201204: [AtCoder]ARC120 E - 1D Party
Description
Score : $700$ points
Problem Statement
There are $N$ people, Person $1$ through Person $N$, standing on a number line. Person $i$ is now at coordinate $A_i$ on the number line.
Here, $A_1 < A_2 < A_3 < \dots < A_N$ and all $A_i$ are even numbers.
We will now hold a party for $k$ seconds.
During the party, each person can freely move on the number line at a speed of at most $1$ per second. (It is also allowed to move in the negative direction within the speed limit.)
The people request that the following condition be satisfied for every integer $i$ such that $1 \le i \lt N$:
- there is at least one moment during the party (including the moment it ends) when Person $i$ and Person $i + 1$ are at the same coordinate.
Find the minimum $k$ for which there is a strategy that the people can take to satisfy all the conditions.
We can prove that the answer is an integer under the Constraints of this problem.
Constraints
- $2 \le N \le 2 \times 10^5$
- $0 \le A_i \le 10^9$
- $A_1 < A_2 < A_3 < \dots < A_N$
- $A_i$ is an even number.
Input
Input is given from Standard Input in the following format:
$N$ $A_1$ $A_2$ $A_3$ $\dots$ $A_N$
Output
Print the answer as an integer.
Sample Input 1
3 0 6 10
Sample Output 1
5
During the party lasting $5$ seconds, each person should move as follows:
- Person $1$: always moves in the positive direction at a speed of $1$.
- Person $2$: moves in the positive direction at a speed of $1$ during the first $2$ seconds, then in the negative direction at a speed of $1$ during the remaining $3$ seconds.
- Person $3$: always moves in the negative direction at a speed of $1$.
If they move in this way, Person $2$ and Person $3$ will be at the same coordinate exactly $2$ seconds after the beginning, and Person $1$ and Person $2$ will be at the same coordinate at the end of the party.
Thus, they can satisfy all the conditions for $k = 5$. For smaller $k$, there is no strategy to satisfy all of them.
Sample Input 2
5 0 2 4 6 8
Sample Output 2
3
During the party lasting $3$ seconds, each person should, for example, move as follows:
- Person $1$: always moves in the positive direction at a speed of $1$.
- Person $2$: moves in the positive direction at a speed of $1$ during the first $2$ seconds, then in the negative direction at a speed of $1$ during the remaining $1$ second.
- Person $3$: does not move at all.
- Person $4$: moves in the negative direction at a speed of $1$ during the first $2$ seconds, then in the positive direction at a speed of $1$ during the remaining $1$ second.
- Person $5$: always moves in the negative direction at a speed of $1$.
Sample Input 3
10 0 2 4 6 8 92 94 96 98 100
Sample Output 3
44