Codeforces815C Karen And SuperMarket 解题报告【树上DP/树上背包(?)】

解决一项购物优惠券使用问题,通过构建树形结构并利用动态规划算法,确定在有限预算内能够购买的最大商品数量。

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On the way home, Karen decided to stop by the supermarket to buy some groceries.

She needs to buy a lot of goods, but since she is a student her budget is still quite limited. In fact, she can only spend up to b dollars.
The supermarket sells n goods. The i-th good can be bought for ci dollars. Of course, each good can only be bought once.
Lately, the supermarket has been trying to increase its business. Karen, being a loyal customer, was given n coupons. If Karen purchases the i-th good, she can use the i-th coupon to decrease its price by di. Of course, a coupon cannot be used without buying the corresponding good.
There is, however, a constraint with the coupons. For all i ≥ 2, in order to use the i-th coupon, Karen must also use the xi-th coupon (which may mean using even more coupons to satisfy the requirement for that coupon).
Karen wants to know the following. What is the maximum number of goods she can buy, without exceeding her budget b?
Input
The first line of input contains two integers n and b (1 ≤ n ≤ 5000, 1 ≤ b ≤ 109), the number of goods in the store and the amount of money Karen has, respectively.
The next n lines describe the items. Specifically:
The i-th line among these starts with two integers, ci and di (1 ≤ di < ci ≤ 109), the price of the i-th good and the discount when using the coupon for the i-th good, respectively.
If i ≥ 2, this is followed by another integer, xi (1 ≤ xi < i), denoting that the xi-th coupon must also be used before this coupon can be used.
Output
Output a single integer on a line by itself, th

引用\[1\]中提到了一种动态规划的方法来解决CodeForces - 982C问题。在这个问题中,subtree指的是子连通块,而不是子。为了使cnt_white - cnt_black尽可能大,可以使用两次动态规划来求解。第一次是自底向上的过程,维护一个dp数组,表示以每个节点为根的子中的最大连通块。第二次是自顶向下的过程,处理自底向上过程中无法包含的链所代表的子。在第二次遍历中,需要维护一个sum变量,用于存储链所代表的子的贡献。根据ans\[u\]的正负,决定是否能对相邻的子节点做出贡献。如果ans\[u\]为正,则减去dp\[v\]就是链所代表的子的权值。最终,ans\[u\]代表包含节点u在内的子连通块的最大权值。\[1\] 问题: CodeForces - 982C DP是什么问题?如何解决? 回答: CodeForces - 982C是一个动态规划问题。在这个问题中,需要求解子连通块的最大权值和,使得cnt_white - cnt_black尽可能大。解决这个问题的方法是使用两次动态规划。第一次是自底向上的过程,维护一个dp数组,表示以每个节点为根的子中的最大连通块。第二次是自顶向下的过程,处理自底向上过程中无法包含的链所代表的子。在第二次遍历中,需要维护一个sum变量,用于存储链所代表的子的贡献。根据ans\[u\]的正负,决定是否能对相邻的子节点做出贡献。最终,ans\[u\]代表包含节点u在内的子连通块的最大权值。\[1\] #### 引用[.reference_title] - *1* *2* [CodeForces - 1324F Maximum White Subtree(dp)](https://blog.youkuaiyun.com/qq_45458915/article/details/104831678)[target="_blank" data-report-click={"spm":"1018.2226.3001.9630","extra":{"utm_source":"vip_chatgpt_common_search_pc_result","utm_medium":"distribute.pc_search_result.none-task-cask-2~all~insert_cask~default-1-null.142^v91^koosearch_v1,239^v3^insert_chatgpt"}} ] [.reference_item] [ .reference_list ]
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