在两个数组中找到和最小的 k 对

给定两个按升序排序的整数数组 arr1[] 和 arr2[] 以及一个整数 k。找到k个和最小的对，使得一对中的一个元素属于arr1[]，另一个元素属于arr2[]
例子：

Input :  arr1[] = {1, 7, 11}
         arr2[] = {2, 4, 6}
         k = 3
Output : [1, 2],
         [1, 4],
         [1, 6]
Explanation: The first 3 pairs are returned 
from the sequence [1, 2], [1, 4], [1, 6], 
[7, 2], [7, 4], [11, 2], [7, 6], [11, 4], 
[11, 6]

方法1（简单）

1. 找到所有对并存储它们的总和。此步骤的时间复杂度为 O(n1 * n2)，其中 n1 和 n2 是输入数组的大小。
2. 然后根据总和对对进行排序。这一步的时间复杂度是 O(n1 * n2 * log (n1 * n2))

总体时间复杂度：O(n1 * n2 * log (n1 * n2))
方法2（高效）：
我们从最小和对开始，一一找到k个最小和对。这个想法是跟踪 arr2[] 的所有元素，这些元素已经为每个元素 arr1[i1] 考虑过，以便在迭代中我们只考虑下一个元素。为此，我们使用索引数组 index2[] 来跟踪另一个数组中下一个元素的索引。它只是意味着在每次迭代中将第二个数组的哪个元素与第一个数组的元素相加。我们在索引数组中为形成下一个最小值对的元素增加值。

C++

// C++ program to prints first k pairs with least sum from two
// arrays.
#include
 
using namespace std;
 
// Function to find k pairs with least sum such
// that one element of a pair is from arr1[] and
// other element is from arr2[]
void kSmallestPair(int arr1[], int n1, int arr2[],
                                   int n2, int k)
{
    if (k > n1*n2)
    {
        cout << "k pairs don't exist";
        return ;
    }
 
    // Stores current index in arr2[] for
    // every element of arr1[]. Initially
    // all values are considered 0.
    // Here current index is the index before
    // which all elements are considered as
    // part of output.
    int index2[n1];
    memset(index2, 0, sizeof(index2));
 
    while (k > 0)
    {
        // Initialize current pair sum as infinite
        int min_sum = INT_MAX;
        int min_index = 0;
 
        // To pick next pair, traverse for all elements
        // of arr1[], for every element, find corresponding
        // current element in arr2[] and pick minimum of
        // all formed pairs.
        for (int i1 = 0; i1 < n1; i1++)
        {
            // Check if current element of arr1[] plus
            // element of array2 to be used gives minimum
            // sum
            if (index2[i1] < n2 &&
                arr1[i1] + arr2[index2[i1]] < min_sum)
            {
                // Update index that gives minimum
                min_index = i1;
 
                // update minimum sum
                min_sum = arr1[i1] + arr2[index2[i1]];
            }
        }
 
        cout << "(" << arr1[min_index] << ", "
             << arr2[index2[min_index]] << ") ";
 
        index2[min_index]++;
 
        k--;
    }
}
 
// Driver code
int main()
{
    int arr1[] = {1, 3, 11};
    int n1 = sizeof(arr1) / sizeof(arr1[0]);
 
    int arr2[] = {2, 4, 8};
    int n2 = sizeof(arr2) / sizeof(arr2[0]);
 
    int k = 4;
    kSmallestPair( arr1, n1, arr2, n2, k);
 
    return 0;
}

Java

// Java code to print first k pairs with least
// sum from two arrays.
import java.io.*;
  
class KSmallestPair
{
    // Function to find k pairs with least sum such
    // that one element of a pair is from arr1[] and
    // other element is from arr2[]
    static void kSmallestPair(int arr1[], int n1, int arr2[],
                                            int n2, int k)
    {
        if (k > n1*n2)
        {
            System.out.print("k pairs don't exist");
            return ;
        }
      
        // Stores current index in arr2[] for
        // every element of arr1[]. Initially
        // all values are considered 0.
        // Here current index is the index before
        // which all elements are considered as
        // part of output.
        int index2[] = new int[n1];
      
        while (k > 0)
        {
            // Initialize current pair sum as infinite
            int min_sum = Integer.MAX_VALUE;
            int min_index = 0;
      
            // To pick next pair, traverse for all
            // elements of arr1[], for every element, find
            // corresponding current element in arr2[] and
            // pick minimum of all formed pairs.
            for (int i1 = 0; i1 < n1; i1++)
            {
                // Check if current element of arr1[] plus
                // element of array2 to be used gives
                // minimum sum
                if (index2[i1] < n2 &&
                    arr1[i1] + arr2[index2[i1]] < min_sum)
                {
                    // Update index that gives minimum
                    min_index = i1;
      
                    // update minimum sum
                    min_sum = arr1[i1] + arr2[index2[i1]];
                }
            }
      
            System.out.print("(" + arr1[min_index] + ", " +
                            arr2[index2[min_index]]+ ") ");
      
            index2[min_index]++;
            k--;
        }
    }
 
    // Driver code
    public static void main (String[] args)
    {
        int arr1[] = {1, 3, 11};
        int n1 = arr1.length;
      
        int arr2[] = {2, 4, 8};
        int n2 = arr2.length;
      
        int k = 4;
        kSmallestPair( arr1, n1, arr2, n2, k);
    }
}
/*This code is contributed by Prakriti Gupta*/

Python3

# Python3 program to prints first k pairs with least sum from two
# arrays.
 
import sys
# Function to find k pairs with least sum such
# that one element of a pair is from arr1[] and
# other element is from arr2[]
def kSmallestPair(arr1, n1, arr2, n2, k):
    if (k > n1*n2):
        print("k pairs don't exist")
        return
 
    # Stores current index in arr2[] for
    # every element of arr1[]. Initially
    # all values are considered 0.
    # Here current index is the index before
    # which all elements are considered as
    # part of output.
    index2 = [0 for i in range(n1)]
 
    while (k > 0):
        # Initialize current pair sum as infinite
        min_sum = sys.maxsize
        min_index = 0
 
        # To pick next pair, traverse for all elements
        # of arr1[], for every element, find corresponding
        # current element in arr2[] and pick minimum of
        # all formed pairs.
        for i1 in range(0,n1,1):
            # Check if current element of arr1[] plus
            # element of array2 to be used gives minimum
            # sum
            if (index2[i1] < n2 and arr1[i1] + arr2[index2[i1]] < min_sum):
                # Update index that gives minimum
                min_index = i1
 
                # update minimum sum
                min_sum = arr1[i1] + arr2[index2[i1]]
         
        print("(",arr1[min_index],",",arr2[index2[min_index]],")",end = " ")
 
        index2[min_index] += 1
 
        k -= 1
 
# Driver code
if __name__ == '__main__':
    arr1 = [1, 3, 11]
    n1 = len(arr1)
 
    arr2 = [2, 4, 8]
    n2 = len(arr2)
 
    k = 4
    kSmallestPair( arr1, n1, arr2, n2, k)
 
# This code is contributed by
# Shashank_Sharma

C#

// C# code to print first k pairs with
// least with least sum from two arrays.
using System;
 
class KSmallestPair
{
    // Function to find k pairs with least
    // sum such that one element of a pair
    // is from arr1[] and other element is
    // from arr2[]
    static void kSmallestPair(int []arr1, int n1,
                        int []arr2, int n2, int k)
    {
        if (k > n1 * n2)
        {
            Console.Write("k pairs don't exist");
            return;
        }
     
        // Stores current index in arr2[] for
        // every element of arr1[]. Initially
        // all values are considered 0. Here
        // current index is the index before
        // which all elements are considered
        // as part of output.
        int []index2 = new int[n1];
     
        while (k > 0)
        {
            // Initialize current pair sum as infinite
            int min_sum = int.MaxValue;
            int min_index = 0;
     
            // To pick next pair, traverse for all
            // elements of arr1[], for every element, 
            // find corresponding current element in
            // arr2[] and pick minimum of all formed pairs.
            for (int i1 = 0; i1 < n1; i1++)
            {
                // Check if current element of arr1[]
                // plus element of array2 to be used 
                // gives minimum sum
                if (index2[i1] < n2 && arr1[i1] +
                    arr2[index2[i1]] < min_sum)
                {
                    // Update index that gives minimum
                    min_index = i1;
     
                    // update minimum sum
                    min_sum = arr1[i1] + arr2[index2[i1]];
                }
            }
     
        Console.Write("(" + arr1[min_index] + ", " +
                        arr2[index2[min_index]] + ") ");
     
            index2[min_index]++;
            k--;
        }
    }
 
    // Driver code
    public static void Main (String[] args)
    {
        int []arr1 = {1, 3, 11};
        int n1 = arr1.Length;
     
        int []arr2 = {2, 4, 8};
        int n2 = arr2.Length;
     
        int k = 4;
        kSmallestPair( arr1, n1, arr2, n2, k);
    }
}
 
// This code is contributed by Parashar.

Javascript

// javascript program to prints first k pairs with least sum from two
// arrays.
// Function to find k pairs with least sum such
// that one element of a pair is from arr1[] and
// other element is from arr2[]
function kSmallestPair(arr1,n1,arr2,n2,k)
{
    if (k > n1*n2)
    {
        document.write("k pairs don't exist");
        return ;
    }
 
    // Stores current index in arr2[] for
    // every element of arr1[]. Initially
    // all values are considered 0.
    // Here current index is the index before
    // which all elements are considered as
    // part of output.
    let index2 = new Array(n1);
    index2.fill(0);
 
    while (k > 0)
    {
        // Initialize current pair sum as infinite
        let min_sum = Number.MAX_VALUE;
        let min_index = 0;
 
        // To pick next pair, traverse for all elements
        // of arr1[], for every element, find corresponding
        // current element in arr2[] and pick minimum of
        // all formed pairs.
        for (let i1 = 0; i1 < n1; i1++)
        {
            // Check if current element of arr1[] plus
            // element of array2 to be used gives minimum
            // sum
            if (index2[i1] < n2 &&
                arr1[i1] + arr2[index2[i1]] < min_sum)
            {
                // Update index that gives minimum
                min_index = i1;
 
                // update minimum sum
                min_sum = arr1[i1] + arr2[index2[i1]];
            }
        }
 
        document.write("(" + arr1[min_index] + ", "
            + arr2[index2[min_index]] + ") ");
 
        index2[min_index]++;
 
        k--;
    }
}
 
    let arr1 = [1, 3, 11];
    let n1 = arr1.length;
 
    let arr2 = [2, 4, 8];
    let n2 = arr2.length;
 
    let k = 4;
    kSmallestPair( arr1, n1, arr2, n2, k);
 
     
 

输出

(1, 2) (1, 4) (3, 2) (3, 4) 

时间复杂度： O(k*n1)
方法3：使用排序、最小堆、映射
我们应该找到一种方法将我们的搜索空间限制为可能的候选和组合，而不是暴力破解所有可能的总和组合。

1. 对数组数组 A 和数组 B 进行排序。
2. 在 C++ 中创建一个最小堆，即 priority_queue 来存储总和组合以及构成总和的数组 A 和 B 中的元素的索引。堆按总和排序。
3. 使用最小可能的总和组合（即 (A[0] + B[0])）和两个数组中元素的索引 (0, 0) 来初始化堆。最小堆内的元组将是 (A[0] + B[0], 0, 0)。堆按第一个值排序，即两个元素的总和。
4. 弹出堆以获取当前最小的总和以及构成总和的元素的索引。让元组为 (sum, i, j)。
   • 接下来将 (A[i + 1] + B[j], i + 1, j) 和 (A[i] + B[j + 1], i, j + 1) 插入最小堆，但要确保最小堆中尚不存在一对索引，即 (i + 1, j) 和 (i, j + 1)。要检查这一点，我们可以在 C++ 中使用 set。
   • 回到 4 直到 K 次。

C++

// C++ program to Prints
// first k pairs with
// least sum from two arrays.
 
#include 
using namespace std;
 
// Function to find k pairs
// with least sum such
// that one element of a pair
// is from arr1[] and
// other element is from arr2[]
void kSmallestPair(vector A, vector B, int K)
{
    sort(A.begin(), A.end());
    sort(B.begin(), B.end());
 
    int n = A.size();
 
    // Min heap which contains tuple of the format
    // (sum, (i, j)) i and j are the indices
    // of the elements from array A
    // and array B which make up the sum.
 
    priority_queue >,
                   vector > >,
                   greater > > >
        pq;
 
    // my_set is used to store the indices of
    // the  pair(i, j) we use my_set to make sure
    // the indices doe not repeat inside min heap.
 
    set > my_set;
 
    // initialize the heap with the minimum sum
    // combination i.e. (A[0] + B[0])
    // and also push indices (0,0) along
    // with sum.
 
    pq.push(make_pair(A[0] + B[0], make_pair(0, 0)));
 
    my_set.insert(make_pair(0, 0));
 
    // iterate upto K
    int flag=1;
    for (int count = 0; count < K && flag; count++) {
 
        // tuple format (sum, i, j).
        pair > temp = pq.top();
        pq.pop();
 
        int i = temp.second.first;
        int j = temp.second.second;
 
        cout << "(" << A[i] << ", " << B[j] << ")"
             << endl; // Extracting pair with least sum such
                      // that one element is from arr1 and
                      // another is from arr2
 
        // check if i+1 is in the range of our first array A
        flag=0;
        if (i + 1 < A.size()) {
            int sum = A[i + 1] + B[j];
            // insert (A[i + 1] + B[j], (i + 1, j))
            // into min heap.
            pair temp1 = make_pair(i + 1, j);
 
            // insert only if the pair (i + 1, j) is
            // not already present inside the map i.e.
            // no repeating pair should be present inside
            // the heap.
 
            if (my_set.find(temp1) == my_set.end()) {
                pq.push(make_pair(sum, temp1));
                my_set.insert(temp1);
            }
          flag=1;
        }
        // check if j+1 is in the range of our second array
        // B
        if (j + 1 < B.size()) {
            // insert (A[i] + B[j + 1], (i, j + 1))
            // into min heap.
 
            int sum = A[i] + B[j + 1];
            pair temp1 = make_pair(i, j + 1);
 
            // insert only if the pair (i, j + 1)
            // is not present inside the heap.
 
            if (my_set.find(temp1) == my_set.end()) {
                pq.push(make_pair(sum, temp1));
                my_set.insert(temp1);
            }
          flag=1;
        }
    }
}
 
// Driver Code.
int main()
{
    vector A = { 1 };
    vector B = { 2, 4, 5, 9 };
    int K = 3;
    kSmallestPair(A, B, K);
    return 0;
}
 
// This code is contributed by Dhairya.

输出

(1, 2)
(1, 4)
(1, 5)