[pull] master from wisdompeak:master #364

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	class Solution {
	public:
	vector<bool> subsequenceSumAfterCapping(vector<int>& nums, int k) {
	int n = nums.size();
	vector<int>dp(k+1,0);
	dp[0] = 1;
	vector<bool>rets(n, 0);

	sort(nums.begin(), nums.end());
	int i = 0;
	for (int x=1; x<=n; x++) {
	while (i<n && nums[i]<x) {
	for (int c = k; c>=1; c--) {
	if (c<nums[i]) break;
	dp[c] \|= dp[c-nums[i]];
	}
	i++;
	}
	int m = n-i;
	for (int j=0; j<=m && j*x<=k; j++) {
	if (dp[k-j*x]) {
	rets[x-1] = 1;
	break;
	}
	}
	}
	return rets;
	}
	};

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	### 3685.Subsequence-Sum-After-Capping-Elements

	对于n个元素,问是否可以组合出和为k的方案,就是一个典型的有界背包问题,用o(n*k)的时间复杂度可解。大致算法是:
	```cpp
	for (int i=0; i<nums.size(); i++) {
	for (int c=k; c>=1; c--) {
	dp[c] \|= dp[c-nums[i];
	}
	}
	```

	对于此题而言,对于每一个给定的x,我们要用nums里所有小于x的元素,以及剩余的元素当做x使用,问是否能组合出和为k的方案。我们暂时先不考虑第二种用法,即只用nums里小于x的元素。我们发现,随着x的增大,其实第一个for循环里可选的元素种类的上界也在单调增大,故总体这依然是一个o(n*k)可解的问题。

	然后更深入地思考,我们将x从小到大遍历的时候,可以不断提升i,从而加入所有不超过x的新nums[i],就可以不断更新dp。同时我们还需要考虑等于x的元素:因为capping的缘故,这样的元素有n-i个。此时我们需要考虑这额外的n-i个元素能否对于组成k有帮助。显然我们可以用同样的背包思想:
	```cpp
	for (int j=1; j<n-i; j++) {
	if (dp[k-x*j]) {
	return true;
	break;
	}
	}
	```
	注意,这个过程中我们只是判断是否能组合成k,不会去更新dp。我们始终保持dp[c]表示"能否用所有小于x的元素组成c"。这样,即使x变大,dp的定义依然适用。

1 change: 1 addition & 0 deletions Readme.md

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		[2902.Count-of-Sub-Multisets-With-Bounded-Sum](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/2902.Count-of-Sub-Multisets-With-Bounded-Sum) (H)
		[3489.Zero-Array-Transformation-IV](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/3489.Zero-Array-Transformation-IV) (H-)
		[3592.Inverse-Coin-Change](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/3592.Inverse-Coin-Change) (H)
	[3685.Subsequence-Sum-After-Capping-Elements](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/3685.Subsequence-Sum-After-Capping-Elements) (H)
		* ``键盘型``
		[650.2-Keys-Keyboard](https://github.com/wisdompeak/LeetCode/blob/master/Dynamic_Programming/650.2-Keys-Keyboard) (M+)
		[651.4-Keys-Keyboard](https://github.com/wisdompeak/LeetCode/tree/master/Dynamic_Programming/651.4-Keys-Keyboard) (M+)
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