给定一个整型数组，请打印出元素和为0的所有子数组。
例如，
输入:
{ 4, 2, -3, -1, 0, 4 }
输出:
Sub-arrays with 0 sum are
{ -3, -1, 0, 4 }
{ 0 }

输入:
{ 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 }
输出:
Sub-arrays with 0 sum are
{ 3, 4, -7 }
{ 4, -7, 3 }
{ -7, 3, 1, 3 }
{ 3, 1, -4 }
{ 3, 1, 3, 1, -4, -2, -2 }
{ 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 }

1. 傻瓜式方法

傻瓜式方法的想法很简单，遍历数组的所有子数组，并计算它们的和。如果子数组的和等于0，那么就打印它。它的时间复杂度是 O(n³)，因为一共有 n(n+1)/2(注：原文n²不对)个子数组，花费的时间是O(n²)，然后还要花费 O(n)的时间求元素的和。当然，如果在找子数组的同时计算元素的和，那么这种方法可以优化成O(n²)时间。

C++语言实现：

#include <iostream>
#include <unordered_map>
using namespace std;
 
// Function to print all sub-arrays with 0 sum present
// in the given array
void printallSubarrays(int arr[], int n)
{
    // consider all sub-arrays starting from i
    for (int i = 0; i < n; i++)
    {
        int sum = 0;
 
        // consider all sub-arrays ending at j
        for (int j = i; j < n; j++)
        {
            // sum of elements so far
            sum += arr&#91;j&#93;;
 
            // if sum is seen before, we have found a sub-array 
            // with 0 sum
            if (sum == 0)
                cout << "Subarray &#91;" << i << ".." << j << "&#93;\n";
        }
    }
}
 
// main function
int main()
{
    int arr&#91;&#93; = { 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 };
    int n = sizeof(arr)/sizeof(arr&#91;0&#93;);
 
    printallSubarrays(arr, n);
 
    return 0;
}
&#91;/cpp&#93;


Java语言实现：
&#91;java&#93;class PrintallSubarrays
{
    // Function to print all sub-arrays with 0 sum present
    // in the given array
    public static void printallSubarrays(int arr&#91;&#93;)
    {
        // n is length of the array
        int n = arr.length;
 
        // consider all sub-arrays starting from i
        for (int i = 0; i < n; i++)
        {
            int sum = 0;
     
            // consider all sub-arrays ending at j
            for (int j = i; j < n; j++)
            {
                // sum of elements so far
                sum += arr&#91;j&#93;;
     
                // if sum is seen before, we have found a sub-array with 0 sum
                if (sum == 0)
                    System.out.println("Subarray &#91;" + i + ".." + j + "&#93;");
            }
        }
    }
 
    // main function
    public static void main (String&#91;&#93; args)
    {
        int arr&#91;&#93; = { 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 }; 
        printallSubarrays(arr);
    }
}&#91;/java&#93;

<strong>输出：</strong><span style="font-family: consolas; font-size: small;">
Subarray [0..2]
Subarray [0..9]
Subarray [1..3]
Subarray [2..5]
Subarray [3..9]
Subarray [5..7]</span>

<br>
<div style="background-color: #99CCFF; height: 1px;"></div>
<h3>2. 使用map打印子数组</h3>
我们可以使用map打印所有和为0的子数组，子数组信息存放在另外的数组中并作为map的value。开始，我们创建一个空map存放所有和等于给定值的子数组的的结束下标。其中，key是和，value就是之前提到的另外一个数组，此数组中存放的就是和等于key的所有子数组的结束下标。我们遍历给定数组，在遍历程中维护目前所遍历过的元素的和。如果元素的和在之前见过，那就意味着至少存在一个子数组和为0，子数组的结束下标就是当前遍历的下标，这样我们就能打印出所有的子数组。

C++语言实现：
[cpp]
#include <iostream>
#include <unordered_map>
using namespace std;
 
// Function to print all sub-arrays with 0 sum present
// in the given array
void printallSubarrays(int arr[], int n)
{
    // create an empty multi-map to store ending index of all
    // sub-arrays having same sum
    unordered_multimap<int, int> map;
 
    // insert (0, -1) pair into the map to handle the case when
    // sub-array with 0 sum starts from index 0
    map.insert(pair<int, int>(0, -1));
 
    int sum = 0;
    
    // traverse the given array
    for (int i = 0; i < n; i++)
    {
        // sum of elements so far
        sum += arr&#91;i&#93;;
 
        // if sum is seen before, there exists at-least one 
        // sub-array with 0 sum
        if (map.find(sum) != map.end())
        {
            auto it = map.find(sum);
 
            // find all sub-arrays with same sum
            while (it != map.end() && it->first == sum)
            {
                cout << "Subarray &#91;" << (it->second + 1) << ".." << i << "&#93;\n";
                it++;
            }
        }
 
        // insert (sum so far, current index) pair into multi-map
        map.insert(pair<int, int>(sum, i));
    }
}
 
// main function
int main()
{
    int arr[] = { 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 };
    int n = sizeof(arr)/sizeof(arr[0]); 
    printallSubarrays(arr, n); 
    return 0;
}

java语言实现：

import java.util.HashMap;
import java.util.Map;
import java.util.ArrayList;
 
class PrintallSubarrays
{
    // Utility function to insert <key, value> into the Multimap
    private static void insert(Map<Integer, ArrayList> hashMap, 
                                Integer key, Integer value) 
    {
        // if key is already present
        if (hashMap.containsKey(key))
        {
            hashMap.get(key).add(value);
        }else
        // key is seen for the first time
        {
            ArrayList<Integer> list = new ArrayList<Integer>();
            list.add(value); 
            hashMap.put(key, list);
        }
    }
 
    // Function to print all sub-arrays with 0 sum present
    // in the given array
    public static void printallSubarrays(int arr[])
    {
        // n is length of the array
        int n = arr.length;
        
        // create an empty Multi-map to store ending index of all
        // sub-arrays having same sum
        Map<Integer, ArrayList> hashMap = new 
                                    HashMap<Integer, ArrayList>();
 
        // insert (0, -1) pair into the map to handle the case when
        // sub-array with 0 sum starts from index 0
        insert(hashMap, 0, -1);
     
        int sum = 0;
        
        // traverse the given array
        for (int i = 0; i < n; i++)
        {
            // sum of elements so far
            sum += arr&#91;i&#93;;
     
            // if sum is seen before, there exists at-least one 
            // sub-array with 0 sum
            if (hashMap.containsKey(sum))
            {
                ArrayList<Integer> list = hashMap.get(sum);
 
                // find all sub-arrays with same sum
                for (Integer value: list) {
                    System.out.println("Subarray [" + (value + 1) + ".." +  i + "]");
                }
            }
 
            // insert (sum so far, current index) pair into the Multi-map
            insert(hashMap, sum, i);
        }
    }
 
    // main function
    public static void main (String[] args)
    {
        int A[] = { 3, 4, -7, 3, 1, 3, 1, -4, -2, -2 }; 
        printallSubarrays(A);
    }
}

输出:
Subarray [0..2]
Subarray [1..3]
Subarray [2..5]
Subarray [5..7]
Subarray [3..9]
Subarray [0..9]

这样，只需遍历一次数组就能求出所有的子数组。

练习： 打印出和为非0的所有子数组。