Explorations into the Lawn Mowing Problem: Specific Cases and General Trends

Transcription

1 Explorations into the Lawn Mowing Problem: Specific Cases and General Trends *, Vanessa Kleckner, and Amelia Stonesifer St. Olaf College September 25, 2012

2 Outline 1 Introduction 2 Results 3 Conclusions

3 The General Lawn Mowing Problem Suppose that we have a lawn, L,

4 The General Lawn Mowing Problem Suppose that we have a lawn, L, and a mower, M.

5 The General Lawn Mowing Problem Suppose that we have a lawn, L, and a mower, M. What is the most efficient way to mow L by moving M?

6 Our Lawn Mowing Problem Suppose that a lawn, L, is given, say a rectangular area in the plane

7 Our Lawn Mowing Problem Suppose that a lawn, L, is given, say a rectangular area in the plane and that M is a smaller oriented region, say a circle, with radius r.

8 Our Lawn Mowing Problem Suppose that a lawn, L, is given, say a rectangular area in the plane and that M is a smaller oriented region, say a circle, with radius r. Is there a most efficient path to cover L by moving M? If so, what is it?

9 Also called

10 Also called Geometric Milling Problem

11 Also called Geometric Milling Problem Covering Tour problem

12 Also called Geometric Milling Problem Covering Tour problem Applications in

13 Also called Geometric Milling Problem Covering Tour problem Applications in Automatic tool paths

14 Also called Geometric Milling Problem Covering Tour problem Applications in Automatic tool paths Automatic inspection systems

15 Polynomial Time Problems in the set P can be solved in polynomial time relative to the size of the problem.

16 Polynomial Time Problems in the set P can be solved in polynomial time relative to the size of the problem. These problems are tractable.

17 Polynomial Time Problems in the set P can be solved in polynomial time relative to the size of the problem. These problems are tractable. In P, solving and checking a solution in polynomial time are equivalent.

18 NP Hard NP problems have solutions which can be checked in polynomial time, but it can take longer to solve the problem.

19 NP Hard NP problems have solutions which can be checked in polynomial time, but it can take longer to solve the problem. NP-Hard problems can be algorithmically reduced to a NP-Complete problem.

20 NP Hard NP problems have solutions which can be checked in polynomial time, but it can take longer to solve the problem. NP-Hard problems can be algorithmically reduced to a NP-Complete problem. The Lawn Mowing problem can be simplified to the Traveling Salesman Problem, which is NP-Hard.

21 NP Hard NP problems have solutions which can be checked in polynomial time, but it can take longer to solve the problem. NP-Hard problems can be algorithmically reduced to a NP-Complete problem. The Lawn Mowing problem can be simplified to the Traveling Salesman Problem, which is NP-Hard. We must focus on specific cases rather than trying to find a general solution.

22 Definitions Mowing: a path that covers the lawn such that all points on the lawn are within a given width of the path. A mowing trajectory, denoted by m, is a 3-dimensional path that includes the angle turned as the height.

23 Definitions (a) A 2D mowing (b) Corresponding mowing trajectory

24 Conjectures 1 A mowing whose mowing trajectory achieves minimum height is also most efficient.

25 Conjectures 1 A mowing whose mowing trajectory achieves minimum height is also most efficient. 2 For a lawn with width 2r, the most efficient mowing is one that goes straight down the middle of the lawn.

26 Outline 1 Introduction 2 Results 3 Conclusions

27 Preliminary Fact Consider all mowings that use a fixed, finite set of turning angles θ = {θ 1, θ 2, θ 3,..., θ n }

28 Preliminary Fact Consider all mowings that use a fixed, finite set of turning angles θ = {θ 1, θ 2, θ 3,..., θ n } Theorem: There exists a mowing, m 0, such that h(m0 ) is minimal.

29 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory}

30 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory} 2 Let A be the finite set of all angles turned in the mowing.

31 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory} 2 Let A be the finite set of all angles turned in the mowing. Let θ 0 = inf(a).

32 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory} 2 Let A be the finite set of all angles turned in the mowing. Let θ 0 = inf(a). 3 Let D be the set of all differences of angles, so D = {d = θ i θ j : 1 i < j n}.

33 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory} 2 Let A be the finite set of all angles turned in the mowing. Let θ 0 = inf(a). 3 Let D be the set of all differences of angles, so D = {d = θ i θ j : 1 i < j n}. Let d 0 = inf(d) > 0

34 Existence of Minimum Height Proof: 1 Let h 0 = inf{h(m ) : m is a mowing trajectory} 2 Let A be the finite set of all angles turned in the mowing. Let θ 0 = inf(a). 3 Let D be the set of all differences of angles, so D = {d = θ i θ j : 1 i < j n}. Let d 0 = inf(d) > 0 4 By definition of inf, there must be a mowing m 0 such that h 0 h(m0 ) < h 0 + d 0 2.

35 Existence of Minimum Height We want to show that h(m 0 ) = h 0

36 Existence of Minimum Height We want to show that h(m 0 ) = h 0 1 If m 1 is any mowing other than m 0, h 0 h(m 1 ).

37 Existence of Minimum Height We want to show that h(m 0 ) = h 0 1 If m 1 is any mowing other than m 0, h 0 h(m 1 ). 2 If h(m 1 ) < h 0 + d 0 2, then h(m 1 ) h(m 0 ) < d 0 2.

38 Existence of Minimum Height We want to show that h(m 0 ) = h 0 1 If m 1 is any mowing other than m 0, h 0 h(m 1 ). 2 If h(m 1 ) < h 0 + d 0 2, then h(m 1 ) h(m 0 ) < d But, d 0 is minimal, so h(m 1 ) = h(m 0 ) and h 0 = h(m 0 ).

39 Existence of Most Efficient Mowing Similar to minimum height proof.

40 Existence of Most Efficient Mowing Similar to minimum height proof. Uses a subsequence that converges in the Hausdorff metric.

41 Old Conjectures Revisited Conjecture: A mowing whose mowing trajectory achieves minimum height is also most efficient.

42 Old Conjectures Revisited Conjecture: A mowing whose mowing trajectory achieves minimum height is also most efficient.

43 Old Conjectures Revisited Counterexample: A mowing that achieves minimum height but not maximum efficiency

44 Old Conjectures Revisited Counterexample: A mowing that achieves minimum height but not maximum efficiency

45 Old Conjectures Revisited Conjecture: For a lawn L with width 2r, the most efficient mowing is the straight-line path, or the path goes straight down the middle of the lawn.

46 Old Conjectures Revisited Conjecture: For a lawn L with width 2r, the most efficient mowing is the straight-line path, or the path goes straight down the middle of the lawn. Clearly, this conjecture is true.

47 Old Conjectures Revisited Conjecture: For a lawn L with width 2r, the most efficient mowing is the straight-line path, or the path goes straight down the middle of the lawn.

48 Old Conjectures Revisited This T-path is a better solution for low turn cost.

49 Old Conjectures Revisited This T-path is a better solution for low turn cost.

50 Old Conjectures Revisited This T-path is a better solution for low turn cost. It takes a bit of a proof to show that this is the best solution for the one-strip lawn.

51 Two-Strip Lawn We suggested several possible mowings. We conjectured when each mowing would be most efficient.

52 Outline 1 Introduction 2 Results 3 Conclusions

53 Expanding our problem Potential future research:

54 Expanding our problem Potential future research: 1 Find relationship between length and width in two-strip lawn

55 Expanding our problem Potential future research: 1 Find relationship between length and width in two-strip lawn 2 Determine minimal height and maximal efficiency relationship

56 Expanding our problem Potential future research: 1 Find relationship between length and width in two-strip lawn 2 Determine minimal height and maximal efficiency relationship 3 Investigate most efficient starting points for lawn mower

57 Expanding our problem Ambitious research possibilities:

58 Expanding our problem Ambitious research possibilities: 1 Do not restrict turns to only π 2

59 Expanding our problem Ambitious research possibilities: 1 Do not restrict turns to only π 2 2 More strips of lawn and non-rectangular lawns

60 Thank you!