LeetCode DayDynamic Programming Part1

509. Fibonacci Number

The Fibonacci numbers, commonly denoted F(n) form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,

F(0) = 0, F(1) = 1
F(n) = F(n - 1) + F(n - 2), for n > 1.
Given n, calculate F(n).

Example 1:

Input: n = 2
Output: 1
Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.
Example 2:

Input: n = 3
Output: 2
Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.
Example 3:

Input: n = 4
Output: 3
Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.

Constraints:

0 <= n <= 30
Original Page

Recursion Method

    public int fib(int n) {
        if(n==0){
            return 0;
        }     
        else if(n==1){
            return 1;
        }   
        else{
            return fib(n-1) + fib(n-2);
        }
    }

The Recursion Method is like DFS to go deep and then do the backtracking to get the final answer.
time: O(2^n)
space: O(1)

    private int[] dp = new int[31];
    public int fib(int n) {
        if(n<2){
            dp[n] = n;
            return n;
        }  

        if(n>=2 && dp[n]!=0){
            return dp[n];
        }

        dp[n] = fib(n-1) + fib(n-2);
        return dp[n];
    }




</p>
<p>we can use a global array to save the result to avoid re-recursion of the same elements. e.g. the figure below displays that f(17) and f(18) are two different recursion routes and if we use the normal recursion method we have to calculate them more than once. </p>

<p><img src="https://img.php.cn/upload/article/000/000/000/172127798860260.png" alt="Image description" loading="lazy"    style="max-width:90%"  style="max-width:90%"></p>

<p>time: O(n), space: O(n)</p>

<h2>
  
  
  Dynamic Programming
</h2>



<pre class="brush:php;toolbar:false">    public int fib(int n) {
        if(n<2){
            return n;
        }
        int[] dp = new int[n+1];
        dp[0] = 0;
        dp[1] = 1;
        for(int i=2; i<=n; i++){
            dp[i] = dp[i-1] + dp[i-2];
        }
        return dp[n];
    }

The recursion works from top to bottom and then backtracking, the memory recursion will save the recursion results to avoid double calculation. Now the dynamic programming works from the bottom to the top and saves each step's result to the dp array.
time: O(n)
space: O(n)

Also we can dynamically update the limit num instead of an array. This will save space complexity, especially for a huge number of elements.

    public int fib(int n) {
        if(n<2){
            return n;
        }
        int start = 0;
        int pre = 1;
        int res = pre;

        for(int i=2; i<=n; i++){
            res = start + pre;
            start = pre;
            pre = res;
        }
        return res;
    }

70. Climbing Stairs

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

70. Climbing Stairs

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
Original Page

    public int climbStairs(int n) {
        if(n<3){
            return n;
        }
        int[] dp = new int[n+1];
        dp[0] = 0;
        dp[1] = 1;
        dp[2] = 2;
        for(int i=3; i<=n; i++){
            dp[i] = dp[i-1] + dp[i-2];
        } 
        return dp[n];       
    }

    public int climbStairs(int n) {
        if(n<3){
            return n;
        }

        int prepre = 1;
        int pre = 2;
        int res = 0;
        for(int i=3; i<=n; i++){
            res = prepre + pre;
            prepre = pre;
            pre = res;
        } 
        return res;       
    }

746. Min Cost Climbing Stairs

You are given an integer array cost where cost[i] is the cost of ith 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
Original Page

    public int minCostClimbingStairs(int[] cost) {
        if(cost.length < 2){
            return 0;
        }

        int[] dp = new int[cost.length+1];
        dp[0] = 0;
        dp[1] = 0;

        for(int i=2; i<dp.length; i++){
            dp[i] = Math.min(dp[i-1]+cost[i-1], dp[i-2]+cost[i-2]);
        }
        return dp[dp.length-1];

    }
the key thing of this problem is the `init array` and `the meaning of the array` and the `Recurrence relation`

Be Careful that we should the question has told us that we can start from index 0 and index 1, which implies if the number of stairs is less than 2, we will cost 0 because we can start at that point and the behaviour of start will cost 0, only the behaviour of move cost.

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