By Elevator or Stairs?

You are planning to buy an apartment in a $$$n$$$-floor building. The floors are numbered from $$$1$$$ to $$$n$$$ from the bottom to the top. At first for each floor you want to know the minimum total time to reach it from the first (the bottom) floor.

Let:

• $$$a_i$$$ for all $$$i$$$ from $$$1$$$ to $$$n-1$$$ be the time required to go from the $$$i$$$-th floor to the $$$(i+1)$$$-th one (and from the $$$(i+1)$$$-th to the $$$i$$$-th as well) using the stairs;
• $$$b_i$$$ for all $$$i$$$ from $$$1$$$ to $$$n-1$$$ be the time required to go from the $$$i$$$-th floor to the $$$(i+1)$$$-th one (and from the $$$(i+1)$$$-th to the $$$i$$$-th as well) using the elevator, also there is a value $$$c$$$ — time overhead for elevator usage (you need to wait for it, the elevator doors are too slow!).

In one move, you can go from the floor you are staying at $$$x$$$ to any floor $$$y$$$ ($$$x \ne y$$$) in two different ways:

• If you are using the stairs, just sum up the corresponding values of $$$a_i$$$. Formally, it will take $$$\sum\limits_{i=min(x, y)}^{max(x, y) - 1} a_i$$$ time units.
• If you are using the elevator, just sum up $$$c$$$ and the corresponding values of $$$b_i$$$. Formally, it will take $$$c + \sum\limits_{i=min(x, y)}^{max(x, y) - 1} b_i$$$ time units.

You can perform as many moves as you want (possibly zero).

So your task is for each $$$i$$$ to determine the minimum total time it takes to reach the $$$i$$$-th floor from the $$$1$$$-st (bottom) floor.