The Snowblower Problem

1. The Snowblower Problem Valentin Polishchuk joint work with Estie Arkin, Michael Bender, Joseph Mitchell Applied Math and Statistics, Computer Science Stony Brook University

2. Backyard

3. The Morning After

4. Uniformly Covered

5. snowblower

6. Snowblower

7. SNOWBLOWER

8. snowblowing

9. Snowblowing

10. That simple? • How SB works • No, not the inside

11. Shifting Piling

12. Respect Max Depth (Height) D

13. Pile Up Too High

14. Stuck

15. Where to Dispose of Snow?

16. Piled Arbitrarily High

17. Boundary – Cliff

18. Infinite Capacity

19. Not that Unreasonable

20. In this City

21. snow melter

22. Snow Melter

23. SNOW MELTER

24. Formally • Polygonal domain P • integral-orthogonal • pixeled • no holes • SB 1 pixel (garage)

25. SB motion • From pixel to pixel • picks up all snow • throws onto aneighbor pixel • or away from P • if on the bd • dispose of • D ≥ 2 • max depth (height) s s

26. Objective: Shortest Tour City of Danville Public Works Department Danville, VA 24540

27. TSP-like • Milling/lawnmowing[AFM’00, AHS’00, H’91] • visit = remove • never re-visit in NP • Material handling[pushing blocks; extensive OR literature] • visit = move • may need to re-visit a lot in NP? • The Snowblower Problem (SBP) • visit = move • stacking ≤D allowed • visit a bd pixel = remove • our algs in NP

28. Throw Direction On which pixel can snow be placed? s

29. Left Right

30. Forward?

31. Yes - if adjustable

32. Adjustable-Throw Model Right, left, or fwd s s

33. Fixed-Throw Model Right only s s

34. Default Model • Throwback allowed • For ease of exposition • Not intended to be realistic • Algorithms easy to describe • Other models reduce to it s s

35. Results • O(1)-apx algorithms • NP-complete

36. A Key Idea • Voronoi decomposition • closest bd pixel edge • tie-breaking • clear Voronoi-cell-by-Voronoi-cell • Lower Bounds • snow amount • distance to bd

37. Lower Bounds • snow LB • s(R) -- # of pixels of R with snow • distance LB • d(R) = [distance pixel to the bd] pixelR

38. Combs Voronoi Cells • Lines e handle tooth tooth e

39. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s e

40. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s e

41. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s e

42. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s e

43. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s 2 e

44. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s 3 e

45. Line-clearing • Backthrows • up by D • U-turn • Forward throw • down s e

46. Cost of Line L • Each D-full pass cost ≤ 4 · d(cleared) cost = 2(h+D) d(cleared) = (h+1+…+h+D)/D ~ h+D/2 • Pass that is not D-full cost≤ 2 · s(L) • Cost of L c(L) ≤ 4 · d(L) + 2 · s(L) D h e

47. Clearing a Comb • While line L • snow(L) ≥ D • line-clearing • D-full passes • c(L) ≤ 4 · d(L) • Brush-ready s

48. Brush • “Capacitated DFS” • tooth by tooth • until D units of snow moved • clear a tooth • move down

49. Cost • A brush • through handle • through teeth • Each tooth is visited ≤ 2 s(tooth)< D • brush-ready • For all brushes c(red)≤ 4 · s(teeth)

50. Cost • t – lowest tooth pixel • dist to bd > t • if D-full d(cleared) = [distance to the bd] > t • Next brush (blue) ≤ 2 · t • c(blue) ≤ 2 · d(cleared) + 4 · s(handle) t