Approximating Shortest Paths in Large-Scale Networks with an

Approximating Shortest Paths in Large-Scale Networks withan Application to Intelligent Transportation Systems

YU-LI CHOU y T.J. Watson Research Center, IBM, Yorktown Heights, NY, Email: [email protected]

H. EDWIN ROMEIJN y Rotterdam School of Management, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam,The Netherlands, Email: [email protected]

ROBERT L. SMITH y Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI48109-2117, Email: [email protected]

(Received: January 1996; revised: January 1997; accepted: February 1998)

We propose a hierarchical algorithm for approximating shortestpaths between all pairs of nodes in a large-scale network. Thealgorithm begins by extracting a high-level subnetwork of rela-tively long links (and their associated nodes) where routingdecisions are most crucial. This high-level network partitionsthe shorter links and their nodes into a set of lower-level sub-networks. By fixing gateways within the high-level network forentering and exiting these subnetworks, a computational sav-ings is achieved at the expense of optimality. We explore themagnitude of these tradeoffs between computational savingsand associated errors both analytically and empirically with acase study of the Southeast Michigan traffic network. An order-of-magnitude drop in computation times was achieved with anon-line route guidance simulation, at the expense of less than6% increase in expected trip times.

O ur interest in this article is directed toward solving verylarge-scale shortest path problems, motivated by the prob-lem of finding minimum travel time paths within an on-lineroute guidance system. Route guidance within the context ofIntelligent Transportation Systems is the task of providingroutes between origins and destinations that promise tominimize the trip times experienced. The link travel timesthat are provided as an input to this function are time-dependent forecasts based upon current and anticipatedtraffic congestion (see e.g., Kaufman and Smith,[7] Wunder-lich and Smith[9]). Because of the rapid change of link traveltimes caused by time-varying travel demands and laneblockage resulting from incidents, the data used in comput-ing the shortest paths information is updated periodically,ideally every 5 to 10 minutes. During the time intervalbetween data updates, a shortest path must be provided forevery origin/destination (O/D) associated with trips thatbegin during that time slice. Thus, the calculation of shortestpaths must be efficient enough to respond in a timely way totrip requests on a real-time basis. Because a realistic problemmay have hundreds of thousands of nodes, all of which maybe potential origins and destinations, a fast heuristic that canprovide good approximations in a limited amount of timemay be preferred to exact methods.

We present here a hierarchical approach for finding ap-

proximate solutions for shortest path problems. A key ideabehind our development is the imposition of a hierarchicalnetwork structure. Our approach is motivated by the obser-vation that traffic networks have a hierarchical structure thatdivides links (and nodes corresponding to intersections ofthese links) into two or more classes (Yagyu et al.[10]). Thehigher level corresponds to longer links (i.e., highways)where the frequency of decision opportunities is low, withrouting decisions being correspondingly more important.The lower level corresponds to links of limited duration (i.e.,surface streets) and, correspondingly, more opportunities tocorrect routing decision errors. The links and nodes com-prising the higher level subnetwork correspond to a macro-scopic representation of the whole network. This high-levelsubnetwork partitions the original network into a set ofsubnetworks at the lower level in a way to be describedlater. By solving for all pairs of shortest paths in the higherlevel subnetwork and interfacing with gateways into thelower level subnetworks, we achieve significant economicsof computation, albeit at the expense of a loss of optimality.We explore in this article the magnitudes of the computa-tional gains and associated errors from optimality.

A similar approach is taken by Shapiro, Waxman, andNir,[8] who classify each of the arcs in the network, andapproximate shortest paths based on the assumption that itis desirable to spend as little time as possible on lower levelarcs. Their approach can be very efficient, especially whenonly a few shortest paths need to be computed. Moreover, ithas the advantage that it is not necessary to explicitly con-struct a hierarchy of subnetworks. Alternatively, our ap-proach has the advantage that much of the preprocessingcomputations can be done in parallel, so that the computa-tion of an additional shortest path is very cheap. Our ap-proach thus seems more suitable for cases where at least amoderate number of shortest paths have to be computed.

Habbal, Koutsopoulos and Lerman[6] propose a paralleldecomposition method for solving the all-pairs shortest pathproblem. Their algorithm, however, is an exact one, and

Subject classifications: Distance algorithmsOther key words: Networks, graphs, traffic models.

163INFORMS Journal on Computing 0899-1499y 98 y0902-0163 $05.00Vol. 10, No. 2, Spring 1998 © 1998 INFORMS

seems to be less suitable for use in an on-line route guidancesystem, which is the motivation for our study.

We begin by introducing the hierarchical algorithm (HA)in Section 1. We show how the nature of traffic networksinduces a natural choice for how the hierarchical networkshould be modeled. In Section 2, we discuss the issue of theefficiency of HA for solving the all-pairs shortest path prob-lem, as well as the case where only a limited number ofnodes can occur as origin or destination. Furthermore, con-ditions for efficiently implementing HA are presented. InSection 3, the efficiency of HA when applied to an on-lineroute guidance system is investigated. A numerical experi-ment is reported in Section 4, which is based upon a realtraffic network, the Southeast Michigan road network.

1. A Hierarchical Approach1.1 OverviewIn this section, we introduce a hierarchical approach tosolving for shortest paths in a large-scale network. The mainidea is to decompose the network into several smaller (low-level) subnetworks. When a shortest path needs to be foundbetween two nodes in the same subnetwork, we approxi-mate this shortest path by the shortest path having theproperty that all nodes on the path are contained in thatsame subnetwork. The resulting path is an approximateshortest path in that the true shortest path could leave thesubnetwork at some point, and re-enter it before arriving atthe destination.

To provide approximations of shortest paths betweennodes not contained in the same subnetwork, we define a(high-level) network, which is actually a subnetwork of theoriginal network, whose nodes intersect all (low-level) sub-networks. The nodes that are present in both the high-levelnetwork and in one or more low-level subnetworks arecalled macronodes. The high-level network is called the ma-cronetwork, reflecting the fact that it gives a macroscopicview of the original network. Correspondingly, the low-level subnetworks are called microsubnetworks, reflectingthe fact that they give a microscopic view of a part of theoriginal network. Given this decomposition, we can approx-imate a shortest path between two arbitrary nodes of theoriginal network by constraining the path to pass out of themicrosubnetwork containing the origin, through the ma-cronetwork, and into the microsubnetwork containing thedestination. Variants of this general procedure are consid-ered, based upon the departing macronode out of the originsubnetwork and the entering macronode into the destina-tion subnetwork chosen.

1.2 The Hierarchical Algorithm1.2.1 DefinitionLet G 5 (V, A, C) be a strongly connected directed graphwith a set of nodes V, a set of arcs A # V 3 V, and anon-negative arc length function C: A 3 R. (A directedgraph is strongly connected if there exists a directed pathfrom each node to each node of the network.) Our objectiveis to approximate shortest paths within the network G.

We will first describe the HA in its most general form,

consisting of the following three phases. In Sections 1.2.2and 1.2.3, we will briefly discuss some of the choices that canbe made within this general framework. These choices willthen be illustrated in Section 4.

Phase 0: DecompositionExtract a strongly connected, but not necessarily fully

dense, macronetwork G 5 (V, A, C), where V # V, A # V 3 V,and C: A3 R. The nodes in V will be called macronodes, andcorrespondingly, the arcs in A will be called macroarcs. Eachmacroarc (I, J) [ A corresponds to some directed micropathfrom I to J in the original network G where the correspond-ing arc length C(I, J) is defined as the length of this path.Note that the micropath corresponding to an arc (I, J) [ A isnot necessarily the shortest path between I and J in the graphG, but rather depends on the structural properties of thenetwork. Finally, note that all macronodes are micronodesas well and that there exists such a subnetwork associatedwith every choice of macronode set V # V because G is, byassumption, strongly connected.

Divide the original network G into H strongly connected,but not necessarily disjoint, microsubnetworks Gh 5(Vh, Ah, Ch), where Vh # V, V 5 øh Vh, Ah 5 A ù (Vh 3 Vh),and V ù Vh Þ A for h 5 1, . . . , H. Note that every cover (Vh,h 5 1, 2, . . . , H) of V (i.e., a set of subsets satisfying V 5øh Vh) is allowed as long as each Vh in the cover contains atleast one macronode.

Phase I: Constrained Shortest PathsFind all required shortest paths (which can either be all

shortest paths or a suitable subset thereof, depending on thetype of shortest path problem that needs to be solved) in themacronetwork G and in the microsubnetworks Gh, h 51, . . . , H. Let the function

f : V 3 V 3 R

return the shortest path lengths in the macronetwork G.Similarly, let

fh: Vh 3 Vh 3 R

return the shortest path lengths in microsubnetwork Gh, h 51, . . . , H.

Phase II: CombinationConsider a pair of micronodes i [ Vh, j [ V,. If h 5 , (i.e.,

the micronodes are contained in the same microsubnet-work), then the corresponding shortest path between i and jis approximated by the shortest path from i to j computed inPhase I with all intermediate nodes contained in this sub-network Vh. Otherwise, the shortest path is approximated bycombining three parts computed in Phase I:

(i) the shortest path (in Gh) from i to some macronodeI [ Vh ù V;

(ii) the shortest path (in the macronetwork G) from I tosome macronode J [ V, ù V; and

(iii) the shortest path (in G,) from macronode J to j.

Note that there always exist macronodes I and J in steps (i)and (iii) that are also micronodes in Gh and G,, respectively,by construction in Phase 0. Variants of HA depend upon

164Chou, Romeijn, and Smith

how these nodes I and J are selected when there are multiplecandidates. Note also that the combination phase can besimplified somewhat when the origin node i and/or thedestination node j are macronodes. In particular, in thosecases we can skip step (i) and/or (iii) in Phase II. However,depending on the particular choice of macronode made inthese steps, it may be fruitful to treat an origin or destinationmacronode as any other micronode. We will come back tothis later when discussing strategies for choosing the exitingand entering macronetwork macronodes for a given O/Dpair.

1.2.2 Comments on the Decomposition PhaseIn the decomposition phase of the Hierarchical Algorithm,there are obviously many possible ways of choosing themacronodes, macroarcs, and microsubnetworks. However,in a specific application, there usually is a natural hierarchywithin the arcs. For example, in a traffic network, the ma-cronetwork can be chosen to be the network consisting ofhighways and freeways, with the macronodes being a suit-ably chosen subset of the entrances to and exits from these.The macroarcs will then be micropaths consisting only ofhighways and freeways, and the microsubnetworks couldbe chosen in a natural way as the subnetworks enclosed bythe macroarcs. In Section 4, we will illustrate this procedureof forming the macro- and micronetworks using the actualSoutheast Michigan road network.* We will also see therethat usually some adjustments are necessary to obtain amacronetwork that is connected and sufficiently dense in theoriginal network for the HA algorithm to be successful.

1.2.3 Comments on the Combination PhaseIf, in the combination phase, different microsubnetworkscontaining the origin and destination contain only onemacronode each, it is unambiguous as to how to constructthe approximating path from the description of HA. How-ever, often there will be more than one macronode that canbe used to connect the microsubnetworks to the macronet-work. In that case, we need to make a choice among thesemacronodes. An intuitively appealing choice would be tochoose the macronodes that allow us to enter the macronet-work as soon as possible. In other words, we choose theclosest macronode that can be reached from the origin, andthe macronode from which the destination can be reached asquickly as possible. More precisely, if i [ Vh is the origin,and j [ V, (, Þ h) is the destination, then the connectingmacronodes are chosen to be

I* 5 arg minI[VhùV

fk~i , I!

and

J* 5 arg minJ[V,9V

f,~ J , j! .

The version of HA corresponding to this rule for the selec-tion of macronodes will be called Nearest HA.

Obviously the best possible selection rule among all HAvariants in terms of quality of the solution obtained (but alsothe most expensive one in terms of computation time) is toselect the pair of macronodes that yields the shortest approx-imate path. In other words, choose the pair of connectingmacronodes as follows:

~I*, J*! 5 arg min~I, J![~VhùV!3~V,ùV!

$ fh~i , I! 1 f ~I , J! 1 f,~ J , j!% .

This variant of HA will be called Best HA.At times there will be more than one microsubnetwork

containing the origin (and/or destination). In that case wewould adjust the choice of macronodes as follows. For Near-est HA, we set

I* 5 arg minI[øh:i[VhVkùV

fk~i , I!

and similarly for J*. Similar adjustments would be made forfinding (I*, J*) for Best HA.

2. Computational Complexity Analysis2.1 Notation and definitionsFor the heuristic HA to be useful, there obviously needs tobe time savings to balance the loss in precision in computingthe approximate shortest paths produced by it when com-pared to an exact algorithm. In this section we investigatethe relative efficiency of both Nearest HA and Best HAunder various conditions. Before analyzing the complexityof HA, we first review the concepts of 2, V, and Q functions(see e.g., Aho, Hopcroft and Ullman[1]).

Definition 2.1. Let f and g be functions from the non-negativeintegers to the non-negative reals.

(a) f(n) is said to be 2(g(n)) (or f(n) 5 2(g(n))) if there existpositive constants c and n0 such that

f~n! < cg~n! for all n > n0 .

Note that the 2-notation is used to specify an upper bound on therate of growth of some function. Analogously, the V-notation isused to provide a lower bound on the rate of growth, whereas theQ-notation is used if the upper and lower bounds coincide.

(b) The function f(n) is said to be V(g(n)) (or f(n) 5V(g(n))) if there exist positive constants c9 and n0 suchthat

f~n! > c9g~n! for all n > n0 .

(c) The function f(n) is said to be Q(g(n)) (or f(n) 5Q(g(n))) if there exist positive constants c, c9 and n0 suchthat

c9g~n! < f~n! < cg~n! for all n > n0 .

Now, assume that the number of nodes uVu in the graph Gis Q(N), where N is an input parameter characterizing thesize of the graph. Because we are focusing on road networks,which are sparse (i.e., the degree of each of the nodes can be

* SEMCOG, 1985. Survey of Regional Traffic Volume Patterns inSoutheast Michigan. Technical Report, Southeast Michigan Councilof Governments, Private Communication.

165Approximating Shortest Paths in Large-Scale Networks

bounded from above by a constant, independent of the sizeof the network), the most efficient algorithm for computingall shortest path lengths exactly is Dijkstra’s algorithm (seee.g., Denardo,[4] Dreyfus and Law[5]). The number of oper-ations needed by this algorithm is 2(N log N) and V(N) forsolving for the shortest paths from a single origin to alldestinations. We will repeatedly make use of this result inthe remainder of this section.

To gain insight into the potential gain in computationtime achievable by HA and how we should decompose theoriginal network to realize this gain, we assume within thissection that we decompose the network in such a way thatthe number of macronodes is Q(Nm) (for some 0 , m , 1),and that the number of microsubnetworks is Q(Nk) (for some0 , k , 1). The actual performance gain under real-worldconditions will be explored in Section 4. Each microsubnet-work is further assumed to be of roughly the same size, sothat the number of nodes in each subnetwork is Q(N12k).Moreover, we assume that each macronode is shared, onaverage, by a constant number of microsubnetworks, so thatthe number of macronodes per microsubnetwork is Q(Nm2k)(so that the number of macronodes is roughly equal for allmicrosubnetworks). Because we need at least one macro-node per microsubnetwork, we obviously need that k ¶ m.The relative magnitudes of m and k to achieve the greatestcomputational savings will be explored under two scenariosin the next two subsections.

2.2 The All-Pairs Shortest Path ProblemWe begin by investigating the efficiency of HA for solvingthe all-pairs shortest path problem. Let CN(N) denote thenumber of operations needed for approximating all shortestpath lengths in a network structure defined above usingNearest HA, CB(N) the corresponding number using BestHA, and CD(N) the number of operations needed for com-puting all shortest paths using Dijkstra’s algorithm in thesame network. The following theorem gives the relativeefficiency of Nearest HA with respect to Dijkstra’s algo-rithm.

Theorem 2.2 The relative efficiency of Nearest HA with respectto Dijkstra’s algorithm for the all-pairs shortest path problemsatisfies

CD~N!

CN~N!5 2~log N! and

CD~N!

CN~N!5 V~1! .

Proof. See the appendix. n

The proof of the above result shows that the combinationphase is the most time-consuming one, and, in fact, that thecomplexity for that phase is independent of the particulardecomposition chosen. Notwithstanding this fact, we couldtry to minimize the effort necessary for Phase I, even thoughthis phase is already much less time-consuming than PhaseII. Recall from the proof of Theorem 2.2 that Phase I takes onthe order of Nmax(22k,2m) operations (disregarding the “logN” term). Recall also that m Ä k. Therefore, it seems optimal(with respect to computation time) to choose m 5 k. Thecomputation time for Phase I then becomes of the order

Nmax(22k,2k), which is minimized if 2 2 k 5 2k, or if k 5 2⁄3.This means that to most efficiently implement Nearest HA,the network should be decomposed in such a way that

(i) the number of subnetworks should be of the sameorder as the number of macronodes, i.e., the numberof macronodes per subnetwork should beindependent of the size of the network; and

(ii) the size of each of the subnetworks should be smallerthan the size of the macronetwork.

However, note that a pure minimization of the computa-tional effort does not take into account any natural structurepresent in the network, or the magnitude of the error in-curred by the corresponding decomposition.

The following theorem is the analogue of Theorem 2.2 forBest HA.

Theorem 2.3 The relative efficiency of Best HA with respect toDijkstra’s algorithm for the all-pairs shortest path problem satis-fies

CD~N!

CB~N!5 2~N2~k2m!log N!

and

CD~N!

CB~N!5 V~N2~k2m!! ,

where m Ä k.


Unlike in the case of Nearest HA, the only instances wheresavings might be obtained occur when k 5 m, i.e., when thenumber of macronodes per subnetwork does not grow withN. Whenever m . k, the complexity “savings” actually be-come a complexity “dissavings” in the sense that Dijkstra’salgorithm is more efficient than Best HA! However, if k 5 m,then the relative efficiency of Best HA is the same as therelative efficiency of Nearest HA.

2.3 The Limited Shortest Path ProblemWe have seen that HA does not always perform more effi-ciently than Dijkstra’s algorithm for solving all shortestpaths in a network defined as above. Fortunately, however,in traffic networks, most nodes cannot actually occur asorigin or destination. Consider, for instance, surface streetintersections or highway entrances or exits. To model thisobservation, we assume that there are Q(Ng) nodes that canoccur as origins or destinations, for a total of Q(N2g) shortestpaths to be computed (where 0 , g ¶ 1). Moreover, weassume that these origins (destinations) are spread out uni-formly over the network, and that every microsubnetworkcontains at least one origin, so that g Ä k.

In this subsection, we will be concerned with computingall shortest paths between the nodes that can occur as originor destination. Note that, in traffic networks, we rarely needto compute all such shortest paths because network data(e.g., link travel times) is being updated periodically and notall possible trips will take place between any two updates.This situation will be analyzed in Section 3.


Similar to the notation in the previous section, letCD(N; g), CN(N; g), and CB(N; g) denote the number of op-erations needed for computing or approximating the re-quired shortest paths in the network structure defined aboveusing Dijkstra’s algorithm, Nearest HA, and Best HA, re-spectively. In particular, it is easy to see that

CD~N ; g! 5 2~N11glog N!

and

CD~N ; g! 5 V~N11g! .

Theorem 2.4. The relative efficiency of Nearest HA with respectto Dijkstra’s algorithm for the limited-pairs shortest path problemsatisfies

CD~N ; g!

CN~N ; g!5 2~N,1~k,m,g!log N!

and

CD~N ; g!

CN~N ; g!5 VSminSN,2~k,m,g!

log N, N12max~g,m2k!D D ,

where

,1~k , m , g! 5 min~k 2 max~0, m 2 g! ,

1 2 g 2 2max~0, m 2 g!!

,2~k , m , g! 5 min~k 2 max~0, m 2 g! ,

1 2 g 2 2~m 2 g!! .


As before, we can investigate the optimal number ofmacronodes and microsubnetworks in terms of computa-tional efficiency. If, for simplicity, we only consider the2-expression for the relative efficiency of Nearest HA, wesee that

CD~N ; g!

CN~N ; g!5 2~Nmin~k,k2m1g,12g,11g22m!log N! .

Clearly, it is optimal to choose k as large as possible. Becausek ¶ m by assumption, we will let k 5 m. The relativeefficiency then becomes

CD~N ; g!

CN~N ; g!5 2~Nmin~m,g,12g,11g22m!log N! .

The optimal relative efficiency can now be obtained if m 51 1 g 2 2m, or m 5 (1 1 g)/3. It is easy to see that (1 1g)/3 Ä min(g, 1 2 g), so that the optimal relative efficiencyis 2(Nmin(g,12g)log N), for k 5 m 5 (1 1 g)/3.

Now consider the case of Best HA. Then the followingtheorem is the analog of Theorem 2.4.

Theorem 2.5. The relative efficiency of Best HA with respect toDijkstra’s algorithm for the limited-pairs shortest path problemsatisfies

CD~N ; g!

CB~N ; g!5 2~N,91~k,m,g!log N!

and

CD~N ; g!

CB~N ; g!5 VSminSN,92~k,m,g!

log N, N12g22~m2k!D D .

where

,91~k , m , g! 5 min~k 2 max~0, m 2 g! ,

1 2 g 2 2~m 2 k!!

,92~k , m , g! 5 min~k 2 max~0, m 2 g! ,

1 2 g 2 2~m 2 g!! .


From Theorem 2.5, we have the following relative efficiencyfor the case of Best HA

CD~N ; g!

CB~N ; g!5 2~Nmin~k,k2m1g,12g22m12k!log N! .

Again, it is optimal to choose k 5 m, yielding

CD~N ; g!

CB~N ; g!5 2~Nmin~m,g,12g!log N! .

So the optimal relative efficiency is again2(Nmin(g,12g)log N), attained for k 5 m Ä min(g, 1 2 g).

Table I summarizes the complexity results derived in thissection.

3. On-Line Shortest Path CalculationsIn this section, we explore the complexity of Best HA andDijkstra’s algorithm when implemented in an on-line short-est path route guidance system. In an on-line route guidancesystem, the arc length function C is updated regularly, onthe basis of recent information concerning the link traversaltimes (see Kaufman and Smith[7]). In such a system, a pathwill have to be provided for a certain O/D pair if a requestis made for that pair during the time slice during which thearc length function is valid. Moreover, such a path will onlyhave to be computed if and when the O/D pair is requestedfor the first time during a time slice. Unlike the problemsdiscussed in the preceding section, where the number ofO/D pairs requiring solutions was assumed known, the

Table I. Summary of Complexity Results

Operation Complexity

Number of shortest pathssolved

2(Ng)

All-pairs shortest paths inmicronetworks

2(Nmax(g,m)112klog N)

All-pairs shortest paths inmacronetworks

2(N2mlog N)

Combination phaseNearest HA 2(Nm2k)Best HA 2(N2(m2k))


relevant question here is: “How many different O/D pairswill be requested in a given time slice?” To help answer thisquestion, we first describe a classical problem, called thecoupon-collector problem.

3.1 Coupon-Collector ProblemThe coupon-collector problem (CCP) is defined as follows.Suppose a set containing , distinct objects is given. A col-lector samples from the set with replacement, where witheach trial, he has a probability pi of drawing the ith object,independently of previous events.

Given a probability vector p, let Yn(p) be the number ofdifferent items observed in the first n trials. The followingresult on the expected value of this number is well known(see e.g., Boneh and Hofri,[2] Caron and McDonald[3]):

E~Yn~ p!! 5 Oi51

,

~1 2 ~1 2 pi!n! . (1)

3.2 Computational Complexity Per Time SliceIn this subsection, we investigate the complexity of Best HA,as well as Dijkstra’s algorithm, in an on-line route guidancesystem. For simplicity, we assume that each microsubnet-work has the same size, i.e., uVhu 5 uVu/H for all h. We startby introducing some additional notation.

Let the expected number of O/D requests per unit time beconstant, given by n. If the length of the time slice is equal tot, then the expected number of O/D requests in a time sliceis nt. We also assume that the probabilities for the O/Drequests are homogeneous over time. In particular, let pij

denote the probability that an O/D request corresponding tothe pair of nodes (i, j) is made, for all i, j [ V. Furthermore,let piz be the probability that an O/D request is made fromorigin i, i.e., piz 5 (j pij. Similarly, let the probability that anO/D request is made to destination j be denoted by pzj 5(i pij. If W , V is a set of nodes, denote the probability thatan O/D pair is requested from an origin in W by pWz 5(i[W piz, and denote the probability that a request is made toa destination in W by pzW 5 (j[W pzj. Finally, let 9i 5øh:i[Vh

Vh be the union of nodes contained in the microsub-networks that share node i [ V.

Note that in this framework, the number of nodes that canoccur as origin or destination is given by the cardinality ofthe set V# [ {i [ V: piz . 0 or pzi . 0}. Now, if uV# u 5 Q(Ng) (forg ¶ 1), then the analysis in Section 2 gives the efficiency ofHA compared to Dijkstra’s algorithm when computing allshortest paths that might be requested. This can be seen asthe limiting case of the analysis in this section, as the dura-tion of the time slices increases.

3.2.1 Complexity of Dijkstra’s AlgorithmWe now consider the expected number of computationsrequired by Dijkstra’s algorithm per time slice. We mayrecall that the shortest paths from an origin, say i, to all thenodes in the network G can be obtained at modest additionalcost (and the same complexity!) by using Dijkstra’s algo-rithm, even if only one O/D path is desired. Therefore, we

assume that we compute all shortest paths from a singleorigin the first time an O/D request is made from that originduring the time slice.

This problem can clearly be viewed as an instance of thecoupon-collector problem. The items are all possible originsi [ V that can be requested, with associated probabilities piz,and each O/D request within a time slice is an independenttrial. Therefore, from Equation 1, the expected number ofone-pair shortest path problems we have to solve over thenetwork G is

Oi51

uVu

~1 2 ~1 2 piz!nt!

per time slice. Thus, the computational complexity of Dijk-stra’s algorithm for the duration of t time units can beobtained by multiplying the expected number of differentorigins observed during a time slice with the complexity ofthe one-pair shortest path problem, yielding

7D~t! 5 2S uV uloguV uzOi51

uVu

~1 2 ~1 2 piz!nt!D . (2)

3.2.2 Complexity of HATo investigate the complexity of the two versions of HA, wedistinguish, as we did before, between the number of oper-ations required for Phase I and Phase II computations.

In Phase I, several shortest path problems over differentsubnetworks need to be solved. These problems can beclassified as: (i) the shortest paths in the macronetwork; (ii)the shortest paths within a subnetwork corresponding toorigins; and (iii) the shortest paths within a subnetworkcorresponding to destinations. We first consider the ex-pected number of computations required for solving theshortest paths in the macronetwork per time slice. During atime slice, a one-to-all shortest paths problem will be solvedin the macronetwork from each macronode that is an ele-ment of the subnetwork containing the origin of some O/Drequest that was made during that time slice. In terms of thecoupon-collector problem, the items are now the macro-nodes, and the item probability corresponding to macro-node I [ V is

qI 5 Oi[9I

Oj[y 9i

p ij .

The expected number of corresponding one-origin shortestpath problems that has to be solved during time slice t is

OI[V

~1 2 ~1 2 qI!nt! .

The product of the expected number of one-pair shortestpath problems and the cost for solving each problem bringsthe complexity of solving the shortest paths in the macronet-work to

2S uV uloguV u z OI[V

~1 2 ~1 2 qI!nt!D . (3)


We next consider the complexity of solving for the secondtype problem, i.e., all shortest paths from an origin, withinthe subnetwork containing that origin, corresponding torequests during a time slice. In other words, for each neworigin, the shortest paths from that origin to all destinationswithin its subnetwork are computed. In this case, the itemsare the origins, with corresponding probability piz as definedabove. The complexity for this phase over the time slicebecomes

2S ~ uV u/K!log~ uV u/K! z Oi51

uVu

~1 2 ~1 2 piz!nt!D . (4)

For the type three problem, it can be seen that the one-pairshortest path problem from a macronode within a givensubnetwork containing that macronode is computed the firsttime a destination of an O/D request is made in that sub-network, and the origin is not. For each O/D request, theprobability that the destination is an element of subnetworkGh (the item probability) is

rh 5 Oi[y Vh

Oj[Vh

pij .

If a destination in Gh is sampled, the shortest paths to thatdestination from all macronodes in Gh needs to be com-puted. Because there are on the order of uVu/H macronodesin each subnetwork (recall that the number of subnetworkssharing a given macronode is roughly a constant), the totalcomputational cost for these computations during the timeslice is

2S S uV uH D logS uV u

H D z S uV uH D z O

h51

H

~1 2 ~1 2 rh!nt!D . (5)

The complexity of Phase I for t units of time, denoted by7I(t), can now be obtained by adding Equations 3–5.

In Phase II, the combination is performed for each newappearance of an O/D pair across subnetworks. For BestHA, on the order of (uVu/H)2 feasible paths have to becompared for each O/D pair. If we define

d ij 5 H 10

if i and j are in different subnetworksotherwise,

for all i, j [ V, then the complexity of Phase II computationsfor Best HA during the time slice is

7BII~t! 5 2S S uV u

H D 2

z Oi51

uVu Oj51

uVu

~1 2 ~1 2 pij z d ij!nt!D . (6)

The total complexity of Best HA for the duration of a timeslice t, denoted by 7B(t), is then simply equal to

7B~t! 5 7 I~t! 1 7BII~t! . (7)

It is clear that the complexity expressions in Equations 2 and7 are extremely difficult to compare. An overall comparisonof the efficiency of HA with Dijkstra’s algorithm cannot bemade. Therefore, in the next section, we will return to this

topic when we consider an experiment using a set of real-lifedata.

4. Application to the Southeast Michigan VehicularTransportation Network

In this section, we illustrate the process of implementing theHierarchical Algorithms for solving shortest path problemsin a large-scale real network. In particular, special attentionis paid to the case of on-line route guidance. The methodol-ogy of implementing HA as well as the numerical results arethe main focus of this section.

4.1 Description of the Network and Data ResourcesThe real network we used to conduct the experiments in thissection is the Southeast Michigan road network. This roadnetwork was generated from the topological data providedby the regional planning authority of the Southeast Michi-gan Council of Governments (SEMCOG).* The region cov-ered by this road network consists of five counties, Macomb,Monroe, Oakland, Washtenaw, and Wayne. Moreover, thisnetwork includes most roadway segments in the region,from freeways to major streets.

There are 3,189 nodes and 5,658 links (11,316 directedarcs) in this road network, where links represent the streetsegments and nodes represent the intersections. Figure 1shows the Southeast Michigan road network. Links are clas-sified by SEMCOG into six different levels, from class 1(major freeways) to class 6 (local streets). Each road class hasan associated speed limit, as shown in Table II. The linktravel time is then defined as the time it takes to traverse thelink at the corresponding speed limit.

Other resources provided by SEMCOG are a set of zonecentroids, and the average number of trips between zonecentroids over a typical 24-hour period. Each zone centroidrepresents the center of the population within a certain area(zone). Standard transportation methods were used by SEM-COG to identify 1,543 zones and corresponding zone cen-troids across the road network as origins and destinationsfor regional travel.

4.2 Aggregation of the NetworkIn this section, we consider the problem of defining anappropriate two-level structure for the Southeast Michigannetwork so that it is suitable for the application of HA.

We will first form a macronetwork in the Southeast Mich-igan network. As we have suggested in Section 1.2.2, anatural way of forming the macronetwork for a traffic net-work is to define the high-speed roads and the major inter-changes as the macrolinks and the macronodes, respectively.Based upon this principle, we start by defining the nodesconnecting either two class 1 links, or one class 1 and oneclass 2 link as macronodes. The initial macronetwork is thenformed by connecting these macronodes through macro-links formed by a path consisting of class 1 links only.

* SEMCOG, 1985. Survey of Regional Traffic Volume Patterns inSoutheast Michigan. Technical Report, Southeast Michigan Councilof Governments, Private Communication.


However, the macronetwork obtained in this way is notconnected, and contains “gaps,” that obviously will causepoor performance of the HA algorithms. Figure 2 shows thisoriginal two-level network where the thick solid lines rep-

resent the macrolinks. To solve these problems, we upgradesome lower level links to class 1 (but retaining the originallink traversal times, so that this upgrading does not have aneffect on the complete network), and form new macrolinksusing these new class 1 links.

The resulting macronetwork is shown in Figure 3. Oncethe macronetwork is defined, we decompose the networkinto a set of microsubnetworks by letting the macrolinkscarve up the network. In this way, each area surrounded bymacrolinks is a microsubnetwork. Figure 4 shows the micro-subnetworks defined for the Southeast Michigan network.

Figure 1. The Southeast Michigan road network.

Table II. Speed Limits in Miles/Hr

Road class 1 2 3 4 5 6

Speed limit 65 50 40 40 35 35


Finally, we add the zone centroids to the network. Thesezone centroids are isolated nodes in the network. To connectthese nodes to the network we add, for each zone centroid,two links which connect the centroid to the two nearestnodes in the network (in Euclidean distance). These centroidconnectors are then assigned a length equal to this Euclideandistance, and are characterized as class 7 links with a lowspeed limit of 20 miles/hour. This models the fact that travelon the connectors (i.e., the low-volume surface streets) be-tween centroids and the arterial network is generally slow.The addition of the zone centroids and their connector links

increases the size of the network from 3,189 nodes to 4,732nodes (and from 11,316 to 14,300 arcs).

4.3 Numerical ResultsIn this section, we report on computational experimentswith both Nearest and Best HA, as well as Dijkstra’s algo-rithm, for solving for shortest paths for all possible trip pairsin the Southeast Michigan road network. The heuristics andDijkstra’s algorithm were coded in C and executed on anIBM RS6000 workstation.

Figure 2. The original two-level Southeast Michigan road network.


As noted before, the only trips possible in the network arebetween zone centroids. In other words, we have to computethe shortest paths between 1,543 3 1,543 5 2,380,849 pairs ofnodes, in a network with 4,732 nodes. The total number oftrips is equal to n 5 12,753,236.

Furthermore, the data provided by SEMCOG on the av-erage number of trips between centroids over a 24-hourperiod is incorporated to weigh the errors resulting from theapproximation algorithms. In particular, the error associatedwith each O/D pair is weighed by the corresponding num-

ber of trips made during 24 hours. The average error result-ing from HA is then calculated as follows:

e# 5

Oi51

n ~ , i 2 , i! z ni

Oi51

n , ini

where, for O/D pair i, ,i denotes the shortest path length,, i denotes the approximated shortest path length, and ni

denotes the number of trips during a 24-hour period.

Figure 3. The modified two-level Southeast Michigan road network.


The weighted results of solving the shortest paths for alltrip pairs are shown in Table III. We can see from this tablethat Nearest HA is about 15 times as efficient as Dijkstra’salgorithm, whereas Best HA is almost twice as efficient asDijkstra’s algorithm. Although the error for Nearest HA isquite high, the error for Best HA is small.

In the next section, we conduct a simulation of on-lineshortest path information queries in the Southeast Michiganroad network. Because Best HA gives very accurate approx-imations at acceptable cost, we will only focus on Best HAand Dijkstra’s algorithm in the next section.

4.4 An Application of On-Line Shortest Path QueriesIn this subsection, we conduct a numerical experiment for anon-line route guidance system in the Southeast Michigan

road network. In this experiment, O/D requests are madeprobabilistically between zone centroids, where the probabilis-tic O/D request matrix is generated based on the data of theaverage number of trips made for a 24-hour period. We assume

Figure 4. The micronetworks of the Southeast Michigan road network.

Table III. Results of Solving the Shortest Paths for allTrips

e#(%)

CPU Time (seconds)

Phase I Phase II Total

Dijkstra 0.0 — — 60.22Nearest HA 21.5 4.25 0.66 4.91Best HA 4.9 4.68 29.32 34.00


that the number of O/D requests is constant over time, andthus the number of requests per minute is equal to 12,753,236/1,440 ' 8,856. The probability of a request made for O/D pairi, denoted by pi, is then computed as pi 5 ni/n for all i.

The main purpose of this experiment is to compare therelative efficiency of Best HA and Dijkstra’s algorithm in anon-line route guidance system through a simulation con-ducted using real-life data.

4.4.1 CPU Times Required by the Algorithms Per Time SliceAs we have discussed above, in an on-line route guidancesystem the complexity of Dijkstra’s algorithm as well as thatof Best HA depends on the duration of the time slice that isbeing considered. Our goal in this simulation is to comparethe efficiency of Best HA with that of Dijkstra’s algorithmusing various time slice durations.

In an on-line route guidance system, it is sufficient to onlyprovide the shortest path information for the O/D paircurrently being requested. Moreover, recall that, usingDijkstra’s algorithm, the shortest path from an origin to adestination node is obtained once the destination node islabeled as a permanent node during the process of buildinga shortest path tree from that origin. However, at modestadditional cost, we can obtain the complete shortest pathtree from the origin. We propose two policies for implement-ing Dijkstra’s algorithm in an on-line route guidance systembased on these properties of Dijkstra’s algorithm. The firstpolicy (Policy 1) we propose is to stop the search process assoon as the destination becomes “permanently labeled”when solving for the shortest path for an O/D pair. Thus,under this policy, only the shortest paths from the origin tonodes that have been labeled permanently are computed.The second policy (Policy 2), on the other hand, is to com-

plete the search until all the nodes have been labeled per-manent. As a result, the shortest paths from the origin to allother nodes in the network are obtained after the search.When solving for the shortest path for one particular O/Dpair, Dijkstra’s algorithm obviously performs more effi-ciently under Policy 1 because the search process is stoppedafter obtaining the necessary information. However, whenapplied to an on-line route guidance system, there is noguarantee of better performance of Policy 1. Note that in anon-line route guidance system, the shortest path informa-tion, once obtained, is stored until an update of the distance(link travel time) matrix takes place. Thus, it may take longerto complete a single shortest path search from an originunder Policy 2, but more shortest path information can beprovided at modest additional cost within the current timeslice when requests are made from the same origin.

In practice, a reasonable time period between the updatesof link traversal times will be no more than 20 minutes.Thus, we run each single simulation for a 20-minute period.Furthermore, the cumulative CPU times required to solvethe shortest paths in response to O/D requests are recordedat different points in time during each simulation. Our majoraim is to simulate the CPU times consumed by Dijkstra’salgorithm and by Best HA per time slice in an on-line routeguidance system using various time slice lengths. Figure 5shows the total CPU times required by both approachesunder different policies over the whole simulation period. Itcan be seen that under either policy, the CPU time requiredby using Dijkstra’s algorithm increases rapidly within a veryshort period of time, whereas that required by Best HAincreases gradually and slowly throughout the simulationperiod. The figure also shows that Dijkstra’s algorithm per-forms more efficiently under Policy 1 if the duration of a

Figure 5. CPU time as a function of time slice length.


time slice is less than 3.7 minutes. However, for a longerperiod of time, Policy 2 outperforms Policy 1. A similarobservation can be made for Best HA. However, becauseonly subnetworks are considered instead of the completenetwork, the critical time slice length where Policy 2 startsdominating Policy 1 is much smaller (about 1 minute).

We can see from Figure 5 that Best HA is much moreefficient than Dijkstra’s algorithm during the entire simula-tion period. The amounts of time savings achieved by BestHA over Dijkstra’s algorithm for various periods of time areshown in Figure 6. To be more specific, Table IV shows theefficiency ratios for a number of time slices. The averagerelative error by Best HA collected at various points in timeduring the simulation are shown in Figure 7. For a 20-minute simulation period, the average relative error is5.78%.

4.4.2 Expected CPU Time Required by the AlgorithmsIn this section, we compute the expected CPU times re-quired by both algorithms, based upon the discussion con-ducted in Section 3.2, for an on-line route guidance system

implemented in the Southeast Michigan road network. Wefurther compare the expected values to the simulation re-sults obtained in the preceding section.

As we have shown in Section 3.2, Equation 2 gives theexpected amount of computational effort required by Dijk-stra’s algorithm to solve the shortest paths in response toO/D requests for t units of time. This expected number isobtained by multiplying the expected computation time re-quired to solve a one-pair shortest path problem over thewhole network with the expected number of shortest pathsthat need to be solved in a t-unit period. We may also recallthat by applying Best HA, there are four types of subprob-lems that need to be solved. Furthermore, different compu-tational effort is required to solve different types of subprob-lems. These subproblems can be categorized as follows:

1. the shortest paths in the macronetwork,2. the shortest paths within a subnetwork corresponding to

origins,3. the shortest paths within a subnetwork corresponding to

destinations,4. the combination of approximate solutions.

Again, the expected CPU time required for solving each typeof subproblem can be obtained by multiplying the expectednumber of subproblems that need to be solved during a timeslice with the corresponding time needed for solving a singlesubproblem. The computation time required by Best HA isthen the sum of the expected times required for solvingthese subproblems. For more details on the expected com-putation time required by the algorithms, we refer to Equa-tions 3–6 in Section 3.2.

We compute the expected CPU times required by bothalgorithms based upon Equations 2 and 7. However, we

Figure 6. The CPU time savings achieved by Best HA over Dijkstra’s algorithm.

Table IV. Efficiency Ratio of Dijkstra’s Algorithm vs.Best HA

TimeSlice Policy I Policy II

5 14.7 11.610 15.8 9.215 15.5 7.620 15.2 6.6


substitute the average CPU time required to solve eachsingle subproblem for the corresponding expected compu-tational effort. The expected CPU times required by bothalgorithms are then computed using various periods of time.Figure 8 shows the expected total CPU time required byboth algorithms compared to the simulation results obtainedin the preceding subsection. Furthermore, the expected CPUtimes required by Best HA for solving subproblems of types

1–4 as mentioned above are shown in Figure 9, where CTi(t)is the (expected) CPU time corresponding to a subproblemof type i in a t-unit time slice.

Computing the expected CPU times required by the al-gorithms as illustrated here is a very cost-effective way ofobtaining an overall picture on the performance of the algo-rithms in an on-line route guidance system. This is, in par-ticular, very helpful in making decisions on the choices of

Figure 7. The average relative error yielded by Best HA during the simulation period.

Figure 8. Expected vs. simulation CPU times.


the algorithm, the duration of a time slice, etc. Although theexpected CPU times computed this way might not be asaccurate as the simulation results, it gives a rather goodestimation without the effort and the computer time andmemory space required in conducting a simulation.

The expected computational costs can also be used toadjust the size of the macronetwork and the microsubnet-works so that an efficient implementation of HA can beachieved in the hierarchical network. As we have men-tioned, the efficiency of HA strongly depends on the choiceof decomposition. The size of the macronetwork and micro-subnetworks affect the amount of computational effort re-quired for solving subproblems of different types. For in-stance, consider the expected CPU times required by BestHA for solving subproblems of types 1–4. A large amount ofCPU time required in Phase II (subproblems of type 4)suggests reducing the number of macronodes within eachmicrosubnetwork, thus reducing the size of the macronet-work. On the other hand, a high computational cost ofsolving subproblems of type 2 or type 3 suggests reducingthe sizes of microsubnetworks, thus increasing the numberof microsubnetworks. These insights can be very helpful forconstructing the hierarchical network structure at an earlystage of implementation.

Acknowledgments

This work was supported in part by the Intelligent TransportationSystems Research Center of Excellence at the University of Michi-gan.

References

1. A.V. AHO, J.E. HOPCROFT, and J.D. ULLMAN, 1983. Data Struc-tures and Algorithms, Addison-Wesley Publishing Company,Boston.

2. A. BONEH and M. HOFRI, 1990. The Coupon-Collector ProblemRevisited. Technical Report CSD-TR-952, Computer SciencesDepartment, Purdue University, West Lafayette, Indiana.

3. R.J. CARON and J.F. MCDONALD, 1989. A New Approach to theAnalysis of Random Methods for Detecting Necessary LinearInequality Constraints. Mathematical Programming 43, 97–102.

4. E.V. DENARDO, 1982. Dynamic Programming: Models and Applica-tions, Prentice-Hall, Englewood Cliffs, NJ.

5. S.E. DREYFUS and A.M. LAW, 1977. The Art and Theory of DynamicProgramming, Academic Press, New York, NY.

6. M.B. HABBAL, H.N. KOUTSOPOULOS, and S.R. LERMAN, 1994. ADecomposition Algorithm for the All-Pairs Shortest Path Prob-lem on Massively Parallel Computer Architectures. Transporta-tion Science 28:4, 292–308.

7. D.E. KAUFMAN and R.L. SMITH, 1993. Fastest Paths in Time-Dependent Networks for IVHS Applications. IVHS Journal 1:1,1–11.

8. J. SHAPIRO, J. WAXMAN, and D. NIR, 1992. Level graphs andapproximate shortest path algorithms. Networks 22, 691–717.

9. K.E. WUNDERLICH and R.L. SMITH, 1992. Large Scale TrafficModeling for Route-Guidance Evaluation: A Case Study. IVHSTechnical Report 92-08, The University of Michigan, Ann Arbor,MI.

10. T. YAGYU, M. FUSHIMI, Y. UEYAMA, and S. AZUMA, 1993. QuickRoute-Finding Algorithm. Technical report, Toyota Motor Corp.and Matsushita Electric Industrial Co., Ltd., Tokyo.

Appendix

Theorem 2.2 The relative efficiency of Nearest HA with respectto Dijkstra’s algorithm for the all-pairs shortest path problemsatisfies

CD~N!

CN~N!5 2~log N!

Figure 9. Expected CPU times for solving the different subproblems in Best HA.


and

CD~N!

CN~N!5 V~1! .

Proof. In Phase I, the shortest paths that need to be solvedby Nearest HA are the shortest paths for all O/D pairswithin every microsubnetwork, and in the macronetwork.The total number of operations required in this phase is thus

Q~Nk! z CD~Q~N12k!! 1 CD~Q~Nm!! ,

which is

2~Nmax~22k,2m!log N! and V~Nmax~22k,2m!! .

In Phase II, the nearest macronode is selected fromQ(Nm2k) surrounding macronodes for each micronode, re-quiring a total of Q(N11m2k) operations. Furthermore, toobtain lengths of the approximate shortest paths that areacross subnetworks, one addition is needed for each path ifthe path connects a macronode and a micronode, whereastwo additions are needed if the path connects two micro-nodes. It is easy to see that the number of paths of the secondtype is much more important, and that, independent of thevalues of m and k, there are Q(N2) such O/D pairs. (Thereare Q(N12k) nodes per subnetwork, and Q(Nk) subnetworks,yielding Q((N12k)2 z (Nk)2) relevant O/D pairs.)

Because k . 0 and m , 1, the combination phase (Phase II)outweighs the shortest path phase (Phase I), so adding upthe number of operations for both phases, we get

CN~N! 5 Q~N2! .

The result now follows easily from the complexity results forDijkstra’s algorithm. n

Theorem 2.3. The relative efficiency of Best HA with respect toDijkstra’s algorithm for the all-pairs shortest path problem satis-fies

CD~N!

CB~N!5 2~N2~k2m!log N!

and

CD~N!

CB~N!5 V~N2~k2m!! ,

where m Ä k.

Proof. The number of operations in Phase I is the same asfor Nearest HA, i.e.,

2~Nmax~22k,2m!log N! and V~Nmax~22k,2m!! .

In Phase II, the best approximating path is selected fromQ(N2(m2k)) possible paths (corresponding to that number ofpossible pairs of macronodes for the originating subnetworkand the destination subnetwork, respectively). Since thereare again Q(N2) relevant O/D-pairs, this phase requires atotal of Q(N212(m2k)) operations.

Again it is clear that the combination phase (Phase II)outweighs the shortest path phase (Phase I), so adding up

the number of operations for both phases, we get

CB~N! 5 Q~N212~m2k!! .

The result now follows easily from the complexity results forDijkstra’s algorithm. n

Theorem 2.4. The relative efficiency of Nearest HA with respectto Dijkstra’s algorithm for the limited-pairs shortest path problemsatisfies

CD~N ; g!

CN~N ; g!5 2~N,1~k,m,g!log N!

and

CD~N ; g!

CN~N ; g!5 VSminSN,2~k,m,g!

log N, N12max~g,m2k!D D ,

where

,1~k , m , g! 5 min~k 2 max~0, m 2 g! , 1 2 g

2 2 max~0, m 2 g!!

,2~k, m, g! 5 min~k 2 max~0, m 2 g!, 1 2 g 2 2~m 2 g!!.

Proof. We will again distinguish between the number ofoperations required by Nearest HA in Phase I and Phase II.

In Phase I, several shortest paths have to be solved invarious subnetworks. First, we solve for all shortest pathswithin a subnetwork from each origin, which requires a totalof 2(Ng112klog N) and V(Ng112k) operations. Next, theshortest paths for all pairs of nodes have to be computed inthe macronetwork, because we have assumed that there is atleast one origin within each microsubnetwork. This requires2(N2m log N) and V(N2m) operations. Finally, within all mi-crosubnetworks containing at least one destination, theshortest paths have to be computed from all macronodescontained in that microsubnetwork. This requires 2(Nm112k

log N) and V(Nm112k) operations. Adding the numbers ofoperations required for solving these subproblems bringsthe number of operations required in Phase I to

2~Nmax~max~g,m!112k,2m!log N!

and

V~Nmax~max~g,m!112k,2m!! .

In Phase II, the nearest macronode is selected fromQ(Nm2k) surrounding macronodes for each micronodewhich can be an origin (or destination), requiring a total ofQ(Ng1m2k) operations. Furthermore, to obtain lengths of theapproximate shortest paths that are across subnetworks, oneaddition is needed for each path if the path connects amacronode and a micronode, whereas two additions areneeded if the path connects two micronodes. It is easy to seethat the number of paths of the second type is much moreimportant, and that, independent of the values of m and k,there are Q(N2g) O/D pairs.

The total number of operations required by Nearest HAnow satisfies

CN~N ; g! 5 2~max~Nmax~max~g,m!112k,2m!log N , Ng1max~g,m2k!!!


and

CN~N ; g! 5 V~Nmax~max~g,m!112k,2m,2g!! .

The result now follows in a straightforward manner fromthe complexity results for Dijkstra’s algorithm. n

Theorem 2.5. The relative efficiency of Best HA with respect toDijkstra’s algorithm for the limited-pairs shortest path problemsatisfies

CD~N ; g!

CB~N ; g!5 2~N,91~k,m,g!log N!

and

CD~N ; g!

CB~N ; g!5 VSminSN,92~k,m,g!

log N, N12g22~m2k!D D .

where

,91~k , m , g! 5 min~k 2 max~0, m 2 g! , 1 2 g 2 2~m 2 k!!

,92~k, m, g! 5 min~k 2 max~0, m 2 g!, 1 2 g 2 2~m 2 g!!.

Proof. The number of operations in Phase I is the same asfor Nearest HA, i.e.,

2~Nmax~max~g,m!112k,2m!log N!

and

V~Nmax~max~g,m!112k,2m!! .

In Phase II, there are Q(N2g) O/D pairs that require com-bining paths obtained from Phase I, at a cost of Q(N2(m2k))per combination. Thus, Q(N2(g1m2k)) operations are nec-essary for Phase II. Thus, the total number of operationsrequired by Best HA satisfies

CB~N ; g! 5 2~max~Nmax~max~g,m!112k,2m!log N , N2~g1m2k!!!

and

CB~N ; g! 5 V~Nmax~max~g,m!112k,2~g1m2k!!! .

The result now follows in a straightforward manner fromthe complexity results from Dijkstra’s algorithm. n


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Approximating Shortest Paths in Large-Scale Networks with an