USACO 2018 December Contest, Platinum

Problem 1. Balance Beam

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In order to save money for a new stall in her barn, Bessie the cow has started performing in the local circus, demonstrating her remarkable sense of balance as she carefully walks back and forth on an elevated balance beam!

The amount of money Bessie earns in her performance is related to where she manages to ultimately jump off the beam. The beam has positions labeled $0, 1, \ldots, N+1$ from left to right. If Bessie ever reaches $0$ or $N+1$ she falls off one of the ends of the beam and sadly gets no payment.

If Bessie is at a given position $k$, she can do either of the following:

1. Flip a coin. If she sees tails, she goes to position $k-1$, and if she sees heads, she goes to position $k + 1$ (i.e. $\frac{1}{2}$ probability of either occurrence).

2. Jump off the beam and receive payment of $f(k)$ $(0 \leq f(k) \leq 10^9)$.

Bessie realizes that she may not be able to guarantee any particular payment outcome, since her movement is governed by random coin flips. However, based on the location where she starts, she wants to determine what her expected payment will be if she makes an optimal sequence of decisions ("optimal" meaning that the decisions lead to the highest possible expected payment). For example, if her strategy earns her payment of $10$ with probability $1/2$, $8$ with probability $1/4$, or $0$ with probability $1/4$, then her expected payment will be the weighted average $10(1/2) + 8(1/4) + 0(1/4) = 7$.


The first line of input contains $N$ ($2 \leq N \leq 10^5$). Each of the remaining $N$ lines contain $f(1) \ldots f(N)$.

OUTPUT FORMAT (file balance.out):

Output $N$ lines. On line $i$, print out $10^5$ times the expected value of payment if Bessie starts at position $i$ and plays optimally, rounded down to the nearest integer.





Problem credits: Franklyn Wang and Spencer Compton

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