## DMPG '16 S5 - Pizza Bag

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Points: 12 (partial)
Time limit: 1.4s
Memory limit: 256M

Author:
Problem type

At a programming competition, there is one very large pizza which is being served for free. You brought a bag to the competition, and intend to take home the best tasting pieces that you can.

The pizza appears to be divided into slices, where some slices have good toppings, and some have bad toppings. You quickly evaluate each section based on its deliciousness, and assign them all some value between and . It should be noted that the slices are connected in a circle — that is, slice is connected to slice , slice to , and so on. Slice is additionally connected to slice .

Since there is only one pizza available, the organizers have limited everyone to at most slices, and only if all slices are connected (that is, only the two pieces at the ends are not connected to two other slices you have taken).

If you are the first person to pick, what is the most delicious sector of pizza you can take? A sector (connected series of slices) has deliciousness equal to the sum of its individual slices. There will always be at least one slice which has a positive deliciousness rating.

#### Constraints

For all subtasks:

##### Subtask 2 [15%]

All deliciousness values given will be strictly positive.

##### Subtask 3 [20%]

Slice will have a deliciousness value of while all other slices will have a value in the range . You may assume that the optimal solution never involves taking slice for this subtask.

##### Subtask 5 [30%]

No additional constraints.

#### Input Specification

On the first line, there will be two integers and .

On the second line, there will be space-separated integers, where the -th integer represents the deliciousness of piece number .

#### Output Specification

Output a single integer, the maximum deliciousness which can be taken. You may assume that the solution is always positive.

It is possible for this value to exceed the limits of a 32-bit integer. It is advised to use 64-bit integers instead, meaning that Java users should use long, and C++ users should use long long.

#### Sample Input 1

5 5
1 3 -8 -2 4

#### Sample Output 1

8

#### Illustration for Sample 1

A diagram of the pizza is given below.

#### Sample Input 2

6 2
0 1 8 1 5 5

#### Sample Output 2

10

## Comments

• commented on Nov. 2, 2019, 11:27 p.m.

If all slices have negative deliciousness, can I select nothing?

• commented on Nov. 2, 2019, 11:42 p.m.

See the last line in the problem statement:

There will always be at least one slice which has a positive deliciousness rating.