Scenario Week 4: Art Gallery Competition
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1 Scenario Week 4: Art Gallery Competition Organisers' Report 26 February 2016
2 The Week 94 participants 24 teams submissions for Part 2 One submissions for Part 1 subtle server bug discovered (non-lethal) floating-point arithmetic is a nasty thing
3 The Problem and its solutions Original formulation is due to Chvátal (1975) Textbook algorithm by Fisk (1978) Triangulation and 3-colouring, delivers a decent n/3 solution Better solutions exist for specific polygons L-partitioning for rectilinear polygons: n/4 solution Detecting convex suqpolygons just one guard required; Even better: detecting star sub-polygons; A good survey: Art Gallery Theorems and Algorithms by O Rourke
4 Initial setup Part 1: 30 polygons for finding the best guards sets 1 5 are trivial to test intuition (small size); 8 13 are rectilinear ( vertices); composed from triangles ( vertices); are quasi-convex with large convex regions; composed from various random shapes. Part 2: 20 polygons/guards to find refutations About 2/3 problems had one node non-covered (easy to find); 6 or 7 problems required a proper algorithms (or a lot of patience).
5 Part 1, polygon 14
6 Checking your solutions Server is written in Scala via Spray framework on servlets (1500 LOC); Run during the week on a single Linux machine with 4 GB RAM; Each team s submissions are processed by a separate actor (non-blocking); All geometric processing is implemented in Scala from scratch, no third-party libraries (1800 LOC, including tests); ~150 unit tests + several randomised testing procedures (bazillions of randomly-generated polygons); still missed one floating-point bug :( Guards checking procedure is a slightly modified version of Joe-Simpson algorithm for visibility polygons (1985).
7 Checking your solutions
8 Detecting grey areas precisely Step 1: compute all individual visibility areas via Joe-Simpson algorithm.
9 Detecting grey areas precisely Step 2: triangulate the initial polygon
10 Detecting grey areas precisely Step 3*: add visibility areas one by one, compute intersections with present triangles and Δ-partition again
11 Detecting grey areas precisely Step 3*: add visibility areas one by one, compute intersections with present triangles and Δ-partition again
12 Detecting grey areas precisely Step 3*: add visibility areas one by one, compute intersections with present triangles and Δ-partition again
13 Detecting grey areas precisely Loop Invariant: at the end of each iteration, each triangle is either fully visible or is fully grey (invisible). Proof By induction on edges of visibility areas.
14 Detecting grey areas precisely Step 4: iterate through all the triangles of the partition and check if a centre of each belongs to some visibility area. If not, return the centre of such triangle as a refutation.
15 Behind the Scenes
16
17 Kareem s Demo
18 Analysing submission patterns Taking data about Part 1 submissions Recording time of successful submissions (green) Propagated submissions (purple) No submission (blank)
19 Experimentators Hour Problems
20 Hard workers Hour Problems
21 Real geeks Hour Problems
22 Late bloomers Hour Problems
23 Parallel computers Hour Problems
24 Part 1 problems: Toughies polygon 10 (338 vertices) polygon 12 (288 vertices) polygon 13 (334 vertices) polygon 17 (360 vertices)
25 Shameless Advertisement
26 MSc Programme by PPLV: Logic, Semantics and Verification of Programs. Analysis of the correctness of large systems; Concurrent and distributed programming; Formal methods and theorem proving (yay!); Dark magic of abstract algebra and category theory to make better software (without actual bugs); Starts next year, apply in 2017!
27 The Competition
28 Ranking solutions Solutions were not ranked based on the total sum of guards; Instead, (1) for each polygon, teams were grouped according to the number of guards, smaller is better (e.g., 5 groups of solutions) Teams that didn t solve a polygon were all put into the last group for this polygon (e.g., group 6 for the previous example) (2) Next, per-polygon rankings were aggregated for each team; (3) Overall ranking is based on a sum of per-polygon rankings; Team B that did worse than team A for some problems might still be ranked above A Teams that solved all 30 problems were ranked first amongst each other.
29 Expectations and Surprises For the first three days results in in Part 1 were consistent with the triangulation-based algorithm. Last-minute results look way better than the baseline Several top-ranked solutions are astonishingly good Although we suspect some of them to be hand-crafted. Part 2 didn t seem to pose too much challenge after all.
30 Finish line
31 The Winners
32 Surprises Part 1, polygon vertices 7 guards (best solution)
33 (4.0, )
34 In conclusion This week was fun to design and even more fun to observe. We hope, it was fun to participate in it. Have a nice weekend and take some time to enjoy art in galleries, which are now well-guarded. Thanks!
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