实验五分支限界法

01背包问题的分治限界法的实现

剪枝函数

限界函数

1.实验目的

1、理解分支限界法的剪枝搜索策略，掌握分支限界法的算法框架

2、设计并实现问题，掌握分支限界算法。

2.实验环境

java

3.问题描述

给定n种物品和一背包。物品i的重量是wi，其价值为vi，背包的容量为C。问应如何选择装入背包的物品，使得装入背包中物品的总价值最大?

4.复杂度

算法整体的时间复杂度主要由回溯操作的时间复杂度决定，在最坏情况下为O(2^n)。然而，在实际应用中，由于剪枝操作的存在，实际的分支数会远远小于2^n，因此算法的执行时间通常会有所缩减。

5.算法实现

package shiyan5;import java.io.File;
import java.io.FileWriter;
import java.io.IOException;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.Scanner;class Item {int weight;int value;double density; // 物品的单位价值（价值重量比）public Item(int weight, int value) {this.weight = weight;this.value = value;this.density = (double) value / weight;}
}public class BranchAndBound {static List<Item> items;static int capacity;static int maxTotalValue;static List<Item> selectedItems;public static void main(String[] args) {readInputData("input.txt");branchAndBound();writeOutputData("output.txt");}private static void readInputData(String filename) {try {File file = new File(filename);Scanner scanner = new Scanner(file);int n = scanner.nextInt();capacity = scanner.nextInt();items = new ArrayList<>();for (int i = 0; i < n; i++) {int weight = scanner.nextInt();int value = scanner.nextInt();items.add(new Item(weight, value));}scanner.close();} catch (IOException e) {e.printStackTrace();}}private static void branchAndBound() {Collections.sort(items, (a, b) -> Double.compare(b.density, a.density)); // 按单位价值从大到小排序maxTotalValue = 0;selectedItems = new ArrayList<>();backtrack(0, 0, 0);// 按照物品的原始顺序排序Collections.sort(selectedItems, (a, b) -> Integer.compare(items.indexOf(a), items.indexOf(b)));}private static void backtrack(int level, int currentWeight, int currentValue) {if (level == items.size() || currentWeight == capacity) {if (currentValue > maxTotalValue) {maxTotalValue = currentValue;selectedItems.clear();for (int i = 0; i < level; i++) {if (items.indexOf(items.get(i)) != -1) {selectedItems.add(items.get(i));}}}return;}if (currentWeight + items.get(level).weight <= capacity) {currentWeight += items.get(level).weight;currentValue += items.get(level).value;backtrack(level + 1, currentWeight, currentValue);currentWeight -= items.get(level).weight;currentValue -= items.get(level).value;}if (bound(level + 1, currentWeight, currentValue) > maxTotalValue) {backtrack(level + 1, currentWeight, currentValue);}}private static double bound(int level, int currentWeight, int currentValue) {double maxBound = currentValue;int currentLevel = level;int currentWeightSum = currentWeight;while (currentWeightSum < capacity && currentLevel < items.size()) {if (currentWeightSum + items.get(currentLevel).weight <= capacity) {currentWeightSum += items.get(currentLevel).weight;maxBound += items.get(currentLevel).value;} else {double remainingWeight = capacity - currentWeightSum;maxBound += (remainingWeight / items.get(currentLevel).weight) * items.get(currentLevel).value;break;}currentLevel++;}return maxBound;}private static void writeOutputData(String filename) {try {FileWriter writer = new FileWriter(filename);writer.write("背包中物品的总价值: " + maxTotalValue);writer.write("\n\n已选物品列表:\n");for (Item item : selectedItems) {writer.write("重量: " + item.weight + ", 价值: " + item.value);writer.write("\n");}writer.close();System.out.println("输出成功！");} catch (IOException e) {e.printStackTrace();}}
}

输入