(joint work withDavid Aldous) Wilfrid Kendall · Introduction Stereology Construction AsymptoticsConclusionReferences Short-length routes in low-cost networks (joint work withDavid

Introduction Stereology Construction Asymptotics Conclusion References

Short-length routes in low-cost networks(joint work with David Aldous)

Wilfrid [email protected]

Colloquium talk

http://www.stat.berkeley.edu/users/aldous/

http://www.warwick.ac.uk/go/wsk

mailto:[email protected]


An ancient optimization problem

A RomanEmperor’sdilemma:

PRO: Roads are needed tomove legions quicklyaround the country;

CON: Roads are expensiveto build and maintain;Pro optimoquod faciendum est?










CON: Roads are expensiveto build and maintain;

Pro optimoquod faciendum est?







Modern variants

British Railwaynetworkbefore Beeching


Modern variants


British Railwaynetworkafter Beeching


Modern variants


British Railwaynetworkafter Beeching

UK Motorways:


A mathematical idealization

Consider N cities x(N) = x1, . . . , xN in square side√

N.

Assess road network G = G(x(N)) connecting cities by:

network total road length len(G)

(minimized by Steiner minimum tree ST(x(N)));

versus

average network distance between two random cities,

average(G) = 1N(N − 1)

∑∑i≠j

distG(xi, xj) ,

(minimized by laying tarmac for complete graph).




N.



(minimized by Steiner minimum tree ST(x(N)));

versus



∑∑i≠j

distG(xi, xj) ,





N.



(minimized by Steiner minimum tree ST(x(N)));versus



∑∑i≠j

distG(xi, xj) ,





N.


network total road length len(G)(minimized by Steiner minimum tree ST(x(N)));

versus



∑∑i≠j

distG(xi, xj) ,





N.


network total road length len(G)(minimized by Steiner minimum tree ST(x(N)));versus



∑∑i≠j

distG(xi, xj) ,





N.


network total road length len(G)(minimized by Steiner minimum tree ST(x(N)));versus



∑∑i≠j

distG(xi, xj) ,



Aldous and Kendall (2008) provide answers for the

First 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 doeslarge-N trade-off between len(G) and average(G) behave?

(And how clever do we have to be to get a good trade-off?)

len(ST(x(N))) is no more than O(N) (Steele 1997, §2.2);

Average Euclidean distance between two randomlychosen cities is at most

√2N;

Perhaps increasing total network length by const×Nα

might achieve average network distance no more thanorder Nβ longer than average Euclidean distance?



First Question




Note:



√2N;





First Question




Note:



√2N;





First Question




Note:



√2N;




Further Questions

Question about fluctuations

Given a good compromise between average(G) and len(G),how might the variance behave?

Question about true geodesics

The upper bound is obtained by controlling non-geodesicpaths. How might true geodesics behave?

Question about flows

Consider a network which exhibits good trade-offs. What canbe said about flows in this network?


Further Questions








Further Questions








First question (I)

Idealize the road network as a low-intensity invariantPoisson line process Π1.

Unit intensity is 12 d r dθ: we will use this and scale.

Pick two cities x and y at distance n =√

N units apart.

Remove lines separating the two cities;

focus on cell Cx,y containing the two cities.


First question (I)




N units apart.




First question (I)




N units apart.




First question (I)




N units apart.




First question (I)




N units apart.




First question (II)

Upper-bound “network distance” between two cities by

mean semi-perimeter of cell, 12 E

[len ∂Cx,y

].

Aldous and Kendall (2008) answer First Question usingthis, and use other methods from stochastic geometry toshow that the resolution is nearly optimal.


First question (II)

Upper-bound “network distance” between two cities bymean semi-perimeter of cell, 1

2 E[len ∂Cx,y

].



First question (II)

Upper-bound “network distance” between two cities bymean semi-perimeter of cell, 1

2 E[len ∂Cx,y

].



Links to random metric spacesThe study of the metric space generated by the line processforms a chapter in the theory of random metric spaces:

Vershik (2004) builds random metric spaces out ofrandom distance matrices (compare MDS in statistics);almost all such metric spaces are isometric to Urysohn’scelebrated universal metric space.

But these spaces aredefinitely not finite-dimensional!

The Brownian map has been introduced as the limit ofrandom quadrangulations of the 2-sphere (for example,Le Gall 2009).

But these spaces are definitely not flat!

A famous conjecture (late 1940’s) by D. G. Kendall isthat large cells in the line process tessellation are nearlycircular.

This is now known to be true (Miles, Kovalenko).

the project builds on a wide range of work: from300-year-old French encyclopaedist to recentcalculations on self-similar random processes.



Vershik (2004) builds random metric spaces out ofrandom distance matrices (compare MDS in statistics);almost all such metric spaces are isometric to Urysohn’scelebrated universal metric space.

But these spaces aredefinitely not finite-dimensional!








Vershik (2004) builds random metric spaces out ofrandom distance matrices (compare MDS in statistics);almost all such metric spaces are isometric to Urysohn’scelebrated universal metric space. But these spaces aredefinitely not finite-dimensional!

















The Brownian map has been introduced as the limit ofrandom quadrangulations of the 2-sphere (for example,Le Gall 2009). But these spaces are definitely not flat!















A famous conjecture (late 1940’s) by D. G. Kendall isthat large cells in the line process tessellation are nearlycircular. This is now known to be true (Miles, Kovalenko).






A famous conjecture (late 1940’s) by D. G. Kendall isthat large cells in the line process tessellation are nearlycircular. This is now known to be true (Miles, Kovalenko).



Georges-Louis Leclerc, Comte de Buffon(7 September, 1707 – 16 April, 1788)

Calculate π by dropping a needlerandomly on a ruled plane andcounting mean proportion of hits,

or (dually)

(H. Steinhaus) compute length ofregularizable curve by countingmean number of hits byunit-intensity invariant Poisson lineprocess.




or (dually)





or (dually)



Tools from stereology and stochastic geometry

Buffon The length of a curve equals the mean number of hits bya unit-intensity Poisson line process;

Slivnyak Condition a Poisson process on placing a “point” z at aspecified location.

The conditioned process is again aPoisson process with added z;

Angles Generate a planar line process from a unit-intensity Poisson point process on a refer-ence line `, by constructing lines throughthe points p whose angles θ ∈ (0, π) to `are independent with density 1

2 sinθ.

Theresult is a unit-intensity Poisson line pro-cess. Intensity measure in these coordi-nates: sinθ

2 d p dθ.




Slivnyak Condition a Poisson process on placing a “point” z at aspecified location.

The conditioned process is again aPoisson process with added z;


2 sinθ.


2 d p dθ.




Slivnyak Condition a Poisson process on placing a “point” z at aspecified location. The conditioned process is again aPoisson process with added z;


2 sinθ.


2 d p dθ.






2 sinθ.


2 d p dθ.






2 sinθ.


2 d p dθ.






2 sinθ. Theresult is a unit-intensity Poisson line pro-cess. Intensity measure in these coordi-nates: sinθ

2 d p dθ.


The key construction

(Remember, line process renormalized to unit intensity.)

Compute mean length of ∂Cx,y

by use of independentunit-intensity invariant Poisson line process Π2, anddetermine the mean number of hits.

It is convenient to form Π∗2 by deleting from Π2 thoselines separating x from y. (Mean number of hits:2|x − y| = 2n.)




Compute mean length of ∂Cx,y by use of independentunit-intensity invariant Poisson line process Π2,

anddetermine the mean number of hits.





Compute mean length of ∂Cx,y by use of independentunit-intensity invariant Poisson line process Π2, anddetermine the mean number of hits.





Compute mean length of ∂Cx,y by use of independentunit-intensity invariant Poisson line process Π2, anddetermine the mean number of hits.



Mean perimeter length as a double integral

Theorem

E[len ∂Cx,y

]− 2|x − y| =

12

∫∫R2(α− sinα)exp

(−1

2 (η− n))

d z

Note that α = α(z) and η = η(z) both depend on z.

Fixed α: locus of z iscircle.

Fixed η: locus of z isellipse.



Theorem

E[len ∂Cx,y

]− 2|x − y| =

12


(−1

2 (η− n))

d z






Theorem

E[len ∂Cx,y

]− 2|x − y| =

12


(−1

2 (η− n))

d z






Theorem

E[len ∂Cx,y

]− 2|x − y| =

12


(−1

2 (η− n))

d z





Asymptotics

Theorem

Careful asymptotics for n→∞ show that

E[

12 len ∂Cx,y

]=

n+ 14


(−1

2 (η− n))

d z ≈

n+ 43

(log n+ γ + 5

3

)where γ = 0.57721 . . . is the Euler-Mascheroni constant.

Thus a unit-intensity invariant Poisson line process is withinO(log n) of providing connections which are as efficient asEuclidean connections.


Asymptotics

Theorem

Careful asymptotics for n→∞ show that

E[

12 len ∂Cx,y

]=

n+ 14


(−1

2 (η− n))

d z ≈

n+ 43

(log n+ γ + 5

3

)where γ = 0.57721 . . . is the Euler-Mascheroni constant.

Thus a unit-intensity invariant Poisson line process is withinO(log n) of providing connections which are as efficient asEuclidean connections.


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 minimum 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 areconnected to the cities).



Use a hierarchy of:1 a (sparse) Poisson line process;














3 the Steiner minimum tree ST(x(N)));

4 a few boxes from a grid at a small length scale, to avoidpotential “hot-spots” where cities are close (boxes areconnected to the cities).














Answering the first question

Theorem

For any configuration x(N) in square side√

N and for any se-quence wN →∞ there are connecting networks GN such that:

len(GN) = len(ST(x(N)))+ o(N)

average(GN) = 1N(N − 1)

∑∑i≠j

‖xi − xj‖ + o(wN log N)

The sequence wN can tend to infinity arbitrarily slowly.


A complementary result

Theorem

Given a configuration of N cities in [0,√

N]2 which isLN = o(

√log N)-equidistributed: random choice XN of city

can be coupled to uniformly random point YN so that

E[min

1,|XN − YN|

LN

]-→ 0 ;

then

any connecting network GN with length boundedabove by a multiple of N connects the cities withaverage connection length exceeding average Euclideanconnection length by at least Ω(

√log N) .



Theorem


N]2 which isLN = o(



E[min

1,|XN − YN|

LN

]-→ 0 ;

then any connecting network GN with length boundedabove by a multiple of N

connects the cities withaverage connection length exceeding average Euclideanconnection length by at least Ω(

√log N) .



Theorem


N]2 which isLN = o(



E[min

1,|XN − YN|

LN

]-→ 0 ;

then any connecting network GN with length boundedabove by a multiple of N connects the cities withaverage connection length exceeding average Euclideanconnection length by at least Ω(

√log N) .


Sketch of proof

Use tension between two facts:

(a) efficient connection of a random pair of cities forces apath which is almost parallel to the Euclidean path, and

(b) the coupling means such a random pair is almost anindependent uniform draw from [0,

√N]2

(equidistribution),

so a random perpendicular to the Euclidean path isalmost a uniformly random line.


Sketch of proof




√N]2

(equidistribution),



Sketch of proof




√N]2

(equidistribution),



Sketch of proof




√N]2

(equidistribution),



Simulations (example)

1000 simulationsat n = 1000000:average 21.22,s.e. 0.23,asymptotic 21.413.

Vertical exaggeration:√n


ConclusionAldous and Kendall (2008) show

the “N cities in [0,√

N]2” connection problem can beresolved using a Poisson line process to gain nearlyEuclidean efficiency at negligible cost;conversely any configuration which is not tooconcentrated cannot be treated much more efficiently.

Poisson line processes are not computationally hard!Relates to Computer Science notion of “spanner graph”,View as a chapter in the theory of random metric spaces.Recent further work:

Random variation of network distance is relatively small.Traffic flow in the network scales well.“near geodesics” are pretty good;Traffic flow.

User equilibrium for flows?Same problem in 3-space or higher dimensions?QUESTIONS?




N]2” connection problem can beresolved using a Poisson line process to gain nearlyEuclidean efficiency at negligible cost;

conversely any configuration which is not tooconcentrated cannot be treated much more efficiently.















Poisson line processes are not computationally hard!

Relates to Computer Science notion of “spanner graph”,View as a chapter in the theory of random metric spaces.Recent further work:







Poisson line processes are not computationally hard!Relates to Computer Science notion of “spanner graph”,

View as a chapter in the theory of random metric spaces.Recent further work:







Poisson line processes are not computationally hard!Relates to Computer Science notion of “spanner graph”,View as a chapter in the theory of random metric spaces.

Recent further work:















Random variation of network distance is relatively small.

Traffic flow in the network scales well.“near geodesics” are pretty good;Traffic flow.







Random variation of network distance is relatively small.Traffic flow in the network scales well.

“near geodesics” are pretty good;Traffic flow.







Random variation of network distance is relatively small.Traffic flow in the network scales well.“near geodesics” are pretty good;

Traffic flow.















User equilibrium for flows?

Same problem in 3-space or higher dimensions?QUESTIONS?







User equilibrium for flows?Same problem in 3-space or higher dimensions?

QUESTIONS?









BibliographyThis is a rich hypertext bibliography. Journals are linked to their homepages, andstable URL links (as provided for example by JSTOR or Project Euclid ) havebeen added where known. Access to such URLs is not universal: in case ofdifficulty you should check whether you are registered (directly or indirectly) withthe relevant provider. In the case of preprints, icons , , , linking tohomepage locations are inserted where available: note that these are less stablethan journal links!.

Aldous, D. J. and W. S. Kendall (2008, March).Short-length routes in low-cost networks via Poisson line patterns.Advances in Applied Probability 40(1), 1–21, , and

http://arxiv.org/abs/math.PR/0701140 .

Böröczky, K. J. and R. Schneider (2008).The mean width of circumscribed random polytopes.Canadian Mathematical Bulletin accepted.Submitted manuscript.

Le Gall, J.-F. (2009).Geodesics in large planar maps and in the Brownian map.Acta Mathematica to appear.

http://links.jstor.org

http://projecteuclid.org

http://www.warwick.ac.uk/go/wsk/ppt/451.pdf

http://arxiv.org/abs/math.PR/0701140


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. Kendall, and J. Mecke (1995).Stochastic geometry and its applications (Second ed.).Chichester: John Wiley & Sons.(First edition in 1987 joint with Akademie Verlag, Berlin).

Vershik, A. M. (2004).Random and universal metric spaces.In Dynamics and randomness II, Volume 10 of Nonlinear Phenom. Complex

Systems, pp. 199–228. Dordrecht: Kluwer Acad. Publ.

Documents

(joint work withDavid Aldous) Wilfrid Kendall · Introduction Stereology Construction AsymptoticsConclusionReferences Short-length routes in low-cost networks (joint work withDavid