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The Snowblower Problem

The Snowblower Problem. Valentin Polishchuk joint work with Esther Arkin, Michael Bender, Joseph Mitchell. Applied Math and Statistics, Computer Science Stony Brook University, USA. Suppose that your backyard looks like this:. One morning you wake up and it is covered with snow….

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The Snowblower Problem

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

  2. Suppose that your backyard looks like this:

  3. One morning you wake up and it is covered with snow…

  4. …uniformly covered

  5. So you pull out your snowblower…

  6. …or your Snowblower

  7. …or your SNOWBLOWER

  8. …and begin snowblowing Depending on your backyard, snowblowing may look like this…

  9. or snowblowing may look like this:

  10. This talk: Algorithmic Aspects of Snowblowing • Introduce the snowblower problemrelated toTraveling Salesperson Problem (TSP) and material-handling problems • Constant-factor approximation algorithms • NP-Completness • Online snowblowing • Robotic vacuum-cleaning

  11. Snowblower:Material-Shifting Machine • lifts snow from one location • piles it on an adjacent location

  12. We must respect max snow depth (height)D,because…

  13. …if we pile snow up too high…

  14. …the snowblower gets stuck

  15. Where to dispose of the snow? (neighbor’s yard)

  16. We can pile snow arbitrarily high on the neighbor’s lawn…

  17. Effectively the neighbor’s lawn has infinite capacity

  18. Alternatively: boundary is “cliff” We dump as much snow as we want

  19. (A bigger cliff)

  20. We can achieve infinite capacity using a snow melter…

  21. Snow Melter

  22. SNOW MELTER

  23. SB Problem Definition • Polygonal domain P • pixelated sb • SB 1 pixel initially in garage

  24. SB Motion • SB moves from pixel to adjacent pixel • picks up all snow • throws onto aneighbor pixel • or over the boundary of region • max depth of snow D • Objective: minimize the length of the tour of the snowblower sb sb

  25. The SBP is TSP-like • Geometric Traveling Salesperson Problem (TSP) • Find shortest tour visiting given set of points • Lawnmowing[Arkin,Fekete,Mitchell00, ArkinHeldSmith00, Held91] • visit = remove • never re-visit) in NP, solution_length = poly(input) • Material handling • visit = move • may need to re-visit a lot ) in NP? solution_length = exp(input)? • The Snowblower Problem • visit = move • stacking ≤D allowed • visit a boundary pixel = remove • our algs ) in NP sb 3 [pushing blocks; extensive OR literature] sb sb sb 2 wk lm

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

  27. Left Right

  28. Forward?

  29. Yes - if adjustable

  30. Backward? Not easy to implement. But it makes the algorithms easier.

  31. Default Model Throwback allowed Not intended to be realistic,but easy to describe other models reduce to it s s

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

  33. Fixed-Throw Model Right only s s

  34. A Key Idea • Voronoi decomposition • closest boundary pixel edge • clear Voronoi-cell-by-Voronoi-cell • Lower Bounds • snow amount • distance to boundary

  35. Lower Bounds • snow LB snowLB(R) = # of pixels of region R with snow • distance LB distLB(R) = [distance from pixel to the boundary] pixelR

  36. Combs Voronoi Cells Are Of Two Types • Lines e handle tooth tooth e

  37. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s e

  38. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s e

  39. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s e

  40. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s e

  41. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s 2 e

  42. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s 3 e

  43. Line-clearing (D = 3) • Move up D,doing back throws • U-turn • Forward throw moving down to boundary s e

  44. Cost of Line L • Each D-full pass (clearing D snow units) distLB(cleared) = [(h+1)+…+(h+D) ]/D ~ h+D/2 cost = 2(h+D)≤ 4 distLB(cleared) • The pass that is not D-full cost≤ 2 snowLB(L) • Total cost to clear line L cost(L) ≤ 4 distLB(L) + 2 snowLB(L) D h e

  45. Clearing a Comb (D = 3) • First clear linesusing D-full passes(D units of snow each) • cost(L) ≤ 4 distLB(L) • Now the comb is ready for another operation “brush-ready” s

  46. Brush Operation for Clearing Combs • “Capacitated DFS” • Proceed tooth by tooth • until D units of snow moved • clear a tooth and move down

  47. A brush • through handle • through teeth

  48. Cost of a Brush (red part) • Each tooth is visited ≤ 2 times (there & back twice) [Because brush-ready we know snow(tooth)< D] • For all brushes cost(red) ≤ 4 snowLB(teeth)

  49. Cost of a Brush (blue part) • Charge blue part of a brush to distLB of the snow cleared in previous brush distLB(cleared) = [distance to the bd] > t • Next brush (blue) ≤ 2 · t • cost(blue) ≤ 2 distLB(cleared) + 4 snowLB(handle) t

  50. All Brushes • cost(brushes) = cost(blue) + cost(red) ≤ 4 snowLB(comb) + 2 distLB(cleared)

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