由字符串数组中的 A 0 和 B 1 组成的最长子集的长度

给定一个由N 个二进制字符串和两个整数A和B组成的数组arr[] ，任务是 找到最多由A 0 s 和B 1 s 组成的最长子集的长度。

例子：

Input: arr[] = {“1”, “0”, “0001”, “10”, “111001”}, A = 5, B = 3
Output: 4
Explanation: 
One possible way is to select the subset {arr[0], arr[1], arr[2], arr[3]}.
Total number of 0s and 1s in all these strings are 5 and 3 respectively.
Also, 4 is the length of the longest subset possible.

Input: arr[] = {“0”, “1”, “10”}, A = 1, B = 1
Output: 2
Explanation: 
One possible way is to select the subset {arr[0], arr[1]}.
Total number of 0s and 1s in all these strings is 1 and 1 respectively.
Also, 2 is the length of the longest subset possible.

朴素的方法：解决上述问题最简单的方法是使用递归。在每次递归调用时，想法是包含或排除当前字符串。一旦考虑了所有的可能性，打印得到的子集的最大长度。

下面是上述方法的实现：

C++14

// C++ program for the above approach
#include 
using namespace std;
 
// Function to count 0's in a string
int count0(string s)
{
    // Stores count of 0s
    int count = 0;
 
    // Iterate over characters of string
    for (int i = 0; i < s.size(); i++) {
 
        // If current character is '0'
        if (s[i] == '0') {
            count++;
        }
    }
    return count;
}
 
// Recursive function to find the length of
// longest subset from an array of strings
// with at most A 0's and B 1's
int solve(vector vec,
          int A, int B, int idx)
{
    // If idx is equal to N
    // or A + B is equal to 0
    if (idx == vec.size() || A + B == 0) {
        return 0;
    }
 
    // Stores the count of 0's in arr[idx]
    int zero = count0(vec[idx]);
 
    // Stores the count of 1's in arr[idx]
    int one = vec[idx].size() - zero;
 
    // Stores the length of the
    // subset if arr[i] is included
    int inc = 0;
 
    // If zero is less than or equal to A
    // and one is less than or equal to B
    if (zero <= A && one <= B) {
        inc = 1 + solve(vec, A - zero,
                        B - one, idx + 1);
    }
 
    // Stores the length of the subset
    // if arr[i] is excluded
    int exc = solve(vec, A, B, idx + 1);
 
    // Returns max of inc and exc
    return max(inc, exc);
}
 
// Function to find the length of the
// longest subset from an array of
// strings with at most A 0's and B 1's
int MaxSubsetlength(vector arr,
                    int A, int B)
{
    // Return
    return solve(arr, A, B, 0);
}
 
// Driver Code
int main()
{
    vector arr = { "1", "0", "10" };
    int A = 1, B = 1;
 
    cout << MaxSubsetlength(arr, A, B);
    return 0;
}

Java

// Java program for the above approach
import java.util.*;
 
class GFG {
// Function to count 0's in a string
static int count0(String s)
{
    // Stores count of 0s
    int count = 0;
 
    // Iterate over characters of string
    for (int i = 0; i < s.length(); i++) {
 
        // If current character is '0'
        if (s.charAt(i) == '0') {
            count++;
        }
    }
    return count;
}
 
// Recursive function to find the length of
// longest subset from an array of strings
// with at most A 0's and B 1's
static int solve(String[] vec,
          int A, int B, int idx)
{
    // If idx is equal to N
    // or A + B is equal to 0
    if (idx == vec.length || A + B == 0) {
        return 0;
    }
 
    // Stores the count of 0's in arr[idx]
    int zero = count0(vec[idx]);
 
    // Stores the count of 1's in arr[idx]
    int one = vec[idx].length() - zero;
 
    // Stores the length of the
    // subset if arr[i] is included
    int inc = 0;
 
    // If zero is less than or equal to A
    // and one is less than or equal to B
    if (zero <= A && one <= B) {
        inc = 1 + solve(vec, A - zero,
                        B - one, idx + 1);
    }
 
    // Stores the length of the subset
    // if arr[i] is excluded
    int exc = solve(vec, A, B, idx + 1);
 
    // Returns max of inc and exc
    return Math.max(inc, exc);
}
 
// Function to find the length of the
// longest subset from an array of
// strings with at most A 0's and B 1's
static int MaxSubsetlength(String[] arr,
                    int A, int B)
{
    // Return
    return solve(arr, A, B, 0);
}
 
 
    public static void main (String[] args) {
    String[] arr = { "1", "0", "10" };
    int A = 1, B = 1;
 
    System.out.print(MaxSubsetlength(arr, A, B));
    }
}
// This code is contributed by offbeat

Python3

# Python3 program for the above approach
 
# Function to count 0's in a string
def count0(s):
     
    # Stores count of 0s
    count = 0
 
    # Iterate over characters of string
    for i in range(len(s)):
         
        # If current character is '0'
        if (s[i] == '0'):
            count += 1
             
    return count
 
# Recursive function to find the length of
# longest subset from an array of strings
# with at most A 0's and B 1's
def solve(vec, A, B, idx):
     
    # If idx is equal to N
    # or A + B is equal to 0
    if (idx == len(vec) or A + B == 0):
        return 0
 
    # Stores the count of 0's in arr[idx]
    zero = count0(vec[idx])
 
    # Stores the count of 1's in arr[idx]
    one = len(vec[idx]) - zero
 
    # Stores the length of the
    # subset if arr[i] is included
    inc = 0
 
    # If zero is less than or equal to A
    # and one is less than or equal to B
    if (zero <= A and one <= B):
        inc = 1 + solve(vec, A - zero,
                             B - one, idx + 1)
 
    # Stores the length of the subset
    # if arr[i] is excluded
    exc = solve(vec, A, B, idx + 1)
 
    # Returns max of inc and exc
    return max(inc, exc)
 
# Function to find the length of the
# longest subset from an array of
# strings with at most A 0's and B 1's
def MaxSubsetlength(arr, A, B):
     
    # Return
    return solve(arr, A, B, 0)
 
# Driver Code
if __name__ == '__main__':
     
    arr = [ "1", "0", "10" ]
    A = 1
    B = 1
     
    print(MaxSubsetlength(arr, A, B))
 
# This code is contributed by SURENDRA_GANGWAR

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
 
class GFG{
     
// Function to count 0's in a string
static int count0(string s)
{
     
    // Stores count of 0s
    int count = 0;
 
    // Iterate over characters of string
    for(int i = 0; i < s.Length; i++)
    {
         
        // If current character is '0'
        if (s[i] == '0')
        {
            count++;
        }
    }
    return count;
}
 
// Recursive function to find the length of
// longest subset from an array of strings
// with at most A 0's and B 1's
static int solve(List vec, int A, int B,
                 int idx)
{
     
    // If idx is equal to N
    // or A + B is equal to 0
    if (idx == vec.Count || A + B == 0)
    {
        return 0;
    }
 
    // Stores the count of 0's in arr[idx]
    int zero = count0(vec[idx]);
 
    // Stores the count of 1's in arr[idx]
    int one = vec[idx].Length - zero;
 
    // Stores the length of the
    // subset if arr[i] is included
    int inc = 0;
 
    // If zero is less than or equal to A
    // and one is less than or equal to B
    if (zero <= A && one <= B)
    {
        inc = 1 + solve(vec, A - zero,
                             B - one, idx + 1);
    }
 
    // Stores the length of the subset
    // if arr[i] is excluded
    int exc = solve(vec, A, B, idx + 1);
 
    // Returns max of inc and exc
    return Math.Max(inc, exc);
}
 
// Function to find the length of the
// longest subset from an array of
// strings with at most A 0's and B 1's
static int MaxSubsetlength(List arr, int A,
                           int B)
{
     
    // Return
    return solve(arr, A, B, 0);
}
 
// Driver Code
public static void Main()
{
    List arr = new List{ "1", "0", "10" };
    int A = 1, B = 1;
 
    Console.WriteLine(MaxSubsetlength(arr, A, B));
}
}
 
// This code is contributed by ukasp

Javascript

 
// JavaScript program to implement
// the above approach
 
// Function to count 0's in a string
function count0(s)
{
    // Stores count of 0s
    let count = 0;
  
    // Iterate over characters of string
    for (let i = 0; i < s.length; i++) {
  
        // If current character is '0'
        if (s[i] == '0') {
            count++;
        }
    }
    return count;
}
  
// Recursive function to find the length of
// longest subset from an array of strings
// with at most A 0's and B 1's
function solve(vec, A, B, idx)
{
    // If idx is equal to N
    // or A + B is equal to 0
    if (idx == vec.length || A + B == 0) {
        return 0;
    }
  
    // Stores the count of 0's in arr[idx]
    let zero = count0(vec[idx]);
  
    // Stores the count of 1's in arr[idx]
    let one = vec[idx].length - zero;
  
    // Stores the length of the
    // subset if arr[i] is included
    let inc = 0;
  
    // If zero is less than or equal to A
    // and one is less than or equal to B
    if (zero <= A && one <= B) {
        inc = 1 + solve(vec, A - zero,
                        B - one, idx + 1);
    }
  
    // Stores the length of the subset
    // if arr[i] is excluded
    let exc = solve(vec, A, B, idx + 1);
  
    // Returns max of inc and exc
    return Math.max(inc, exc);
}
  
// Function to find the length of the
// longest subset from an array of
// strings with at most A 0's and B 1's
function MaxSubsetlength(arr, A, B)
{
    // Return
    return solve(arr, A, B, 0);
}
 
 
// Driver code
 
    let arr = [ "1", "0", "10" ];
    let A = 1, B = 1;
  
    document.write(MaxSubsetlength(arr, A, B));
           

输出：

2

时间复杂度： O(2 ^N )
辅助空间： O(1)

高效方法：基于以下观察，可以使用 Memoization 优化上述方法：

Recursion Tree

• It can be observed that there exists a overlapping subproblem and optimal substructure property.
  Therefore, the idea is to use dynamic programming to optimize the above approach.
• The idea is to use the 3D-state dynamic programming.
• Suppose Dp(i, A, B) represents the maximum length of the subset with at most A 0s and B 1s.
• Then the transition of states can be defined by either selecting a string at i^th index or not,
  • Dp(i, A, B) = Max(1+Dp(i+1, A- Z, B-O), Dp(i+1, A, B)),  
    where Z is the count of Os and O is the count of 0s.

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to count number
// of 0s present in the string
int count0(string s)
{
    // Stores the count of 0s
    int count = 0;
 
    // Iterate over characters of string
    for (int i = 0; i < s.size(); i++) {
 
        // If current character is '0'
        if (s[i] == '0') {
            count++;
        }
    }
    return count;
}
// Recursive Function to find the length of
// longest subset from given array of strings
// with at most A 0s and B 1s
int solve(vector vec, int A,
          int B, int idx,
          vector > >& dp)
{
    // If idx is equal to N or
    // A + B is equal to 0
    if (idx == vec.size() || A + B == 0) {
        return 0;
    }
    // If the state is already calculated
    if (dp[A][B][idx] > 0) {
        return dp[A][B][idx];
    }
 
    // Stores the count of 0's
    int zero = count0(vec[idx]);
 
    // Stores the count of 1's
    int one = vec[idx].size() - zero;
 
    // Stores the length of longest
    // by including arr[idx]
    int inc = 0;
 
    // If zero is less than A
    // and one is less than B
    if (zero <= A && one <= B) {
        inc = 1
              + solve(vec, A - zero,
                      B - one, idx + 1, dp);
    }
 
    // Stores the length of longest subset
    // by excluding arr[idx]
    int exc = solve(vec, A, B, idx + 1, dp);
 
    // Assign
    dp[A][B][idx] = max(inc, exc);
 
    // Return
    return dp[A][B][idx];
}
 
// Function to find the length of the
// longest subset of an array of strings
// with at most A 0s and B 1s
int MaxSubsetlength(vector arr,
                    int A, int B)
{
    // Stores all Dp-states
    vector > > dp(
        A + 1,
        vector >(B + 1,
                             vector(arr.size() + 1,
                                         0)));
    // Return
    return solve(arr, A, B, 0, dp);
}
 
// Driver Code
int main()
{
    vector arr = { "1", "0", "10" };
    int A = 1, B = 1;
 
    cout << MaxSubsetlength(arr, A, B);
    return 0;
}

输出：

2

时间复杂度： O(N*A*B)
辅助空间： O(N*A*B)

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