01-背包

此为本蒟蒻第n次发文，若有错误或不足之处，还请各位牛犇多多指出。

目录

引入

01背包

时间复杂度 $O(n^2)$ 空间复杂度 $O(n^2)$ 代码

滚动数组优化

空间复杂度 $O(n)$ 代码

练习与拓展

资源来源与参考

0. 引入

在正式开始讲『01背包』之前，我们先来看看下面的例题。

[USACO07DEC]Charm Bracelet S

题目描述

Bessie has gone to the mall’s jewelry store and spies a charm bracelet. Of course, she’d like to fill it with the best charms possible from the N (1 ≤ N ≤ 3,402) available charms. Each charm i in the supplied list has a weight Wi (1 ≤ Wi ≤ 400), a ‘desirability’ factor Di (1 ≤ Di ≤ 100), and can be used at most once. Bessie can only support a charm bracelet whose weight is no more than M (1 ≤ M ≤ 12,880).

Given that weight limit as a constraint and a list of the charms with their weights and desirability rating, deduce the maximum possible sum of ratings.

有 $N$ 件物品和一个容量为 $M$ 的背包。第 $i$ 件物品的重量是 $W_i$，价值是 $D_i$。求解将哪些物品装入背包可使这些物品的重量总和不超过背包容量，且价值总和最大。

输入格式

* Line 1: Two space-separated integers: N and M

* Lines 2…N+1: Line i+1 describes charm i with two space-separated integers: Wi and Di

第一行：物品个数 $N$ 和背包大小 $M$。

第二行至第 $N+1$ 行：第 $i$ 个物品的重量 $W_i$