] can be transferred from f[i−1]f[i - 1] and f[i−2]f[i - 2], that is:

f[i]=f[i−1]+f[i−2]f[i] = f[i - 1] + f[i - 2]

The initial conditions are f[0]=1f[0] = 1 and f[1]=1f[1] = 1, that is, the number of ways to climb to the 0th step is 1, and the number of ways to climb to the 1st step is also 1.

The answer is f[n]f[n].

Since f[i]f[i] is only related to f[i−1]f[i - 1] and f[i−2]f[i - 2], we can use two variables aa and bb to maintain the current number of ways, reducing the space complexity to O(1)O(1).

The time complexity is O(n)O(n), and the space complexity is O(1)O(1).

Python3

class Solution:
    def climbStairs(self, n: int) -> int:
        a, b = 0, 1
        for _ in range(n):
            a, b = b, a + b
        return b

Java

class Solution {
    public int climbStairs(int n) {
        int a = 0, b = 1;
        for (int i = 0; i < n; ++i) {
            int c = a + b;
            a = b;
            b = c;
        }
        return b;
    }
}

C++

class Solution {
public:
    int climbStairs(int n) {
        int a = 0, b = 1;
        for (int i = 0; i < n; ++i) {
            int c = a + b;
            a = b;
            b = c;
        }
        return b;
    }
};

Go

func climbStairs(n int) int {
	a, b := 0, 1
	for i := 0; i < n; i++ {
		a, b = b, a+b
	}
	return b
}

TypeScript

function climbStairs(n: number): number {
    let p = 1;
    let q = 1;
    for (let i = 1; i < n; i++) {
        [p, q] = [q, p + q];
    }
    return q;
}

Rust

impl Solution {
    pub fn climb_stairs(n: i32) -> i32 {
        let (mut p, mut q) = (1, 1);
        for i in 1..n {
            let t = p + q;
            p = q;
            q = t;
        }
        q
    }
}

JavaScript

/**
 * @param {number} n
 * @return {number}
 */
var climbStairs = function (n) {
    let a = 0,
        b = 1;
    for (let i = 0; i < n; ++i) {
        const c = a + b;
        a = b;
        b = c;
    }
    return b;
};

PHP

class Solution {
    /**
     * @param Integer $n
     * @return Integer
     */
    function climbStairs($n) {
        if ($n <= 2) {
            return $n;
        }
        $dp = [0, 1, 2];
        for ($i = 3; $i <= $n; $i++) {
            $dp[$i] = $dp[$i - 2] + $dp[$i - 1];
        }
        return $dp[$n];
    }
}

Solution 2: Matrix Quick Power to Accelerate Recursion

We set Fib(n)Fib(n) to represent a $1 \times 2matrixmatrix\begin{bmatrix} F_n & F_{n - 1} \end{bmatrix},where, where F_nandandF_{n - 1}aretheare then−thand-th and (n - 1)$-th Fibonacci numbers respectively.

We hope to derive Fib(n)Fib(n) based on Fib(n−1)=[Fn−1Fn−2]Fib(n-1) = \begin{bmatrix} F_{n - 1} & F_{n - 2} \end{bmatrix}. That is to say, we need a matrix basebase, so that Fib(n−1)×base=Fib(n)Fib(n - 1) \times base = Fib(n), that is:

[Fn−1Fn−2]×base=[FnFn−1]\begin{bmatrix} F_{n - 1} & F_{n - 2} \end{bmatrix} \times base = \begin{bmatrix} F_n & F_{n - 1} \end{bmatrix}

Since Fn=Fn−1+Fn−2F_n = F_{n - 1} + F_{n - 2}, the first column of the matrix basebase is:

[11]\begin{bmatrix} 1 \\ 1 \end{bmatrix}

The second column is:

[10]\begin{bmatrix} 1 \\ 0 \end{bmatrix}

Therefore:

[Fn−1Fn−2]×[1110]=[FnFn−1]\begin{bmatrix} F_{n - 1} & F_{n - 2} \end{bmatrix} \times \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix} = \begin{bmatrix} F_n & F_{n - 1} \end{bmatrix}

We define the initial matrix res=[11]res = \begin{bmatrix} 1 & 1 \end{bmatrix}, then FnF_n is equal to the first element of the first row of the result matrix of resres multiplied by basen−1base^{n - 1}. We can solve it using matrix quick power.

The time complexity is O(log⁡n)O(\log n), and the space complexity is O(1)O(1).

Python3

import numpy as np


class Solution:
    def climbStairs(self, n: int) -> int:
        res = np.asmatrix([(1, 1)], np.dtype("O"))
        factor = np.asmatrix([(1, 1), (1, 0)], np.dtype("O"))
        n -= 1
        while n:
            if n & 1:
                res *= factor
            factor *= factor
            n >>= 1
        return res[0, 0]

Java

class Solution {
    private final int[][] a = {{1, 1}, {1, 0}};

    public int climbStairs(int n) {
        return pow(a, n - 1)[0][0];
    }

    private int[][] mul(int[][] a, int[][] b) {
        int m = a.length, n = b[0].length;
        int[][] c = new int[m][n];
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                for (int k = 0; k < a[0].length; ++k) {
                    c[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        return c;
    }

    private int[][] pow(int[][] a, int n) {
        int[][] res = {{1, 1}, {0, 0}};
        while (n > 0) {
            if ((n & 1) == 1) {
                res = mul(res, a);
            }
            n >>= 1;
            a = mul(a, a);
        }
        return res;
    }
}

C++

class Solution {
public:
    int climbStairs(int n) {
        vector<vector<long long>> a = {{1, 1}, {1, 0}};
        return pow(a, n - 1)[0][0];
    }

private:
    vector<vector<long long>> mul(vector<vector<long long>>& a, vector<vector<long long>>& b) {
        int m = a.size(), n = b[0].size();
        vector<vector<long long>> res(m, vector<long long>(n));
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                for (int k = 0; k < a[0].size(); ++k) {
                    res[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        return res;
    }

    vector<vector<long long>> pow(vector<vector<long long>>& a, int n) {
        vector<vector<long long>> res = {{1, 1}, {0, 0}};
        while (n) {
            if (n & 1) {
                res = mul(res, a);
            }
            a = mul(a, a);
            n >>= 1;
        }
        return res;
    }
};

Go

type matrix [2][2]int

func climbStairs(n int) int {
	a := matrix{{1, 1}, {1, 0}}
	return pow(a, n-1)[0][0]
}

func mul(a, b matrix) (c matrix) {
	m, n := len(a), len(b[0])
	for i := 0; i < m; i++ {
		for j := 0; j < n; j++ {
			for k := 0; k < len(a[0]); k++ {
				c[i][j] += a[i][k] * b[k][j]
			}
		}
	}
	return
}

func pow(a matrix, n int) matrix {
	res := matrix{{1, 1}, {0, 0}}
	for n > 0 {
		if n&1 == 1 {
			res = mul(res, a)
		}
		a = mul(a, a)
		n >>= 1
	}
	return res
}

TypeScript

function climbStairs(n: number): number {
    const a = [
        [1, 1],
        [1, 0],
    ];
    return pow(a, n - 1)[0][0];
}

function mul(a: number[][], b: number[][]): number[][] {
    const [m, n] = [a.length, b[0].length];
    const c = Array(m)
        .fill(0)
        .map(() => Array(n).fill(0));
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            for (let k = 0; k < a[0].length; ++k) {
                c[i][j] += a[i][k] * b[k][j];
            }
        }
    }
    return c;
}

function pow(a: number[][], n: number): number[][] {
    let res = [
        [1, 1],
        [0, 0],
    ];
    while (n) {
        if (n & 1) {
            res = mul(res, a);
        }
        a = mul(a, a);
        n >>= 1;
    }
    return res;
}

JavaScript

/**
 * @param {number} n
 * @return {number}
 */
var climbStairs = function (n) {
    const a = [
        [1, 1],
        [1, 0],
    ];
    return pow(a, n - 1)[0][0];
};

function mul(a, b) {
    const [m, n] = [a.length, b[0].length];
    const c = Array(m)
        .fill(0)
        .map(() => Array(n).fill(0));
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            for (let k = 0; k < a[0].length; ++k) {
                c[i][j] += a[i][k] * b[k][j];
            }
        }
    }
    return c;
}

function pow(a, n) {
    let res = [
        [1, 1],
        [0, 0],
    ];
    while (n) {
        if (n & 1) {
            res = mul(res, a);
        }
        a = mul(a, a);
        n >>= 1;
    }
    return res;
}