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2742. Painting the Walls
Description
You are given two 0indexed integer arrays, cost
and time
, of size n
representing the costs and the time taken to paint n
different walls respectively. There are two painters available:
 A paid painter that paints the
i^{th}
wall intime[i]
units of time and takescost[i]
units of money.  A free painter that paints any wall in
1
unit of time at a cost of0
. But the free painter can only be used if the paid painter is already occupied.
Return the minimum amount of money required to paint the n
walls.
Example 1:
Input: cost = [1,2,3,2], time = [1,2,3,2] Output: 3 Explanation: The walls at index 0 and 1 will be painted by the paid painter, and it will take 3 units of time; meanwhile, the free painter will paint the walls at index 2 and 3, free of cost in 2 units of time. Thus, the total cost is 1 + 2 = 3.
Example 2:
Input: cost = [2,3,4,2], time = [1,1,1,1] Output: 4 Explanation: The walls at index 0 and 3 will be painted by the paid painter, and it will take 2 units of time; meanwhile, the free painter will paint the walls at index 1 and 2, free of cost in 2 units of time. Thus, the total cost is 2 + 2 = 4.
Constraints:
1 <= cost.length <= 500
cost.length == time.length
1 <= cost[i] <= 10^{6}
1 <= time[i] <= 500
Solutions
Solution 1: Memorization
We can consider whether each wall is painted by a paid painter or a free painter. Design a function $dfs(i, j)$, which means that from the $i$th wall, and the current remaining free painter working time is $j$, the minimum cost of painting all the remaining walls. Then the answer is $dfs(0, 0)$.
The calculation process of function $dfs(i, j)$ is as follows:
 If $n  i \le j$, it means that there are no more walls than the free painter’s working time, so the remaining walls are painted by the free painter, and the cost is $0$;
 If $i \ge n$, return $+\infty$;
 Otherwise, if the $i$th wall is painted by a paid painter, the cost is $cost[i]$, then $dfs(i, j) = dfs(i + 1, j + time[i]) + cost[i]$; if the $i$th wall is painted by a free painter, the cost is $0$, then $dfs(i, j) = dfs(i + 1, j  1)$.
Note that the parameter $j$ may be less than $0$. Therefore, in the actual coding process, except for the $Python$ language, we add an offset $n$ to $j$ so that the range of $j$ is $[0, 2n]$.
Time complexity $O(n^2)$, space complexity $O(n^2)$. Where $n$ is the length of the array.

class Solution { private int n; private int[] cost; private int[] time; private Integer[][] f; public int paintWalls(int[] cost, int[] time) { n = cost.length; this.cost = cost; this.time = time; f = new Integer[n][n << 1  1]; return dfs(0, n); } private int dfs(int i, int j) { if (n  i <= j  n) { return 0; } if (i >= n) { return 1 << 30; } if (f[i][j] == null) { f[i][j] = Math.min(dfs(i + 1, j + time[i]) + cost[i], dfs(i + 1, j  1)); } return f[i][j]; } }

class Solution { public: int paintWalls(vector<int>& cost, vector<int>& time) { int n = cost.size(); int f[n][n << 1  1]; memset(f, 1, sizeof(f)); function<int(int, int)> dfs = [&](int i, int j) > int { if (n  i <= j  n) { return 0; } if (i >= n) { return 1 << 30; } if (f[i][j] == 1) { f[i][j] = min(dfs(i + 1, j + time[i]) + cost[i], dfs(i + 1, j  1)); } return f[i][j]; }; return dfs(0, n); } };

class Solution: def paintWalls(self, cost: List[int], time: List[int]) > int: @cache def dfs(i: int, j: int) > int: if n  i <= j: return 0 if i >= n: return inf return min(dfs(i + 1, j + time[i]) + cost[i], dfs(i + 1, j  1)) n = len(cost) return dfs(0, 0)

func paintWalls(cost []int, time []int) int { n := len(cost) f := make([][]int, n) for i := range f { f[i] = make([]int, n<<11) for j := range f[i] { f[i][j] = 1 } } var dfs func(i, j int) int dfs = func(i, j int) int { if ni <= jn { return 0 } if i >= n { return 1 << 30 } if f[i][j] == 1 { f[i][j] = min(dfs(i+1, j+time[i])+cost[i], dfs(i+1, j1)) } return f[i][j] } return dfs(0, n) }

impl Solution { #[allow(dead_code)] pub fn paint_walls(cost: Vec<i32>, time: Vec<i32>) > i32 { let n = cost.len(); let mut record_vec: Vec<Vec<i32>> = vec![vec![1; n << 1  1]; n]; Self::dfs(&mut record_vec, 0, n as i32, n as i32, &time, &cost) } #[allow(dead_code)] fn dfs( record_vec: &mut Vec<Vec<i32>>, i: i32, j: i32, n: i32, time: &Vec<i32>, cost: &Vec<i32> ) > i32 { if n  i <= j  n { // All the remaining walls can be printed at no cost // Just return 0 return 0; } if i >= n { // No way this case can be achieved // Just return +INF return 1 << 30; } if record_vec[i as usize][j as usize] == 1 { // This record hasn't been written record_vec[i as usize][j as usize] = std::cmp::min( Self::dfs(record_vec, i + 1, j + time[i as usize], n, time, cost) + cost[i as usize], Self::dfs(record_vec, i + 1, j  1, n, time, cost) ); } record_vec[i as usize][j as usize] } }