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main.cpp
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//
// main.cpp
// 动态规划(dynamic programming)
// 01背包问题
// 给定ItemCount个物品, 物品i的重量是Weights[i], 价值Values[i], 背包的容量为TotalWeight
// 问应该如何选择装入背包的物品，使得背包内物品的总价值最大？
// Created by vincent on 2017/10/17.
// Copyright © 2017年 vincent. All rights reserved.
//
#include <iostream>
const int ItemCount = 4; // 物品数量
const int TotalWeight = 10; // 背包容量
int Values[ItemCount] = {111, 222, 332, 444}; // 物品i的价值
int Weights[ItemCount] = {1, 3, 5, 6}; // 物品i的重量
// 方式1：二维数组算法
// max[i][j]表示前i个物品放在j容量的背包里的最大价值
int max[ItemCount + 1][TotalWeight + 1];
int GetMax(int totalWeight, int targetIndex) {
// 0行和0列全部填充为0，用于辅助计算
for (int i = 0; i <= TotalWeight; ++i) {
max[0][i] = 0;
printf("%d\t", i);
}
for(int i = 0; i <= ItemCount; ++i) {
max[i][0] = 0;
}
printf("\n");
for (int i = 1; i <= ItemCount; ++i) {
int index = i - 1;
printf("%d\t", i);
for (int j = 1; j <= TotalWeight; ++j) {
// 容量j的背包能装i号物品
if (j >= Weights[index]) {
// [最优子结构]容量够，考虑装与不装index号物品两种情况，取最大值
int doPut = max[i - 1][j - Weights[index]] + Values[index];
int doNotPut = max[i - 1][j];
max[i][j] = std::max(doPut, doNotPut);
printf("%d\t", max[i][j]);
} else {
// [最优子结构]容量不够，仅考虑不装入index号物品的情况
max[i][j] = max[i - 1][j];
printf("%d\t", max[i][j]);
}
}
printf("\n");
}
return max[targetIndex + 1][totalWeight];
}
// 回溯装入的物品
int Result[ItemCount];
void Find(int i, int j) {
if (i >= 0) {
if (max[i][j] == max[i - 1][j]) {
// 与上一行价值相等，说明没有装入
Result[i - 1] = 0;
Find(i - 1, j);
} else {
int value = j - Weights[i - 1];
int doPut = max[i - 1][j - Weights[i - 1]] + Values[i - 1];
if (value > 0 && max[i][j] == doPut) {
Result[i - 1] = 1; // 标记被装入
Find(i - 1, j - Weights[i - 1]);
}
}
}
}
// 方式2：递归算法
int GetMaxRecursive(int lastWeight, int index) {
int retMaxGold;
if(index == 0) {
// [边界] 装最后一个物品
if(lastWeight >= Weights[index]) {
// 容量够，获得最后一个物品的价值
retMaxGold = Values[index];
} else {
// 容量不够，获得到0
retMaxGold = 0;
}
} else if(lastWeight >= Weights[index]) {
// [最优子结构]容量够，考虑装与不装index号物品两种情况，取最大值
int doPut = GetMaxRecursive(lastWeight - Weights[index], index - 1) + Values[index];
int doNotPut = GetMaxRecursive(lastWeight, index - 1);
retMaxGold = std::max(doPut, doNotPut);
} else {
// [最优子结构]容量不够，仅考虑不装入index号物品的情况
retMaxGold = GetMaxRecursive(lastWeight, index - 1);
}
return retMaxGold;
}
int main(int argc, const char * argv[]) {
printf("方式1：背包内最大价值：%d\n", GetMax(TotalWeight, ItemCount - 1));
printf("选中物品(1表示选中)：");
Find(ItemCount, TotalWeight);
for (int i = 0; i < ItemCount; ++i) {
if(Result[i] == 1) {
printf("%d ", i);
}
}
printf("\n方式2：背包内最大价值：%d\n", GetMaxRecursive(TotalWeight, ItemCount - 1));
return 0;
}