基礎揹包問題 - 0-1 揹包問題 - 深度優先搜索 (遞歸)

基礎揹包問題 - 0-1 揹包問題 - 深度優先搜索 (遞歸)

1. 基礎揹包問題

有 N 件物品和一個承受最大重量爲 W 的揹包。第 i 件物品的重量是 w[i]，價值是 v[i]。求解將哪些物品裝入揹包可使這些物品的重量總和不超過揹包容量，且價值總和最大。

揹包問題 (Knapsack problem) 是一種組合優化的 NP 完全問題。給定一組物品，每種物品都有自己的重量和價格，在限定的總重量內，我們如何選擇，才能使得物品的總價格最高。

如果限定每種物品只能選擇 0 個或 1 個，則問題稱爲 0-1 揹包問題 (0-1 knapsack problem) 。
如果限定物品 j 最多隻能選擇 maxj 個，則問題稱爲有界揹包問題 (bounded knapsack problem，BKP)。
如果不限定每種物品的數量，每種物品都就可以選擇任意多個，則問題稱爲無界揹包問題或完全揹包問題 (unbounded knapsack problem，UKP)。

2. 0-1 揹包問題

2.1 基本思路

最基礎的揹包問題，每種物品僅有一件，可以選擇放或不放。利用遞歸針對每個物品是否放入揹包進行兩種情況下的搜索。0-1 揹包問題在選第 i 個物品時，只有 2 種狀態可選，放還是不放。

2    // 測試用例個數
4 5 // N = 4 物品的類別數，W = 5 揹包承受的最大重量，最大價值 7
2 3 // w[0] = 2，v[0] = 3
1 2 // w[1] = 1，v[1] = 2
3 4 // w[2] = 3，v[2] = 4
2 2 // w[3] = 2，v[3] = 2
5 8 // N = 5 物品的類別數，W = 8 揹包承受的最大重量，最大價值 10
3 4
5 5
1 2
2 1
2 3

2
4 5
2 3
1 2
3 4
2 2
5 8 
3 4
5 5
1 2
2 1
2 3

2.2 dfs_v1

算法遍歷的組合數爲 $2^n$，計算複雜度大。

/*
============================================================================
Name        : yongqiang.cpp
Author      : Yongqiang Cheng
Version     : Version 1.0.0
Copyright   : Copyright (c) 2020 Yongqiang Cheng
Description : Hello World in C++, Ansi-style
============================================================================
*/

#define _CRT_SECURE_NO_WARNINGS
#include <iostream>
#include <time.h>

using namespace std;

int computation = 0;

void dfs_v1(int map[100][2], int N, int W, int sumweight, int sumvalue, int *max_value, int count, int book[100])
{
	if (count >= N)
	{
		computation++;

		if ((sumweight <= W) && (sumvalue > *max_value))
		{
			*max_value = sumvalue;
		}

		return;
	}

	if (0 == book[count])
	{
		int addvalue = 0, addweight = 0;

		// select
		addvalue = map[count][1];
		addweight = map[count][0];

		book[count] = 1;
		dfs_v1(map, N, W, sumweight + addweight, sumvalue + addvalue, max_value, count + 1, book);
		book[count] = 0;

		// unselect
		addvalue = 0;
		addweight = 0;

		book[count] = 1;
		dfs_v1(map, N, W, sumweight, sumvalue, max_value, count + 1, book);
		book[count] = 0;
	}

	return;
}

int inference(int map[100][2], int N, int W)
{
	int ret = 0;
	int max_value = INT_MIN;
	int count = 0;
	int sumweight = 0;
	int sumvalue = 0;
	int book[100] = { 0 };

	dfs_v1(map, N, W, sumweight, sumvalue, &max_value, count, book);

	ret = max_value;

	return ret;
}

int main()
{
	clock_t start = 0, end = 0;
	double cpu_time_used = 0;

	const char input_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\input.txt";
	const char output_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\output.txt";

	freopen(input_txt, "r", stdin);
	// freopen(output_txt, "w", stdout);

	start = clock();
	printf("Start of the program, start = %ld\n", start);
	printf("Start of the program, start = %ld\n\n", start);

	int case_num;
	cin >> case_num;

	for (int i = 1; i <= case_num; i++)
	{
		int ret = 0;
		int	N = 0, W = 0;
		int map[100][2] = { 0 };

		computation = 0;

		cin >> N >> W;

		for (int h = 0; h < N; h++)
		{
			cin >> map[h][0] >> map[h][1]; // weight - value
		}

		ret = inference(map, N, W);

		cout << "computation " << i << " = " << computation << endl;
		cout << "CASE #" << i << " = " << ret << endl;
	}

	end = clock();
	printf("\nEnd of the program, end = %ld\n", end);
	printf("End of the program, end = %ld\n", end);

	cpu_time_used = ((double)(end - start)) / CLOCKS_PER_SEC;
	printf("Total time taken by CPU: %f\n", cpu_time_used);

	printf("Exiting of the program...\n");

	fclose(stdin);
	// fclose(stdout);

	return 0;
}

Start of the program, start = 9
Start of the program, start = 9

computation 1 = 16
CASE #1 = 7
computation 2 = 32
CASE #2 = 10

End of the program, end = 15
End of the program, end = 15
Total time taken by CPU: 0.006000
Exiting of the program...
請按任意鍵繼續. . .

2.3 dfs_v2

/*
============================================================================
Name        : yongqiang.cpp
Author      : Yongqiang Cheng
Version     : Version 1.0.0
Copyright   : Copyright (c) 2020 Yongqiang Cheng
Description : Hello World in C++, Ansi-style
============================================================================
*/

#define _CRT_SECURE_NO_WARNINGS
#include <iostream>
#include <time.h>

using namespace std;

int computation = 0;

void dfs_v2(int map[100][2], int N, int W, int sumweight, int sumvalue, int *max_value, int count, int book[100])
{
	if (sumweight > W)
	{
		return;
	}

	if (count >= N)
	{
		computation++;

		if (sumvalue > *max_value)
		{
			*max_value = sumvalue;
		}

		return;
	}

	if (0 == book[count])
	{
		int addvalue = 0, addweight = 0;

		// select
		addvalue = map[count][1];
		addweight = map[count][0];

		book[count] = 1;
		dfs_v2(map, N, W, sumweight + addweight, sumvalue + addvalue, max_value, count + 1, book);
		book[count] = 0;

		// unselect
		addvalue = 0;
		addweight = 0;

		book[count] = 1;
		dfs_v2(map, N, W, sumweight, sumvalue, max_value, count + 1, book);
		book[count] = 0;
	}

	return;
}

int inference(int map[100][2], int N, int W)
{
	int ret = 0;
	int max_value = INT_MIN;
	int count = 0;
	int sumweight = 0;
	int sumvalue = 0;
	int book[100] = { 0 };

	dfs_v2(map, N, W, sumweight, sumvalue, &max_value, count, book);

	ret = max_value;

	return ret;
}

int main()
{
	clock_t start = 0, end = 0;
	double cpu_time_used = 0;

	const char input_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\input.txt";
	const char output_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\output.txt";

	freopen(input_txt, "r", stdin);
	// freopen(output_txt, "w", stdout);

	start = clock();
	printf("Start of the program, start = %ld\n", start);
	printf("Start of the program, start = %ld\n\n", start);

	int case_num;
	cin >> case_num;

	for (int i = 1; i <= case_num; i++)
	{
		int ret = 0;
		int	N = 0, W = 0;
		int map[100][2] = { 0 };

		computation = 0;

		cin >> N >> W;

		for (int h = 0; h < N; h++)
		{
			cin >> map[h][0] >> map[h][1]; // weight - value
		}

		ret = inference(map, N, W);

		cout << "computation " << i << " = " << computation << endl;
		cout << "CASE #" << i << " = " << ret << endl;
	}

	end = clock();
	printf("\nEnd of the program, end = %ld\n", end);
	printf("End of the program, end = %ld\n", end);

	cpu_time_used = ((double)(end - start)) / CLOCKS_PER_SEC;
	printf("Total time taken by CPU: %f\n", cpu_time_used);

	printf("Exiting of the program...\n");

	fclose(stdin);
	// fclose(stdout);

	return 0;
}

Start of the program, start = 8
Start of the program, start = 8

computation 1 = 12
CASE #1 = 7
computation 2 = 23
CASE #2 = 10

End of the program, end = 12
End of the program, end = 12
Total time taken by CPU: 0.004000
Exiting of the program...
請按任意鍵繼續. . .

2.4 dfs_v3

/*
============================================================================
Name        : yongqiang.cpp
Author      : Yongqiang Cheng
Version     : Version 1.0.0
Copyright   : Copyright (c) 2020 Yongqiang Cheng
Description : Hello World in C++, Ansi-style
============================================================================
*/

#define _CRT_SECURE_NO_WARNINGS
#include <iostream>
#include <time.h>

using namespace std;

int computation = 0;

void dfs_v3(int map[100][2], int N, int W, int sumweight, int sumvalue, int *max_value, int count, int book[100])
{
	if (sumweight > W)
	{
		return;
	}

	if (count >= N)
	{
		computation++;

		if (sumvalue > *max_value)
		{
			*max_value = sumvalue;
		}

		return;
	}

	if (0 == book[count])
	{
		int addvalue = 0, addweight = 0;

		// select
		addvalue = map[count][1];
		addweight = map[count][0];

		if ((sumweight + addweight) <= W)
		{
			book[count] = 1;
			dfs_v3(map, N, W, sumweight + addweight, sumvalue + addvalue, max_value, count + 1, book);
			book[count] = 0;
		}

		// unselect
		addvalue = 0;
		addweight = 0;

		book[count] = 1;
		dfs_v3(map, N, W, sumweight, sumvalue, max_value, count + 1, book);
		book[count] = 0;
	}

	return;
}

int inference(int map[100][2], int N, int W)
{
	int ret = 0;
	int max_value = INT_MIN;
	int count = 0;
	int sumweight = 0;
	int sumvalue = 0;
	int book[100] = { 0 };

	dfs_v3(map, N, W, sumweight, sumvalue, &max_value, count, book);

	ret = max_value;

	return ret;
}

int main()
{
	clock_t start = 0, end = 0;
	double cpu_time_used = 0;

	const char input_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\input.txt";
	const char output_txt[] = "D:\\visual_studio_workspace\\yongqiangcheng\\yongqiangcheng\\output.txt";

	freopen(input_txt, "r", stdin);
	// freopen(output_txt, "w", stdout);

	start = clock();
	printf("Start of the program, start = %ld\n", start);
	printf("Start of the program, start = %ld\n\n", start);

	int case_num;
	cin >> case_num;

	for (int i = 1; i <= case_num; i++)
	{
		int ret = 0;
		int	N = 0, W = 0;
		int map[100][2] = { 0 };

		computation = 0;

		cin >> N >> W;

		for (int h = 0; h < N; h++)
		{
			cin >> map[h][0] >> map[h][1]; // weight - value
		}

		ret = inference(map, N, W);

		cout << "computation " << i << " = " << computation << endl;
		cout << "CASE #" << i << " = " << ret << endl;
	}

	end = clock();
	printf("\nEnd of the program, end = %ld\n", end);
	printf("End of the program, end = %ld\n", end);

	cpu_time_used = ((double)(end - start)) / CLOCKS_PER_SEC;
	printf("Total time taken by CPU: %f\n", cpu_time_used);

	printf("Exiting of the program...\n");

	fclose(stdin);
	// fclose(stdout);

	return 0;
}

Start of the program, start = 9
Start of the program, start = 9

computation 1 = 12
CASE #1 = 7
computation 2 = 23
CASE #2 = 10

End of the program, end = 12
End of the program, end = 12
Total time taken by CPU: 0.003000
Exiting of the program...
請按任意鍵繼續. . .