1335. Minimum Difficulty of a Job Schedule
Description
You want to schedule a list of jobs in d
days. Jobs are dependent (i.e To work on the ith
job, you have to finish all the jobs j
where 0 <= j < i
).
You have to finish at least one task every day. The difficulty of a job schedule is the sum of difficulties of each day of the d
days. The difficulty of a day is the maximum difficulty of a job done on that day.
You are given an integer array jobDifficulty
and an integer d
. The difficulty of the ith
job is jobDifficulty[i]
.
Return the minimum difficulty of a job schedule. If you cannot find a schedule for the jobs return -1
.
Example 1:
Input: jobDifficulty = [6,5,4,3,2,1], d = 2 Output: 7 Explanation: First day you can finish the first 5 jobs, total difficulty = 6. Second day you can finish the last job, total difficulty = 1. The difficulty of the schedule = 6 + 1 = 7
Example 2:
Input: jobDifficulty = [9,9,9], d = 4 Output: -1 Explanation: If you finish a job per day you will still have a free day. you cannot find a schedule for the given jobs.
Example 3:
Input: jobDifficulty = [1,1,1], d = 3 Output: 3 Explanation: The schedule is one job per day. total difficulty will be 3.
Constraints:
1 <= jobDifficulty.length <= 300
0 <= jobDifficulty[i] <= 1000
1 <= d <= 10
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ as the minimum difficulty to finish the first $i$ jobs within $j$ days. Initially $f[0][0] = 0$, and all other $f[i][j]$ are $\infty$.
For the $j$-th day, we can choose to finish jobs $[k,..i]$ on this day. Therefore, we have the following state transition equation:
$$ f[i][j] = \min_{k \in [1,i]} {f[k-1][j-1] + \max_{k \leq t \leq i} {jobDifficulty[t]}} $$
The final answer is $f[n][d]$.
The time complexity is $O(n^2 \times d)$, and the space complexity is $O(n \times d)$. Here $n$ and $d$ are the number of jobs and the number of days respectively.
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