410. Split Array Largest Sum

Description

Solution

Pick One

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Given an array which consists of non-negative integers and an integer m, you can split the array into m non-empty continuous subarrays. Write an algorithm to minimize the largest sum among these m subarrays.

Note:
If n is the length of array, assume the following constraints are satisfied:

• 1 ≤ n ≤ 1000
• 1 ≤ m ≤ min(50, n)

Examples:

Input:
 nums = [7,2,5,10,8]
 m = 2
 
 
Output:
 18
 
 
Explanation:
 There are four ways to split nums into two subarrays.
 The best way is to split it into [7,2,5] and [10,8],
 where the largest sum among the two subarrays is only 18.

Solution

DFS, time O(n ^ m) (TLE), space O(n)

直观的解法，但会超时。

class Solution {
    private int maxSum;
    
    public int splitArray(int[] nums, int m) {
        maxSum = Integer.MAX_VALUE;
        dfs(nums, 0, Integer.MIN_VALUE, m);
        return maxSum;
    }
    
    public void dfs(int[] nums, int start, int max, int m) {
        if (start == nums.length) {
            if (m == 0) {
                maxSum = Math.min(maxSum, max);
            }
            return;
        }
        
        if (nums.length - start < m) {
            return;
        }
        
        int sum = 0;
        for (int i = start; i < nums.length; ++i) {
            sum += nums[i];
            max = Math.max(max, sum);
            dfs(nums, i + 1, max, m - 1);
        }
    }
}

DP, time O(n ^ 2 * m), space O(n * m)

这道题让我想起了算法导论上的一道切绳子？的题。可以用二维dp去做，dp[i][j]表示将nums中前i个元素split成j段的maxSum。

注意dp[][]的初始化！

class Solution {
    public int splitArray(int[] nums, int m) {
        int n = nums.length;
        int[][] dp = new int[n + 1][m + 1];
        for (int [] row : dp) {
            Arrays.fill(row, Integer.MAX_VALUE);    // important
        }
        
        dp[0][0] = 0;
        
        for (int i = 1; i <= n; ++i) {
            for (int j = 1; j <= Math.min(i, m); ++j) {
                int sum = 0;
                for (int k = i; k >= j; --k) {
                    sum += nums[k - 1];
                    dp[i][j] = Math.min(Math.max(sum, dp[k - 1][j - 1]), dp[i][j]);
                }
            }
        }
        
        return dp[n][m];
    }
}

Binary search + Greedy, time O(n∗log(sum of array)), space O(1)

我还能说什么呢，这个解法是真牛逼。。

The answer is between maximum value of input array numbers and sum of those numbers.
Use binary search to approach the correct answer. We have l = max number of array; r = sum of all numbers in the array;Every time we do mid = (l + r) / 2;
Use greedy to narrow down left and right boundaries in binary search.

1. Cut the array from left.
2. Try our best to make sure that the sum of numbers between each two cuts (inclusive) is large enough but still less than mid.
3. We'll end up with two results: either we can divide the array into more than m subarrays or we cannot.

• If we can, it means that the mid value we pick is too small because we've already tried our best to make sure each part holds
  as many non-negative numbers as we can but we still have numbers left.
  So, it is impossible to cut the array into m parts and make sure each parts is no larger than mid. We should increase m.
  This leads to mid = l + 1;
• If we can't, it is either we successfully divide the array into m parts and the sum of each part is less than mid,
  or we used up all numbers before we reach m. Both of them mean that we should lower mid because we need to find the minimum one. This leads to r = mid - 1;

class Solution {
    public int splitArray(int[] nums, int m) {
        long sum = 0;   // use long to avoid overflow
        int max = 0;
        for (int num : nums) {
            sum += num;
            max = Math.max(num, max);
        }
        
        long left = max;
        long right = sum;
        
        while (left <= right) {
            long mid = left + (right - left) / 2;
            
            if (!isValid(nums, m, mid)) {
                left = mid + 1;
            } else {
                right = mid - 1;
            }
        }
        
        return (int) left;
    }
    
    public boolean isValid(int[] nums, int m, long target) {
        long sum = 0;
        int count = 1;
        
        for (int num : nums) {
            sum += num;
            if (sum <= target) {
                continue;
            }
            
            sum = num;
            if (++count > m) {
                return false;
            }
        }
        
        return true;
    }
}