Workout Workout
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English

Sam has prepared a fitness program so that he can become stronger! The program is made of N sessions. During the i-th session, Sam will do a certain amount of pushups. The number of pushups he does in each session is strictly increasing.

The difficulty of his fitness program is equal to the maximum difference in the number of pushups between any two consecutive training sessions.

To make his program less difficult, Sam has decided to add up to K additional training sessions to his fitness program. He can add these sessions anywhere in his fitness program, and do any number of pushups in each of them. After the additional training sessions are added, the number of pushups he does in each session must still be strictly increasing. What is the minimum possible difficulty?

This mission was proposed by Yossi Matsumoto and Wajeb Saab for Google Kickstart Round A 2020.

Input:

Two arguments

  • the first one is a list of positive integers - the number of pushups in each training session;
  • the second is a positive integer - additional sessions that Sam may add to his fitness program.

Output:

An integer - the minimum difficulty possible after up to K additional training sessions are added.

Example:

workout([100, 200, 230], 1) == 50
workout([10, 13, 15, 16, 17], 2) == 2

In the first case Sam can add up to one session. The added sessions are marked in bold: 100, 150, 200, 230. The difficulty is now 50.

In the second case Sam can add up to two sessions. Then his optimal fitness program may be as follows: 10, 11, 13, 15, 16, 17, 18. The difficulty is now 2.

Precondition:
2 ≤ N ≤ 1000
1 ≤ K ≤ 500