3013. Divide an Array Into Subarrays With Minimum Cost II

Posted on Edited on In

3013. Divide an Array Into Subarrays With Minimum Cost II

Description

You are given a 0-indexed array of integers nums of length n, and two positive integers k and dist.

The cost of an array is the value of its first element. For example, the cost of [1,2,3] is 1 while the cost of [3,4,1] is 3.

You need to divide nums into k disjoint contiguous subarrays, such that the difference between the starting index of the second subarray and the starting index of the kth subarray should be less than or equal to dist. In other words, if you divide nums into the subarrays nums[0..(i1 - 1)], nums[i1..(i2 - 1)], …, nums[ik-1..(n - 1)], then ik-1 - i1 <= dist.

Return the minimum possible sum of the cost of these subarrays.

Example 1:

1
2
Input: nums = [1,3,2,6,4,2], k = 3, dist = 3
Output: 5

Explanation: The best possible way to divide nums into 3 subarrays is: [1,3], [2,6,4], and [2]. This choice is valid because ik-1 - i1 is 5 - 2 = 3 which is equal to dist. The total cost is nums[0] + nums[2] + nums[5] which is 1 + 2 + 2 = 5.
It can be shown that there is no possible way to divide nums into 3 subarrays at a cost lower than 5.

Example 2:

1
2
Input: nums = [10,1,2,2,2,1], k = 4, dist = 3
Output: 15

Explanation: The best possible way to divide nums into 4 subarrays is: [10], [1], [2], and [2,2,1]. This choice is valid because ik-1 - i1 is 3 - 1 = 2 which is less than dist. The total cost is nums[0] + nums[1] + nums[2] + nums[3] which is 10 + 1 + 2 + 2 = 15.
The division [10], [1], [2,2,2], and [1] is not valid, because the difference between ik-1 and i1 is 5 - 1 = 4, which is greater than dist.
It can be shown that there is no possible way to divide nums into 4 subarrays at a cost lower than 15.

Example 3:

1
2
Input: nums = [10,8,18,9], k = 3, dist = 1
Output: 36

Explanation: The best possible way to divide nums into 4 subarrays is: [10], [8], and [18,9]. This choice is valid because ik-1 - i1 is 2 - 1 = 1 which is equal to dist.The total cost is nums[0] + nums[1] + nums[2] which is 10 + 8 + 18 = 36.
The division [10], [8,18], and [9] is not valid, because the difference between ik-1 and i1 is 3 - 1 = 2, which is greater than dist.
It can be shown that there is no possible way to divide nums into 3 subarrays at a cost lower than 36.

Constraints:

  • 3 <= n <= 10^5
  • 1 <= nums[i] <= 10^9
  • 3 <= k <= n
  • k - 2 <= dist <= n - 2

Hints/Notes

Solution

Language: C++

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
class Solution {
public:
    // so, we need to pick nums[0], with another k - 1 numbers
    // the best choice is pick the minimum k - 1 numbers
    // but we have the constraint that ik-1 - i1 <= dist
    // i1 = len(subArray1), so ik-1 <= dist + len(subArray1)
    long long minimumCost(vector<int>& nums, int k, int dist) {
        long long cur = nums[0], res = LLONG_MAX;
        int n = nums.size();
        // starting from i, we need to pick k - 1 numbers,
        // so the last number we can iterate  is n - k + 1
        multiset<int> mx, mi;
        int right = 1;
        // we are always trying to find the k - 1 value within the window [i, i + dist]
        for (int i = 1; i <= n - k + 1; i++) {
            // feeding numbers until i + dist
            // now we need to get the minimum k - 1 numbers
            // use 2
            while (right <= min(i + dist, n - 1)) {
                mi.insert(nums[right]);
                cur += nums[right];
                if (mi.size() > k - 1) {
                    auto it = prev(mi.end());
                    int tmp = *it;
                    cur -= tmp;
                    mi.erase(it);
                    mx.insert(tmp);
                }
                right++;
            }
            res = min(res, cur);
            // now we need to erase nums[i] from the sliding window
            if (mi.contains(nums[i])) {
                auto it = mi.lower_bound(nums[i]);
                cur -= *it;
                mi.erase(it);
                if (!mx.empty()) {
                    it = mx.begin();
                    mi.insert(*it);
                    cur += *it;
                    mx.erase(it);
                }
            } else {
                auto it = mx.lower_bound(nums[i]);
                mx.erase(it);
            }
        }
        return res;
    }
};