1 / 81

The Snowblower Problem

The Snowblower Problem. Estie Arkin, Michael Bender, Joseph Mitchell, Valentin Polishchuk. Stony Brook University. Suppose that your backyard looks like this:. One morning you wake up and it is covered with snow…. …uniformly covered. So you pull out your snowblower ….

zeno
Download Presentation

The Snowblower Problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Snowblower Problem Estie Arkin, Michael Bender, Joseph Mitchell, Valentin Polishchuk Stony Brook University

  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 snowblowing problem(intermediate between TSP and material-handling problems) • Give O(1) approximation algorithms for several versions of the problem • Prove NP-hardness for some versions of the problem

  11. Snowblower:Material-Shifting Machine • It lifts snow from one location, and 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. Where to dispose of the snow? (neighbor’s yard)

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

  18. Effectively the neighbor’s lawn has infinite capacity

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

  20. (A bigger cliff)

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

  22. Snow Melter

  23. SNOW MELTER

  24. SB Problem Definition - Driveway • Polygonal domain P • integral-orthogonaland pixelated • no holes • SB 1 pixel (initially in garage)

  25. SB Problem Definition • 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 ≥ 2 • Objective: minimize the length of the path of the snowblower sb sb

  26. The SBP is TSP-like • Milling/lawnmowing[Arkin,Fekete,Mitchell00, ArkinHeldSmith00, Held91] • 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 boundary pixel = remove • our algs in NP

  27. We are not the first to consider pixel environments for snowblowing… City of Danville Public Works Department Danville, VA 24540

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

  29. Left Right

  30. Forward?

  31. Yes - if adjustable

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

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

  34. Default Model • Throwback allowed • Not intended to be realistic,but easy to describe and other models reduce to it 8-APX s s

  35. Adjustable-Throw Model Right, left, or fwd 9-APX s s

  36. Fixed-Throw Model Right only 106-APX s s

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

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

  39. Combs Boundary Cells Are Of Two Types (by our tiebreaking rule) • Lines e handle tooth tooth 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 e

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

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

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

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

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

  47. Cost of Line L • Each D-full pass (clearing D snow units) distLB(cleared) = (h+1+…+h+D)/D ~ h+D/2cost = 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

  48. Clearing a Comb (D = 3) • First clear whatever lineswe can using D-full passes(where clear D units of snow) • cost(L) ≤ 4 distLB(L) • Now the comb is ready for another operation: “brush-ready”. s

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

  50. Cost of a Brush (red part) • A brush • through handle • through teeth • 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)

More Related