An Efficient Label Setting/Correcting Shortest Path Algorithm

Transcription

1 An Effcent Label Settng/Correctng Shortest Path Algorthm Antono Sedeño-Noda, Carlos González-Martín Departamento de Estadístca, Investgacón Operatva y Computacón (DEIOC) Unversdad de La Laguna, La Laguna, Tenerfe (España) Abstract: We desgn a new label shortest path algorthm by applyng the concept of a pseudo permanent label. Ths approach allows an algorthm to partton the set of nodes nto two new sets: pseudo permanently labeled nodes and ts complementary set. From ths pont of vew, ths new label method can be consdered as label settng and s also a Dkstra (1959) method. Moreover, durng the executon of each teraton of the algorthm, at least one node becomes permanently labeled when some nodes whch belong to the set of pseudo permanently labeled nodes are scanned. In the case of networks wth non-negatve length arcs s easy to prove that ths node has the mnmum dstance label among the non-pseudo permanently labeled nodes. On the other hand whch nodes belongng to the set of pseudo permanently labeled nodes that are permanently labeled nodes s unknown. Therefore, all dstance labels are temporary and the algorthm becomes a label correctng method. Nevertheless, the proposed algorthm exhbts some nce features, such as: (1) the tme bound for the runnng of the algorthm for a network wth n nodes and m arcs s O(nm) (2) The theoretcal number of scanned arcs n each teraton of the algorthm s less than the number of scanned arcs n each pass of the prevous label correctng algorthms (3) The algorthm ncorporates two new rules whch allow easy detecton of a negatve cycle n the network. (4) The algorthm s qute smple and very easy to mplement, and does not requre sophstcated data structures. (5) The algorthm exhbts flexblty n the order n whch the new pseudo permanently labeled nodes are scanned. The above features are possble through the applcaton of the pseudo permanent label concept. Keywords Network/Graphs, dstance algorthms, desgn and analyss of algorthms, pseudo permanent labels 1. Introducton In a drected network wth arbtrary lengths, the shortest path (SP) problem conssts n fndng drected paths of shortest length from a source node to all other nodes, or detectng a drected cycle of negatve length. The SP problem s one of the most fundamental n network 1

2 optmzaton. Numerous algorthms for the solutons of the SP problem and several real applcatons are revewed n Ahua et al. [1]. Algorthms for the SP problem can generally be dvded n two maor groups: (1) Shortest path smplex algorthms that try to determne the optmal tree from the ntal feasble tree, makng smplex pvots wth non-tree arcs whch do not satsfy Bellman s optmalty condton, and (2) labelng methods that try to make the dstance labels permanent (optmal) for all nodes. Labelng methods are also parttoned n label settng and label correctng methods. Label settng methods are characterzed by settng one dstance label of a known node as permanent n each stage of the method, whle the label correctng methods consder all dstance labels as temporary untl the end of the algorthm. The frst-n frst-out (ffo) mplementaton of Bellmam-Ford-Moore (Bellman [2], Ford [7], Moore [14]) label correctng algorthm acheves the best strongly polynomal runnng tme of O(nm) for solvng the SP problem on a network of n nodes and m arcs. Several varants of ths early label correctng algorthm exst n the lterature, all of whch try to mprove ts practce performance. These papers nclude, n chronologcal order: Pape [16] and Levt [13], Taran [18], Pallotno [15], Glover et al. [10], Bertksekas [3] and Goldberg and Radzk [12]. Wth the addtonal assumpton that arc lengths are ntegers bounded below by N 2, the O( nmlog N ) bound of Goldberg [11] mproves the Bellmam-Ford-Moore bound unless N s very large. Note that we address the shortest path problem n a drected network wth arbtrary lengths where negatve cycles can exst. Therefore, we do not consder label settng algorthms, as the Dkstra [6] algorthm, because they do not work well wth negatve length arcs (see Cherkassky et al. [5]). Moreover, the above algorthms need to ncorporate tools to detect the presence of a negatve cycle. The classcal modfcatons of the label correctng algorthm to dentfy negatve cycles are avalable n Ahua et al. [1], p A number of others ways of dentfyng negatve cycles are commented n Cherkassky and Goldberg [4]. In ths paper, we ntroduce a new label correctng algorthm based on the concept of a pseudo permanent label. Ths new dea was ntroduced n Sedeño-Noda and González-Martín [17] to desgn a new shortest path smplex algorthm. The noton of a pseudo permanent label lets an algorthm partton the set of nodes n two subsets, smlarly to how permanent label are allowed to be parttoned n all label settng algorthms. From ths vewpont, the proposed algorthm can be consdered as a label settng algorthm. Obvously, f the network has negatve Correspondng author. e-mal address: asedeno@ull.es 2

3 arc lengths, then t s not possble to know f the label of a node s permanent. Thus, t s mpossble to know whch nodes are permanently labeled n our algorthm. But, any permanently labeled node belongs to the set of pseudo permanently labeled nodes. Therefore, our algorthm mantans an upper bound of the number of permanently labeled nodes n each teraton. In any case, the proposed algorthm becomes a label correctng algorthm snce all labels are consdered temporary untl the algorthm ends. But, f the network has non-negatve length arcs, then the new pseudo permanently labeled node wth mnmum label dstance becomes permanently labeled. Thus, we obtan a label settng algorthm from ths concept. To date, the ffo mplementaton of the Bellmam-Ford-Moore algorthm, Taran algorthm [18], threshold algorthm n Glover et al. [10] and Goldberg and Radzk algorthm [12] have been proven to run n O(nm) tme for networks wth arbtrary lengths. The proof of ths bound s based on each pass (teraton), these algorthms scan all arcs emanatng from nodes whose dstances were mproved n the prevous pass (or also n the current pass). The proposed algorthm has an O(nm) tme bound n the worse case. However, the concept of usng a pseudo permanent label n an algorthm mples that ths algorthm only scans a subset of arcs n the network. In other words, theoretcally our algorthm must scan fewer arcs n each pass than prevous label correctng algorthms. Ths affrmaton wll be clear from an expermental study that we carred out. Our algorthm reduces the number of node scan operatons snce only new pseudo permanently labeled nodes are scanned n each pass. The computatonal overhead cost to establsh f a node s pseudo permanently labeled becomes rrelevant when the nstances of the problem are not smple. Node scan operatons are a tme consumng operatons for label correctng algorthms and ts number s notably reduced n our algorthm. If the nstances problems are easy, the behavour of our algorthm s not sgnfcantly dfferent from the behavour of other label correctng algorthms. An analyss of the expermental results from the proposed algorthm reveals ts robustness n practce. Moreover, the order n whch the new pseudo permanently labeled nodes are scanned has sgnfcance nfluence n the emprcal behavour of the algorthm. Therefore, ths property provdes an addtonal degree of freedom n our algorthm. We performed an emprcal test that consders the next structures to store the new pseudo permanently labeled nodes: stack, queue and heap where the dstance label s the key. A revew of these varants n practce showed that the fastest algorthm uses the heap structure, though ths verson has a theoretcal complexty of O( nmlog n ) (when a Fbonacc heap s used then 2 O( nm n log n) + ). In the case of a network wth non-negatve length arcs, 3

4 we can obtan for ths verson an O( m + nlog n) tme label settng algorthm. Therefore, we have a label settng/correctng algorthm n only one approach. In addton, our algorthm ncorporates two new rules to effcently detect (not drectly dentfy) a negatve cycle n the network. These new rules are computatonally faster n dentfyng negatve cycles than the classcal rules ponted out n Ahua et al. [1] snce we do not need to wat for the label of a node to change n tmes. Moreover, all practcal effcent rules to detect negatve cycles studed n Cherkassky and Goldberg [4] requres a computatonal effort superor to O(n) wth the excepton of the amortzed search that s executed every tme the label correctng algorthm performs Ω(n) work. The rules ncorporated n the algorthm requres O(1) tme. Once agan, the source for these two rules s the pseudo permanent label. Fnally, the algorthm presented n ths paper s easy and rather smple to mplement wthout sophstcated data structures or addtonal procedures to fnd cycles (see for example Goldberg and Radzk [12]). An ntroducton to the defntons and notatons related to the SP problem and ts formulaton are gven n secton 2. The next secton ntroduces the new label settng/correctng algorthm and ts correspondng features. contans a detaled pseudo code and an explanaton of the proposed algorthm. Moreover, the worse case theoretcal complexty of the algorthm s proven. Secton 4 also offers a dscusson on the several varants of the proposed algorthm n relaton wth the order n whch the new pseudo permanently labeled nodes are scanned. Secton 5 ncludes a report on the computer results of our method when compared wth other label correctng methods from a subset of the experment carred out n Cherkassky et al. [5] and Cherkassky and Goldberg [4]. Fnally, n secton 6, we offer our conclusons and lnes of future research. 2. The Sp Problem: Defntons And Notaton Gven a drected network G = (V, A), let V { 1,..., n} = be the set of n nodes and A be the set + of m arcs. We denote by Γ = { (, ) } and { (, ) } V A Γ = V A V. For each arc (, ) A, let c R be ts length. The network has a dstngushed node s, called the source. The length of a drected path s the sum of the lengths of arcs n the path. The SP problem conssts n dentfyng a shortest path from node s to node for each non-source node V, or to fnd a negatve cycle. A negatve cycle s a drected cycle of negatve length. The SP problem s sad be feasble f G does not have a negatve cycle. 4

5 A paths tree of G s a spannng tree T rooted at s such that the unque path n the tree from node root s to every other node s a drected path. We refer to these spannng trees as drected out-spannng trees (herenafter tree). Note that n ths knd of tree, each node V \ { s} has only one node predecessor, that s, the n-degree of all non-root nodes n a drected outspannng tree s one. For all V, let B ( T ) be the set of nodes n the drected path tree from node s to node. Clearly, a shortest paths tree (herenafter optmal tree) T of G s a tree where each path n the tree from node s to node V \ { s} s a shortest path n G from s to node V \ { s}. Assumpton 1. The network contans a drected path from source node s to all non-source node V. For each non-source node V, f ( s, ) A then we mpose assumpton 1 by addng the artfcal arc ( s, ) wth nfnte length to A. Note that these arcs appear n the optmal soluton f and only f the SP problem does not have a feasble soluton wthout artfcal arcs. Moreover, we denote by T 0 the tree consstng of arcs (, ) s, V { s}. The dstance labels correspondng to a tree T are obtaned from the system of equatons c + d ( T ) d ( T ) = 0 (, ) T by settng d ( T ) = 0. Thus, gven a tree T, we defne the reduced cost c ( T ) = c + d ( T ) d ( T ) (, ) A. The dstance label of node, d ( T ), represents an upper bound of the length of the shortest path from node s to node. In other s words, d T for each node V. For an optmal tree ( ) d( T ) T, we obtan the followng optmalty condtons: c + d T d T A or, equvalently, í ( ) ( ) (, ) c T A ( ) 0 (, ) The above nequaltes are called Bellman s optmalty condtons. The equalty n the Bellman nequatons holds for all arcs n Gven a tree T, the predecessor ndex T. Pred of node s the prevous node to node n the drected path from node s to node n T. As such, gven the node root s and the vector ndex Pred, we can determne the paths n the tree T. Thus we can choose to store T or vector Pred when desgnng an algorthm, or both, whch s the approach used n ths paper. Moreover, the general labelng method uses the vector Pred nstead of T and, therefore, the dstance labels are denoted by d d ( T) dscountng of T. 5

6 Defnton 1. Gven a paths tree T, a node V s sad to be permanently labeled n T f d T =. ( ) d( T ) 3. A New Label Settng/Correctng Algorthm. Ths paper ntroduces a method whch smultaneously satsfes the characterstcs of a label settng method and a label correctng method. In step of the algorthm the dstance label of a node s made permanent (settng), but the set of permanently labeled nodes s unknown. Therefore, all dstance labels are consdered as temporary (correctng) untl the end of the algorthm. We acheve ths dualty by usng the method whch ncorporates the concept of pseudo permanent dstance labels ntroduced n Sedeño-Noda and González-Martín [17]: Defnton 2. A node V s a pseudo permanently labeled node n T, f and only f, ck ( T ) 0 ( ck + dk ( T ) d ( T) ), k Γ, B ( T ). That s, gven a tree T, a node V s pseudo permanently labeled node, f and only f, all nodes n the tree path from node s to node are pseudo permanently labeled nodes and all arcs leadng to have non-negatve reduced costs. The above concept allows us to partton the set of nodes n two subsets of nodes. Thus, gven a tree T, let L( T ) be the set of pseudo permanently labeled nodes and L( T) = V \ L( T ). Note that f G has no negatve cycles that nclude node s then, s L( T) ; otherwse, L( T ) =. In addton, by defnton 2, there are no arcs ( k, l ) n T that satsfy k L( T ) and l L( T ). For each node L( T ), we defne the subsets L ( T ) { L( T )} = { Γ } and then, we denote (, ) arg mn { c ( T ) L ( T) } L ( T ) L( T) the set of arcs = Γ and =. Note that (, ) wth L( T ) does not have a cycle. We defne = { < } H ( T) { L( T) c d ( T) d ( T) } H ( T ) L( T ) c ( T ) 0 ( ) L( T) \ H ( T) then, c ( T ) 0 ( c + d ( T) d ( T) ). When the arc = + <. Note that f (, ) s replaced by the tree arc ( pred, ) n the tree T and the dstance label of the node H ( T) becomes equal to c + d ( T) (or ncrease n c ( T ) unts) H ( T), 6

7 the next property, proved n Sedeño-Noda and González-Martín [17], assures us that at least one node of H ( T ) becomes pseudo permanently labeled. Lemma 1. Gven a tree T, f G has no negatve cycles, then at least one node n H ( T ) s pseudo permanently labeled by replacng all arcs ( pred, ) by arcs (, ), H ( T). Therefore, Lemma 1 provdes a rule to update the dstance labels of a lmted set of nodes. We name ths rule the Next Pseudo Permanently Labeled Nodes (NPPLN) rule. Note that f T s optmal then H ( T ) =. Moreover, from Lemma 1, n Sedeño-Noda and González-Martín [17] we arrve at the next result: Corollary 1. Gven a tree T, f L( T ) < n and H ( T ) =, then G has a negatve cycle. The above result descrbes a rule to ndentfy negatve cycles connectng only nodes n the set L( T ). In ths sense, ths rule predcts the exstence of a negatve cycle wthout trackng the nodes n the cycle. Gven a tree T, the replacement of arc ( pred, ) by arc (, ) for each node H ( T) s called a stage. If G has no negatve cycle, then at least one node H ( T ) becomes permanently labeled n each stage, whch leads to the next lemma (for the proof, see Sedeño-Noda and González-Martín [17]): Lemma 2. Gven a tree T, f G has no negatve cycles then at least one node n H ( T ) becomes permanently labeled replacng all arcs ( pred, ) by arcs (, ), H ( T). In short, each permanently labeled node s a pseudo permanently labeled node but the nverse s not always true. Thus a non pseudo permanently labeled node s a non-permanently labeled node. Therefore, label correctng algorthms only must to scan new pseudo permanently labeled nodes. Note that n prevously proposed SP algorthms (see, for nstance, Ahua et al [1], p. 142), the proof that n each teraton, at least one node becomes permanently labeled s carred out knowng that any label correctng algorthm scans, n the current pass, all arcs emanatng from nodes whose dstance were mproved n the prevous pass, ncludng non pseudo permanently labeled nodes. However, note that Lemma 1 mples a scan at most 7

8 L( T) L( T ) arcs n each pass of an algorthm usng the NPPLN rule. On the other hand, note that n a stage of the proposed algorthm, some nodes whch were pseudo permanently labeled may be non pseudo permanently labeled at the end of the stage. That s, the number of pseudo permanently labeled nodes may go up or down n a stage. However, from lemma 2, we know from the NPPLN rule that one of the nodes n H ( T ) s pseudo permanently labeled and permanently labeled. Therefore, n the begnnng of stage k, the number of pseudo permanently labeled nodes s at least L( T) k ( L( T) n k ) snce n each stage at least one addtonal node s (pseudo) permanently labeled. Furthermore, f a stage k fnshes wth L( T) < k then, the network contans a negatve cycle. Ths s the second rule to detect negatve cycles contanng nodes. In prevous label correctng algorthms, the applcaton of ths rule requres that the dstance labels of the nodes n stage k correspond to the length of shortest paths usng at most k arcs. In the case of a shortest path smplex algorthm, ths cycle s detected when a smplex pvot wth the enterng arc (, ) satsfes the condton that node belongs to the set of descendant nodes of node. The prevous results n ths secton were used to desgn the smplex shortest path algorthm found n Sedeño-Noda and González-Martín [17]. In ths secton, we wll cover some of the mprovements n the most recent developments n order to desgn a label settng/correctng shortest path algorthm. than We begn by askng a queston. Is t possble to scan even a fewer number of arcs + Γ? The answer s yes! We have dentfed two ways (so far): L ( T ) (1) Frst, when the NPPLN rule s appled, only the nodes n H ( T ) have modfed ther dstance label. Thus, only the arcs emanatng from the set of nodes H ( T ) need be scanned, that s, + Γ. In Sedeño-Noda and González-Martín [17], all nodes hangng of each node H ( T ) H ( T) n the new bass tree have modfed ther dstance label, therefore the algorthm needs to scan a number of nodes greater than or equal to H ( T ). Therefore, n the new algorthm, we need to know whose nodes n H ( T ) becomes pseudo permanently labeled. (2) Second, t s possble that c ( T ) s greater than c ( ) k T for some k L ( T ) for some node n H ( T ). Thus, f the tree arc ( pred, ) s replaced by the arc k (, ) n the tree T then, d ( T ) = d ( T) + c ( T ) > d ( T) + c ( T). In other words, node does not become (pseudo) 8

9 permanently labeled. Therefore, we do not brng n H ( T ) as follows: (, ) n tree T. That s, we redefne the set { k } H ( T ) = L ( T ) c ( T ) < 0 and c ( T ) c ( T ) k Γ or { } k k H ( T ) = L( T ) c + d ( T ) < d ( T ) and c + d ( T ) c + d ( T ) k Γ (ξ ) We now proceed to reproduce the proof of Lemma 1 for the new set H ( T ) : Lemma 3. Gven a tree T, f G has no negatve cycles then at least one node n the new set H ( T ) becomes pseudo permanently labeled replacng all arcs ( pred, ) by arcs (, ) H ( T). Proof. Let the tree T represent the one obtaned from tree T when the replacements wth the enterng arcs (, ), H ( T), were made. Then, d ( T ) = d ( T ) + c ( T ) for all H ( T) and each node V \ H ( T ) satsfes d ( T ) = d ( T). Thus, H ( T) and L ( T ), the reduced cost c ( T ) = c + d ( T ) d ( T ) = c + d ( T ) d ( T ) c ( T ) = c ( T ) c ( T ) 0. Moreover, L( T) \ H ( T) and L ( T ), the reduced cost c ( T ) = c + d ( T ) d ( T ) = c + d ( T ) d ( T ) = c ( T ) c ( T). In addton, each node L( T) \ H ( T) satfes one of the followng three cases: (1) a node k L ( T ) must exst such that c ( T) = c ( T ) < c ( T ) wth c ( T ) < 0; (2) k k node hangs n T from a node k L( T ) satsfyng case (1); (3) node smultaneously satsfes cases (1) and (2). All these stuatons mply that each node L( T) \ H ( T) s a no pseudo permanently labeled node n T. In order to conclude the proof, we proceed by contradcton. Assume that n T, each node H ( T ) s a no pseudo permanently labeled node. From the above arguments, each node L( T ) s connected from some node k L( T ) wth c ( T ) < 0 or t has an ascendant V n T wth ck ( T ) < 0 for some k L( T ) (otherwse, L( T ) ). Therefore, t s easy to see that a negatve cycle exsts n G connectng a subset of the nodes n L( T ), contradctng the k k 9

10 hypothess of the lemma. Thus, some node L( T ) must be pseudo permanently labeled. Fnally, snce no node L( T) \ H ( T) becomes a pseudo permanently labeled node, at least one node H ( T) must be a pseudo permanently labeled node n T. Lemma 3 and smlar arguments found n Sedeño-Noda and González-Martín [17] allow us to easly prove that Lemma 2 and Corollary 1 hold when the new set H ( T ) s consdered. Moreover, Lemma 2 assures that the NPPLN rule s appled at most n-1 tmes, snce n any ntal tree there s at least one permanently labeled node (the source node). Therefore, n the begnnng of stage k, the number of pseudo permanently labeled nodes s at least ( L( T) n k ). Furthermore, f a stage k fnshes wth L( T) < L( T) k k then, the network contans a negatve cycle. In addton, f a negatve cycle only connectng nodes n the set L( T ) exsts then the algorthm wll be able to detect t by Corollary 1. That s, f durng a stage of our algorthm nether node n H ( T ) becomes pseudo permanently labeled, then we have detected a negatve cycle. In addton, the algorthm needs to count the number of nodes n L( T ) to detect a negatve cycle or to fnsh wth an optmal tree. We ntroduce an algorthmc element to effcently check the cardnalty of the set L( T ). The dea conssts n concurrently performng the scan node operatons of the set H ( T ) and the verfcaton f a node becomes non pseudo permanently labeled. To do so, the number of pseudo permanently labeled nodes n the begnnng of a teraton s a upper bound that s equal to the number of pseudo permanently labeled nodes n the prevous teraton plus the number of nodes n the current set H ( T ). Note that the nodes n H ( T ) are potental new pseudo permanently labeled nodes. Therefore, when the algorthm scan the nodes n H ( T ), only needs to determne whch nodes become non pseudo permanently labeled nodes from those whch belong to sets H ( T ) and L( T ). The algorthm mantans a sub-tree of T contanng only nodes H ( T ) and L( T ). Note that, n any stage, no node that hangs from any branch of nodes n H ( T ) n T s pseudo permanently labeled. After the scan node operatons of the set H ( T ) n any stage, ths sub-tree only contans pseudo permanently labeled nodes and the algorthm has computed the number of pseudo permanently labeled nodes. It s at ths pont where we found an addtonal way to reduce the number of node scan operatons. When the algorthm performs the scan operaton of some node node n H ( T ), 10

11 any other node n H ( T ) can becomes non pseudo permanently labeled ( c ( T ) < 0 ). If stll has not been scanned then the algorthm mproves ts label dstance to d ( T ) = d ( T ) + c ( T ) and adds the arc (, ) to the sub-tree of pseudo permanently labeled nodes. Ths operaton reduces by at least one the scan node operatons snce node wll be scanned wth a most approprate label dstance. In ths sense the order n whch the nodes n H ( T ) are scanned can nfluence n the practce. Also, note that ths modfcaton does not affect lemma 3 snce at least one node n H ( T ) becomes pseudo permanently labeled and the remander nodes n H ( T ) can becomes pseudo permanently labeled or mantan ther current state. Fnally, n the case of networks wth non-negatve length arcs we obtan the next result: Lemma 4. Gven a tree T, f G has no negatve length arcs then the node H ( T ) wth mnmum d ( T ) + c ( T ) becomes permanently labeled replacng the arc ( pred, ) by the arc (, ). Proof. Lemma 4 holds from lemma 3 and by nducton. Note that n a fxed teraton of any label settng algorthm, any node n the sub-tree of permanently labeled nodes s a pseudo permanently labeled node. Therefore, the non-permanently labeled node wth mnmum dstance label wll belong to the set H ( T ). 4. Detaled Descrpton of the Algorthm We now contnue wth a presentaton of the new SP algorthm. In the algorthm, each node V has the followng elements: (a) The label dstance d[ ] ; (output of the algorthm) (b) The predecessor ndex Pred[ ] that stores the prevous node to node n the current best path from node s to node ; (output of the algorthm) (c) The label Status[ ] that can take on values n the set { 0,1, 2,3,4 }. Status[ ] s 4 (or PPL_IN_H) f node s pseudo permanently labeled and belongs to H ( T ) ; Status[ ] s 3 (PPL) f node s pseudo permanently labeled; Status[ ] = 2 (IN_NPPL) f s non pseudo permanently labeled and belongs to NPPL (see below); Status[ ] = 1 (IN_H) f s non pseudo permanently labeled and belongs to H ( T ) and Status[ ] = 0 (NIN) when s non pseudo permanently labeled. 11

12 (d) The node ndex [ ] that stores arg mn { c : c 0} Γ <, f each ncomng arc n node has a non-negatve cost then [ ] s equal to NULL; (e) The reduced cost of the arc ( [ ], ), c [ ] when [ ] NULL. Otherwse, we set c [ ] = 0 ; The algorthm uses the followng elements: (f) In each stage of the algorthm, the sub-tree T that stores the current path tree contanng only pseudo permanently labeled nodes and nodes n H ( T ). In the pseudo coded s used the notaton T { V (, ) T} + =. (g) A dynamc structure to store H (for example a stack, queue, or a heap) that stores all nodes n H ( T ) n each stage; (h) The counter nodes_n_h that stores the number of nodes n H; () The counter ppl that stores the number of pseudo permanently labeled nodes at the end of any stage n the algorthm. At the begnnng of any stage of the algorthm, the value of the ppl s the number of pseudo permanently labeled nodes n the prevous stage plus nodes_n_h. () The current stage k (t s also a lower bound of the number of permanently labeled nodes). (k) A dynamc structure to store NPPL (for example a stack or queue) that n each stage stores any node L( T) such that [ ] H ( T), that s, the set to buld the next set H; Usng the above gudelnes, we can now ntroduce the new label settng/correctng algorthm whch s reproduced n Fgure 1. An explanaton of the proposed algorthm s as follows: The algorthm starts by buldng the tree T 0 wth the dstance label of any non-source node equal to nfnte, c [ ] = 0 and, therefore, [ ] = NULL. Therefore, ntally the set H = { s}, nodes_n_h = 1. Under the assumpton that the network has no negatve cycle contanng node s, the number of permanent labeled nodes n T 0 s one. But, we set k=0 and ppl = n snce all non-source nodes can be pseudo permanently labeled (there s no drect arc from node s to any non-source node wth fnte length). Thus, any non-source node has status[ ] = 4 and the source node has status[ s ] = 5. Now, whle new pseudo permanently labeled nodes exst, the loop n the algorthm starts. In Lne 8, an upper bound of pseudo permanently labeled nodes s calculated 12

13 Effcent Label Settng/Correctng Shortest Path (ELSCSP) Algorthm; (1) T = ; H = ; NPPL = ; d[s] = 0; Status[ s] = PPL _ IN _ H ; Insert s n H; (2) For all V do (3) c [ ] = 0; [ ] = NULL ; (4) If s then (5) T T {( s, ) } = + ; d[] = ; Pred[] = s; Status[ ] = PPL ; (6) k = 0; ppl = n - 1; Nodes_n_H = 1 (7) Whle ( H ) do / loop / (8) ppl = ppl + Nodes _ n _ H ; (9) Whle ( H ) do / SCAN all node n H / (10) Select a node H ; Delete from H; Status[ ] = Status[ ] 1; (11) For all + Γ do / SCAN / (12) If (( c = c + d[ ] d[ ] < 0 ) && ( c < c [ ])) then; (13) c [ ] = c ; [ ] = ; (14) If ( Status[ ] >= PPL ) then (15) DFS(, Status, T, ppl); (16) If (( Status[ ] == PPL ) && ( Status[ ] == IN _ H ))then (17) ppl = ppl +1; d[] = d[] + c [ ]; Pred[] = [ ]; c [ ] = 0; (18) = + {( [ ], )} T T ; [ ] = NULL ; Status[ ] = PPL _ IN _ H ; (19) If (( Status[ ] == PPL ) && ( Status[ ] == NIN ))then (20) Status[ ] = IN _ NPPL ; Insert n NPPL; (21) nodes_n_h = 0; k = k + 1; (22) If (( [ s] == NULL ) && (k ppl <n))then (23) Whle ( NPPL ) do (24) Select a node NPPL ; Delete from NPPL; Status[ ] = NIN ; (25) If ( Status[ [ ]] == PPL ) then (26) = + {( [ ], )} T T ; (27) d[] = d[] + c [ ]; Pred[] = [ ]; (28) c [ ] = 0; [ ] = NULL ; Status[ ] = PPL _ IN _ H ; (29) Insert n H; nodes_n_h = nodes_n_h + 1; / end of the loop / (30) If (( ppl < (31) Else T s an optmal tree; k ) ((nodes_n_h == 0) && (ppl < n))) then G has a negatve cycle Fgure 1: ELSCSP algorthm 13

14 and stored n ppl. Therefore, the algorthm n the scan operaton must nvestgate whch nodes become non pseudo permanently labeled nodes. Lnes 9 to 20 n the algorthm summarze the scannng operaton of all nodes n lst H. In other words, the arcs whose reduced costs probably become negatve, when the dstance labels of the nodes n H are consdered to update [ ] and c [ ]. Note that n the algorthm f the dstance label of any node changes, all arcs emanatng from ths node are scanned. Therefore, n the algorthm, c [ ] always equals the most negatve reduced cost of all arcs arrvng n node for each node. Lnes 14 and 15 dentfy the new non pseudo permanent labeled nodes. That s, f status[ ] >= PPL then the procedure DFS(, Status, T, ppl) s called. The arguments n the procedure are the parameters, Status, and the varables PPL and ppl. Ths procedure traverses the pseudo permanently labeled nodes that are descendants of node, changng the status of these nodes and deletng the correspondng arcs of the tree T of pseudo permanently labeled nodes. Obvously, the number of pseudo permanently labeled nodes s also modfed. The procedure s reproduced n Fgure 2. Note that f node has Status equal to PPL_IN_H, after of the executon of the procedure DFS, ts Status becomes IN_H. In Lnes 16 and 17, any node wth Status IN_H s made pseudo permanently labeled (PPL_IN_H) snce node stll belongs to H and wll be scanned later. To do so, ts dstance label, Pred label Status are approprately changed and the arc ( [ ], ) s added to T. Clearly, ppl ncreases n one unt. Instead, f node has Status equal to NIN (Lnes 19-20), ths node s nserted n NPPL and ts Status becomes IN_NPPL snce ths node satsfes the condtons to belong to the set H for the next stage. Clearly, the last modfcaton s only made f the Status of node s PPL. Note that the Status of a node k n H can be equal to IN_H snce when another node n H ( T ) was scanned and made any ascendants of node k non pseudo permanently labeled. At ths pont n the algorthm, the set H has been scanned and the pseudo permanently labeled nodes have been dentfed. Therefore, the ndex of the stage s ncreased, that s, k = k + 1. Next, f [ s] NULL then, the algorthm detects a negatve cycle contanng node s (ppl = 0 < k). Also, f ppl < k then, addtonal nodes become non-pseudo permanently labeled nodes and the algorthm detects a negatve cycle. Otherwse, lnes (25) to (29) n the algorthm reproduce (ξ ). In other words, NPPL, f node belongs to the next H ( T ) ( Status [ [ ]] == PPL ) then, ts dstance label change n c [ ] unts, ts Status becomes PPL_IN_H and the arc 14

15 ( [ ], ) s added to T. Snce, c [ ] s the mnmum reduced cost of all arcs arrvng n node and the dstance label of node has mproved n c [ ] unts, the new value of c [ ] becomes equal to zero and [ ] = NULL. Obvously, all these nodes are nserted n H and the value of nodes_n_h s updated. Note that when Status[ [ ]] s not equal to PPL the dstance label of the node s not updated snce [ ] s not a pseudo permanently labeled node. Thus nether current non-pseudo permanently labeled node s ntroduced n H. The algorthm ends when all nodes are pseudo permanently labeled, that s ppl equals n, or, when a negatve cycle n G s detected. Ths last stuaton s easly verfed by ppl < k n Lne 22 or by nodes_n_h = 0 when ppl < n (nether node becomes pseudo permanently labeled n a stage). Procedure DFS(, Status, Pred, T, ppl); (1) If Status[ ] == PPL_ IN _ H then (2) Status[ ] = IN _ H ; (3) Else (4) Status[ ] = NIN ; (5) ppl = ppl -1; (6) T = T { pred[ ], }; (7) For all + T do (8) DFS(, Status, Pred, T, ppl); Now we can state the followng theorem: Fgure 2: DFS procedure Theorem 1. Startng from the tree T 0, the ELSCSP algorthm solves the SP problem n at most n 1 stages and O(nm) tme. Proof. The ntal tree T 0 can be obtaned n O(n) tme. We assume that the Intalzaton, Insert and Delete operatons n the structure H requre an O(1) tme. In each stage k, the bottleneck operaton scans all nodes n H. Ths operaton requres O( + Γ ) tme. The H ( T ) recursve procedure DFS s called from each H vstng n the worst case all nodes n the sub-tree T. In ths worst case all nodes n the sub-tree T becomes non-pseudo permanently labeled. Therefore, n a fxed stage, the total effort n all the calls to the DFS procedure s 15

16 O( + Γ + k). Fnally, The loop to establsh the new set H (Lnes (22)-(29)) requres O(n- H ( T ) k) tme. In addton, note that each replacement between arcs can be done n O(1). Therefore, each stage k s performed usng O( + Γ + n) < O(m) tme. If G does not have any negatve H ( T ) cycle then Lemma 2 assures us that at least the label of one node s made permanent n each stage. Snce the number of permanent labeled nodes n T 0 s at least 1, the algorthm performs at most n 1 stages. In any case, the ELSCSP algorthm executes at most n 1 teratons and runs n O(nm) tme. We ponted out n the ntroducton that the order n whch the nodes n the set H ( T ) are scanned has sgnfcant nfluence on the emprcal behavour of the algorthm. We consdered the structures assocated to H as stack, queue and heap where the dstance label s the key. For the bnary heap structure n the algorthm Lnes (10) and (24) are changed by ExtractMnKey(H, ) and Insert(H, ), respectvely. Both operatons requre O( log n ) tme. Note that n Lne 17, the dstance label of a node n H s modfed. In ths case, we can consder two possbltes: the operaton of DecreaseKey on node s not made or ths operaton s made. It s easy to prove that the complexty of the algorthm usng a heap n the frst case s n 1 O( nm + ( n k)log( n k)), that s, k = 1 2 O( nm n log n) + and n the second case, O( nm log n ). When a Fbonacc heap s used, the operaton DecreaseKey uses constant tme and therefore the complexty of the algorthm s 2 O( nm n log n) +. From lemma 4, t s easy to obtan a label settng algorthm when the length arcs are non-negatve by elmnatng the loop of lne (9) (only the node n H wth mnmum label s scanned). In ths case the algorthm runs n O( m + nlog n) tme. Ths modfcaton can be made by usng a scan-node counter lmtng the number of nodes n H that can be scanned n the loop of lne (9). We denote by ELSCSPS, ELSCSPQ, ELSCSPH and ELSCSPHU, the proposed algorthm usng a stack, a queue, a bnary heap wthout dstance updates and a heap wth dstance updates, respectvely. The mplementaton of the DFS procedure s also teratve. The partal tree T s stored as a double lnked lst usng two ponters (next and prevous) assocated wth each pseudo permanently node. These ponters let to traversal the partal tree T n depth frst search order. 16

17 5. Computatonal Results. In the state-of-art of shortest path problem, several expermental studes on the behavor of the exstent algorthms arse. Our start pont was the experments carred out n Cherkassky et al. [5] and Cherkassky and Goldberg [4]. In the second study, and takng nto the account the results from the frst study, the authors only consder the compettve codes of the followng label correctng algorthms: ffo mplementaton of Bellmam-Ford-Moore algorthm wth parent-checkng heurstc (BFM), ffo mplementaton of Bellmam-Ford-Moore algorthm wth subtree dsassembly rule to detect negatve cycle, that s, Taran [18] algorthm (BFCT), Pallotno [15] algorthm (TWO_Q), Pallotno [15] algorthm wth subtree dsassembly rule (PALT), network smplex method wth subtree traversal rule to detect negatve cycle (SIMP) and Goldberg and Radzk [12] algorthm (GORC). These codes were wrtten n C and compled wth the Lnux gcc compler usng the O4 optmzaton opton. The BFP, TWO_Q codes are contaned n the SPLIB-1.4 lbrary and the BFCT, PALT, SIMP and GORC codes belong to the SPC-1.2 lbrary. Both lbrares are avalable n the personal web page of A. V. Goldberg ( These mplementatons use the adacency lst representaton of the nput graph, smlar to that of Gallo and Pallotno [8]. Snce the commented studes are an mportant reference n the lterature of label correctng algorthms, we ncorporate four versons of our algorthm usng the programmng style, the data structures n the commented lbrares. In other words, the programmng envronment s mantaned. The ELSCSPS, ELSCSPQ, ELSCSPH and ELSCSPHU codes are added to the study. The enumerated codes were tested on a Intel Pentum M wth 2 GHz processor 760 wth 1Gb RAM runnng Red Hat Lnux. As n the reference studes, we report the user CPU tmes n seconds, averaged over several nstances generated wth the same parameters takng nto account the followng ten seeds: , , , , , , , , , and In each cell of a table appear: the average runnng tme (n bold) and the average number of scans per node.we ran these algorthms on a subset of shortest path problem famles from Cherkassky et al. [5]. We used the SPRAND generator attrbuted to Cherkassky et al. [5]. We present results for random graphs wth unform arc lengths at random from nterval [ ] 0,10000 wth m = 4n (Rand-4 famly, Table 1), m = n 2 / 4 (Rand1-4 famly, Table 2) and, addtonally, m = 1000n (Rand-1000 famly, Table 3). 17

18 Table 1: Rand-4 famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Table 2: Rand1-4 famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU The graphs n Rand-4 famly are sparse. In Table 1, t s observed that our algorthms are slower than the other codes. However our algorthms are not notably slower. These dfferences are only sgnfcant when the number of nodes s bg large. In addton, note that ELSCSPH and ELSCSPHU perform a fewer number of scans per node than other codes, but snce the graphs are sparse, ths fact s not translated nto an mprovement n the CPU tme. However, when graphs are dense (Table 2) as n Rand1-4 famly data, the best algorthm s ELSCSPHU, followed by ELSCSPH. Moreover, the next best algorthms are BFM, BFCT, SIMP, ELSCSPS and ELSCSPQ. GORC s the worst algorthm. Note, that ELSCSPH and 18

19 ELSCSPHU perform less number of scans per node than other codes. In ths famly, the number of nodes s not very large. Thus, the dfferences among the CPU tmes are not that large. For these reasons, we consder the Rand-1000 famly where the graphs reflect ntermedate denseness. For ths famly, the results equal the results for the Rand1-4 famly, but the dfferences are easly apprecated. For example, GORC s three tmes slower than ELSCSPHU. In summary, when the SPRAND generator s consdered and the graphs are not sparse the best algorthm s ELSCSPHU snce t requres the least CPU tme and makes the smallest number of scan per nodes operatons. The second best under the same condtons s ELSCSPH. The behavor of the algorthms BFM, BFCT, SIMP, ELSCSPS and ELSCSPQ are smlar for dense graphs. Table 3: Rand-1000 famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU We used the SPGRID generator attrbuted to Cherkassky et al. [5]. We present results for Grd-SSquare (square grds) famly data (Table 4), Grd-SSquare-S (square grds wth an artfcal source) famly data (Table 5), Grd-PHard (layered graphs wth non-negatve arc lengths) famly data (Table 6) and Grd-PHard (layered graphs wth arbtrary arc lengths) famly data (Table 7). For the Grd-SSquare famly data the best algorthm s TWO_Q, followed by PALT. The performance of the BFCT, GORC, SIMP and ELSCSPx algorthms s also good. Among the ELSCSPx algorthms, ELSCSPS and ELSCSPQ are the best. When the Grd-SSquare-S famly data s consdered, the best algorthms are BFCT and SIMP. The performance of the GORC and ELSCSPx algorthms s also good. Once agan, among ELSCSPx algorthms, ELSCSPS and ELSCSPQ are the best. However, note that for both famles of grd graphs, the number of scans operatons per node performed by ELSCSPHU 19

20 algorthm s In other words, the behavor of ths algorthm matches the label settng algorthms. Table 4: Grd-SSquare famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Table 5: Grd-SSquare-S famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Also note that the graphs n both later grd famles are sparse. Therefore, the behavor ELSCSPHU algorthm can take advantage when the grds graphs become layered graphs. Ths stuaton s reflected n the followng tables. For the Grd-PHard famly data, the best algorthm s ELSCSPHU, and the performance of BFCT, SIMP and the ELSCSPx algorthms s also good. Moreover, when the experment wth Grd-NHard famly data s consdered, the better algorthms are PALT, TWO_Q, ELSCSPH and ELSCSPHU. BFCT, SIMP, ELSCSPS and ELSCSPQ are also good algorthms. In any case, n all famles of grd/layered graphs the ELSCSPHU algorthm, followed by ELSCSPH perform the least number of scan nodes operatons. 20

21 Table 6: Grd-PHard famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Table 7: Grd-NHard famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Fnally, we also used the SPACYC generator n Cherkassky et al. [5]. We present results for Acyc-Pos famly data (acyclc graphs wth non-negatve arc lengths) (Table 8) and Acyc- Neg famly data (acyclc graphs wth negatve arc lengths) (Table 9). For Acyc-Pos famly data the best algorthms are SIMP, BFCT, TWO_Q and BFM. The ELSCSPS, ELSCSPQ, PALT and ELSCSPH algorthms follow n order of performance. Fnally the worst algorthms are GORC and ELSCSPHU. However, once agan ELSCSPHU makes the fewest number of nodes scan operatons. The best algorthm for Acyc-Neg famly data s clearlygorc. It also makes the less number of nodes scan operatons. The next best algorthm s SIMP. The performance of the BFCT and ELSCSPx algorthms s smlar. The worst (very bad) algorthms are TWO_Q, PALT and BFM. 21

22 Note that the graphs n ths famly are sparse ( m < 4n n Table 8 and m < 5n n Table 9). Therefore our algorthms can not take advantage of the small number of nodes scan operatons that they perform. Table 8: Acyc-Pos famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Table 9: Acyc-Neg famly data. nodes/arcs BFM BFCT GORC SIMP PALT TWO_Q ELSCSPS ELSCSPQ ELSCSPH ELSCSPHU Conclusons. We address a novel label settng/correctng shortest path algorthm based on a new concept. The algorthm exhbts fresh characterstc n relaton wth the theoretcal complexty (least number of arcs to scan n each pass of the algorthm) and wth the dentfcaton of negatve cycles (two new effcent rules are reported). Moreover, a detaled scheme of ths algorthm s gven. Ths scheme offers evdence that the algorthm scans a subset of nodes together n each pass. Therefore, the orders n these nodes are consdered to be rrelevant from the theoretcal vew of pont. However, n the emprcal performance of the algorthm, the order that the new 22

23 pseudo permanently labeled nodes are scanned becomes relevant. In partcular, when the new pseudo permanently labeled nodes are consdered n non-decreasng order of ts label dstance, the algorthm performs the smallest number of scan nodes operatons among all exstent label correctng algorthms for all of famly data n our experment wth the excepton of acyclc graphs wth negatve arc lengths. In a future, we wll try to dentfy emprcal complexty as a functon of network parameters (for each generator) of these algorthms by regresson analyss, prevously dentfyng the bottleneck operatons of each algorthm n the practce. As n Cherkassky and Goldberg [4], we wll also consder studyng the effcency of the proposed negatve cycle rules. It wll also be necessary to study how the complexty of non-scan operatons n our algorthm can be reduced to take advantage of the mnmum number of nodes scan operatons that t makes. Acknowledgments We are very grateful to professor A. V. Goldberg by provde free access to the SPLIB-1.4 and SPC-1.2 lbrares n hs personal web page. Ths work has been partally supported by Spansh Government Research Proect MTM , and also receved contrbutons from European Funds of Regonal Development, and Canary Islands Autonomy Government Research Proect PI042004/078. References [ 1 ] R Ahua, T. Magnant, J. B. Orln, Network Flows, Prentce-Hall, [ 2 ] R. Bellman, On a Route Problem, Quart. Appl. Math. 16 (1958), [ 3 ] D. P. Bertsekas, A smple and fast label correctng algorthm for shortest paths, Networks 23 (1993), [ 4 ] B. V. Cherkassky, A. V. Goldberg, Negatve-Cycle Detecton Algorthms, Math. Program. 85 (1999), [ 5 ] B. V. Cherkassky, A. V. Goldberg, T. Radzk, Shortest paths algorthms: Theory and expermental evaluaton, Math. Program. 73 (1996), [ 6 ] E. W. Dkstra, A note on two problems n connecton wth graphs, Numer. Math. 1 (1959), [ 7 ] L. R. Ford, Network Flow Theory, The rand Corporaton Report P-923, Santa Monca, Calf,

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