01 knapsack problem

0 1 knapsack problem
01 Backpack (Zeroonepack): there are n items and a backpack with a capacity of V. ( only one piece per item ) The cost of item I is c[i] and the value is w[i]. The sum of the values is maximized by solving which items are loaded into the backpack.   TopicsThere are n items and a backpack with a capacity of V. The cost of article I is c[i], the value is w[i]. The sum of the values is maximized by solving which items are loaded into the backpack. Basic IdeasThis is the most basic knapsack problem, characterized by: only one piece of each item, you can choose to put or not put. Define the state with sub-question: F[i][v] Indicates the maximum value that the first I item can get in a backpack with a capacity of V. The state transfer equation is:F[i][v]=max{f[i-1][v],f[i-1][v-c[i]]+w[i]}  

Initialize:  f?0for (I=1 to  N) {for      (j=1 to V)      {             f[i][j]=f[i-1][j];             if (J>=c[i])             {                  f[i][j]=max{f[i-1][j], F[i-1][j-c[i]]+w[i]};             }        }}

第一种是第i件不放进去，这时所得价值为:f[i-1][v]

第二种是第i件放进去，这时所得价值为：f[i-1][v-c[i]]+w[i]

最后比较第一种与第二种所得价值的大小，哪种相对大，f[i][v]的值就是哪种。


Use the table to illustrate:

back containment	0	1	2	3	4	5	6	7	8	9	1 0	1 1	1 2	1 3	1 4	1 5
5 Items	0	0	0	0	0	0	6	12	12	15	15	18	22	22	25	25
4 Items	0	0	3	3	3	3	3	12	12	15	15	18	22	22	25	25
3 Items	0	0	0	0	0	0	0	12	12	15	15	15	22	22	22	22
2 Items	0	0	0	0	3	12	12	12	12	15	15	15	15	15	15	15
1 Items	0	0	0	0	0	12	12	12	12	12	12	12	12	12	12	12

Examples:5 items, (weight, value) are: (5,a), (4, 3), (7,ten), (2, 3 ), ( 6 , 6 ) The backpack has a capacity of Code:

#include <stdio.h> #include <string.h> #include <algorithm>using namespace Std;int main () {int f[6][16 ];int cos[6]={0,5,4,7,2,6};int Val[6]={0,12,3,10,3,6};memset (f,0,sizeof (f)); for (int. i=1;i<=5;i++) {for (int j=1;j <=15;j++) {f[i][j]=f[i-1][j];if (J>=cos[i]) {F[i][j]=max (f[i-1][j],f[i-1][j-cos[i]]+val[i]);}}} for (int i=1;i<=5;i++) {for (int j=1;j<=15;j++)  printf ("%d", f[i][j]);  

this is stored in a two-bit array that optimizes space and stores it with one array! Code: 

Optimization: Convert f array to one-dimensional array #include<stdio.h> #include <string.h> #include <algorithm>using namespace Std;int Main () {int f[16];int cos[6]={0,5,4,7,2,6};int val[6]={0,12,3,10,3,6};memset (f,0,sizeof (f)); for (int i=1;i<=5;i++ {for (int j=15;j>=1;j--) {if (J>=cos[i]) {F[j]=max (f[j],f[j-cos[i]]+val[i]);}}} for (int i=1;i<=15;i++) {  

 

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01 knapsack problem