Java-POJ1014-Dividing

多重背包问题的特点是物品数量可以大于1但是有限制。状态定义与01背包一致。

多重背包的解法有多种，复杂度也各不相同。

对于物品数Ci较大的数据，可以采取二进制数进行优化（就是这样，别问就是baidu！）

如何理解多重背包的二进制优化？

使得每种物品的转移次数由O(MxCi)变为O(Mxlog(Ci))

f[i]=f[i] | f[i-a[i]] | …| f[i-a[i]xc[i]]（a[i]为物品价值，c[i]为物品数量）

二进制优化后令W[i]=a[i]xc[i](c[i]被二进制分拆)

/*
Memory: 3508K        Time: 329MS
Language: Java        Result: Accepted
*/

实现见代码：

 package poj.ProblemSet;

 import java.util.Scanner;

 public class poj1014 {
     public static final int MAXN = 400000;
     public static boolean[] f = new boolean[MAXN];
     public static int[] w = new int[100];

     public static void main(String[] args) {
         Scanner cin = new Scanner(System.in);
         for (int Case = 0; cin.hasNext(); ) {
             int value = 0, cnt = 0;
             boolean flag = false;
             for (int i = 1; i <= 6; i++) {
                 int val = cin.nextInt();
                 value += val * i;
                 for (int j = 1, x = 0; val > 0; j *= 2) {
                     x = Math.min(j, val);
                     w[++cnt] = i * x;
                     val -= x;
                 }
             }
             if (value == 0) break;
             System.out.println("Collection #" + (++Case) + ":");
             if (value % 2 == 0) {
                 f[0] = true;
                 for (int i = 1; i < MAXN; i++) f[i] = false;
                 for (int i = 1; i <= cnt; i++)
                     for (int j = value / 2; j >= w[i]; j--)
                         f[j] |= f[j - w[i]];
                 flag = true;
             }
             System.out.println((!flag?"Can't":(f[value/2]?"Can":"Can't"))+" be divided.");
             System.out.println();
         }
     }
 }

 PS:另外一种O(VN)的方法是用数据结构单调队列优化！！！Orz

/*
Memory: 6064K        Time: 282MS
Language: Java        Result: Accepted
*/

 package poj.ProblemSet;

 import java.util.Scanner;

 public class poj1014 {
     public static final int MAXN = 400000;
     public static int[] queue = new int[MAXN];
     public static boolean[] f = new boolean[MAXN];
     public static int[] c = new int[7];
     public static void main(String[] args) {
         Scanner cin = new Scanner(System.in);
         for (int Case = 0; cin.hasNext(); ) {
             int value = 0;
             boolean flag = false;
             for (int i = 1; i <= 6; i++) {
                 c[i] = cin.nextInt();
                 value += i * c[i];
             }
             if (value == 0) break;
             System.out.println("Collection #" + (++Case) + ":");
             if (value % 2 == 0) {
                 f[0] = true;
                 for (int i = 1; i <= value / 2; i++) f[i] = false;
                 for (int i = 1, x = 0; i <= 6; i++) {
                     x = i * c[i];
                     for (int j = 0; j < i; j++)
                         if (f[j]) queue[j] = j;
                         else queue[j] = -MAXN;
                     for (int j = i; j <= value / 2; j++)
                         if (f[j]) queue[j] = j;
                         else {
                             queue[j] = queue[j - i];
                             if (queue[j - i] + x >= j) f[j] = true;
                         }
                 }
                 flag = true;
             }
             System.out.println((!flag?"Can't":(f[value/2]?"Can":"Can't"))+" be divided.");
             System.out.println();
         }
     }
 }