Short-length routes in low-cost networks via Poisson line patterns

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1 Short-length routes in low-cost networks via Poisson line patterns (joint work with David Aldous) Wilfrid Kendall Stochastic processes and algorithms workshop, Hausdorff Research Institute for Mathematics Bonn, 3-7 September September 2007

2 An ancient optimization problem A Roman Emperor s dilemma:

3 An ancient optimization problem A Roman Emperor s dilemma: PRO: Roads are needed to move legions quickly around the country;

4 An ancient optimization problem A Roman Emperor s dilemma: PRO: Roads are needed to move legions quickly around the country; CON: Roads are expensive to build and maintain;

5 An ancient optimization problem A Roman Emperor s dilemma: PRO: Roads are needed to move legions quickly around the country; CON: Roads are expensive to build and maintain; Pro optimo quod faciendum est?

6 British Railway network before Beeching Modern variants

7 Modern variants British Railway network before Beeching British Railway network after Beeching

8 Modern variants British Railway network before Beeching British Railway network after Beeching UK Motorways:

9 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N.

10 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess network G = G(x (N) ) of roads connecting cities by:

11 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess network G = G(x (N) ) of roads connecting cities by: network total road length len(g)

12 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess network G = G(x (N) ) of roads connecting cities by: network total road length len(g) (which is minimized by the Steiner tree ST(x (N) ));

13 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess network G = G(x (N) ) of roads connecting cities by: network total road length len(g) (which is minimized by the Steiner tree ST(x (N) )); versus average network distance between two randomly chosen cities, average(g) = 1 N(N 1) dist G (x i, x j ), i j

14 A mathematical idealization Consider N cities x (N) = {x 1,..., x N } in square side N. Assess network G = G(x (N) ) of roads connecting cities by: network total road length len(g) (which is minimized by the Steiner tree ST(x (N) )); versus average network distance between two randomly chosen cities, average(g) = 1 N(N 1) dist G (x i, x j ), (minimized by laying tarmac for complete graph). i j

15 Aldous and K. (2007) provide answers for the following Question Consider a configuration x (N) of N cities in [0, N] 2 as above, and a well-chosen connecting network G = G(x (N) ). How does large-n trade-off between len(g) and average(g) behave? And how clever do we have to be to get a good trade-off?

16 Today s focus (I) Idealize the road network as a low-intensity invariant Poisson line process Π 1.

17 Today s focus (I) Idealize the road network as a low-intensity invariant Poisson line process Π 1. Pick two cities x and y at distance n units apart.

18 Today s focus (I) Idealize the road network as a low-intensity invariant Poisson line process Π 1. Pick two cities x and y at distance n units apart. Remove lines separating the cities

19 Today s focus (I) Idealize the road network as a low-intensity invariant Poisson line process Π 1. Pick two cities x and y at distance n units apart. Remove lines separating the cities and identify the cell C x,y which then contains the two cities.

20 Today s focus (II) Upper-bound the network distance between the two cities by the mean semi-perimeter of this cell, 1 2 E [ len C x,y ].

21 Today s focus (II) Upper-bound the network distance between the two cities by the mean semi-perimeter of this cell, 1 2 E [ len C x,y ].

22 Today s focus (II) Upper-bound the network distance between the two cities by the mean semi-perimeter of this cell, 1 2 E [ len C x,y ]. Aldous and K. (2007) show how to apply this to resolve our Question, and how to use other methods from stochastic geometry to show that the resolution is nearly optimal.

23 Georges-Louis Leclerc, Comte de Buffon (September 7, April 16, 1788) Calculate π by dropping a needle randomly on a ruled plane and counting mean proportion of hits,

24 Georges-Louis Leclerc, Comte de Buffon (September 7, April 16, 1788) Calculate π by dropping a needle randomly on a ruled plane and counting mean proportion of hits, or (dually)

Georges-Louis Leclerc, Comte de Buffon (September 7, 1707 - April 16, 1788) Calculate π by dropping a needle randomly on a ruled plane and counting mean proportion

25 Georges-Louis Leclerc, Comte de Buffon (September 7, April 16, 1788) Calculate π by dropping a needle randomly on a ruled plane and counting mean proportion of hits, or (dually) (H. Steinhaus) compute length of a regularizable curve by counting mean number of hits of curve by a unit-intensity invariant Poisson line process.

26 Tools from stereology and stochastic geometry Buffon The length of a curve equals the mean number of hits by a unit-intensity Poisson line process;

27 Tools from stereology and stochastic geometry Buffon The length of a curve equals the mean number of hits by a unit-intensity Poisson line process; Slivynak Condition a Poisson process on placing a point z at a specified location.

28 Tools from stereology and stochastic geometry Buffon The length of a curve equals the mean number of hits by a unit-intensity Poisson line process; Slivynak Condition a Poisson process on placing a point z at a specified location. The conditioned process is again a Poisson process with added z;

29 Tools from stereology and stochastic geometry Buffon The length of a curve equals the mean number of hits by a unit-intensity Poisson line process; Slivynak Condition a Poisson process on placing a point z at a specified location. The conditioned process is again a Poisson process with added z; Angles Generate a planar line process from a unit-intensity Poisson point process on a reference line l, by constructing lines through the points whose angles θ (0, π) to l are independent with density 1 2 sin θ.

30 Tools from stereology and stochastic geometry Buffon The length of a curve equals the mean number of hits by a unit-intensity Poisson line process; Slivynak Condition a Poisson process on placing a point z at a specified location. The conditioned process is again a Poisson process with added z; Angles Generate a planar line process from a unit-intensity Poisson point process on a reference line l, by constructing lines through the points whose angles θ (0, π) to l are independent with density 1 2 sin θ. The result is a unit-intensity Poisson line process.

31 The key construction For simplicity, renormalize to a unit-intensity line process. Compute mean length of C x,y

32 The key construction For simplicity, renormalize to a unit-intensity line process. Compute mean length of C x,y by use of independent unit-intensity invariant Poisson line process Π 2,

33 The key construction For simplicity, renormalize to a unit-intensity line process. Compute mean length of C x,y by use of independent unit-intensity invariant Poisson line process Π 2, and determine the mean number of hits.

34 The key construction For simplicity, renormalize to a unit-intensity line process. Compute mean length of C x,y by use of independent unit-intensity invariant Poisson line process Π 2, and determine the mean number of hits. It is convenient to form Π 2 by deleting from Π 2 those lines separating x from y. (Mean number of hits: 2 x y = 2n.)

35 Some stochastic geometry (I) We have E [ len C x,y ] 2 x y = E [ # ( Π 2 C x,y )].

36 Some stochastic geometry (I) We have E [ ] len C x,y 2 x y = [ ( )] E # Π 2 C x,y. This is the total intensity of the intersection point process Π 1 Π 2 thinned by removing z Π 1 Π 2 when z is separated from both x and y by Π 1.

37 Some stochastic geometry (I) We have E [ len C x,y ] 2 x y = E [ # ( Π 2 C x,y )]. This is the total intensity of the intersection point process Π 1 Π 2 thinned by removing z Π 1 Π 2 when z is separated from both x and y by Π 1. We can appeal to a variant of Slivynak s theorem: the retention probability for z is the probability that no line of Π 1 hits both of segments xz and yz, namely exp ( 1 2 ( x z + y z x y )) = exp ( 1 2 (η n)), where η = x z + y z.

38 Some stochastic geometry (II) The intensity of the completely unthinned intersection process Π 1 Π 2 is π 2.

39 Some stochastic geometry (II) The intensity of the completely unthinned intersection process Π 1 Π 2 is π 2. The intensity of Π 1 Π 2 is obtained by careful computation of the probability that the intersection lines of a point of Π 1 Π 2 do not hit xy, using the Angle construction from above. The resulting intensity is: π 2 α sin α π = α sin α 2 where α is the exterior angle of the triangle xyz at z.,

40 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( ) ( intersection intensity R 2 intersection at z: lines don t hit xy )( retention: z not sep from xy ) d z

41 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( ) ( intersection intensity R 2 intersection at z: lines don t hit xy )( retention: z not sep from xy ) d z

42 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( π 2 R 2 intersection at z: lines don t hit xy )( retention: z not sep from xy ) d z

43 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( π 2 R 2 intersection at z: lines don t hit xy )( retention: z not sep from xy ) d z

44 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( π 2 R 2 intersection at z: lines don t hit xy ) exp ( 1 2 (η n)) d z

45 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = ( π 2 R 2 intersection at z: lines don t hit xy ) exp ( 1 2 (η n)) d z

46 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = π 2 α sin α exp ( 1 R 2 π 2 (η n)) d z

47 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = 1 (α sin α) exp ( 1 2 R 2 2 (η n)) d z Note that α = α(z) and η = η(z) both depend on z.

48 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = 1 (α sin α) exp ( 1 2 R 2 2 (η n)) d z Note that α = α(z) and η = η(z) both depend on z. Fixed α: locus of z is circle.

49 Mean perimeter length as a double integral Theorem E [ ] len C x,y 2 x y = 1 (α sin α) exp ( 1 2 R 2 2 (η n)) d z Note that α = α(z) and η = η(z) both depend on z. Fixed α: locus of z is circle. Fixed η: locus of z is ellipse.

50 Asymptotics Theorem Careful asymptotics for n show that E [ 1 2 len C ] x,y = n (α sin α) exp ( 1 R 2 2 (η n)) d z n + 4 ( log n + γ + 5 ) 3 3 where γ = is the Euler-Mascharoni constant.

51 Asymptotics Theorem Careful asymptotics for n show that E [ 1 2 len C ] x,y = n (α sin α) exp ( 1 R 2 2 (η n)) d z n + 4 ( log n + γ + 5 ) 3 3 where γ = is the Euler-Mascharoni constant. Thus a unit-intensity invariant Poisson line process is within O(log n) of providing connections which are as efficient as Euclidean connections.

52 Simulations (example) 1000 simulations at n = 10: average 6.74, s.e. 0.41, asymptotic Vertical exaggeration: n

53 Illustration of the final construction Use a hierarchy

54 Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process;

55 Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process; 2 a rectangular grid at a moderately large length scale;

56 Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process; 2 a rectangular grid at a moderately large length scale; 3 the Steiner tree ST(x (N) ));

Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process; 2 a rectangular grid at a moderately large length scale; 3 the

57 Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process; 2 a rectangular grid at a moderately large length scale; 3 the Steiner tree ST(x (N) )); 4 a few boxes from a grid at a small length scale, to avoid potential hot-spots where cities are close (boxes are connected to the cities).

58 Illustration of the final construction Use a hierarchy of: 1 a (sparse) Poisson line process; 2 a rectangular grid at a moderately large length scale; 3 the Steiner tree ST(x (N) )); 4 a few boxes from a grid at a small length scale, to avoid potential hot-spots where cities are close (boxes are connected to the cities).

59 The result Theorem For any configuration x (N) in square side N and for any sequence w N there are connecting networks G N such that: len(g N ) = len(st(x (N) )) + o(n) average(g N ) = 1 x i x j + o(w N log N) N(N 1) The sequence {w N } can tend to infinity arbitrarily slowly. i j

60 A complementary result Theorem Given a configuration of N cities in [0, N] 2 which is L N = o( log N)-equidistributed: random choice X N of city can be coupled to uniformly random point Y N so that [ { E min 1, X }] N Y N 0 ; L N

61 A complementary result Theorem Given a configuration of N cities in [0, N] 2 which is L N = o( log N)-equidistributed: random choice X N of city can be coupled to uniformly random point Y N so that [ { E min 1, X }] N Y N 0 ; L N then any connecting network G N with length bounded above by a multiple of N

62 A complementary result Theorem Given a configuration of N cities in [0, N] 2 which is L N = o( log N)-equidistributed: random choice X N of city can be coupled to uniformly random point Y N so that [ { E min 1, X }] N Y N 0 ; L N then any connecting network G N with length bounded above by a multiple of N connects the cities with average connection length exceeding average Euclidean connection length by at least Ω( log N).

63 Sketch of proof Use tension between two facts: (a) efficient connection of a random pair of cities forces a path which is almost parallel to the Euclidean path, and (b) the coupling means such a random pair is almost an independent uniform draw from [0, N] 2 (equidistribution), so a random perpendicular to the Euclidean path is almost a uniformly random line.

64 Conclusion Aldous and K. (2007) show

65 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost;

66 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently.

67 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. Poisson line processes are not computationally hard!

68 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. Poisson line processes are not computationally hard! What about random variation of network distance?

69 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. Poisson line processes are not computationally hard! What about random variation of network distance? What about distances in 3-space or even higher dimensions?

70 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. Poisson line processes are not computationally hard! What about random variation of network distance? What about distances in 3-space or even higher dimensions? View as a chapter in the theory of random metric spaces?

71 Conclusion Aldous and K. (2007) show the N cities in [0, N] 2 connection problem can be resolved using a Poisson line process to gain nearly Euclidean efficiency at negligible cost; conversely any configuration which is not too concentrated cannot be treated much more efficiently. Poisson line processes are not computationally hard! What about random variation of network distance? What about distances in 3-space or even higher dimensions? View as a chapter in the theory of random metric spaces? QUESTIONS?

72 Bibliography This is a rich hypertext bibliography. Journals are linked to their homepages, and stable URL links (as provided for example by JSTOR or Project Euclid ) have been added where known. Access to such URLs is not universal: in case of difficulty you should check whether you are registered (directly or indirectly) with the relevant provider. In the case of preprints, icons,,, linking to homepage locations are inserted where available: note that these are less stable than journal links!. Aldous, D. J. and W. S. K. (2007). Short-length routes in low-cost networks via Poisson line patterns. Research Report 451,, and Steele, J. M. (1997). Probability theory and combinatorial optimization, Volume 69 of CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Stoyan, D., W. S. K., and J. Mecke (1995). Stochastic geometry and its applications (Second ed.). Chichester: John Wiley & Sons. (First edition in 1987 joint with Akademie Verlag, Berlin).

Introduction The Poissonian City Variance and efficiency Flows Conclusion References. The Poissonian City. Wilfrid Kendall.

Introduction The Poissonian City Variance and efficiency Flows Conclusion References. The Poissonian City. Wilfrid Kendall. The Poissonian City Wilfrid Kendall w.s.kendall@warwick.ac.uk Mathematics of Phase Transitions Past, Present, Future 13 November 2009 A problem in frustrated optimization Consider N cities x (N) = {x 1,...,