[LeetCode] 548. Split Array with Equal Sum 分割数组成和相同的子数组

Given an array with n integers, you need to find if there are triplets (i, j, k) which satisfies following conditions:

1. 0 < i, i + 1 < j, j + 1 < k < n - 1
2. Sum of subarrays (0, i - 1), (i + 1, j - 1), (j + 1, k - 1) and (k + 1, n - 1) should be equal.

where we define that subarray (L, R) represents a slice of the original array starting from the element indexed L to the element indexed R.

Example:

Input: [1,2,1,2,1,2,1]
Output: True
Explanation:
i = 1, j = 3, k = 5.
sum(0, i - 1) = sum(0, 0) = 1
sum(i + 1, j - 1) = sum(2, 2) = 1
sum(j + 1, k - 1) = sum(4, 4) = 1
sum(k + 1, n - 1) = sum(6, 6) = 1

Note:

1. 1 <= n <= 2000.
2. Elements in the given array will be in range [-1,000,000, 1,000,000].

给一个数组，找出三个位置，使得数组被分为四段，使得每段之和相等，问存不存在这样的三个位置，注意三个位置上的数字不属于任何一段。

解法1: 暴力法，那就是三重循环，时间复杂度是O(n^3)，空间复杂度是O(n)。过不了大数据, TLE。

解法2: 采用空间换时间，从中间进行分割，然后在前半部分进行搜索，看看是不是可以找到和相同的划分，如果找到了，就将和加入哈希表中；然后再在后半部分进行搜索，如果找到了和相同的划分并且该和也存在于哈希表中，这说明找到了合适的i，j，k，可以将数组划分为和相同的四个部分，返回true。这样时间复杂度就降低成了O(n^2)。

解法3: 建立一个长度为数组长度的sum[i]，然后计算出每一个位置的它前面所有数字的和，这样就避免了以后大量的重复计算和的运算。然后计算i-j是否有满足1-i的和等于i-j, 有的话存到set里，避免重复计算。然后在计算位置k，是否也等于之前set里的值，如果有就返回True.

解法4: 用数组sums记录前n项和，在用字典idxs统计sums元素对应的下标列表，根据sums和idxs枚举满足(0, i - 1) == (i + 1, j - 1)条件的i，j。利用字典jlist记录子数组和对应的j值列表。最后遍历k，枚举jlist中子数组和(k + 1, n - 1)对应的j值，然后判断是否存在 (j + 1, k - 1) 与 (k + 1, n - 1) 相等

Java: 暴力， TLE

public class Solution {

    public int sum(int[] nums, int l, int r) {
        int summ = 0;
        for (int i = l; i < r; i++)
            summ += nums[i];
        return summ;
    }

    public boolean splitArray(int[] nums) {
        if (nums.length < 7)
            return false;
        for (int i = 1; i < nums.length - 5; i++) {
            int sum1 = sum(nums, 0, i);
            for (int j = i + 2; j < nums.length - 3; j++) {
                int sum2 = sum(nums, i + 1, j);
                for (int k = j + 2; k < nums.length - 1; k++) {
                    int sum3 = sum(nums, j + 1, k);
                    int sum4 = sum(nums, k + 1, nums.length);
                    if (sum1 == sum2 && sum3 == sum4 && sum2 == sum4)
                        return true;
                }
            }
        }
        return false;
    }
}　　

Java: 暴力，TLE

public class Solution {
    public boolean splitArray(int[] nums) {
        if (nums.length < 7)
            return false;
        int[] sum = new int[nums.length];
        sum[0] = nums[0];
        for (int i = 1; i < nums.length; i++) {
            sum[i] = sum[i - 1] + nums[i];
        }
        for (int i = 1; i < nums.length - 5; i++) {
            int sum1 = sum[i - 1];
            for (int j = i + 2; j < nums.length - 3; j++) {
                int sum2 = sum[j - 1] - sum[i];
                for (int k = j + 2; k < nums.length - 1; k++) {
                    int sum3 = sum[k - 1] - sum[j];
                    int sum4 = sum[nums.length - 1] - sum[k];
                    if (sum1 == sum2 && sum3 == sum4 && sum2 == sum4)
                        return true;
                }
            }
        }
        return false;
    }
}

Java: TLE

public class Solution {
    public boolean splitArray(int[] nums) {
        if (nums.length < 7)
            return false;

        int[] sum = new int[nums.length];
        sum[0] = nums[0];
        for (int i = 1; i < nums.length; i++) {
            sum[i] = sum[i - 1] + nums[i];
        }
        for (int i = 1; i < nums.length - 5; i++) {
            int sum1 = sum[i - 1];
            for (int j = i + 2; j < nums.length - 3; j++) {
                int sum2 = sum[j - 1] - sum[i];
                if (sum1 != sum2)
                    continue;
                for (int k = j + 2; k < nums.length - 1; k++) {
                    int sum3 = sum[k - 1] - sum[j];
                    int sum4 = sum[nums.length - 1] - sum[k];
                    if (sum3 == sum4 && sum2 == sum4)
                        return true;
                }
            }
        }
        return false;
    }
}　　

Java: Accepted

public class Solution {
    public boolean splitArray(int[] nums) {
        if (nums.length < 7)
            return false;
        int[] sum = new int[nums.length];
        sum[0] = nums[0];
        for (int i = 1; i < nums.length; i++) {
            sum[i] = sum[i - 1] + nums[i];
        }
        for (int j = 3; j < nums.length - 3; j++) {
            HashSet < Integer > set = new HashSet < > ();
            for (int i = 1; i < j - 1; i++) {
                if (sum[i - 1] == sum[j - 1] - sum[i])
                    set.add(sum[i - 1]);
            }
            for (int k = j + 2; k < nums.length - 1; k++) {
                if (sum[nums.length - 1] - sum[k] == sum[k - 1] - sum[j] && set.contains(sum[k - 1] - sum[j]))
                    return true;
            }
        }
        return false;
    }
}　　

Python:

# Time:  O(n^2)
# Space: O(n)

class Solution(object):
    def splitArray(self, nums):
        """
        :type nums: List[int]
        :rtype: bool
        """
        if len(nums) < 7:
            return False

        accumulated_sum = [0] * len(nums)
        accumulated_sum[0] = nums[0]
        for i in xrange(1, len(nums)):
            accumulated_sum[i] = accumulated_sum[i-1] + nums[i]
        for j in xrange(3, len(nums)-3):
            lookup = set()
            for i in xrange(1, j-1):
                if accumulated_sum[i-1] == accumulated_sum[j-1] - accumulated_sum[i]:
                    lookup.add(accumulated_sum[i-1])
            for k in xrange(j+2, len(nums)-1):
                if accumulated_sum[-1] - accumulated_sum[k] == accumulated_sum[k-1] - accumulated_sum[j] and \
                   accumulated_sum[k - 1] - accumulated_sum[j] in lookup:
                    return True
        return False　　

Python:

class Solution(object):
    def splitArray(self, nums):
        """
        :type nums: List[int]
        :rtype: bool
        """
        size = len(nums)
        sums = [0] * (size + 1)
        for x in range(size):
            sums[x + 1] += sums[x] + nums[x]

        idxs = collections.defaultdict(list)
        for x in range(size):
            idxs[sums[x + 1]].append(x)

        jlist = collections.defaultdict(list)
        for i in range(1, size):
            for j in idxs[2 * sums[i] + nums[i]]:
                if i < j < size:
                    jlist[sums[i]].append(j + 1)

        for k in range(size - 2, 0, -1):
            for j in jlist[sums[size] - sums[k + 1]]:
                if j + 1 > k: continue
                if sums[k] - sums[j + 1] == sums[size] - sums[k + 1]:
                    return True
        return False　　

C++:

class Solution {
public:
    bool splitArray(vector<int>& nums) {
        if (nums.size() < 7) return false;
        int n = nums.size();
        vector<int> sums = nums;
        for (int i = 1; i < n; ++i) {
            sums[i] = sums[i - 1] + nums[i];
        }
        for (int j = 3; j < n - 3; ++j) {
            unordered_set<int> s;
            for (int i = 1; i < j - 1; ++i) {
                if (sums[i - 1] == (sums[j - 1] - sums[i])) {
                    s.insert(sums[i - 1]);
                }
            }
            for (int k = j + 1; k < n - 1; ++k) {
                int s3 = sums[k - 1] - sums[j], s4 = sums[n - 1] - sums[k];
                if (s3 == s4 && s.count(s3)) return true;
            }
        }
        return false;
    }
};

C++:

class Solution {
public:
    bool splitArray(vector<int>& nums) {
        if (nums.size() < 7) return false;
        int n = nums.size(), target = 0;
        int sum = accumulate(nums.begin(), nums.end(), 0);
        for (int i = 1; i < n - 5; ++i) {
            if (i != 1 && nums[i] == 0 && nums[i - 1] == 0) continue;
            target += nums[i - 1];
            if (helper(nums, target, sum - target - nums[i], i + 1, 1)) {
                return true;
            }
        }
        return false;
    }
    bool helper(vector<int>& nums, int target, int sum, int start, int cnt) {
        if (cnt == 3) return sum == target;
        int curSum = 0, n = nums.size();
        for (int i = start + 1; i < n + 2 * cnt - 5; ++i) {
            curSum += nums[i - 1];
            if (curSum == target && helper(nums, target, sum - curSum - nums[i], i + 1, cnt + 1)) {
                return true;
            }
        }
        return false;
    }

C++: 暴力+优化　　

class Solution {
public:
    bool splitArray(vector<int>& nums) {
        int n = nums.size();
        vector<int> sums = nums;
        for (int i = 1; i < n; ++i) {
            sums[i] = sums[i - 1] + nums[i];
        }
        for (int i = 1; i <= n - 5; ++i) {
            if (i != 1 && nums[i] == 0 && nums[i - 1] == 0) continue;
            for (int j = i + 2; j <= n - 3; ++j) {
                if (sums[i - 1] != (sums[j - 1] - sums[i])) continue;
                for (int k = j + 2; k <= n - 1; ++k) {
                    int sum3 = sums[k - 1] - sums[j];
                    int sum4 = sums[n - 1] - sums[k];
                    if (sum3 == sum4 && sum3 == sums[i - 1]) {
                        return true;
                    }
                }
            }
        }
        return false;
    }
};

　　

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Split an array into two equal Sum subarrays

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