面试OR笔试41——01背包

1 题目及要求

1.1 题目描述

有n 个物品，它们有各自的重量和价值，现有给定容量的背包，如何让背包里装入的物品具有最大的价值总和？

解法解析见博客动态规划解决01背包问题

这里给出三种解法。

2 解答

2.1 代码

#include <iostream>#include <string>#include <vector>#include <algorithm>#include <iomanip>using namespace std;// 一维动态规划, 无方案输出int package(const vector<int> &w, const vector<int> &v, int c) {int wn(w.size()), vn(v.size());wn < vn ? wn : wn = vn;if (c < 0 || wn < 1) return 0;vector<int> dt(c + 1, 0);for (int k1(0);k1 < wn;++k1) {for (int k2(c);w[k1] <= k2;--k2) {dt[k2] = max(dt[k2], dt[k2 - w[k1]] + v[k1]);}}return dt[c];}// 递归实现, 复杂度略高, 有方案输出int package1(const vector<int> &w, const vector<int> &v, int wn, int k, int c, int val, vector<bool> &status) {if (wn == k) return val;status[k] = false;if (c < w[k]) {return package1(w, v, wn, k + 1, c, val, status);}int valtmp = package1(w, v, wn, k + 1, c, val, status);status[k] = true;val = package1(w, v, wn, k + 1, c - w[k], val + v[k], status);if (val < valtmp) {status[k] = false;val = valtmp;package1(w, v, wn, k + 1, c, val, status);}return val;}int package1(const vector<int> &w, const vector<int> &v, int c, vector<int> &out) {int wn(w.size()), vn(v.size());wn < vn ? wn : wn = vn;if (c < 0 || wn < 1) return 0;vector<bool> status(wn, false);int val(package1(w, v, wn, 0, c, 0, status));out.clear();for (int k1(0);k1 < wn;++k1) if(status[k1]) out.push_back(k1);return val;}// 二维动态规划, 有方案输出int package2(const vector<int> &w, const vector<int> &v, int c, vector<int> &out) {int wn(w.size()), vn(v.size());wn < vn ? wn : wn = vn;if (c < 0 || wn < 1) return 0;vector<vector<int> > dt(wn+1, vector<int>(c + 1, 0));for (int k1(0);k1 < wn;++k1) {for (int k2(0);k2 <= c;++k2) {k2 < w[k1] ? dt[k1 + 1][k2] = dt[k1][k2] : dt[k1 + 1][k2] = max(dt[k1][k2], dt[k1][k2 - w[k1]] + v[k1]);}}out.clear();for (int k1(wn), k2(c);0 < k1;--k1) {if (dt[k1][k2] != dt[k1 - 1][k2]) {out.push_back(k1-1);k2 -= w[k1-1];}}reverse(out.begin(), out.end());return dt[wn][c];}int main() {vector<int> w = { 2,3,4,5 }, v = { 3,4,5,6 };vector<int> o;int c = 8;cout << "weight  : ";for (auto i : w) cout << setw(4) << i << ' ';cout << endl << "value   : ";for (auto i : v) cout << setw(4) << i << ' ';cout << endl;for (int k1(0);k1 < c+1;++k1) {cout << "capacity:" << setw(4) << k1 << endl;cout << "maxvalue:" << setw(4) << package2(w, v, k1, o) << endl;cout << "scheme  :";for (auto i : o) cout << setw(4) << i;cout << endl << endl;}return 0;}