Again this year, I just missed the cut-off for moving past Round 2. I finished at position #623 with only the top 500 advancing. I was actually tied with #315 with about 400 other participants in points, but didn’t do well in the tie-breaker: time.

The first problem, Swinging Wild, involved crossing a juggle by swinging on vines with various positions and lengths. I found the description a bit long and confusing and skipped it at first, but I knew it must be relatively easy since it was the first problem, so I came back to it later. My first solution was rejected, and I realized my approach was bad and I had to start over. I considered moving to another problem, but at least now I understood this one. Eventually I got it by using dynamic programming and starting at the end vine, but it took way too long as I had several wrong submissions working out bugs. For each vine, I worked out the minimum length of vine needed to reach the other side from that vine. Working backwards, the minimum for each vine depended on the values for the vines already computed. I still completely missed the possibility that you might want to swing backward.

The second problem, Aerobics, was to place circles in a rectangular area without any overlap. In general, that’s the famous packing problem with no efficient solution, but in this case you’re told that there is plenty of room: over five times the area of the circles. So it comes down to find a heuristic for placing circles. I was lucky that mine worked, which was to sort the circles and place them in rows. The Google solution was even simpler: just place the circles randomly and pick a new position when there is overlap.

With less than 30 minutes left, I took a stab at the third problem, Mountain View. Given constraints about a set of mountain heights, you have to generate a set of heights consistent with the constraints. I didn’t have any code handy for solving a system of linear inequalities, but I tried a simple solution of adjusting each height to meet the constraints where it needed to be the tallest and repeating until all constraints were satisfied or some limit was reached. Unfortunately the contest ended before I finished. I did submit my solution 10 minutes later even though the site had gone in practice mode. My dumb solution was good enough to pass the small test test but not the large set.

I realized later that if I knew Mathematica better I could have used it to solve the inequalities. Later I hand-coded one of the simpler test cases (the last one on the problem page) in Mathematica:

FindInstance[{a >= 0, b >= 0, c >= 0, d >= 0,

(d – a) * 1 > (b – a) * 3, (d – a) * 2 > (c – a) * 3,

(c – b) * (3 – 1 ) >= (d – b) * (2 – 1)},

{a, b, c, d}, Integers]

It quickly returned a valid answer:

{{a -> 7, b -> 0, c -> 1, d -> 0}}

I guess I need to learn how to do file I/O in Mathematica before next year.

*Update:* I did get the general solution working in Mathematica (posted on StackExchange for comments), but it can’t handle the 1000+ variables and ~1M constraints of the large problem, though I guess the constraints could be pruned first since often one constraint is redundant with two others.