使用动态编程找到子集和的解决方案

Question

我想做的事

我想找到一个与目标T相加的数组的子集。 我还想使用动态编程方法（以及自下而上的解决方案）来做到这一点。

我现在有什么

目前我只找到一种方法来查看是否在所有大小为N子集中，是否存在至少一个具有所需总和的子集。 见下面的代码。

public boolean solve(int[] numbers, int target) {

    //Safeguard against invalid parameters
    if ((target < 0) || (sum(numbers) < target)){
        return false;
    }

    boolean [][] table = new boolean [target + 1] [numbers.length + 1]  ;

    for (int i = 0; i <= numbers.length; ++i) {
        table[0][i] = true;
    }


    /* Base cases have been covered. 
     * Now look set subsets [1..n][target] to be true or false.
     * n represents the number of elements from the start that have a subset 
     * that sums to target
     */
    for (int i = 1; i <= target; ++i){
        for (int j = 1; j <= numbers.length; ++j){
            /* Mark index j as one of the numbers in the array
             *  which is part of the solution with the given subtarget */
            table [i][j] = table[i][j-1];
            if (i >= numbers[j-1])
                table[i][j] = table[i][j] || table[i - numbers[j-1]] [j-1];
        }
    }

    return table[target][numbers.length];
}

我被卡住了

现在，我知道，如果有一个解决方案，但我不能想办法实际输出的解决方案。

我不是在寻找任何人为我提供特定的代码，但我们欢迎伪代码以及如何保存解决方案的提示。

Answer 1

您提供的算法可以保持不变，除了DP表格table[][]之外，您不需要存储任何其他内容。 您只需要一个额外的后处理阶段，在该阶段中，您可以通过table[][] “向后”步进以获得解决方案集。

回忆一下：

你已经计算了表table[i][j] ，它存储每个值0 <= i <= t（：= target ）和每个0 <= j <= n（：= = numbers.length ）是否存在numbers[0..j-1]中的numbers[0..j-1]子集，总和为i。

考虑对应于table[i][j]的子集S（，这是真的）。 注意：

• 仅当table[ i-numbers[j] ][j-1]为真时，子集S才包含数字numbers[j] 。

  （证明：递归地获取table[ i-numbers[j] ][j-1]的解集子S'，并添加numbers[j] ）

• 另一方面，仅当table[ i-numbers[j] ][j-1]为假时，该子集S才包含数字numbers[j] 。

  （证明：假定S包含numbers[j]特罗numbers[j]输出S的，这意味着table[ i-numbers[j] ][j-1]矛盾）

因此，要获取子集，只需使用上面的属性来检查numbers[n-1]是否在子集中总和为t。

• 如果是这样，递归计算numbers[n-2]是否在子集中加到t- numbers[n-1] ，
• 否则递归地计算numbers[n-2]是否在子集中总和为t

Answer 2

以下是子集求和问题的两个Java解决方案。
首先使用递归方法。
其次使用动态规划方法。

/*
Question: Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set 
with sum equal to given sum.

Examples: set[] = {3, 34, 4, 12, 5, 2}, sum = 9
Output:  True  //There is a subset (4, 5) with sum 9.
Let isSubSetSum(int set[], int n, int sum) be the function to find whether there is a subset of set[] with 
sum equal to sum. n is the number of elements in set[].


*/
package SubsetSumProblem;

import java.util.Scanner;

public class UsingResursiveAndDPApproach {
public static void main(String[] args) {
    Scanner in = new Scanner(System.in);
    try{
        System.out.println("Enter the number of elements in the array");
        int n =in.nextInt();
        System.out.println("Enter the elements of the array");
        int[] a=new int[n];
        for(int i=0;i<n;i++)
            a[i]=in.nextInt();
        System.out.println("Enter the sum, which you need to find");
        int sum = in.nextInt();
        System.out.println("Using recursion, the result is: "+usingRecursion(a,a.length,sum));
        System.out.println("Using Dynamic Programming, the result is: "+usingDP(a,sum));
    }
    finally{
        in.close();
    }
}


private static boolean usingRecursion(int[] a,int length, int sum) {

    // 1. Base Cases
    if(sum==0)
        return true;
    if(length==0 && sum!=0)
        return false;

    // 2. To avoid unnecessary steps, we will optimize the recursion method by avoiding 
    //    recursive calls to areas where we are definite that we can SAFELY ignore the case since
    //    the SOLUTION does not exist there.

    // If last element is greater than sum, then ignore it
    if(a[a.length-1]>sum)
        return usingRecursion(a,length-1,sum);

    // 3. This is the recursion step where we will call the method again and again
    /* else, check if sum can be obtained by any of the following
    (a) including the last element
    (b) excluding the last element   */
    return (usingRecursion(a, length-1, sum-a[length-1])|| usingRecursion(a, length-1, sum));

}
/*
Analysis:
    Time Complexity = O(2^n)
    Space Complexity =       // Don't know
*/

private static boolean usingDP(int[] a, int sum) {
    // using boolean matrix for DP
    boolean dp[][] = new boolean[a.length+1][sum+1];  // +1 in row and column


    // if the length of the array is variable (and sum is 0) then fill TRUE, since the SUM=0 
    for(int row=0;row<dp.length;row++){
        dp[row][0] = true;    // NOTE: dp[length=VARIABLE][sum=0], thus we satisfy the condition where length is VARIABLE
                              // and the SUM=0
    }

    // if the SUM is variable and length is 0 then FALSE, since (sum=variable && length=0)
    for(int column=1;column<dp[0].length;column++){
        dp[0][column] = false;  // NOTE: dp[length=0][sum=VARIABLE], thus we satisfy the condition where 
                                // (length=0 && sum=variable)
    }

    for(int i=1;i<dp.length;i++){
        for(int j=1;j<dp[0].length;j++){


            /* Check if sum can be obtained by any of the following
              (a) including the last element
              (b) excluding the last element   */


            // VERY VERY IMP: This is same as "excluding the last element" which is represented in DP 
            dp[i][j] = dp[i-1][j]; // the current position[i][j] would be same as previous position.
                                   // the previous position means that SUM is ACHIEVED OR NOT-ACHIEVED
                                   // int the previous position then it will ofcourse be ACHIEVED or NOT-ACHIEVED
                                   // in the current position.


            // VERY VERY IMP: This is same as "including the last element" which is represented in DP 
            // if the column[ sum is represented in column of the matrix i.e this sum exist] > = sum-a[last_index]
            // then decrease the sum
            if(j>=a[i-1])   // i.e sum >= array[last index element]. If it is true then include this last element by
                            // deducting it from the total sum
                dp[i][j] = dp[i][j] || dp[i-1][j-a[i-1]];  // VERY VERY IMP NOTE: Here dp[i][j] on R.H.S represent
                            // dp[i-1][j] which we have assigned in the previous step

        }
    }
    return dp[a.length][sum];
}
/*
Analysis:
    Time Complexity = O(a.length*sum)
    Space Complexity = O(a.length*sum)
*/
}

Answer 3

这是我的解决方案是一个迭代的dp，但只有一个维度：希望它可以帮助你。

#include <iostream>
#include <cstring>

using namespace std;

const int maxN=1000;
int memo[maxN];
int pi[maxN];

int main(){
    int a[]={7,8,5,1,4};
    memset(memo,-1,sizeof memo);
    memset(pi,-1,sizeof pi);
    int n;
    cin>>n;
    memo[0]=0;
    pi[0]=0;
    for(int i=0;i<(int)sizeof(a)/4;i++){
        for(int num=n;num>=0;num--){
            if(num-a[i]>=0 and memo[num-a[i]]!=-1 and (memo[num]==-1 or memo[num]>1+memo[num-a[i]])){
                memo[num]=1+memo[num-a[i]]; 
                pi[num]=num-a[i];           
            }
        }
    }   
    int N=n;
    while(N!=0){
        cout<<N-pi[N]<<" ";
        N=pi[N];
    }
    cout<<endl;
    cout<<memo[n]<<endl;
    return 0;
}