Home Coming

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Jessica is going to see her family overseas today, so she goes to the local gift shop to buy some souvenirs. There are N souvenirs on the shelf, numbered 1, 2, ..., N from left to right. Souvenirs come in different types, which are denoted by positive integers. Jessica is in a hurry, so she can take only a consecutive interval of souvenirs. Formally, Jessica selects two indices, l and r , and takes all of the souvenirs numbered l, l+1, ..., r-1, r . Also, due to tax rules, airport security will throw away all souvenirs of a particular type if Jessica has more than S of that type in the chosen interval.

For example, suppose S = 2, and Jessica brings six souvenirs: one of type 0, two of type 1, and three of type 2. She will be allowed to keep the souvenir of type 0 and both souvenirs of type 1, but she will lose all of the souvenirs of type 2!

Jessica needs to choose l and r such that she can take the maximum number of souvenirs for her family. What is the maximum number of souvenirs she can bring?

This mission was proposed by Himanshu Jaju and Sadia Nahreen for Google Kickstart Round B 2019.

Input:

Two arguments:

• a list of positive integers, which represents the types of souvenirs on the shelf from left to right;
• a positive integer, the maximum amount of souvenirs of a single type in the chosen interval.

Output:

An integer, the maximum number of souvenirs Jessica can bring to her family.

Example:

```home_coming([1, 1, 4, 1, 4, 4], 2) == 4
home_coming([1, 2, 5, 3, 4, 5, 6, 7], 1) == 6
home_coming([1, 2, 8, 8, 8, 8, 8, 3, 4, 1], 1) == 4
```

In case #1, Jessica should choose l = 2 and r = 5. This allows her to take 4 souvenirs to the airport of types 1, 4, 1 and 4. None of them are thrown away by airport security, so she is able to bring 4 souvenirs to her family.

In case #2, Jessica should choose l = 1 and r = 8. This allows her to take all 8 souvenirs to the airport. Her souvenirs of type 5 are thrown away since she has more than S = 1 of them, so she is able to bring a total of 6 souvenirs to her family.

In case #3, Jessica should choose l = 1 and r = 9. This allows her to take 9 souvenirs to the airport of types 1, 2, 8, 8, 8, 8, 8, 3 and 4. Her souvenirs of type 8 are thrown away since she has more than S = 1 of them, so she is able to bring a total of 4 souvenirs to her family.

Precondition: 2 ≤ N ≤ 1000 1 ≤ S ≤ 100

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