Problem 1: Railway Catering Contracts, *(K Narayan Kumar, CMI)*

The government of Siruseri has just commissioned one of the longest and most modern railway routes in the world. This route runs the entire length of Siruseri and passes through many of the big cities and a large number of small towns and villages in Siruseri.

The railway stations along this route have all been constructed keeping in mind the comfort of the travellers. Every station has big parking lots, comfortable waiting rooms and plenty of space for eateries. The railway authorities would like to contract out the catering services of these eateries.

Unfortunately, catering contractors are not philanthropists and would like to maximise their profits. The Siruseri Economic Survey has done a through feasibility study of the different stations and documented the expected profits (or losses) for the eateries in all the railway stations on this route. Every contractor would like to run the catering service only in the profitable stations and stay away from the loss making ones.

On the other hand the authorities would like to ensure that
every station was catered to. Towards this end, authorities
offered to contract out stations in groups. They would fix a
minimum size *K* and a contractor was only allowed to bid for any
**contiguous** sequence of *K* or more stations.

For example suppose there are 8 stations along the line and their profitability is as follows:

Station 1 2 3 4 5 6 7 8 Expected Profits -20 90 -30 -20 80 -70 -60 125

If *K* was fixed to be 3, a contractor could pick
stations 3, 4, 5 and 6. This would give him a total profit of -40
(i.e. a loss of 40). He could have picked 3, 4 and 5 giving him a
profit of 30. On the other hand if he picked stations 2, 3, 4
and 5, he would make a profit of 120. You can check that this is
the best possible choice when *K* = 3.

If *K* = 1, then the best choice would be to bid for
just station 8 and make a profit of 125.

You have been hired by a contractor. Your task is to help him identify the segment of stations to bid for so at to maximize his expected profit.

Input format

The first line of the input contains two integers *N*
and *K*, where *N* is the number of stations and
*K* is the minimum number of contiguous stations that must
form part or the bid. The next *N*+1 lines (lines
2,...,*N*+1) describe the profitability of the *N*
stations. Line *i*+1 contains a single integer denoting
the expected profit at station *i*.

Output format

A single integer *P*, indicating the maximum possible
profit.

Test Data:

You may assume that 1 ≤ *N* ≤ 1000000 and 1 ≤
*K* ≤ N. You may further assume that
in 50% of the inputs *N* ≤ 5000.

Example:

We illustrate the input and output format using the above example:

Sample Input 1:

8 3 -20 90 -30 -20 80 -70 -60 125

Sample Output 1:

120

Sample Input 2:

8 1 -20 90 -30 -20 80 -70 -60 125

Sample Output 2:

125

Copyright (c) IARCS 2003-2020; Last Updated: 29 Mar 2005