[2007]Algorithms for central-median paths with bounded length on trees

Algorithms for central-median pathswith bounded length on treesRonald I.Becker a ,Isabella Lari b ,Andrea Scozzaric,*aDepartment of Mathematics,University of Cape Town,Rondebosh 7700,South Africa b Dipartimento di Statistica,Probabilita´e Statistiche Applicate,Universita ´di Roma ‘‘La Sapienza’’P.le A.Moro 5,00185Roma,Italy c Dipartimento di Matematica per le Decisioni Economiche,Finanziarie ed Assicurative,Universita´di Roma ‘‘La Sapienza’’Via del Castro Laurenziano 9,00161Roma,Italy Received 15December 2003;accepted 15September 2005Available online 2May 2006AbstractThe location of path-shaped facilities on trees has been receiving a growing attention in the specialized literature in the recent years.Examples of such facilities include railroad lines,highways and public transit lines.Most of the papers deal with the problem of locating a path on a tree by minimizing either the maximum distance from the vertices of the tree to the facility or of minimizing the sum of the distances from all the vertices of the tree to the path.However,neither of the two above criteria alone capture all essential elements of a location problem.The sum of the distances criterion alone may result in solutions which are unacceptable from the point of view of the service level for the clients who are located far away from the facilities.On the other hand,the criterion of the minimization of the maximum distance,if used alone,may lead to very costly service systems.In the literature,there is just one paper that considers the problem of finding an optimal location of a path on a tree using combinations of the two above criteria,and efficient algorithms are provided.In par-ticular,the cases where one criterion is optimized subject to a restriction on the value of the other are considered and linear time algorithms are presented.However,these problems do not consider any bound on the length or cost of the facility.In this paper we consider the two following problems:find a path which minimizes the sum of the distances such that the maximum distance from the vertices of the tree to the path is bounded by a fixed constant and such that the length of the path is not greater than a fixed value;find a path which minimizes the maximum distance with the sum of the distances being not greater than a fixed value and with bounded length.From an application point of view the constraint on the length of the path may refer to a budget constraint for establishing the facility.The restriction on the length of the path complicates the two problems but for both of them we give O(n log 2n )divide-and-conquer algorithms.Ó2006Elsevier B.V.All rights reserved.Keywords:Facility location;Central path;Median path0377-2217/$-see front matter Ó2006Elsevier B.V.All rights reserved.doi:10.1016/j.ejor.2005.09.049*Corresponding author.E-mail addresses:rib@maths.uct.ac.za (R.I.Becker),ri@uniromal.it (ri),andrea.scozzari@uniromal.it (A.Scozzari).European Journal of Operational Research 179(2007)1208–1220/locate/ejorR.I.Becker et al./European Journal of Operational Research179(2007)1208–12201209 1.IntroductionNetwork facility location is concerned with the optimal selection of a site or of a set of facility sites in a network in order to supply a set of costumers.The objective is either minimizing the distance from the furthest client in the network to the facilities(centre criterion)or minimizing the sum of the distances from the clients to the selected facilities(median criterion).These problems have a wide variety of applications in economy as the problem of locating bank accounts to optimizefloat[6]or of locating warehouses or depots.However, these problems traditionally deal with the optimal location of points located at either a vertex or along an arc of the network[7,8,12,13,15,23,25].Several authors extended the theory to include sites that are not merely single points but paths or trees [16,24].There are many practical applications,arising for instance in the design of public transit lines,where the facility to be located is too large to be modelled as a single point.In fact,the public transit line problem is formulated as the problem of locating a path on a road transportation network in order to minimize either the centre or the median criterion.Path-shaped and tree-shaped facilities are also called extensive facilities.Slater firstly[24]solved the problem offinding a core of a tree,that is,finding a path which minimizes the sum of the distances from all vertices of the tree to the path.Linear time algorithms forfinding a core of a tree have been presented in Morgan and Slater[22]and Becker[3].Peng and Lo[20]and Becker et al.[5]both provide algo-rithms that solved an extended core problem:finding a core with specified length or cost.The best result for finding a core with bounded length on trees can be found in[1].A parallel algorithm was also proposed for this problem in[28].More recently,some authors[27,29]considered the study offinding the conditional location of a path shaped facility of limited length on trees.That is,the problem of locating a path of limited length on a tree which minimizes the sum of the distances and under the condition that some existing facilities are already located.The problem offinding an optimal location of a path on a tree minimizing the maximal distance is consid-ered in Hedetniemi et al.[14]where a linear time algorithm is developed.Minieka[21]later solved the problem with a restriction on the length of the path to be located.A comprehensive survey on the problem of locating paths on trees can be found in[11,19].However,some authors have questioned the pertinence of the median and the centre criterion for these problems.Neither of the two above criteria alone capture all essential elements of a location problem. The median criterion alone may result in solutions which are unacceptable from the point of view of the service level for the clients who are located far away from the facilities.On the other hand,the centre criterion if used alone may lead to very costly service systems.Additionally,the median objective function tends to favor clients who are clustered in population centers to the detriment of clients who are spatially dispersed[18].On the other hand,the centre criterion may yield a significant increase in the total distance travelled by the clients.This had led Halpern[9,10]to model the corresponding trade offas a bicriterion problem in which a combination of total distance and maximal distance is minimized.In[17,26]efficient algorithms for the location of central-median point-shaped facilities are provided.In case of a path-shaped facility,such a dual objective is appropriate in many real world locational decisions.Indeed,fair service considerations in locating a public transit line may call for the service(i.e.the path)to be located not too far from any customer without the average distance travelled by all prospective customers being not too large.Averbakh and Berman[2]introduced the problem offinding a path-shaped facility on a tree network by considering both the centre and the median criterion.They defined the following problems:find a path which minimizes the sum of the distances such that the maximum distance from the furthest vertex of the tree to the path is bounded by afixed constant;find a path which minimizes the maximum distance with the sum of the distances being not greater than afixed value;find the set of all the Pareto-Optimal paths.They provided a two-phase procedure for all the three problems.Itfirstfinds a set M of cardinality at most n,where n is the number of vertices in the tree,which includes the set of Pareto-Optimal paths along with some extra paths in O(n)time.In a second phase,this procedurefinds the optimal path for thefirst two problems by searching among the paths in M.Since the cardinality of M is of at most n,the overall two-phase procedure has time complexity of O(n)for thefirst two problems.Extracting from M the Pareto-Optimal paths set requires O(n log n)time instead.In this paper we consider the first two problems introduced in [2]with an additional constraint,namely that the two optimal paths must have length (or cost )bounded by a fixed constant.In real applications,this addi-tional constraint on the length or on the cost may refer to a budget constraint for establishing the facility.In particular,we denote the problem of finding a path which minimizes the sum of the distances with the max-imum distance not greater than a fixed constant and with bounded length by Bounded Cent-Median path prob-lem,while we denote the problem of finding a path which minimizes the maximum distance with sum of the distances less than a fixed value and with bounded length by the Bounded Medi-Central path problem.The Bounded Cent-Median problem was first presented in [4]while,to the best of our knowledge,the Bounded Medi-Central problem does not seem to have been considered elsewhere in the literature.We notice that both the two problems are NP-hard on general graphs.Indeed,they contain as special case the problem of finding a path of bounded length which minimizes the sum of the distances and the problem of finding a path of bounded length which minimizes the maximum distance,respectively.It is well known that these two problems are NP-hard on general graphs [11].For both the problems we provide O(n log 2n )divide-and-conquer algorithms.The idea for both the proce-dures is that a middle vertex in the tree is computed;then an optimal path through this vertex is found.If it is not the optimal path for the problem under consideration,then this optimal path must lie entirely in one of the subtrees rooted at the adjacent vertices of the middle vertex.The algorithm is recursively applied to these sub-trees.An appropriate choice of the middle vertex ensures that the depth of the recursion is O(log n )[5,20].In Section 2,we provide notation and definitions,as well as an account of a preprocessing phase which calculates several quantities needed for the two algorithms.Section 3gives the algorithm for the Bounded Cent-Median path problem,while Section 4provides the procedure for the Bounded Medi-Central path prob-lem.In Appendix are presented some technical considerations related to the two algorithms.2.Notation and definitionsGiven a tree T =(V ,E ),with j V j =n ,let a (e )be a positive weight representing the length of each edge e =(v ,w )2E .Suppose also that to each vertex v is assigned a non-negative weight h (v ).Let P be a path in T ,the length of P is L ðP Þ¼P e 2P a ðe Þ.Given two vertices v and u ,we denote the unique path from v to u by P vu .We define the distance d (v ,u )between v and u of V as the length of P vu .Given a path P in T ,the sum of the distances from P to all the vertices v 2V is D ðP Þ¼P v 2V h ðv Þd ðv ;P Þ,where d (v ,P )is the minimum distance from v 2V to a vertex of P (see [22]).We call D (P )the DISTSUM of P .If P ={v }then we write D (v )instead of D ({v }).A path P which minimizes DISTSUM in T is called a median path or,following the definition in [22,24],a core of the tree T .We define E (P )the ECCENTRICITY of P where E (P )=max v 2V {d (v ,P )}.The shortest path P among those paths that minimizes ECCENTRICITY is the central path of T (see [24]).Let T be rooted at some vertex r ,and denote by T r the rooted tree.We define p (v )as the parent of v in T r and Son (v )as the set of children of v .Denote by T B v ¼ðV B v ;E B v Þthe subtree of T r rooted at vertex v and byT U v ¼ðV U v ;E U v Þthe subtree induced by vertices ðV ÀV B v Þ[f v g .Let deg B (v )be the degree of vertex v in T B v .For both the problems presented in this paper,we need a preprocessing phase that allows us to compute the quantities that will be used in the two algorithms.In the following we give a description of the recursive for-mulas calculated in this preprocessing phase.The quantities corresponding to T B v are often referred as belowquantities,while the ones corresponding to T U v are referred as upper quantities.Let D B (v )be the sum of the distances of the vertices in T B v to vertex v ,and let sum B (v )be the sum of theweights of the vertices in T B v .By using the standard bottom up approach,and proceeding level by level fromthe leaves to the root,we compute D B (v )and sum B (v )as follows (see also [5,4,16,25])sum B ðv Þ¼h ðv Þif v is a leaf of T r ;sum B ðv Þ¼h ðv ÞþP w 2Son ðv Þsum B ðw Þ;8<:ð1ÞD B ðv Þ¼0if v is a leaf of T r ;D B ðv Þ¼P w 2Son ðv Þ½D B ðw Þþsum B ðw Þa ðv ;w Þ .8<:ð2Þ1210R.I.Becker et al./European Journal of Operational Research 179(2007)1208–1220The time needed for the computation of(1)and(2)is O(n).Also,in the following algorithms,we need to cal-culate"v2V,D(v),that is,the sum of the distances of all the vertices in T r to v.Then,D(r)=D B(r)and by proceeding from the root r to the leaves we haveDðvÞ¼DðpðvÞÞþaðv;pðvÞÞ½HÀ2sum BðvÞ ;ð3Þwhere H¼Pv2VhðvÞ(see also[16]).Given a vertex v we denote by E B(v)and E U(v)the ECCENTRICITY of v in T Bvand in T Uv ,respectively.We proceed bottom up and top down to compute these quantities efficiently at eachvertex of T r.In particular,in a bottom up visit of the tree wefirst associate to each vertex v the label E1B ðvÞwhich is the maximum distance from v to a vertex u in T BvE1 B ðvÞ¼0if v is a leaf of T r;maxw2SonðvÞf E1BðwÞþaðv;wÞg otherwise.(ð4ÞLet u1B ðvÞbe a son of v that gives the value of E1BðvÞ.In the preprocessing phase,we also compute the labelsE2 B ðvÞthat represent the maximum distance from v to a vertex u in T Bvn T Bu1BðvÞE2BðvÞ¼0if deg BðvÞ<2;maxw2SonðvÞf E1BðwÞþaðv;wÞj w¼u1BðvÞg otherwise.(ð5ÞFinally,we calculate E3B ðvÞwhich is the maximum distance from v to a vertex u in T Bvn T Bu1BðvÞ[T Bu2BðvÞ,whereu2 B ðvÞis a son of v that gives E2BðvÞE3BðvÞ¼0if deg BðvÞ<3;maxw2SonðvÞf E1BðwÞþaðv;wÞj w¼u1BðvÞand w¼u2BðvÞg otherwise.(ð6ÞThe above three formulas can be computed in O(n)time for all the vertices of T.Let us consider the eccen-tricity E U(v)of a vertex v in T Uv ,then by proceeding from the root toward its leaves we distinguish:if v=r then E U(v)=0,if v is adjacent to the root r:E UðvÞ¼E1BðrÞþaðr;vÞif v¼u1BðrÞ;E2BðrÞþaðr;vÞif v¼u1BðrÞ.(ð7Þif v is not adjacent to r:E UðvÞ¼max fðE UðpðvÞÞþaðpðvÞ;vÞÞ;ðE1BðpðvÞÞþaðpðvÞ;vÞÞg if v¼u1BðpðvÞÞ;max fðE UðpðvÞÞþaðpðvÞ;vÞÞ;ðE2BðpðvÞÞþaðpðvÞ;vÞÞg if v¼u1BðpðvÞÞ.(ð8ÞIn a top down visit of the tree the above formulas can be computed in O(n)time.We now introduce a defi-nition relevant to the formulation of the algorithms of the following sections.Definition1.Given a path P vu and a path P uw with edges disjoint from P vu,the distance saving of P uw with respect to P vu,is the reduction of DISTSUM obtained by adding P uw to P vu(see[22]),that is savðP vu;P uwÞ¼DðP vuÞÀDðP vwÞ.ð9ÞIf thefirst path consists of only one vertex v,we simply write sav(v,P vw).If v=w,then sav(v,P vv)=0.Given a path P vw with w in T Bvand w5v,we can compute sav(v,P vw)in linear time by proceeding top down: savðv;P vwÞ¼savðv;P vpðwÞÞþsum BðwÞaðpðwÞ;wÞ.ð10ÞIn particular,for the problem offinding a path P which minimizes the ECCENTRICITY with bounded DISTSUM, another additional value is used,that is,upecc(v)representing the eccentricity of the path P rv excluding thevertices in T Bv and in the other subtrees incident to the root r(see Fig.1).The recursive equation for this valueisR.I.Becker et al./European Journal of Operational Research179(2007)1208–12201211upecc ðv Þ¼0if v ¼r or v is adjacent to the root r ;max f E 1B ðp ðv ÞÞ;upecc ðp ðv ÞÞg otherwise and v ¼u 1B ðp ðv ÞÞ;max f E 2B ðp ðv ÞÞ;upecc ðp ðv ÞÞg otherwise and v ¼u 1B ðp ðv ÞÞ.8><>:ð11ÞLet T r 0be the subtree of T r rooted at vertex r 0,and let P vu be a path passing through r 0.Let w 1and w 2be the two sons of r 0along the paths P r 0v and P r 0u ,respectively.Define (see Fig.2)E r 0ðv ;u Þ¼max E U ðr 0Þ;max x 2Son ðr 0Þf E 1B ðx Þþa ðr 0;x Þj x ¼w 1and x ¼w 2g &'.ð12ÞNotice that E U (r )=0in the given tree T r ,while in a subtree T r 0rooted at vertex r 0,E U (r 0)may be different from 0.See Appendix for furtherdetails.1212R.I.Becker et al./European Journal of Operational Research 179(2007)1208–1220Hence,by referring to formulas(4),(11)and(12),the ECCENTRICITY of any path P uv belonging to T r0and passing through r0,with u5r0and v5r0is(see Fig.2)EðP uvÞ¼max E1B ðuÞ;E1BðvÞ;upeccðuÞ;upeccðvÞ;E r0ðv;uÞÈÉ.Finally,by a top down scan of T r0,we canfind the lengths of the paths P r0v from the root r0to each vertex v as follows:LðP r0vÞ¼LðP r0pðvÞÞþaðpðvÞ;vÞ.ð13ÞThe length and the DISTSUM of any path P uv passing through the root r0,with u5r0and v5r0are LðP uvÞ¼LðP r0uÞþLðP r0vÞ;DðP uvÞ¼Dðr0ÞÀsavðr0;P r0uÞÀsavðr0;P r0vÞ.The two recursive procedures presented in this paper are based on the following remark.Remark1[5].Given a tree T=(V,E),a path P and a vertex v2V,we have two cases:•P contains v;•P is fully contained in one of the h subtrees,T1,T2,...,T h,obtained by removing v from T.Definition2[5,23].Given a weighted tree T,a middle vertex v of T is a vertex which minimizes the maximum of the number of vertices of the subtrees obtained by removing v.Remark2[15].A middle vertex has maximum subtree cardinality less than or equal to n/2,since,in fact,it is the centroid of the corresponding tree T if we assume that to all its vertices are assigned weights equal to one. Computing the middle vertex requires O(n)time with the algorithm in[7].For both the following algorithms,we root the tree at a middle vertex m,and then we check if the optimal path passes through m,otherwise we search for the optimal path in one of the subtrees obtained by deleting m. We notice that,when wefind the subtrees T1,T2,...,T h,we need to update the below and upper quantities of these subtrees,before the root m of the original T is deleted.See Appendix for further details on the Update procedure.3.Bounded Cent-Median problemIn this section we consider the problem offinding a path which minimizes DISTSUM with ECCENTRICITY at most R and with length less than or equal to‘.Remark3.Given a tree T,the following three cases with respect to the eccentricity may occur:(1)there exists a vertex u in T such that either E2B ðuÞ>R and E U(u)>R,or E3BðuÞ>R.In this case no pathin T is feasible with respect to the ECCENTRICITY;(2)if case(1)does not hold,at most one vertex w2T with E2B ðwÞ>R,E3BðwÞ6R and E U(w)6R may exist.In this case,all the paths feasible with respect to the ECCENTRICITY must contain the unique path P con-taining w such that E(P)6R and of minimum length;(3)if vertex w in case(2)does not exist,then,given the middle vertex m and the tree T m,we may have eitherE1 B ðmÞ>R or E1BðmÞ6R.In thefirst case,all the feasible paths passing through m must contain a givenpath P with E(P)6R and of minimum length,which has m as an endpoint.In the second case,all the paths passing through m are feasible with respect to the eccentricity.In cases(1)and(2)the algorithm checks the feasibility andfinds the optimal path,if it exists,working in the whole tree T.In case(3),given a tree T rooted at a middle vertex,the algorithm proceeds by searching for an optimal path by recursively visiting the subtrees of T rooted at the sons of the middle vertex.R.I.Becker et al./European Journal of Operational Research179(2007)1208–12201213Hence,in case(2)and at each iteration of the recursion in case(3),we have to solve the following problem: given a path P of ECCENTRICITY at most R,find a path P containing P of minimum DISTSUM and length less than or equal to‘.Forfinding P we use a modified version of the algorithm described in the paper by Becker et al.[5]whichfinds a path of minimum DISTSUM and length less than or equal to‘containing a given vertex v.We will give a description of this algorithm in Appendix.Algorithm1(Bounded Cent-Median)Input:a weighted tree TOutput:a path in T of ECCENTRICITY6R,length6‘and of minimum DISTSUM,if such a path exists,or FAIL otherwisebeginfind a middle vertex m of T and root T at m"u2T compute the quantities E1B ðuÞ,E2BðuÞ,E3BðuÞ,E U(u),L(P mu),sav(m,P mu)and D(u)/*see Section2*/ CASE1if$u2T such that E3B ðuÞ>R or(E2BðuÞ>R and E U(u)>R)thenoutput FAIL and Stop/*The problem is infeasible*/ CASE2else if$a vertex w2T such that E2B ðwÞ>R,E3BðwÞ6R and E U(w)6R/*the feasible paths in T pass through w*/consider the two sons u1B ðwÞand u2BðwÞof wlet t¼u1B ðwÞwhile E1B ðtÞ>R do t¼u1BðtÞlet t0¼u2B ðwÞwhile E1B ðt0Þ>R do t0¼u1Bðt0Þif LðP tt0Þ>‘output FAIL and Stop/*The problem is infeasible*/elsefind P tt0containing P tt0(see Appendix)output PüP tt0and DüDðP tt0ÞCASE3elseD*:¼+1,P*:¼;SUBTREE(T)output P*and D*endProcedure SUBTREE(T0)Input:a subtree T0=(V0,E0)of T with n0vertices and the best current DISTSUM D* Output:the best current DISTSUM D*and the corresponding path P*beginfind a middle vertex m0of T0and root T0at m0"u2T0compute the quantities E1B ðuÞ,E2BðuÞ,E3BðuÞ,E U(u),L(P m0u),sav(m0,P m0u)/*see Section2and Appendix*/if E1B ðm0Þ>Rconsider the son u1B ðm0Þof m0let t¼u1B ðm0Þwhile E1B ðtÞ>R do t¼u1BðtÞif LðP m0tÞ>‘/*There is no feasible path passing through m0*/elsefind P m0t containing P m0t(see Appendix)if DðP m0tÞ<DÃthen DÃ:¼DðP m0tÞand PÃ:¼P m0t1214R.I.Becker et al./European Journal of Operational Research179(2007)1208–1220if E U ðu 1B ðm 0ÞÞ6R then SUBTREE(T 0B u 1B ðm 0Þ)return P *and D *else/*E 1B ðm 0Þ6R then,in T 0all the paths passing through m 0are feasiblewith respect to the ECCENTRICITY */let P be the path formed by only m 0find P containing P (see Appendix )if D ðP Þ<D Ãthen D Ã:¼D ðP Þand P Ã:¼Pfor each son s of m 0doif E U (s )6R then SUBTREE(T B s )return P *and D*end Theorem 1(Correctness).Algorithm Bounded Cent-Median correctly finds an optimal solution,if it exists,or output FAIL otherwise.Proof.An optimal path,if it exists,must contain a path P such that E (P )6R ,otherwise,if every path in T has ECCENTRICITY greater than R the problem will result infeasible.Thus,it is convenient to firstly search for such a path P and then to try to find a path P which contains P of minimum DISTSUM and L (P )6‘.This reduces the search for an optimal path by means of considering the three cases described in Remark 3.For each subtree T 0considered in the recursion,a feasible path of minimum DISTSUM containing the middle vertex m 0of T 0is found.Then,the search for an optimal solution is recursively done in all the subtrees obtained by deleting m 0.More precisely,the recursive algorithm proceeds by working on the subtrees rooted at the sons sof m 0such that E U (s )6R .In particular,if E 1B ðm 0Þ>R the algorithm proceeds only on the son u 1B ðm 0Þ.In thisway the algorithm considers only subtrees where it is possible to find paths feasible with respect to the ECCEN-TRICITY .Hence,the algorithm correctly finds an optimal solution if it exists or fail otherwise.hTheorem 2(Complexity).Algorithm Bounded Cent-Median terminates in O(n log 2n )time.Proof.The computation of the below and upper quantities in the main procedure requires O(n )time in T .When computing these quantities,it is possible to directly check which of the three cases listed in Remark 3holds.Finding vertices t and t 0in case 2requires O(n )time.The path P tt 0can be found in time O(n log n )by using the procedure described in Appendix .Let us now consider the recursive procedure SUBTREE .At a given level of the recursion there are a number k of subproblems to be solved.Let n i with i =1,...,k be the number of vertices of the i th subproblem.As in the main procedure,also in SUBTREE the computation of the below and the upper quantities requires O(n i )time (see Appendix and Section 2)as well as the time needed for the computation of t is O(n i ).Finding P m 0t or P requires O(n i log n i )time.Since P ki ¼1ðn i Þ<n ,weobtain,for each level of the recursion,a time complexity of O(n log n ).The depth of the recursion is O(log n )since m is a middle vertex of the current subtree and the cardinalities of the subtrees obtained by removing m are at most n /2(see Remark 2).Hence,the overall time complexity of the algorithm is O(n log 2n ).hNotice that if the weights associated to the edges of the tree are all equal to one,finding a feasible path of minimum DISTSUM containing the middle vertex m of T can be done in O(n )time.Hence,the overall time com-plexity of our algorithm will be O(n log n ).4.Bounded Medi-Central problemIn this section we deal with the problem of finding a path P which minimizes ECCENTRICITY with DISTSUM at most k and with length less than or equal to ‘.Similar to the algorithm for the Bounded Cent-Median problem,we have to search for a path passing through the middle vertex m of T that has minimum ECCENTRICITY with DISTSUM at most k and with length less than or equal to ‘.We call such a path a best path through the middle vertex m .By Remark 1,if it is not the R.I.Becker et al./European Journal of Operational Research 179(2007)1208–12201215。