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Steiner minimal trees and urban service networks
STEINER MINIMAL TREES AND URBAN SERVICE NETWORKS JAMES MACGREGOR SMITH? and MEIR GROSSS Universityof Massachusetts,Amherst,MA01003, U.S.A. (Received10Apt2 1981;in revisedform23 J&y 1981) Abstract--Theoptimalcon~gu~t~on of urban service networks has recently been shown to be a compu~tionalfy difficult problem. However, there are efficient and effective techniques by which this optimal configuration of urban service networks can be approximated. In this paper, we analyze the Lp Steiner Network problem in the plane RZ and demonstrate its app~cab~ity to the urban service network problem. We present a simple algorithm for estimating the Lp metric parameterfor random points in the plane, then utilize it to find the Lp values for four different American cities. Finally, we apply the L$MT ~gorithm described within the text to one of the cities in order to demonstrate the effectiveness of our algorithm for determining optimaI network configurations.
many factors related to the objectives of the networkand to the physical characteristicsof the nodes and the links. However, a basic criterion in the selection of an appropriate design for almost all systems is the minimization of totai cost which is directly related, in part, to the spatialconfigurationof the network. The ~rnportantfactors determini~ service network costs are the construction costs which generally depend on the length of the system, and the movement of flow costs which are a function of both the length of the system and the magnitudeof the flow through it[4, 51. These cost characteristicsdictate two extreme solutions to the probIem of network spatial con~ration with the objective of minimizingtotal transportation costs. If only ffow costs are minim~ed then each Row demandis solved by a straightline connectingoriginand destination. This solution implies build~g the network with the longest path and therefore the highest const~ction costs. If, on the other hand, the flow costs are ignored and only construction costs are considered and mi~mized, then the solutionis the one which minimizes total length. In the design of a transit system, e.g. this dichotomy entails the resolution of the conflict between construction economy, i.e. a minimumlength network, and traveler convenience, i.e. minimum travei time (requiringdirect links between all origin and destination nodes). The procedure of designingthe opt~al network in most instances involves first finding the minimum length solution, and then subject to various constraint (i.e. budget), improve this network (i.e. add links) so as to optimize a given objective. Thus the first step can be expressed as follows: given a network select the minimum spanningtree (MST) which is the subset of links that minimizesthe sum of the shortest routes betweenall nodes. In the paper that follows, we focus on the first problem of the optimal network problem. The .initial problem is crucial to the overall network designproblem for many of the previously discussed applications.The problem we address in the paper is often referred to as the Steiner tree problem[~9]. In the followingpaper, we develop an efficient O(n log n) heuristic for the generalized Steiner tree problem in the plane R2. Our solutionalgorithmis not restricted to just the Euclidean metric but also can be used for network designproblems on the rectilinear metric and those metrics in between.
Transportation,utility and communicationnetworks are conspicuousfeatures of man’s uses of the land. ‘Roads, streets and highways, railways, waterways and airroutes, and utility, electric and communication lines permeate our landscapewith increasingfrequency. The design and conflation of transpo~tion, utility and communicationnetworks are of critical concern to urban and regional planners because of their potential long-termimpact on the region. It is commonknowledge that strong and direct relationshipsexist between transportationand utility systemsand patterns of land use. For example,Putnam[l], Berecham[2]and others show how the metropolitan transportation system affects the intensity and spatial dis~ibution of land use activities, which in turn, influencesthe size, location, and utilization of future transportationfacilities. Similarly,the impact of sewer and water lines on urban sprawl is also well known and documented [3]. More importantly,the changes in spatial relations between land uses directly affect other economic and en~ronmental factors such as energy consumption, air and water pollution, and indirectly, all facets of humanactivity in the region. More immediateand direct concern to the planner is the cost of constructing and operating the service network. This cost in turn has a direct relationship to the network’scon~g~ation or spatiallayout. The general purpose of a transportation, utility, or communicationnetwork is to satisfy the demands for se&ice. Graphicallythe network is represented by a set of origin nodes interlinkedto a set of destinationnodes. The node-linkinterpretationand the flow units, or commodities transferred, are a function of the network in question. For example, in an urban traffic system the nodes are pop~a~on centroids, the links are freeways, arterialsand streets, atid the flowunits are commuters;in a network designed for solid waste disposal, the nodes are the collection, treatment, and discharge points, the links are pipelines and the flow unit is solid waste; and in a communicati& network the nodes are message centers, the links are t~nsmission lines and the flow units are messages. The design of a particular service nework involves tDept.of Industrial E~ineering and Operations Research. $De.pt.of Landscape Architecture and Regional Planning. SEPS Vol. 16. No. 1-D
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J. MAC~R~~R
SMITH and
This generalized Steiner network aIgorithmis desirable because the distance between randomlylocated points in an urban region often requires a distance metric whichis in between the Euclidean and rectilinear metrics. After we complete the development of our algorithm, we examine the ern~~ic~ distance for random points in the plane of four di-tferentAmerican cities and then apply our ~gorithm to one city in p~tic~ar to demonstratethe a~orithm’s speed and effectiveness. As a finalprefatory remark, the ultimaterealizationof the optima1network con~guration will necessarily be altered due to lo&alinthrences:zoning, easement rig&, physical obstacles, social and economic concerns. The algo~t~s we present, however, should serve as valuable guidelinesduring the network design process, and are not meant as replacements of existing design practices.
If we are given a finite set of vertices, also called points, v = (v,, 02,. * ., vn] in the plane R2 withCartesian coordinates (Xi,yi) for (i = 1, 2, . . ., n) where ra2 3, we say that a Steiner -MinimalTree @MT)for the n given points is a ne~ork that interconnects all n points with minimumtotal length. In order to achieve this minimal Iength, additional vertices (Steiner points) somet~es must be added to the network. We will.assume that the distance function is an LP metric[8, IO].in which the distance between any two points Uiand vi in the plane R2 where p E [l, 23is given by
M.GROSS
Another sub-optimalalgorithmfor locating LP Steiner points in the plane is Morrisand Verdini’s[l 11.However, their algorithmis only for locating LP points and not for the L$MT problem nor do they present worst case results for the computational complexity of their algori~m. Before e~borating on the details of our algorithm, let us posit some importantd~~nitions[l&-14]. Bisector
The locus of points between two given vertices ai and vj which is equidistantbetween them:
This is the locus of points closer to 2rIthan to any other Vi:
A polygonal region that encompasses the locus of points closer to oi than to any other point in V.Formally defined as the intersection of half-planesdeterminedby the bisectors of z+ and all other points in V* often denoted as VP(v,):
The collection of Voronoipolygons VP(Q) for each vi in V.Often denoted as VLy 8,
Whenp = 2, we have the familarEuclideanmetric; when V~runoipoint Each vertex in the Voronoidiagramis a Voronoipoint. p = 1, the well-known rectilinear (Ma~hat~n) metric. Each Voronoipoint is the circumcenter of at most three When p E [0, 11,the distance function no longer remains convex. When p E [2, co], its meaningfulnessto real given vertices in R2. The degenerate case of four or worid.applicationsbecomes questionable.When p E [I, more cocircular points is excluded. 21,tt ts a weIi-behavedconvex func~on~l~].The axioms normallyes~bI~shedfor a distancemetric all hold for the The planar straight line graph which is dual to the LP metric p E ]1,2]. Since the construction cost of the L,SMT network Voronoi diagramis a t~an~iation, often referred to as solutinncan be assumedto vary linearly with the length the Delaunaytriangulationand denoted as DT( V). Examples of a Voronoi diagram and its Delaunay of the overall network, the objective function for the t~anguIation for a given set of points and the Euclidean problem is: metric appear in Figs. I and 2 respectively.
As far as is known,little research has been ~o~d~ted on the L,SMT network designproblem. One oPthe first persons to mentionthe L,SMT problem was Mel~k[8]* L, space is a particular subclass of ~inkows~ space. Melzak only briefly examines the L,SMT problem, and tfm proceeds to co~cen~ate his attention on the posz sibility of solving Steiner trees in ~~~kowski space. Since distancein Minkowskispace behaves differentlyin differentdirections(Minkowskispace has a non-isotropic metric) it seems to have Iittle practical use for our Minimize2 = purposes. ifi The underlyi~ algorithmfor cons~ct~g SMT’s on the &, metric operates in a manner similar to then the necessary and su~cient conditions for a mini- Kruskal’s1151~i~urn SpanningTree (MST)algorithm, mum are: However, in our case, we establish a priority queue of Delaunay triangles sorted on their reduction ratio (SAGEST) for each triangle, and, then, systematic~y where (Xi,yi) = the Cartesian coordinates of the Steiner points (i = 1, 2, . . ., m); {Xj, Xi) = the Cartesian coordinatesof the givenvertices in R2(j= 1,2.. , ., n); S = the set of Steiner points; and V = the given set of vertices. Although the objective function is convex, it is not di~er~ntiable.One can transform the objective function so that it is differentiableas follows:
Steiner minimal trees and urban service networks
Fig. 1. L2 Voronoi diagram for N = 18 (after [12]).
Fig. 2. & Delaunay ~ia~ulation for N = 18 (after [12lf.
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J. MACGREGORSMITHand M. GROSS
construct the SMT by a fast disjoint set-union procedure. The reason for this is that each triangle has a potential disjoint SMT in the forest of SMT’s of V. Constructing the Delaunay triangulation assists us in defining the triangles, and information from the Voronoi diagram assists us in determining how to concatenate these triangles into a SMT for V. Before we delve into the details of L, bisectors and L, Steiner triangles, let’s present a systematic overview of the steps of the O(n log n) L,SMT algorithm.
Step 1. (Delaunay triangulation) Presort all the vertices of V by their x-coordinate. Step 1.1. Divide the set V into two disjoint sets L and R by way of the sorted list of x-coordinates. Step 1.2. Recursively construct the Delaunay triangulation of sets L and R; call the triangulation DI and 4 respectively; Step 1.3. Construct, M, the merge curve, equidistant from a point in D1 and 0,. Unify D1 and D, through the merge curve as it threads a zig-zag path between D, and Dr; the L; bisectors play a crucial role in determining the merge edges; Step 2. (Concatenation process) Compute the Minimum Spanning Tree (MST) of the Delaunay Triangulation; mark those triangles which share two edges of the MST and place these doubly marked triangles in a priority queue called Q; Step 3. Where possible, concatenate pairs of triangles (t;, tj); if
this is not possible, add the singleton triangle ti with its Steiner point and edges to the SMT of V Step 4. End the concatenation process when the priority queue, Q, is empty. The general algorithmic process of the L,SMT problem is very similar to that ofthe Euclidean and Rectilinear algorithms discussed elsewhere[l&lS]. What is unique, however, to the L,SMT problem is the generation of L,, bisector, the location of Voronoi points in L, triangles, and, finally, the location of Steiner points in L, triangles. We shall spend some time on L, bisectors since they are crucial to the merge process of the Delaunay triangulation and eventually the SMT configuration. As an example of the peculiar qualities the L, metric p E [l, 21 has on the locus of Steiner points within a triangle, Fig. 3(a) and 3(b) illustrate how the locus of Steiner points is in general a curve and how the curve changes under a rotation of the coordinate axes. As we shall demonstrate in the applications section of the paper, the location of the origin from which all the empirical distance measures are made becomes an optimization problem in its own right since the orientation of the point set to the L,, metric affects the overall Steiner network length interconnecting V. LpBISECTORS In order to demonstrate the behavior of the L, bisector it is useful to examine an example in R’. Assume that two points Ui and aj are located at (0, 0) and (1, 2)
V3:(2 236, 01
Fig. 3. LJMT
Steiner loci under different orientations.
Steiner minims trees and urban service networks
respectively. Then, the equation of the perpendicular bisector for z is given by the following,where z, and z, are the respective n and y-coordinates of a point on the perpendicularbisector:
25
As one may have surmisedfrom the previous equations, the graph of the L,, bisector function is a curve for p values (1, w).The L, metric is defined as follows:
(fzx-0Jp + 12,-~~))i’P = (/l-&/p + /Z-2$)“? Because of the peculiar nature of the L, bisector p ~(1, 2), if we try to solve exactly for the Voronoi points (the circumcentersof the Delaunaytriangles),we will destroy the O(n log n) behavior of the L,SMT airIz$ f l2,1”3= [I1- ZXl” + 12- &)“I. gorithm. Therefore, we must resort to a heuristic proThis, however, is not sufficientto obtain a solutiondue cedure for determiningthe Voronoi points in this situato the presence of the absolute values of the terms. The tion in order to preserve our O(n log ra)~go~thrn. This foflowi~ algebraic manip~ation may be used to eli- means that the merge process must be designed so that it is sufficientlyrobust to handle near-optimallocations minate the absolute values: of Voronoi points. It is also apparent that heuristic solutions for the Steiner points in the LP triangles must be ((zX)2)p’2 f ((2,)“)“‘”= ((1- zX)2)p’2 t ((2- z$)P’2. used. In fact, the heuristicalgorithmwe will use to locate Whileremovingthe absolute value signswillallowone the Voronoi points parallels the L,SMT Steiner point to solve for the correct values numeric~ly, the equation location procedure. Therefore, before actually presenting the heuristic for locating Voronoi points, let us itself cannot be further simplified. We do know that the bisector z will pass through the examinesomepropertiesof the locus of Steinerpointsfor midpoint of the line segment connecting (U, ui)[19]. L, triangles. Figure 5 describes the locus of Steiner points for a Figure 4 illustrates the graph of the ~5, bisector for different valuesof the L, metric for two given point sets, triangle ti where p was varied over the open interval (1, Raisingboth sides to the power p, we have:
Fig.4. Lp bisectors.
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J. MACGREGORSMITHand M. GROSS
Fig. 5. LP locus for a triangle(where1 s p c 2). 2). As expected, the locus of points follows a curve;the general equation for this curve is given by the following:, Minimize 2 = {[lx1- vi,” ’ lx1 - oiklp s]xI
- VtJP + IYI - QyIp + IYl - ujylp •t (YI -
~kylpll’p~
Now the distance follows: dp(qO,
Lemma 1 If we have an isosceles triangle ri with its L, and Lz Steiner points collinear with the apex of the triangle, then the locus of Steiner points for the L, metric, p E (1, 21, is a straight line between the L1 and L2 Steiner points Of ti. Proof Given an isosceles triangle ti with a line collinear with the L1 and the t, points passing through the apex of ri, vi, pick a point, 40, not on this line segment which has a length larger than the distance from the L2 point to Vi and less than the length to L1. Establish this length as the Lp distance from q. to ai (see Fig. 6).
=
[IqO,
-
VjkIp
Uj
+
and tik is defined as
140,
Vjylpl”’
-
and dp(qO,
where again the necessary conditions for a minimum are simultaneous, nonlinear equations. No geometrical technique appears likely to solve the general Steiner point location problem- without using heuristics. There are basically two properties useful for locating L, Steiner points: (1) the deviation of the curve from the straight line connecting L, and the L2 Steiner points in ti; and (2) the acceleration of the Lp point as it travels between the L, and the Lz points. The deviation of the curve appears to be directly related to the degree to which the triangle ti is scalene and its orientation with respect to the origin of the Lp plane. Equilateral and isosceles triangles tend to have a locus of Steiner points that lie along the straight line connecting the L1 and L2 points depending on their orientation to the origin of the L, plane. The following lemma described the well-behavied nature of certain L, Steiner loci, when the triangle ti is isosceles.
+)
is from
q.
vk)
=
f(qOk,
-
vj,‘p
t
/qO,-
z)ky/p]l'pa
If we translate q. parallel to the line segment (vi, vi), the sum of the distances between Vj, Vkk, and q. will remain constant. This is so because the altitude of the triangle vi, qo, vk remains constant and so does the base of the triangle. As we translate qo+ qh clearly the distance between and vi reaches a minimum when qh is collinear with the bisector of (Vi, lik). This is so because the pythogorean theorem holds on this metric[20, 211. The distance between q/, and v, is: q6
dp(q&
Vi)= [lq&- Vi,lp + IqAy - QyPll’P.
If we construct a circle of radius (qo, vi) it will intersect the perpendicular bisector of (v,, vi) at a point q; different from qb. The distance between (q& vi) is less than the distance between (qg, vi) by the difference between the two points 4: and qh. Therefore, as is seen in the diagram, the sum of the distances: tdp(Vj,
< [dp(%
q@ q0)
+
dp(q& +
dp(qO,
Vi) + ui)
dp(q& +
dp(qO,
VP)1 ok)]
Thus, q;1 is the minimum point location where p ranges on the open interval (1,2). The locus of L, points would thus form a straight line between the L1 and L2 Steiner points. Q.E.D. The reader may argue that, if the isosceles triangle were rotated in the L, plane, then the above situation would, not hold. It is true that under rotation, the L, metric is not invariant; however, as the conditions of the
Steiner minimal trees and urban service networks
21
“k
LI
Fig. 6. Location of LP Steiner points in an isosceles triangle.
Lemma 1 state, if the L1 and L2 Steiner points are collinear with the apex of ri, then the same construction that was carried out in the proof would hold true. Figure 7 illustrates this for an isoceles triangle t which has been rotated an angle 135”to the origin. From the dratving the same method of proof could be used to show that the locus of Steiner points was indeed a straight line. Thus, for a special case of an isosceles triangle, the locus of L, Steiner points is a straight line. There appear to be no other properties of L,, triangles which exhibit this well-behaved property of the locus of Steiner points. Although various experiments have been conducted with the deviation from a straight line locus between the L2 and L1 Steiner points, no other substantive results have been obtained. However, Lemma 1 has actually been quite important even for scalene triangles, since it has yielded valuable evidence as to where the approximate locations of the L, Steiner points lie. The other property of major interest to us is the acceleration of the< &,. Steiner point as it moves between the L, and L2 end-points. From many experiments conducted with L, triangles, the acceleration of the L, point appears to vary logarithmically along the locus.
However, since the equation of the locus is different for each triangle, there is no general equation to guide us in defining the acceleration of the Steiner points along the locus. If there were a general equation, we could take second derivatives of this equation and find an expression for the acceleration of the Steiner point along the curve. Since each equation is different, we would have to generate each equation separately, and this is not a computationally efficient solution procedure. Given this behavior and the result of Lemma 1, we are led to a heuristic search procedure for locating the Steiner point in an L, triangle described in the next section. ALGORITHMFOR LOCATINGSTEINERPOINTSJ.NLp TRIANGLES
Step 1 Define the L1 and L2 Steiner points for the ti in Delaunay triangulation; Step 2. Compute the equation of the line passing through the L1 and the Lz Steiner points. Step 3. Employ a golden section search along the line defined
J. MANOR
SMITH and M. GROSS
Fig.7. Locationof Lp Steinerpointsin a rotatedisoscelestriangle. at S&p 2 till the min~um value results. The strictly convex nature of the objective function will guarantee a unique niinimum along this line of search. Call this minimumpoint L,” E IL,, L,]. Step 4 Define the equation of the line ~e~endicular to the line defined at Step 2, through LP . Defin$ e a bounded interval [u, b]. Step 5.
Search this new line through L,,’ until LT,is found; L$ is the point having the minimum objective function value; LT, is then the estimated Steiner point for the Lp triangle. Defining the L, and LZ points creates a bounded interval along which the golden-sectionor some other one dimensionalsearch technique can be utilized. Since our objective function is strictly convex, we will find a minimumalong the interval [L,, Lz]; and with, a suitably chosen t~eshold value, 8, a finite numberof iterationsin two searches will be necessary to converge to L$ Hence, we are able to maintaina constant time algorithm to locate the Steiner point in the L, triangle, thus preserving the O(n log n) complexity of our +,SMT algorithm.The procedure for locatingthe Voronol point,
V$ is carried out in a similar manner for the ti in Delaunay triangulation.We first define the L, and 15, Voronoi points of ti, then search for the minimumvalue along the straightline connectingthe L, and ~5,Voronoi points till we find, Vd)E [L,, L,]. Finally, we search perpendicularto the line (L,, L2) and through VP0to find VP*.
Although Lemma 1 does not directly apply to the process of locating I$,, a similar argument can easily show that the ~5~Voronoi point must, by its very definition,lie on a curve between the L, and Lz Voronoi points. Also,because of the nature of the Voronoipoint being the unique cir~umcenterof three nearest neighborpoints of a Delaunay triangle for LP E (L,, LZ), using a one dimensionalsearch technique can be used to estimate thelocationof V$. COMPLEXKY OF TKE L+WT ALGORITHM
Cons~ction of the ~launay ~ang~ation in Step 1 and its substeps takes O(n log n) time in the worst case on the L, metric, p E [I, 2][131.The key to its efficiency is the data structure which insures an O(n) time merge step. Even though the determination of the Voronoi points is approximate,a constant time procedure is used,
Steinerminimal trees and urban service networks thus, the overall worst-case running time for the L, Delaunaytria~lation is stilt O(n log 8). The ~ncatenation process which serves to guide the construction of the SMT from the indi~dual Steiner point in each Steiner triangle can be done in O(n log n) time. Step 2, construction of the MST for a planar graph takes, U(n) time[22].Constructionof the priority queue Q takes O(logn) time. Creation of the list of nearest triangle neighborsfor Step 3 can be done in O(n) time. The disjoint-set-unionprocedure used to piece together the forest of Steiner trees into a single Steiner tree for the point set is linear in the number of edges of the Delaunay triangulation[23].In the worst case, ail the triangles of the Delanunay t~a~lation would be in Q. Since the number of Delaunay triangles is 2(n-l)c, where c equals the number of vertices on the perimeter of the Delanunay ~a~ation[l2], the overall concatenationprocess is O(nlog n). The total amountof storage required is proportionalto the numberof edges in the Delaunaytriangulation.Since there are exactly 3(n-1)-c edges in the triangulation[l2], the total amount of storage is O(n). Finally,to concludethe algorithmsection of this paper on the LJMT problem, let us briefly examine a technique for determi~ng an empiricalvalue of the expected LP distancemetric parameterp. This proposed technique appears to be partic~~ly appropriate for developinga new network design in instances when the actual links interconnecting q in V are not yet constructed. A method is already available for evaluatingthe most appropriate hP distance function on an existinginterurban network, m&ding an estimate for the value of parameter p[lO], The tech~que which follows was primarily designed for use in intra-~ban network design, especially new network construction, and is relatively straightfo~ard and ~mputationalIy e~cient. First let us assume that the nodes to be interconnected in the network have x-y coordinates in R2. Then the following iterative procedure Unix a shortest path algorithmand regressionanalysisto empiricallyestimate p iS CaFFied Out:
end;
29
Step 2.0:Given SUM(p)for all p valuesp E [l, 21,define
the regressioncurve Y = ~sum~)].
Step 3.0:Compute the empirical matrix of all shortest begin;
paths for the given point set. Step 3.1: DefineD’[i, JI= empiricalmatrixof all
shortest paths Step 3.2:Computethe empiricalsum of all shortest paths:
end;
Step 4.0: Given the regressioncurve Y and SUM (e), compute 4.
end;
In what follows,we will empiricallydeterminean LP metric Valuefor four different Ameri~n CitkS by selectingat random 1.5points in the plane and utilizing the algorithmjust described to determine the Lp value for these 15points. We feel that it is very importantnot to assume a priori, an Lp value because as the experiments that follow demonstrate,such assumptionscan be extremely inaccurate.We will then take one of the cities and apply the L,SMT algorithm to determine its suboptimal network ~onfi~ation. Four cities, considered typical U.S. cities, were chosen for empirical testing of the previous estimation procedure and L,SMT algo~thm,The cities were chosen for their diverse respective topo~aphies and street/highway networks, and included Philadelphia, Pennsylvaniain the Northeast (area 129sq, miles,population 1,9.50,000), Washington,D.C., in the Mid-Eastern Seaboard (area 62 sq, miles, population 760,000),Lexington, Kentucky in the Southeast-Central(area 68sq. miles, population 110,~~, and Portland, Oregon in.the Northwest (area 89 sq, milespopulation3~,~). The distance matrix for each of the four cities were prepared in the following way. A rectangle coordinate system was definedon a 1:24,000scale (1” equals 2900’) map of the city with the origin at the south-westcorner. The maps used were the standard 7g quandrangle Step 1.1: far p = 2.0 Step INC do: llINC is a topo~aphi~ maps publishedby the U.S. Geolo~cal Surdecrement parameter, e.g, INC = vey. Next, fifteen points were randomly selected on the -O.lO~/ street/highway network. The X and Y coordinates of’ S&p 1.2:Computethe distancematrix,giventhe each of the points were determinedby pickinga pair of (x, y) coordinatesof the point set. numbersfrom a random number table and scalingthem Step 1.3:Computethe set of multi-terminalshorto fit the X and Y range of each city respectively. If a test paths between all pairs of points point fell outside the city boundaries,or was placed more us~g~oyd’sor~ant~ig’salgorithm[24]. than l/4 of an inch from the closest street, it was This requires O(N’)time in the worst disregardedand a new pair of randomnumberspicked.If case/l a point fell within l/4 of an inch of a street it was M[i, JI= matrix of all shortest paths repositioned on the street and its coordinates adjusted Step 1.3’(options): For a new network design a~cordi~ly, It is impo~ant to stress that these uniformly dis~ibuted points be drawn within a bounded with no underIy~ trafficnetwork,use the edge lengthsof the Delaunaytrian- region[26]. The distancesbetween the points were measuredwith gulationto compute M[i, Jl. a distance measurer (opisometer). The shortest route Step 1.4:Compute the sum of all shortest paths between each pair of points was determined by in//This is sometimesreferred to as the spection. In those instances where the shortest route “aspersion” value[25]1/ could not readily be determined,a numberof alternative routes were measuredto find the shortest one. (It should SUM (p) = T T dP& vi) be noted that the distance.matrix thus derived assumes d&nCe
30
J. MACGREGOR SMITHand M. GROSS
free flowing conditions on the street/highway network without considering factors such as capacity constraints, congestion, and speed, which usually affect route selection.) The first city examined was Lexington, Kentucky. Based on our algorithm for estimating the L, parameter and with the chosen random points, we computed an L, value of p = 1.029, see Fig. 8. For each of the other cities, Philadelphia, Portland, and Washington, we computed L, values of 1.176, 1.100, and 1.2670respectively, see also Figs. 9-11. Certainly we could collect data on additional random points to compute an expected L, value across the entire urban region but this was felt to
Cl-lYUMll3
be unnescessary for the purposes of this present paper. In view of the applications of our algorithm, we seek the expected L, metric for a given subset of points within the urban region from which we can correctly compute the L, Steiner network for that given point set, not for all possible points in R2 of the urban region. At this stage, let’s take Lexington, Kentucky as an example and show, after we established its L, value, what the L,SMT solution would have been if we had assumed an Euclidean distance metric for the fifteen random points. After demonstrating this, we will then show the L,SMT solutions for the L, metric in which the L, value for Lexington is most closely associated.
----
Fig. 8. Lexington, Kentucky.
---_
Fig. 9. Philadelphia, Pennsylvania.
Steiner minimal trees and urban service networks
Fig. 10. Washington, D.C.
Fig. Il. Portland, Oregon.
32
J. MACGREGOR SMITH and M. GROSS
LEXINGTON .LzANALYSIS Figure 12 illustrates the fifteen random points together with the L, metric Voronoi diagram for the given points. Figure 13 illustrates the L, Delaunay triangulation for the given points which also contains the Minimum Spanning Tree (MST) as a subgraph of its edge set. Figure 14 illustrates the MST together with a suboptimal Steiner Minimal Tree (SMT) solution computed by our previously described algorithm. We will indicate subopjimal SMT’s by a caret above the expression, i.e. SMT. The abandoned edges of the MST are indicated by the dotted lines in the diagram and the three darkened circles represent Steiner points. The overall length of the MST solution’i; 62,136ft or 11.768 miles. The overall length of the SMT solution is 60,832ft or 11.521miles, a
savings of approximately 2.11%. This overall savings is within the expected value of between 2 and 4% as shown by [U] for uniformly distributed points in the plane. It should also be pointed out that the solution times for computing the SMT solution were approx. 0.20 seconds on a CYBER 175 housed at the University of Massachusetts, Amherst Campus. Thus, the algorithm is very fast as expected. Now given these results for the Lz metric, let’s examine the L, metric solution. LEXINGTON L, ANALYSIS Figure 15 illustrates the L, Voronoi diagram for the given point set wherein the given points where translated and rotated about the origin in order for the orthogonal Pattern of the streets to coincide with the orthogonal
Fig. 12.Lz VoronoidiagramLexington,Kentucky.
Steiner minimal trees and urban service networks
33
Fig.13.Lz Delaunay triangulation Lexington, Kentucky. pattern of the Cartesiansystem. Without this translation, the L, SMT solutionwould ,beawkwardto interpret. As was stated earlier, the Lz SMT solution is unaffected by the location of the origin or the orthogonality of the coordinate axes. Figure 16 depicts the L, ~launay triangulationfor the fifteen random points. The rectilinear metric serves as an appropriate approximation to our eqpirical 4 value of 1.029.Figure 17represents the L, SMT solution computed according to our algorithm.In this same figure are indicated the abandonedMST edges and the six Steiner points by the darkened circles. The overall Iength of jhe MST solution is 76,360ft or 14.46 mileswhilethe SMT solu$on is 71,800ftor approx. 13.59 miles.The ratio of the SMT/MSTrepresents a savingsof 5.972%.‘I&isis a larger reduction than that achieved by the LISMT solution but is perhaps not as large as expected for L,SMT solutions since they often range between 8 and 10%[17,281.Nevertheless, the solution
represents a potentiallysignificantcost savingswhen one accounts for the cost per mile of a given network. The paper has introduced the reader to some of the curious and fascinating features of L, bisectors, triangles, and Steiner point location problems and their potential impacton urban service network problems.We have demonstratedthe complexityof findingLp Veronoi and Steiner points loci within triangles and have shown that approximationtechniques are necessary if efficient running time aIgorithms are to be achieved for ‘the LpSMT problemfor large point sets. In the generalcase, one may questionthe configuration of the links when the L, value is not closely approximated by the L1 or L2 metrics. We argue that the optimal confi~ation of the links would closely follow the street pattern or else the edges of the Lp Delaunay
J. MACGREGOR SMITH and M. GROSS
Fig. 14. I&iT
solution Lexington, Kentucky.
Steiner minimal trees and urban service networks
35
J. MACGREGOR SMITH and M. GROSS
31
Steiner minimal trees and urban service networks
Fig. 17. L, SMT solution Lexington, Kentucky.
triangulation as an approximation to the ~5, curves which interconnect the L, Steiner points and the given point set V. This is arguable given that the street network segments are piecewise approximations to the Lp curve segments which generated the Lp value in the first place. Nevertheless, even in the Lexington, Kentucky example, the L, value does not exactly fit the underlying street pattern although the empirical value is very close to the L, metric value. Modifications would be necessary for the final location of the L, Steiner points and L,, link segments. Finally, we have not presented bounds for our algorithm simply because there is no known bound on the ratio of the lengths of heuristic Steiner minimal trees and optimal Steiner trees, either for the L1 or L2 metrics[9, 281 nor for those metrics ranging between these two benchmark metrics. This is a question for further research.
REFERENCES
S. H. Putnam, Urban land use and transportation models: A state-of-the-art summary. Transpn.Rex 9, 187-202(1975). J. Berechman, Interfacing the urban land-use activity system and the transportation system. 1. Reg. Sci. 16, 183-194 (1976). Real Estate Resarrch Corporation (RERC), The Cost of Sprawl. Prepared for he Council of Environmental Quality; The Office of Policy Development and Research, Department of Housing and Urban Development; and the Office of Planning and Management Environmental Protection Agency, U.S. Government Printing Office, Washington, DC. (1974). C. Werner. The role of topology and geometry in optimal network design. Papers of the Reg. Sci Assoc. 21, 173-189 (1%8). C. Werner, Networks of minimum length. Con. Geographer 23,47-69 (1%9). Z. A. Melzak, On the problem of Steiner. Can. Math. Bull. 4, 143-148(l%l).
38
J. MACGREGOR SMITH and M. GROSS
I. Z. A. Melzak, Companion to Concrete Maihema~jcs.Wiley, New York (1973). 8. Z. A. Melzak, Mathematical ideas, Modeling and App~ica~ions.Wiley, New York (1976). 9. F. K. Hlang, The rectilinear Steiner tree problem. J. Design Automation and Fault Tolerance Anal. 2,303-310 (1978). 10. R. F. Love and J. G. Morris, Modelling inter-city road distances by mathematical functions. Op. Res. Quart. 23, 61-71 (1972). 11. .I.G. Morris and W. M. A. Verdini, Minisum1, distance location problems solved via a perturbed problem and Weiszfeld’s algorithm. Op. Res. 27, 1180-l188(1979). 12. D. T. Lee, On finding k nearest nei~bors in the plane. Techn. Rep., apartment of Computer Science, University of Illinois (1976). 13. D. T. Lee, Two dimensional Voronoi diagrams in the Lpmetric. JA%M 27 604-618(1980). 14. M. I. Sham& Comnutational eeometrv. Unuublished Ph.D. _ * Thesis, Yale University (1978).1.5.J. B. Kruskal, On the shortest spanning subtree of a graph, Proc. Am. Math. Sot. 7,48-50 (1956). 16. J. MacGregor Smith, Algorithms for generalized Steiner network orobjems. Ph.D. Thesis. Deo~tment of Mechanical and Ind~st~al Enginee~ng, University of Iflinois, Urbana Campus (1978),~published. 17. J. MacGregor Smith, D. T. Lee and J. S. Liebman, An O(N log N) heuristic algorithm for the rectilinear Steiner minimal tree problem. Enpg Optimization 4,179-192 (1980). 18. J. MacGregor Smith, D. T. Lee and J. S. Liebman, An O(N log N) heuristic for Steiner minimal trees on the eucli-
dean metric. Networks 11; 23-39 (1981). 19. D. T. Lee and C. K. Wong, V~ron~i diagrams in I&,,) metrics with 2~imensiona~ storage annlications. SIAM ’ J. Comput. 19(I), 200-Z 1 (1980). _‘ 20. M. Blumenthal, Theory and Applicafions of Distance Geometn? Oxford Universitv Press. London 119531. 21. J. E. Valentine, Angles in metric spaces. The-biometry of Metric and Linear Spaces: Proc. Conf. Held at Michigan State Uniuersity,East Lansing, June 17-19, 19%. SprinierVerlag, Berlin, pp. 68-73 (1974). 22‘ D. R. Cheriton and R. E. Tarian. Finding minimum scanning L trees. SIAMJ. Compuf. 5,7%-i42 (1976). 23. A. H. Aho, J. E. Hopcrift and 3. D. Ullman, The Design and Analysis of Computer A~go~fhms. Addiso~Wesley, Reading, Mass. (1974). 24. E. Minieka, Optimization Algorithms for Networks and Graphs. Marcel Dekker, New York (1978). 25. K. J. Kansky, Structure of TransportationNetworks: Relationships Between Network Geometry and Regional Characteristics. The University of Chicago, Chicago. Illinois, Department of Geography, Research Paper No. 84 (1963). 26. J. L. Bentley and M. I. Shamos, Divide and conquer for dines-expected time. Info. Prac. Lett. 7,87-91 (1978). 27. E. N. Gilbert and H. 0. Poliak, Steiner minimal trees. SIAM J. Appi. Math. 16, l-29 (1968). 28. F. K. Hwang, An O(N log N) algorithm for suboptimal rectilinear Steiner trees. IEEE Circuits and Systems CAS-26(l) (1979).
many factors related to the objectives of the networkand to the physical characteristicsof the nodes and the links. However, a basic criterion in the selection of an appropriate design for almost all systems is the minimization of totai cost which is directly related, in part, to the spatialconfigurationof the network. The ~rnportantfactors determini~ service network costs are the construction costs which generally depend on the length of the system, and the movement of flow costs which are a function of both the length of the system and the magnitudeof the flow through it[4, 51. These cost characteristicsdictate two extreme solutions to the probIem of network spatial con~ration with the objective of minimizingtotal transportation costs. If only ffow costs are minim~ed then each Row demandis solved by a straightline connectingoriginand destination. This solution implies build~g the network with the longest path and therefore the highest const~ction costs. If, on the other hand, the flow costs are ignored and only construction costs are considered and mi~mized, then the solutionis the one which minimizes total length. In the design of a transit system, e.g. this dichotomy entails the resolution of the conflict between construction economy, i.e. a minimumlength network, and traveler convenience, i.e. minimum travei time (requiringdirect links between all origin and destination nodes). The procedure of designingthe opt~al network in most instances involves first finding the minimum length solution, and then subject to various constraint (i.e. budget), improve this network (i.e. add links) so as to optimize a given objective. Thus the first step can be expressed as follows: given a network select the minimum spanningtree (MST) which is the subset of links that minimizesthe sum of the shortest routes betweenall nodes. In the paper that follows, we focus on the first problem of the optimal network problem. The .initial problem is crucial to the overall network designproblem for many of the previously discussed applications.The problem we address in the paper is often referred to as the Steiner tree problem[~9]. In the followingpaper, we develop an efficient O(n log n) heuristic for the generalized Steiner tree problem in the plane R2. Our solutionalgorithmis not restricted to just the Euclidean metric but also can be used for network designproblems on the rectilinear metric and those metrics in between.
Transportation,utility and communicationnetworks are conspicuousfeatures of man’s uses of the land. ‘Roads, streets and highways, railways, waterways and airroutes, and utility, electric and communication lines permeate our landscapewith increasingfrequency. The design and conflation of transpo~tion, utility and communicationnetworks are of critical concern to urban and regional planners because of their potential long-termimpact on the region. It is commonknowledge that strong and direct relationshipsexist between transportationand utility systemsand patterns of land use. For example,Putnam[l], Berecham[2]and others show how the metropolitan transportation system affects the intensity and spatial dis~ibution of land use activities, which in turn, influencesthe size, location, and utilization of future transportationfacilities. Similarly,the impact of sewer and water lines on urban sprawl is also well known and documented [3]. More importantly,the changes in spatial relations between land uses directly affect other economic and en~ronmental factors such as energy consumption, air and water pollution, and indirectly, all facets of humanactivity in the region. More immediateand direct concern to the planner is the cost of constructing and operating the service network. This cost in turn has a direct relationship to the network’scon~g~ation or spatiallayout. The general purpose of a transportation, utility, or communicationnetwork is to satisfy the demands for se&ice. Graphicallythe network is represented by a set of origin nodes interlinkedto a set of destinationnodes. The node-linkinterpretationand the flow units, or commodities transferred, are a function of the network in question. For example, in an urban traffic system the nodes are pop~a~on centroids, the links are freeways, arterialsand streets, atid the flowunits are commuters;in a network designed for solid waste disposal, the nodes are the collection, treatment, and discharge points, the links are pipelines and the flow unit is solid waste; and in a communicati& network the nodes are message centers, the links are t~nsmission lines and the flow units are messages. The design of a particular service nework involves tDept.of Industrial E~ineering and Operations Research. $De.pt.of Landscape Architecture and Regional Planning. SEPS Vol. 16. No. 1-D
21
22
J. MAC~R~~R
SMITH and
This generalized Steiner network aIgorithmis desirable because the distance between randomlylocated points in an urban region often requires a distance metric whichis in between the Euclidean and rectilinear metrics. After we complete the development of our algorithm, we examine the ern~~ic~ distance for random points in the plane of four di-tferentAmerican cities and then apply our ~gorithm to one city in p~tic~ar to demonstratethe a~orithm’s speed and effectiveness. As a finalprefatory remark, the ultimaterealizationof the optima1network con~guration will necessarily be altered due to lo&alinthrences:zoning, easement rig&, physical obstacles, social and economic concerns. The algo~t~s we present, however, should serve as valuable guidelinesduring the network design process, and are not meant as replacements of existing design practices.
If we are given a finite set of vertices, also called points, v = (v,, 02,. * ., vn] in the plane R2 withCartesian coordinates (Xi,yi) for (i = 1, 2, . . ., n) where ra2 3, we say that a Steiner -MinimalTree @MT)for the n given points is a ne~ork that interconnects all n points with minimumtotal length. In order to achieve this minimal Iength, additional vertices (Steiner points) somet~es must be added to the network. We will.assume that the distance function is an LP metric[8, IO].in which the distance between any two points Uiand vi in the plane R2 where p E [l, 23is given by
M.GROSS
Another sub-optimalalgorithmfor locating LP Steiner points in the plane is Morrisand Verdini’s[l 11.However, their algorithmis only for locating LP points and not for the L$MT problem nor do they present worst case results for the computational complexity of their algori~m. Before e~borating on the details of our algorithm, let us posit some importantd~~nitions[l&-14]. Bisector
The locus of points between two given vertices ai and vj which is equidistantbetween them:
This is the locus of points closer to 2rIthan to any other Vi:
A polygonal region that encompasses the locus of points closer to oi than to any other point in V.Formally defined as the intersection of half-planesdeterminedby the bisectors of z+ and all other points in V* often denoted as VP(v,):
The collection of Voronoipolygons VP(Q) for each vi in V.Often denoted as VLy 8,
Whenp = 2, we have the familarEuclideanmetric; when V~runoipoint Each vertex in the Voronoidiagramis a Voronoipoint. p = 1, the well-known rectilinear (Ma~hat~n) metric. Each Voronoipoint is the circumcenter of at most three When p E [0, 11,the distance function no longer remains convex. When p E [2, co], its meaningfulnessto real given vertices in R2. The degenerate case of four or worid.applicationsbecomes questionable.When p E [I, more cocircular points is excluded. 21,tt ts a weIi-behavedconvex func~on~l~].The axioms normallyes~bI~shedfor a distancemetric all hold for the The planar straight line graph which is dual to the LP metric p E ]1,2]. Since the construction cost of the L,SMT network Voronoi diagramis a t~an~iation, often referred to as solutinncan be assumedto vary linearly with the length the Delaunaytriangulationand denoted as DT( V). Examples of a Voronoi diagram and its Delaunay of the overall network, the objective function for the t~anguIation for a given set of points and the Euclidean problem is: metric appear in Figs. I and 2 respectively.
As far as is known,little research has been ~o~d~ted on the L,SMT network designproblem. One oPthe first persons to mentionthe L,SMT problem was Mel~k[8]* L, space is a particular subclass of ~inkows~ space. Melzak only briefly examines the L,SMT problem, and tfm proceeds to co~cen~ate his attention on the posz sibility of solving Steiner trees in ~~~kowski space. Since distancein Minkowskispace behaves differentlyin differentdirections(Minkowskispace has a non-isotropic metric) it seems to have Iittle practical use for our Minimize2 = purposes. ifi The underlyi~ algorithmfor cons~ct~g SMT’s on the &, metric operates in a manner similar to then the necessary and su~cient conditions for a mini- Kruskal’s1151~i~urn SpanningTree (MST)algorithm, mum are: However, in our case, we establish a priority queue of Delaunay triangles sorted on their reduction ratio (SAGEST) for each triangle, and, then, systematic~y where (Xi,yi) = the Cartesian coordinates of the Steiner points (i = 1, 2, . . ., m); {Xj, Xi) = the Cartesian coordinatesof the givenvertices in R2(j= 1,2.. , ., n); S = the set of Steiner points; and V = the given set of vertices. Although the objective function is convex, it is not di~er~ntiable.One can transform the objective function so that it is differentiableas follows:
Steiner minimal trees and urban service networks
Fig. 1. L2 Voronoi diagram for N = 18 (after [12]).
Fig. 2. & Delaunay ~ia~ulation for N = 18 (after [12lf.
23
24
J. MACGREGORSMITHand M. GROSS
construct the SMT by a fast disjoint set-union procedure. The reason for this is that each triangle has a potential disjoint SMT in the forest of SMT’s of V. Constructing the Delaunay triangulation assists us in defining the triangles, and information from the Voronoi diagram assists us in determining how to concatenate these triangles into a SMT for V. Before we delve into the details of L, bisectors and L, Steiner triangles, let’s present a systematic overview of the steps of the O(n log n) L,SMT algorithm.
Step 1. (Delaunay triangulation) Presort all the vertices of V by their x-coordinate. Step 1.1. Divide the set V into two disjoint sets L and R by way of the sorted list of x-coordinates. Step 1.2. Recursively construct the Delaunay triangulation of sets L and R; call the triangulation DI and 4 respectively; Step 1.3. Construct, M, the merge curve, equidistant from a point in D1 and 0,. Unify D1 and D, through the merge curve as it threads a zig-zag path between D, and Dr; the L; bisectors play a crucial role in determining the merge edges; Step 2. (Concatenation process) Compute the Minimum Spanning Tree (MST) of the Delaunay Triangulation; mark those triangles which share two edges of the MST and place these doubly marked triangles in a priority queue called Q; Step 3. Where possible, concatenate pairs of triangles (t;, tj); if
this is not possible, add the singleton triangle ti with its Steiner point and edges to the SMT of V Step 4. End the concatenation process when the priority queue, Q, is empty. The general algorithmic process of the L,SMT problem is very similar to that ofthe Euclidean and Rectilinear algorithms discussed elsewhere[l&lS]. What is unique, however, to the L,SMT problem is the generation of L,, bisector, the location of Voronoi points in L, triangles, and, finally, the location of Steiner points in L, triangles. We shall spend some time on L, bisectors since they are crucial to the merge process of the Delaunay triangulation and eventually the SMT configuration. As an example of the peculiar qualities the L, metric p E [l, 21 has on the locus of Steiner points within a triangle, Fig. 3(a) and 3(b) illustrate how the locus of Steiner points is in general a curve and how the curve changes under a rotation of the coordinate axes. As we shall demonstrate in the applications section of the paper, the location of the origin from which all the empirical distance measures are made becomes an optimization problem in its own right since the orientation of the point set to the L,, metric affects the overall Steiner network length interconnecting V. LpBISECTORS In order to demonstrate the behavior of the L, bisector it is useful to examine an example in R’. Assume that two points Ui and aj are located at (0, 0) and (1, 2)
V3:(2 236, 01
Fig. 3. LJMT
Steiner loci under different orientations.
Steiner minims trees and urban service networks
respectively. Then, the equation of the perpendicular bisector for z is given by the following,where z, and z, are the respective n and y-coordinates of a point on the perpendicularbisector:
25
As one may have surmisedfrom the previous equations, the graph of the L,, bisector function is a curve for p values (1, w).The L, metric is defined as follows:
(fzx-0Jp + 12,-~~))i’P = (/l-&/p + /Z-2$)“? Because of the peculiar nature of the L, bisector p ~(1, 2), if we try to solve exactly for the Voronoi points (the circumcentersof the Delaunaytriangles),we will destroy the O(n log n) behavior of the L,SMT airIz$ f l2,1”3= [I1- ZXl” + 12- &)“I. gorithm. Therefore, we must resort to a heuristic proThis, however, is not sufficientto obtain a solutiondue cedure for determiningthe Voronoi points in this situato the presence of the absolute values of the terms. The tion in order to preserve our O(n log ra)~go~thrn. This foflowi~ algebraic manip~ation may be used to eli- means that the merge process must be designed so that it is sufficientlyrobust to handle near-optimallocations minate the absolute values: of Voronoi points. It is also apparent that heuristic solutions for the Steiner points in the LP triangles must be ((zX)2)p’2 f ((2,)“)“‘”= ((1- zX)2)p’2 t ((2- z$)P’2. used. In fact, the heuristicalgorithmwe will use to locate Whileremovingthe absolute value signswillallowone the Voronoi points parallels the L,SMT Steiner point to solve for the correct values numeric~ly, the equation location procedure. Therefore, before actually presenting the heuristic for locating Voronoi points, let us itself cannot be further simplified. We do know that the bisector z will pass through the examinesomepropertiesof the locus of Steinerpointsfor midpoint of the line segment connecting (U, ui)[19]. L, triangles. Figure 5 describes the locus of Steiner points for a Figure 4 illustrates the graph of the ~5, bisector for different valuesof the L, metric for two given point sets, triangle ti where p was varied over the open interval (1, Raisingboth sides to the power p, we have:
Fig.4. Lp bisectors.
26
J. MACGREGORSMITHand M. GROSS
Fig. 5. LP locus for a triangle(where1 s p c 2). 2). As expected, the locus of points follows a curve;the general equation for this curve is given by the following:, Minimize 2 = {[lx1- vi,” ’ lx1 - oiklp s]xI
- VtJP + IYI - QyIp + IYl - ujylp •t (YI -
~kylpll’p~
Now the distance follows: dp(qO,
Lemma 1 If we have an isosceles triangle ri with its L, and Lz Steiner points collinear with the apex of the triangle, then the locus of Steiner points for the L, metric, p E (1, 21, is a straight line between the L1 and L2 Steiner points Of ti. Proof Given an isosceles triangle ti with a line collinear with the L1 and the t, points passing through the apex of ri, vi, pick a point, 40, not on this line segment which has a length larger than the distance from the L2 point to Vi and less than the length to L1. Establish this length as the Lp distance from q. to ai (see Fig. 6).
=
[IqO,
-
VjkIp
Uj
+
and tik is defined as
140,
Vjylpl”’
-
and dp(qO,
where again the necessary conditions for a minimum are simultaneous, nonlinear equations. No geometrical technique appears likely to solve the general Steiner point location problem- without using heuristics. There are basically two properties useful for locating L, Steiner points: (1) the deviation of the curve from the straight line connecting L, and the L2 Steiner points in ti; and (2) the acceleration of the Lp point as it travels between the L, and the Lz points. The deviation of the curve appears to be directly related to the degree to which the triangle ti is scalene and its orientation with respect to the origin of the Lp plane. Equilateral and isosceles triangles tend to have a locus of Steiner points that lie along the straight line connecting the L1 and L2 points depending on their orientation to the origin of the L, plane. The following lemma described the well-behavied nature of certain L, Steiner loci, when the triangle ti is isosceles.
+)
is from
q.
vk)
=
f(qOk,
-
vj,‘p
t
/qO,-
z)ky/p]l'pa
If we translate q. parallel to the line segment (vi, vi), the sum of the distances between Vj, Vkk, and q. will remain constant. This is so because the altitude of the triangle vi, qo, vk remains constant and so does the base of the triangle. As we translate qo+ qh clearly the distance between and vi reaches a minimum when qh is collinear with the bisector of (Vi, lik). This is so because the pythogorean theorem holds on this metric[20, 211. The distance between q/, and v, is: q6
dp(q&
Vi)= [lq&- Vi,lp + IqAy - QyPll’P.
If we construct a circle of radius (qo, vi) it will intersect the perpendicular bisector of (v,, vi) at a point q; different from qb. The distance between (q& vi) is less than the distance between (qg, vi) by the difference between the two points 4: and qh. Therefore, as is seen in the diagram, the sum of the distances: tdp(Vj,
< [dp(%
q@ q0)
+
dp(q& +
dp(qO,
Vi) + ui)
dp(q& +
dp(qO,
VP)1 ok)]
Thus, q;1 is the minimum point location where p ranges on the open interval (1,2). The locus of L, points would thus form a straight line between the L1 and L2 Steiner points. Q.E.D. The reader may argue that, if the isosceles triangle were rotated in the L, plane, then the above situation would, not hold. It is true that under rotation, the L, metric is not invariant; however, as the conditions of the
Steiner minimal trees and urban service networks
21
“k
LI
Fig. 6. Location of LP Steiner points in an isosceles triangle.
Lemma 1 state, if the L1 and L2 Steiner points are collinear with the apex of ri, then the same construction that was carried out in the proof would hold true. Figure 7 illustrates this for an isoceles triangle t which has been rotated an angle 135”to the origin. From the dratving the same method of proof could be used to show that the locus of Steiner points was indeed a straight line. Thus, for a special case of an isosceles triangle, the locus of L, Steiner points is a straight line. There appear to be no other properties of L,, triangles which exhibit this well-behaved property of the locus of Steiner points. Although various experiments have been conducted with the deviation from a straight line locus between the L2 and L1 Steiner points, no other substantive results have been obtained. However, Lemma 1 has actually been quite important even for scalene triangles, since it has yielded valuable evidence as to where the approximate locations of the L, Steiner points lie. The other property of major interest to us is the acceleration of the< &,. Steiner point as it moves between the L, and L2 end-points. From many experiments conducted with L, triangles, the acceleration of the L, point appears to vary logarithmically along the locus.
However, since the equation of the locus is different for each triangle, there is no general equation to guide us in defining the acceleration of the Steiner points along the locus. If there were a general equation, we could take second derivatives of this equation and find an expression for the acceleration of the Steiner point along the curve. Since each equation is different, we would have to generate each equation separately, and this is not a computationally efficient solution procedure. Given this behavior and the result of Lemma 1, we are led to a heuristic search procedure for locating the Steiner point in an L, triangle described in the next section. ALGORITHMFOR LOCATINGSTEINERPOINTSJ.NLp TRIANGLES
Step 1 Define the L1 and L2 Steiner points for the ti in Delaunay triangulation; Step 2. Compute the equation of the line passing through the L1 and the Lz Steiner points. Step 3. Employ a golden section search along the line defined
J. MANOR
SMITH and M. GROSS
Fig.7. Locationof Lp Steinerpointsin a rotatedisoscelestriangle. at S&p 2 till the min~um value results. The strictly convex nature of the objective function will guarantee a unique niinimum along this line of search. Call this minimumpoint L,” E IL,, L,]. Step 4 Define the equation of the line ~e~endicular to the line defined at Step 2, through LP . Defin$ e a bounded interval [u, b]. Step 5.
Search this new line through L,,’ until LT,is found; L$ is the point having the minimum objective function value; LT, is then the estimated Steiner point for the Lp triangle. Defining the L, and LZ points creates a bounded interval along which the golden-sectionor some other one dimensionalsearch technique can be utilized. Since our objective function is strictly convex, we will find a minimumalong the interval [L,, Lz]; and with, a suitably chosen t~eshold value, 8, a finite numberof iterationsin two searches will be necessary to converge to L$ Hence, we are able to maintaina constant time algorithm to locate the Steiner point in the L, triangle, thus preserving the O(n log n) complexity of our +,SMT algorithm.The procedure for locatingthe Voronol point,
V$ is carried out in a similar manner for the ti in Delaunay triangulation.We first define the L, and 15, Voronoi points of ti, then search for the minimumvalue along the straightline connectingthe L, and ~5,Voronoi points till we find, Vd)E [L,, L,]. Finally, we search perpendicularto the line (L,, L2) and through VP0to find VP*.
Although Lemma 1 does not directly apply to the process of locating I$,, a similar argument can easily show that the ~5~Voronoi point must, by its very definition,lie on a curve between the L, and Lz Voronoi points. Also,because of the nature of the Voronoipoint being the unique cir~umcenterof three nearest neighborpoints of a Delaunay triangle for LP E (L,, LZ), using a one dimensionalsearch technique can be used to estimate thelocationof V$. COMPLEXKY OF TKE L+WT ALGORITHM
Cons~ction of the ~launay ~ang~ation in Step 1 and its substeps takes O(n log n) time in the worst case on the L, metric, p E [I, 2][131.The key to its efficiency is the data structure which insures an O(n) time merge step. Even though the determination of the Voronoi points is approximate,a constant time procedure is used,
Steinerminimal trees and urban service networks thus, the overall worst-case running time for the L, Delaunaytria~lation is stilt O(n log 8). The ~ncatenation process which serves to guide the construction of the SMT from the indi~dual Steiner point in each Steiner triangle can be done in O(n log n) time. Step 2, construction of the MST for a planar graph takes, U(n) time[22].Constructionof the priority queue Q takes O(logn) time. Creation of the list of nearest triangle neighborsfor Step 3 can be done in O(n) time. The disjoint-set-unionprocedure used to piece together the forest of Steiner trees into a single Steiner tree for the point set is linear in the number of edges of the Delaunay triangulation[23].In the worst case, ail the triangles of the Delanunay t~a~lation would be in Q. Since the number of Delaunay triangles is 2(n-l)c, where c equals the number of vertices on the perimeter of the Delanunay ~a~ation[l2], the overall concatenationprocess is O(nlog n). The total amountof storage required is proportionalto the numberof edges in the Delaunaytriangulation.Since there are exactly 3(n-1)-c edges in the triangulation[l2], the total amount of storage is O(n). Finally,to concludethe algorithmsection of this paper on the LJMT problem, let us briefly examine a technique for determi~ng an empiricalvalue of the expected LP distancemetric parameterp. This proposed technique appears to be partic~~ly appropriate for developinga new network design in instances when the actual links interconnecting q in V are not yet constructed. A method is already available for evaluatingthe most appropriate hP distance function on an existinginterurban network, m&ding an estimate for the value of parameter p[lO], The tech~que which follows was primarily designed for use in intra-~ban network design, especially new network construction, and is relatively straightfo~ard and ~mputationalIy e~cient. First let us assume that the nodes to be interconnected in the network have x-y coordinates in R2. Then the following iterative procedure Unix a shortest path algorithmand regressionanalysisto empiricallyestimate p iS CaFFied Out:
end;
29
Step 2.0:Given SUM(p)for all p valuesp E [l, 21,define
the regressioncurve Y = ~sum~)].
Step 3.0:Compute the empirical matrix of all shortest begin;
paths for the given point set. Step 3.1: DefineD’[i, JI= empiricalmatrixof all
shortest paths Step 3.2:Computethe empiricalsum of all shortest paths:
end;
Step 4.0: Given the regressioncurve Y and SUM (e), compute 4.
end;
In what follows,we will empiricallydeterminean LP metric Valuefor four different Ameri~n CitkS by selectingat random 1.5points in the plane and utilizing the algorithmjust described to determine the Lp value for these 15points. We feel that it is very importantnot to assume a priori, an Lp value because as the experiments that follow demonstrate,such assumptionscan be extremely inaccurate.We will then take one of the cities and apply the L,SMT algorithm to determine its suboptimal network ~onfi~ation. Four cities, considered typical U.S. cities, were chosen for empirical testing of the previous estimation procedure and L,SMT algo~thm,The cities were chosen for their diverse respective topo~aphies and street/highway networks, and included Philadelphia, Pennsylvaniain the Northeast (area 129sq, miles,population 1,9.50,000), Washington,D.C., in the Mid-Eastern Seaboard (area 62 sq, miles, population 760,000),Lexington, Kentucky in the Southeast-Central(area 68sq. miles, population 110,~~, and Portland, Oregon in.the Northwest (area 89 sq, milespopulation3~,~). The distance matrix for each of the four cities were prepared in the following way. A rectangle coordinate system was definedon a 1:24,000scale (1” equals 2900’) map of the city with the origin at the south-westcorner. The maps used were the standard 7g quandrangle Step 1.1: far p = 2.0 Step INC do: llINC is a topo~aphi~ maps publishedby the U.S. Geolo~cal Surdecrement parameter, e.g, INC = vey. Next, fifteen points were randomly selected on the -O.lO~/ street/highway network. The X and Y coordinates of’ S&p 1.2:Computethe distancematrix,giventhe each of the points were determinedby pickinga pair of (x, y) coordinatesof the point set. numbersfrom a random number table and scalingthem Step 1.3:Computethe set of multi-terminalshorto fit the X and Y range of each city respectively. If a test paths between all pairs of points point fell outside the city boundaries,or was placed more us~g~oyd’sor~ant~ig’salgorithm[24]. than l/4 of an inch from the closest street, it was This requires O(N’)time in the worst disregardedand a new pair of randomnumberspicked.If case/l a point fell within l/4 of an inch of a street it was M[i, JI= matrix of all shortest paths repositioned on the street and its coordinates adjusted Step 1.3’(options): For a new network design a~cordi~ly, It is impo~ant to stress that these uniformly dis~ibuted points be drawn within a bounded with no underIy~ trafficnetwork,use the edge lengthsof the Delaunaytrian- region[26]. The distancesbetween the points were measuredwith gulationto compute M[i, Jl. a distance measurer (opisometer). The shortest route Step 1.4:Compute the sum of all shortest paths between each pair of points was determined by in//This is sometimesreferred to as the spection. In those instances where the shortest route “aspersion” value[25]1/ could not readily be determined,a numberof alternative routes were measuredto find the shortest one. (It should SUM (p) = T T dP& vi) be noted that the distance.matrix thus derived assumes d&nCe
30
J. MACGREGOR SMITHand M. GROSS
free flowing conditions on the street/highway network without considering factors such as capacity constraints, congestion, and speed, which usually affect route selection.) The first city examined was Lexington, Kentucky. Based on our algorithm for estimating the L, parameter and with the chosen random points, we computed an L, value of p = 1.029, see Fig. 8. For each of the other cities, Philadelphia, Portland, and Washington, we computed L, values of 1.176, 1.100, and 1.2670respectively, see also Figs. 9-11. Certainly we could collect data on additional random points to compute an expected L, value across the entire urban region but this was felt to
Cl-lYUMll3
be unnescessary for the purposes of this present paper. In view of the applications of our algorithm, we seek the expected L, metric for a given subset of points within the urban region from which we can correctly compute the L, Steiner network for that given point set, not for all possible points in R2 of the urban region. At this stage, let’s take Lexington, Kentucky as an example and show, after we established its L, value, what the L,SMT solution would have been if we had assumed an Euclidean distance metric for the fifteen random points. After demonstrating this, we will then show the L,SMT solutions for the L, metric in which the L, value for Lexington is most closely associated.
----
Fig. 8. Lexington, Kentucky.
---_
Fig. 9. Philadelphia, Pennsylvania.
Steiner minimal trees and urban service networks
Fig. 10. Washington, D.C.
Fig. Il. Portland, Oregon.
32
J. MACGREGOR SMITH and M. GROSS
LEXINGTON .LzANALYSIS Figure 12 illustrates the fifteen random points together with the L, metric Voronoi diagram for the given points. Figure 13 illustrates the L, Delaunay triangulation for the given points which also contains the Minimum Spanning Tree (MST) as a subgraph of its edge set. Figure 14 illustrates the MST together with a suboptimal Steiner Minimal Tree (SMT) solution computed by our previously described algorithm. We will indicate subopjimal SMT’s by a caret above the expression, i.e. SMT. The abandoned edges of the MST are indicated by the dotted lines in the diagram and the three darkened circles represent Steiner points. The overall length of the MST solution’i; 62,136ft or 11.768 miles. The overall length of the SMT solution is 60,832ft or 11.521miles, a
savings of approximately 2.11%. This overall savings is within the expected value of between 2 and 4% as shown by [U] for uniformly distributed points in the plane. It should also be pointed out that the solution times for computing the SMT solution were approx. 0.20 seconds on a CYBER 175 housed at the University of Massachusetts, Amherst Campus. Thus, the algorithm is very fast as expected. Now given these results for the Lz metric, let’s examine the L, metric solution. LEXINGTON L, ANALYSIS Figure 15 illustrates the L, Voronoi diagram for the given point set wherein the given points where translated and rotated about the origin in order for the orthogonal Pattern of the streets to coincide with the orthogonal
Fig. 12.Lz VoronoidiagramLexington,Kentucky.
Steiner minimal trees and urban service networks
33
Fig.13.Lz Delaunay triangulation Lexington, Kentucky. pattern of the Cartesiansystem. Without this translation, the L, SMT solutionwould ,beawkwardto interpret. As was stated earlier, the Lz SMT solution is unaffected by the location of the origin or the orthogonality of the coordinate axes. Figure 16 depicts the L, ~launay triangulationfor the fifteen random points. The rectilinear metric serves as an appropriate approximation to our eqpirical 4 value of 1.029.Figure 17represents the L, SMT solution computed according to our algorithm.In this same figure are indicated the abandonedMST edges and the six Steiner points by the darkened circles. The overall Iength of jhe MST solution is 76,360ft or 14.46 mileswhilethe SMT solu$on is 71,800ftor approx. 13.59 miles.The ratio of the SMT/MSTrepresents a savingsof 5.972%.‘I&isis a larger reduction than that achieved by the LISMT solution but is perhaps not as large as expected for L,SMT solutions since they often range between 8 and 10%[17,281.Nevertheless, the solution
represents a potentiallysignificantcost savingswhen one accounts for the cost per mile of a given network. The paper has introduced the reader to some of the curious and fascinating features of L, bisectors, triangles, and Steiner point location problems and their potential impacton urban service network problems.We have demonstratedthe complexityof findingLp Veronoi and Steiner points loci within triangles and have shown that approximationtechniques are necessary if efficient running time aIgorithms are to be achieved for ‘the LpSMT problemfor large point sets. In the generalcase, one may questionthe configuration of the links when the L, value is not closely approximated by the L1 or L2 metrics. We argue that the optimal confi~ation of the links would closely follow the street pattern or else the edges of the Lp Delaunay
J. MACGREGOR SMITH and M. GROSS
Fig. 14. I&iT
solution Lexington, Kentucky.
Steiner minimal trees and urban service networks
35
J. MACGREGOR SMITH and M. GROSS
31
Steiner minimal trees and urban service networks
Fig. 17. L, SMT solution Lexington, Kentucky.
triangulation as an approximation to the ~5, curves which interconnect the L, Steiner points and the given point set V. This is arguable given that the street network segments are piecewise approximations to the Lp curve segments which generated the Lp value in the first place. Nevertheless, even in the Lexington, Kentucky example, the L, value does not exactly fit the underlying street pattern although the empirical value is very close to the L, metric value. Modifications would be necessary for the final location of the L, Steiner points and L,, link segments. Finally, we have not presented bounds for our algorithm simply because there is no known bound on the ratio of the lengths of heuristic Steiner minimal trees and optimal Steiner trees, either for the L1 or L2 metrics[9, 281 nor for those metrics ranging between these two benchmark metrics. This is a question for further research.
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J. MACGREGOR SMITH and M. GROSS
I. Z. A. Melzak, Companion to Concrete Maihema~jcs.Wiley, New York (1973). 8. Z. A. Melzak, Mathematical ideas, Modeling and App~ica~ions.Wiley, New York (1976). 9. F. K. Hlang, The rectilinear Steiner tree problem. J. Design Automation and Fault Tolerance Anal. 2,303-310 (1978). 10. R. F. Love and J. G. Morris, Modelling inter-city road distances by mathematical functions. Op. Res. Quart. 23, 61-71 (1972). 11. .I.G. Morris and W. M. A. Verdini, Minisum1, distance location problems solved via a perturbed problem and Weiszfeld’s algorithm. Op. Res. 27, 1180-l188(1979). 12. D. T. Lee, On finding k nearest nei~bors in the plane. Techn. Rep., apartment of Computer Science, University of Illinois (1976). 13. D. T. Lee, Two dimensional Voronoi diagrams in the Lpmetric. JA%M 27 604-618(1980). 14. M. I. Sham& Comnutational eeometrv. Unuublished Ph.D. _ * Thesis, Yale University (1978).1.5.J. B. Kruskal, On the shortest spanning subtree of a graph, Proc. Am. Math. Sot. 7,48-50 (1956). 16. J. MacGregor Smith, Algorithms for generalized Steiner network orobjems. Ph.D. Thesis. Deo~tment of Mechanical and Ind~st~al Enginee~ng, University of Iflinois, Urbana Campus (1978),~published. 17. J. MacGregor Smith, D. T. Lee and J. S. Liebman, An O(N log N) heuristic algorithm for the rectilinear Steiner minimal tree problem. Enpg Optimization 4,179-192 (1980). 18. J. MacGregor Smith, D. T. Lee and J. S. Liebman, An O(N log N) heuristic for Steiner minimal trees on the eucli-
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