mirror of
https://github.com/krahets/hello-algo.git
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e720aa2d24
* Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
109 lines
3.5 KiB
C++
109 lines
3.5 KiB
C++
#include <algorithm>
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#include <iostream>
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#include <vector>
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using namespace std;
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/* 0-1 背包：暴力搜索 */
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int knapsackDFS(vector<int> &wgt, vector<int> &val, int i, int c) {
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// 若已选完所有物品或背包无剩余容量，则返回价值 0
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if (i == 0 || c == 0) {
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return 0;
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}
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// 若超过背包容量，则只能选择不放入背包
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if (wgt[i - 1] > c) {
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return knapsackDFS(wgt, val, i - 1, c);
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}
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// 计算不放入和放入物品 i 的最大价值
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int no = knapsackDFS(wgt, val, i - 1, c);
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int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];
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// 返回两种方案中价值更大的那一个
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return max(no, yes);
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}
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/* 0-1 背包：记忆化搜索 */
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int knapsackDFSMem(vector<int> &wgt, vector<int> &val, vector<vector<int>> &mem, int i, int c) {
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// 若已选完所有物品或背包无剩余容量，则返回价值 0
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if (i == 0 || c == 0) {
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return 0;
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}
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// 若已有记录，则直接返回
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if (mem[i][c] != -1) {
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return mem[i][c];
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}
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// 若超过背包容量，则只能选择不放入背包
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if (wgt[i - 1] > c) {
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return knapsackDFSMem(wgt, val, mem, i - 1, c);
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}
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// 计算不放入和放入物品 i 的最大价值
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int no = knapsackDFSMem(wgt, val, mem, i - 1, c);
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int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];
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// 记录并返回两种方案中价值更大的那一个
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mem[i][c] = max(no, yes);
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return mem[i][c];
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}
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/* 0-1 背包：动态规划 */
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int knapsackDP(vector<int> &wgt, vector<int> &val, int cap) {
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int n = wgt.size();
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// 初始化 dp 表
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vector<vector<int>> dp(n + 1, vector<int>(cap + 1, 0));
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// 状态转移
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for (int i = 1; i <= n; i++) {
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for (int c = 1; c <= cap; c++) {
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if (wgt[i - 1] > c) {
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// 若超过背包容量，则不选物品 i
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dp[i][c] = dp[i - 1][c];
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} else {
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// 不选和选物品 i 这两种方案的较大值
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dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);
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}
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}
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}
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return dp[n][cap];
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}
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/* 0-1 背包：空间优化后的动态规划 */
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int knapsackDPComp(vector<int> &wgt, vector<int> &val, int cap) {
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int n = wgt.size();
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// 初始化 dp 表
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vector<int> dp(cap + 1, 0);
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// 状态转移
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for (int i = 1; i <= n; i++) {
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// 倒序遍历
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for (int c = cap; c >= 1; c--) {
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if (wgt[i - 1] <= c) {
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// 不选和选物品 i 这两种方案的较大值
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dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);
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}
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}
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}
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return dp[cap];
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}
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/* Driver Code */
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int main() {
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vector<int> wgt = {10, 20, 30, 40, 50};
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vector<int> val = {50, 120, 150, 210, 240};
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int cap = 50;
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int n = wgt.size();
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// 暴力搜索
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int res = knapsackDFS(wgt, val, n, cap);
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cout << "不超过背包容量的最大物品价值为 " << res << endl;
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// 记忆化搜索
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vector<vector<int>> mem(n + 1, vector<int>(cap + 1, -1));
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res = knapsackDFSMem(wgt, val, mem, n, cap);
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cout << "不超过背包容量的最大物品价值为 " << res << endl;
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// 动态规划
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res = knapsackDP(wgt, val, cap);
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cout << "不超过背包容量的最大物品价值为 " << res << endl;
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// 空间优化后的动态规划
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res = knapsackDPComp(wgt, val, cap);
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cout << "不超过背包容量的最大物品价值为 " << res << endl;
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return 0;
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}
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