Yong-In

Version B
Korea

batch
2002-08-25 20:39

Batch Scheduling

PROBLEM

There is a sequence of N jobs to be processed on one machine. The jobs are numbered
from 1 to N, so that the sequence is 1,2, ..., N. The sequence of jobs must be partitioned
into one or more batches, where each batch consists of consecutive jobs in the sequence.
The processing starts at time 0. The batches are handled one by one starting from the
first batch as follows. If a batch b contains jobs with smaller numbers than batch c, then
batch b is handled before batch c. The jobs in a batch are processed successively on the
machine. Immediately after all the jobs in a batch are processed, the machine outputs
the results of all the jobs in that batch. The output time of a job j is the time when the
batch containing j finishes.

A setup time S is needed to set up the machine for each batch. For each job i, we know
its cost factor Fi and the time Ti required to process it. If a batch contains the jobs x, x+1,
... , x+k, and starts at time t, then the output time of every job in that batch is t + S + (Tx
+ Tx+1 + ... + Tx+k). Note that the machine outputs the results of all jobs in a batch at the
same time. If the output time of job i is Oi, its cost is Oi × Fi. For example, assume that
there are 5 jobs, the setup time S = 1, (T1, T2, T3, T4, T5) = (1, 3, 4, 2, 1), and (F1, F2, F3,
F4, F5) = (3, 2, 3, 3, 4). If the jobs are partitioned into three batches {1, 2}, {3}, {4, 5},
then the output times (O1, O2, O3, O4, O5) = (5, 5, 10, 14, 14) and the costs of the jobs
are (15, 10, 30, 42, 56), respectively. The total cost for a partitioning is the sum of the
costs of all jobs. The total cost for the example partitioning above is 153.

You are to write a program which, given the batch setup time and a sequence of jobs
with their processing times and cost factors, computes the minimum possible total cost.

If your program outputs the correct answer for a test case within the time limit, then you
get full points for the test case, and otherwise you get 0 points.

5
1
1 3
3 2
4 3
2 3
1 4
153
2
50
100 100
100 100
45000

Your program reads from standard input. The first line contains the number of jobs N, 1

N
10000. The second line contains the batch setup time S which is an integer, 0
S

50. The following N lines contain information about the jobs 1, 2, ..., N in that order
as follows. First on each of these lines is an integer Ti, 1
Ti
100, the processing time
of the job. Following that, there is an integer Fi, 1
Fi
100, the cost factor of the job.

Your program writes to standard output. The output contains one line, which contains
one integer: the minimum possible total cost.

Yong-In

Version B
Korea

batch
2002-08-25 20:39

EXAMPLE INPUTS AND OUTPUTS

Example 1: input

Example 2: input

Example 2 is the example in the text.

REMARK

For each test case, the total cost for any partitioning does not exceed 231 − 1.

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출처 IOI 2002