找到所有零和的三元组

给定一系列不同的元素。任务是在总和为零的数组中查找三元组。

例子 ：

Input : arr[] = {0, -1, 2, -3, 1}
Output : (0 -1 1), (2 -3 1)

Explanation : The triplets with zero sum are
0 + -1 + 1 = 0 and 2 + -3 + 1 = 0  

Input : arr[] = {1, -2, 1, 0, 5}
Output : 1 -2  1
Explanation : The triplets with zero sum is
1 + -2 + 1 = 0   

方法1：这是一种简单的方法，需要O(n ³ )时间才能得出结果。

• 方法：天真的方法运行三个循环，并一一检查三个元素的总和是否为零。如果三个元素的总和为零，则显示打印元素，否则找不到打印元素。
• 算法：
  1. 使用循环计数器i ， j ， k运行三个嵌套循环
  2. 第一个循环从0到n-3，第二个循环从i + 1到n-2，第三个循环从j + 1到n-1。循环计数器代表三元组的三个元素。
  3. 检查第i，第j，第k个元素的总和是否等于零。如果是，则打印总和，否则继续。

下面是上述方法的实现：

C++

// A simple C++ program to find three elements
// whose sum is equal to zero
#include
using namespace std;
 
// Prints all triplets in arr[] with 0 sum
void findTriplets(int arr[], int n)
{
    bool found = false;
    for (int i=0; i

---

Java

// A simple Java program to find three elements
// whose sum is equal to zero
class num{
// Prints all triplets in arr[] with 0 sum
static void findTriplets(int[] arr, int n)
{
    boolean found = false;
    for (int i=0; i

---

Python3

# A simple Python 3 program
# to find three elements whose
# sum is equal to zero
 
# Prints all triplets in
# arr[] with 0 sum
def findTriplets(arr, n):
 
    found = False
    for i in range(0, n-2):
     
        for j in range(i+1, n-1):
         
            for k in range(j+1, n):
             
                if (arr[i] + arr[j] + arr[k] == 0):
                    print(arr[i], arr[j], arr[k])
                    found = True
     
             
    # If no triplet with 0 sum
    # found in array
    if (found == False):
        print(" not exist ")
 
# Driver code
arr = [0, -1, 2, -3, 1]
n = len(arr)
findTriplets(arr, n)
 
# This code is contributed by Smitha Dinesh Semwal

C#

// A simple C# program to find three elements
// whose sum is equal to zero
using System;
 
class GFG {
     
    // Prints all triplets in arr[] with 0 sum
    static void findTriplets(int []arr, int n)
    {
        bool found = false;
        for (int i = 0; i < n-2; i++)
        {
            for (int j = i+1; j < n-1; j++)
            {
                for (int k = j+1; k < n; k++)
                {
                    if (arr[i] + arr[j] + arr[k]
                                           == 0)
                    {
                        Console.Write(arr[i]);
                        Console.Write(" ");
                        Console.Write(arr[j]);
                        Console.Write(" ");
                        Console.Write(arr[k]);
                        Console.Write("\n");
                        found = true;
                    }
                }
            }
        }
     
        // If no triplet with 0 sum found in
        // array
        if (found == false)
            Console.Write(" not exist ");
    }
     
    // Driver code
    public static void Main()
    {
        int []arr = {0, -1, 2, -3, 1};
        int n = arr.Length;
        findTriplets(arr, n);
    }
}
 
// This code is contributed by nitin mittal.

Javascript

// A simple Javascript program to find
//three elements whose sum is equal to zero
    arr = [0, -1, 2, -3, 1];
     
// Prints all triplets in arr[] with 0 sum   
    function findTriplets(arr) {
        let found = false;
        for (let i = 0; i < arr.length - 2; i++) {
        for (let j = i + 1; j < arr.length - 1; j++) {
            for (let k = j + 1; k < arr.length; k++) {
                if (arr[i] + arr[j] + arr[k] === 0)
                {
                    document.write(arr[i]);
                    document.write(" ");
                    document.write(arr[j]);
                    document.write(" ");
                    document.write(arr[k]);
                    document.write("<br>");
                    found = true;
                    
                }
            }
        }
        // If no triplet with 0 sum found in array
        if(found === false) {
            document.write(" not exist ");
        }
    }
   }
    findTriplets(arr);
// This code is contributed by Surbhi Tyagi

---

---

---

---

输出

0 -1 1
2 -3 1

复杂度分析：

• 时间复杂度： O(n ³ )。
  由于需要三个嵌套循环，因此时间复杂度为O(n ³ )。
• 辅助空间： O(1)。
  由于不需要额外的空间，因此时间复杂度是恒定的。

方法2：第二种方法使用散列处理得出结果，并在较短的O(n ² )时间求解。

方法：这涉及遍历数组。对于每个元素arr [i]，找到一个总和为“ -arr [i]”的对。该问题简化为成对和，可以使用哈希在O(n)时间内解决。

算法：

1. 创建一个哈希图以存储键值对。
2. 运行带有两个循环的嵌套循环，外部循环从0到n-2，内部循环从i + 1到n-1
3. 检查哈希图中是否存在ith和jth元素的总和乘以-1
4. 如果该元素存在于哈希图中，则打印三元组，否则将第j个元素插入哈希图中。

下面是上述方法的实现：

C++

// C++ program to find triplets in a given
// array whose sum is zero
#include
using namespace std;
 
// function to print triplets with 0 sum
void findTriplets(int arr[], int n)
{
    bool found = false;
 
    for (int i=0; i s;
        for (int j=i+1; j

Java

// Java program to find triplets in a given
// array whose sum is zero
import java.util.*;
 
class GFG
{
 
    // function to print triplets with 0 sum
    static void findTriplets(int arr[], int n)
    {
        boolean found = false;
 
        for (int i = 0; i < n - 1; i++)
        {
            // Find all pairs with sum equals to
            // "-arr[i]"
            HashSet s = new HashSet();
            for (int j = i + 1; j < n; j++)
            {
                int x = -(arr[i] + arr[j]);
                if (s.contains(x))
                {
                    System.out.printf("%d %d %d\n", x, arr[i], arr[j]);
                    found = true;
                }
                else
                {
                    s.add(arr[j]);
                }
            }
        }
 
        if (found == false)
        {
            System.out.printf(" No Triplet Found\n");
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = {0, -1, 2, -3, 1};
        int n = arr.length;
        findTriplets(arr, n);
    }
}
 
// This code contributed by Rajput-Ji

Python3

# Python3 program to find triplets
# in a given array whose sum is zero
 
# function to print triplets with 0 sum
def findTriplets(arr, n):
    found = False
    for i in range(n - 1):
 
        # Find all pairs with sum
        # equals to "-arr[i]"
        s = set()
        for j in range(i + 1, n):
            x = -(arr[i] + arr[j])
            if x in s:
                print(x, arr[i], arr[j])
                found = True
            else:
                s.add(arr[j])
    if found == False:
        print("No Triplet Found")
 
# Driver Code
arr = [0, -1, 2, -3, 1]
n = len(arr)
findTriplets(arr, n)
 
# This code is contributed by Shrikant13

C＃

// C# program to find triplets in a given
// array whose sum is zero
using System;
using System.Collections.Generic;
 
class GFG
{
 
    // function to print triplets with 0 sum
    static void findTriplets(int []arr, int n)
    {
        bool found = false;
 
        for (int i = 0; i < n - 1; i++)
        {
            // Find all pairs with sum equals to
            // "-arr[i]"
            HashSet s = new HashSet();
            for (int j = i + 1; j < n; j++)
            {
                int x = -(arr[i] + arr[j]);
                if (s.Contains(x))
                {
                    Console.Write("{0} {1} {2}\n", x, arr[i], arr[j]);
                    found = true;
                }
                else
                {
                    s.Add(arr[j]);
                }
            }
        }
 
        if (found == false)
        {
            Console.Write(" No Triplet Found\n");
        }
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int []arr = {0, -1, 2, -3, 1};
        int n = arr.Length;
        findTriplets(arr, n);
    }
}
 
// This code has been contributed by 29AjayKumar

输出

-1 0 1
-3 2 1

复杂度分析：

• 时间复杂度： O(n ² )。
  由于需要两个嵌套循环，因此时间复杂度为O(n ² )。
• 辅助空间： O(n)。
  由于需要一个哈希图，因此时间复杂度是线性的。

方法3：该方法使用排序获得正确的结果，并在O(n ² )时间内求解。

方法：以上方法需要额外的空间。这个想法是基于本文的方法2。对于每个元素，请检查是否存在一对总和等于该元素的负值的对。

算法：

1. 以升序对数组进行排序。
2. 从头到尾遍历数组。
3. 对于每个索引i ，创建两个变量l = i + 1和r = n – 1
4. 如果array [l]，array [r]的总和等于零，则运行循环直到l小于r，然后打印三元组并中断循环
5. 如果总和小于零，则增加l的值，通过增加l的值，该总和将随着对数组进行排序而增加，因此array [l + 1]> array [l]
6. 如果总和大于零，则将r的值减小，通过增加l的值，该总和将随着对数组进行排序而减小，因此array [r-1] 。

下面是上述方法的实现：

C++

// C++ program to find triplets in a given
// array whose sum is zero
#include
using namespace std;
 
// function to print triplets with 0 sum
void findTriplets(int arr[], int n)
{
    bool found = false;
 
    // sort array elements
    sort(arr, arr+n);
 
    for (int i=0; i

Java

// Java  program to find triplets in a given
// array whose sum is zero
import java.util.Arrays;
import java.io.*;
 
class GFG {
    // function to print triplets with 0 sum
static void findTriplets(int arr[], int n)
{
    boolean found = false;
 
    // sort array elements
    Arrays.sort(arr);
 
    for (int i=0; i

Python3

# python program to find triplets in a given
# array whose sum is zero
 
# function to print triplets with 0 sum
def findTriplets(arr, n):
 
    found = False
 
    # sort array elements
    arr.sort()
 
    for i in range(0, n-1):
     
        # initialize left and right
        l = i + 1
        r = n - 1
        x = arr[i]
        while (l < r):
         
            if (x + arr[l] + arr[r] == 0):
                # print elements if it's sum is zero
                print(x, arr[l], arr[r])
                l+=1
                r-=1
                found = True
             
 
            # If sum of three elements is less
            # than zero then increment in left
            elif (x + arr[l] + arr[r] < 0):
                l+=1
 
            # if sum is greater than zero than
            # decrement in right side
            else:
                r-=1
         
    if (found == False):
        print(" No Triplet Found")
 
 
# Driven source
arr = [0, -1, 2, -3, 1]
n = len(arr)
findTriplets(arr, n)
 
# This code is contributed by Smitha Dinesh Semwal

C＃

// C#  program to find triplets in a given
// array whose sum is zero
using System;
 
public class GFG{
        // function to print triplets with 0 sum
static void findTriplets(int []arr, int n)
{
    bool found = false;
 
    // sort array elements
    Array.Sort(arr);
 
    for (int i=0; i

Java脚本

 
// Javascript program to find triplets in a given
// array whose sum is zero
 
// function to print triplets with 0 sum
function findTriplets(arr, n)
{
    let found = false;
 
    // sort array elements
    arr.sort((a, b) => a - b);
 
    for (let i=0; i<n-1; i++)
    {
        // initialize left and right
        let l = i + 1;
        let r = n - 1;
        let x = arr[i];
        while (l < r)
        {
            if (x + arr[l] + arr[r] == 0)
            {
                // print elements if it's sum is zero
                document.write(x + " ");
                document.write(arr[l]+ " ");
                document.write(arr[r]+ " " + "<br>");
                l++;
                r--;
                found = true;
            }
 
            // If sum of three elements is less
            // than zero then increment in left
            else if (x + arr[l] + arr[r] < 0)
                l++;
 
            // if sum is greater than zero than
            // decrement in right side
            else
                r--;
        }
    }
 
    if (found == false)
        document.write(" No Triplet Found" + "<br>");
}
 
// Driven source
 
    let arr = [0, -1, 2, -3, 1];
    let n = arr.length;
    findTriplets(arr, n);
 
 
// This code is contributed by Mayank Tyagi
 

输出

-3 1 2
-1 0 1

复杂度分析：

• 时间复杂度： O(n ² )。
  仅需要两个嵌套循环，因此时间复杂度为O(n ² )。
• 辅助空间： O(1)，不需要额外的空间，因此时间复杂度是恒定的。