Lights Out timer 2초 memory 512MB

Farmer John has installed a fancy new milking machine in his barn, but it draws so much power that it occasionally causes the power to go out! This happens so often that Bessie has memorized a map of the barn, making it easier for her to find the exit of the barn in the dark. She is curious though about the impact of power loss on her ability to exit the barn quickly. For example, she wonders how much farther she might need to walk find the exit in the dark.

The barn is described by a simple (non self-intersecting) polygon with integer vertices (x_1, y_1) \ldots (x_n, y_n) listed in clockwise order. Its edges alternate between horizontal (parallel to the x-axis) and vertical (parallel to the y-axis); the first edge can be of either type. The exit is located at (x_1, y_1). Bessie starts inside the barn located at some vertex (x_i, y_i) for i > 1. She can walk only around the perimeter of the barn, either clockwise or counterclockwise, Her goal is to travel a minimum distance to reach the exit. This is relatively easy to do with the lights on, of course, since she will travel either clockwise or counterclockwise from her current location to the exit -- whichever direction is shorter.

One day, the lights go out, causing Bessie to panic and forget which vertex she is standing at. Fortunately, she still remembers the exact map of the barn, so she can possibly figure out her position by walking around and using her sense of touch. Whenever she is standing at a vertex (including at her initial vertex), she can feel the exact interior angle at that vertex, and she can tell if that vertex is the exit. When she walks along an edge of the barn, she can determine the exact length of the edge after walking along the entire edge. Bessie decides on the following strategy: she will move clockwise around the perimeter of the barn until she has felt enough angles and edges to deduce the vertex at which she is currently located. At that point, she can easily figure out how to get to the exit by traveling a minimum amount of remaining distance, either by continuing to move clockwise or by switching direction and moving counter-clockwise.

Please help Bessie determine the largest amount by which her travel distance will increase in the worst case (over all possibilities for her starting vertex) for travel in the dark versus in a lit barn.

Problem credits: Brian Dean