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Optimizing Snow Removal: The Snowblower Problem Explained

This comprehensive exploration addresses the Snowblower Problem (SBP) through a mathematical lens, discussing parameters such as snow depth and disposal strategies. The research highlights algorithms for efficiently managing snow clearing, factoring in Voronoi decomposition and boundary conditions. With implications for urban public works, it offers insights into practical applications like route optimization and material handling. Aimed at improving snow removal strategies, this work examines the balance between efficiency and operational constraints in municipal settings.

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Optimizing Snow Removal: The Snowblower Problem Explained

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  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

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