Discrete Applied Mathematics. Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

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1 Dscrete Appled Mathematcs 7 (9) Contents lsts avalable at ScenceDrect Dscrete Appled Mathematcs journal homepage: Shortest paths n lnear tme on mnor-closed graph classes, wth an applcaton to Stener tree approxmaton Samak Tazar a,, Matthas Müller-Hannemann b a Technsche Unverstät Darmstadt, Department of Computer Scence, Hochschulstraße, 6489 Darmstadt, Germany b Martn-Luther-Unverstät Halle-Wttenberg, Department of Computer Scence, Von-Seckendorff-Platz, 6 Halle (Saale), Germany a r t c l e n f o a b s t r a c t Artcle hstory: Receved October 6 Receved n revsed form Aprl 8 Accepted 7 August 8 Avalable onlne 7 September 8 Keywords: Shortest path Mnor-closed graph classes Planar graph Stener tree Mehlhorn s algorthm We generalze the lnear-tme shortest-paths algorthm for planar graphs wth nonnegatve edge-weghts of Henznger et al. (994) to work for any proper mnor-closed class of graphs. We argue that ther algorthm can not be adapted by standard methods to all proper mnor-closed classes. By usng recent deep results n graph mnor theory, we show how to construct an approprate recursve dvson n lnear tme for any graph excludng a fxed mnor and how to transform the graph and ts dvson afterwards, so that t has maxmum degree three. Based on such a dvson, the orgnal framework of Henznger et al. can be appled. Afterwards, we show that usng ths algorthm, one can mplement Mehlhorn s (988) -approxmaton algorthm for the Stener tree problem n lnear tme on these graph classes. 8 Elsever B.V. All rghts reserved.. Introducton The sngle-source shortest-paths problem wth nonnegatve edge-weghts s one of the most-studed problems n computer scence, because of both ts theoretcal and practcal mportance. Djkstra s classcal algorthm [] has ever snce ts dscovery been one of the best choces n practce. Also from a theoretcal pont of vew, untl very recently, t had the best runnng tme n the addton-comparson model of computaton, namely O(m + n log n) usng Fbonacc heaps [] (we use n to denote the number of vertces of a graph and m for ts number of edges). Pette and Ramachandran [] mproved the theoretcal runnng tme n undrected graphs for the case when the rato r between the largest and smallest edge-weght s not too large. They acheve a runnng tme of O(mα(m, n) + mn{n log n, n log log r}), where α(m, n) s the very slowly growng nverse-ackermann functon. Goldberg [4] proposed an algorthm that runs on average n lnear tme. For the case of nteger edge-weghts, Thorup [] presented a lnear-tme algorthm n the word RAM model of computaton, where the bt-manpulaton of words n the processor s allowed. Hagerup [6] extended and smplfed Thorup s deas to work for drected graphs n nearly lnear tme. But the queston whether the standard addton-comparson model allows shortestpaths computaton n worst-case lnear-tme s stll open. For a farly recent survey about shortest-paths algorthms, see Zwck [7]. For planar graphs, Henznger et al. [8] presented the frst lnear-tme algorthm to calculate shortest-paths wth nonnegatve edge-weghts. Ther algorthm works on drected graphs. It s based on Frederckson s [9,] work who gave an O(n log n)-tme algorthm for ths case and whose dea was n turn based on Lpton and Tarjan s planar separator [] to decompose the graph. Henznger et al. frst decompose the graph nto a recursve dvson and then use ths dvson to relax the edges n a certan order that guarantees lnear runnng tme. They clam that ther algorthm can be adapted to Correspondng author. Tel.: ; fax: E-mal addresses: tazar@algo.nformatk.tu-darmstadt.de (S. Tazar), muellerh@nformatk.un-halle.de (M. Müller-Hannemann). 66-8X/$ see front matter 8 Elsever B.V. All rghts reserved. do:.6/j.dam.8.8.

2 674 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) work for any proper mnor-closed famly of graphs where small separators can be found n lnear tme. Recently, Reed and Wood [] mproved the quadratc-tme separator of Alon et al. [] and showed that all proper mnor-closed graph classes can be separated n lnear tme; so, we should be done. However, both Frederckson s algorthm and Henznger et al. s algorthm assume that the graph has maxmum degree ; whle ths property can be acheved easly for planar graphs, we argue that t can not be acheved by standard methods for arbtrary mnor-closed classes (n partcular, t can not be appled to apex graphs,.e. planar graphs augmented by a super-source ; these graphs have frequent applcaton n the lterature). We show how to buld an approprate recursve dvson of a graph from a proper mnor-closed famly n lnear tme by a nontrval extenson of the algorthm n [8]. Our algorthm works for graphs wth arbtrary degrees. But even after havng the recursve dvson, the shortest paths algorthm n [8] depends on the assumpton that the graph has bounded degree (and contans only a sngle source labeled ntally wth dstance zero, cf. apex graphs). Usng our recursve dvson, we show how to transform the graph and ts dvson to have maxmum degree, so that Henznger et al. s shortest-paths algorthm can be appled. Our modfcatons lead to the frst shortest-paths algorthm for all proper mnor-closed classes of graphs that runs n lnear tme n the addton-comparson model of computaton. We also consder the Stener tree problem, namely fndng the shortest tree that connects a gven set of termnals n an undrected graph. The Stener tree problem s also one of the most fundamental problems n computer scence and of the frst problems shown to be N P -complete by Karp [4]. Bern and Plassmann [] showed that t s even AP X-hard and the best-known nonapproxmablty result s due to Chlebík and Chlebíková [6] who showed a bound of 96/9.. Robns and Zelkovsky [7] presented an algorthm wth approxmaton guarantee + ln + ɛ. + ɛ whch s the best approxmaton algorthm for ths problem known so far. There s a well-known -approxmaton algorthm for ths problem [8,9] that s based on fndng the mnmum spannng tree of the complete dstance network of the set of termnals. Mehlhorn [] mproved the runnng tme of ths algorthm to O(m + n log n). The Stener tree problem n planar graphs s also N P -hard [] but very recently a polynomal tme approxmaton scheme (PTAS) has been found by Borradale et al. [,] for ths case. The runnng tme of the PTAS s O(n log n) wth a constant factor that s exponental n the nverse of the desred accuracy. As an applcaton of our shortest-paths algorthm, we show how to mplement Mehlhorn s [] -approxmaton algorthm n lnear tme on proper mnor-closed graph classes. No better tme bound than Mehlhorn s own mplementaton of O(m + n log n) has prevously been known even for planar graphs. An mportant observaton that we made s that Mehlhorn s dstance network s a mnor of the gven graph and thus, ts mnmum spannng tree can be calculated n lnear tme wth the algorthm of Mares [4] (or that of Cherton and Tarjan [] n the planar case). Our contrbuton and outlne The area of graph mnor theory has been constantly evolvng ever snce the graph mnor theorem of Robertson and Seymour [6] was announced n 988. Many mportant algorthms and meta-algorthms have been presented for large problem famles on mnor-closed graph classes and numerous theoretcal concepts have been developed to handle them. We present the frst lnear-tme algorthms for two fundamental graph-theoretc problems n these classes. Our contrbuton can be summarzed as follows: dentfyng that there s a gap n generalzng Henznger et al. s recursve dvson and shortest paths algorthms to all proper mnor-closed graph classes; argung that the gap can not be closed by standard methods; fllng n the gap usng deep results n graph mnor theory; (re)provng n detal the correctness of the modfed algorthm; showng how to modfy a graph and ts recursve dvson to obtan a bounded-degree graph and a recursve dvson wth the same propertes, resultng n the frst lnear-tme shortest-paths algorthm for proper mnor-closed graph classes; obtanng a useful applcaton, namely, the frst lnear tme Stener tree approxmaton, whch was prevously not even known for planar graphs. In Secton, we revew some needed concepts and prevous work; n Secton, we present our man result about shortest paths and n Secton 4, the applcaton to Stener tree approxmaton.. Prelmnares In ths secton, we revew some concepts and some prevous results that are needed n ths work. These nclude graph mnors, vertex parttonng, graph decomposton, and Henznger et al. s [8] sngle-source shortest-paths algorthm... Graph mnors A mnor of a graph G s a graph that s obtaned from a subgraph of G by contractng a number of edges. A class of graphs s mnor-closed f t s closed under buldng mnors. It s called a proper class f t s nether empty nor the class of all graphs. Examples of proper mnor-closed graph famles are planar graphs, bounded-genus graphs, and apex graphs. The semnal theorem of Robertson and Seymour [6] states that any proper mnor-closed class of graphs can be characterzed by a fnte

3 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) set of excluded mnors. Note that for a proper mnor-closed class of graphs, we can always consder the number of vertces l of the smallest excluded mnor and conclude that the complete graph K l s a partcular excluded mnor of the class. Thus, the class of K l -mnor-free graphs ncludes the consdered mnor-closed class of graphs. In the rest of ths work, we work wth K l -mnor-free graphs, where l s a fxed constant. It follows from a theorem of Mader [7] that K l -mnor-free graphs have constant average degree, for some constant dependng on l. Ths, n turn, mples that these classes of graphs are sparse,.e. we have m = O(n)... Vertex parttonng In [9], Frederckson presented a smple algorthm called FndClusters, based on depth-frst search, that gven a parameter z and an undrected graph wth maxmum degree, parttons ts vertces nto connected components each havng at least z and at most z vertces. Note that snce the algorthm gves us connected components, we can contract each one of them and get a mnor of the nput graph wth at most n/z vertces. Frederckson used ths algorthm to derve fast algorthms for the mnmum spannng tree and shortest-paths [] problems. If a weghted graph does not have maxmum degree, one can apply the followng transformaton: replace a vertex v of degree d(v) wth a zero-weght path of length d(v), such that each edge ncdent to v s now ncdent to exactly one vertex of the path,.e. we can splt v usng zero-weght edges. A smlar transformaton can be appled to drected graphs, too, usng an addtonal zero-weght edge to complete a drected cycle. If the gven graph s embedded n a surface, one can order the edges around the path/cycle n the same way they were ordered around the correspondng vertex n the gven embeddng. Ths way, the transformed graph wll also be embedded n the same surface. However, for an arbtrary mnor-closed class of graphs (e.g. apex graphs), t mght not always be possble to reman n the class after transformng the graph ths way, see Secton. But Frederckson s FndClusters depends on the graph havng bounded degree. Any constant bound would suffce for our purposes but n general such a bound does not exst for arbtrary mnor-closed graph famles. Reed and Wood [] ntroduced an alternatve parttonng concept that can be appled to a graph G = (V, E) wth arbtrary degrees excludng a fxed mnor. Consder some parttonng P = {P,..., P t } of the vertex set V. Let H = (V H, E H ) be the graph obtaned by collapsng every part P of G nto a sngle vertex v V H ( t) and removng loops and parallel edges. Ths way, there s an edge between two vertces v and v j of H f and only f there s an edge between a vertex of P and a vertex of P j n G ( < j t). We say P s a connected H-partton of G f v v j E H f and only f there s an edge of G between every connected component of P and every connected component of P j. Reed and Wood proved the followng lemma : Lemma. ([]). There s a lnear-tme algorthm that gven a constant z and a graph G excludng a fxed K l -mnor, outputs a connected H-partton P = {P,..., P t } of G such that t n/z, and P < c z for all t, where c s a constant dependng only on l. Note that by contractng each connected component of each P n G to a sngle vertex, one gets a graph that contans an somorphc copy of H as a subgraph and so, H s a mnor of G and n partcular, s also K l -mnor-free. Hence, when dealng wth graphs wth no bounded degree, Lemma. can be used nstead of FndClusters to partton the graph and reduce ts sze whle keepng t free of some fxed mnor. Corollary.. Let G be a graph wth n vertces excludng a fxed K l -mnor, and let c = l +l be a fxed constant. There exsts a lnear-tme algorthm H-Partton(G, z, l) wth the followng propertes: t parttons the vertces of G nto at most n/z sets; each set has at most c z vertces; t collapses each set nto a sngle vertex, creatng a new graph G ; G s a mnor of G wth at most n/z vertces... Graph decomposton A balanced vertex-separaton of a graph G = (V, E) s gven by two sets A and B, such that A B = V, there s no edge between A \ B and B \ A, and each one of A and B contans at most an α-fracton of the vertces (for some / α < ). The sze of the separaton s A B. For a functon f, a subgraph-closed class of graphs s sad to be f -separable f every n-vertex graph n the class has an O(f (n))-sze separator. Reed and Wood [] showed that all K l -mnor-free graphs are f -separable n lnear tme for f (n) = O(n / ). For planar graphs, one can use the semnal planar separator theorem of Lpton and Tarjan [] that delvers an O( n)-separator n lnear tme. An (r, s)-dvson of an n-vertex graph s a partton of the edges of the graph nto O(n/r) regons, each contanng r O() vertces and each havng at most s boundary vertces (.e. vertces that occur n more than one regon). For a nondecreasng In ther lemma, we substtute c := l +l and z := k/c.

4 676 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) postve nteger functon f and a postve nteger sequence r = (r, r,..., r k ), an (r, f )-recursve dvson of an n-vertex graph s defned as follows: t contans one regon R G consstng of all of G. If G has more than one edge and r s not empty, then the recursve dvson also contans an (r k, f (r k ))-dvson of G and an (r, f )-recursve dvson of each of ts regons, where r = (r, r,..., r k ). A recursve dvson can be represented compactly by a recursve dvson tree, a rooted tree whose root represents the whole graph and whose leaves represent the edges of the graph. Every nternal node represents a regon, namely, the regon nduced by all the leaves n ts subtree. The chldren of a node of the tree are ts mmedate subregons n the recursve dvson. Usng Frederckson s parttonng [9] and dvson [] methods, Henznger et al. [8] present a lnear-tme algorthm to fnd certan recursve dvsons n planar graphs: they determne a vector r and an (r, cf )-recursve dvson of the graph for some constant c, such that the nequalty ( ) r f (r ) + 8 f (r ) log r + log r j () j= s satsfed for all r s exceedng a constant. The obtaned recursve dvson tree has O(n) nodes and ts depth s roughly O(log n). The dea of the algorthm s as follows: frst, teratvely reduce the sze of the graph by parttonng the vertces of the graph (usng Frederckson s FndClusters) and buldng mnors; then, workng backwards, fnd (r, s)-dvsons of the smaller graphs (for approprate values of r and s), mposng dvsons on the larger graphs and at the same tme buldng the recursve dvson tree. Snce the tme-consumng calculaton of (r, s)-dvsons s done on the smaller graphs, they succeed to prove that the overall tme complexty s lnear..4. Sngle-source shortest-paths on planar graphs Henznger et al. prove the followng theorem: Theorem. ([8]). Let a graph G be gven wth maxmum n-/outdegree and assume that G s equpped wth an (r, cf )-recursve dvson tree, for some constant c, so that nequalty () s satsfed for all r s exceedng a constant. Then, the sngle-source shortestpaths problem wth nonnegatve edge-weghts can be solved on G n lnear tme. To prove ths theorem, they use a complcated chargng scheme that also depends on the graph havng a sngle source and bounded degree. Together wth the result from the prevous subsecton, t follows that sngle-source shortest-paths wth nonnegatve edge-weghts can be calculated n lnear-tme on planar graphs.. Sngle-source shortest paths on mnor-closed graph classes In ths secton, we prove our man theorem about shortest paths: Theorem.. In every proper mnor-closed class of graphs, sngle-source shortest-paths wth nonnegatve edge-weghts can be calculated n lnear tme. Frst, we argue that the degree requrement of Henznger et al. s algorthm can not be fulflled by standard methods for arbtrary mnor-closed classes of graphs. By standard methods we mean splttng a vertex usng zero-weght edges untl the desred degree bound s reached. In Secton. we dscussed a partcular way of splttng vertces that can be appled to embedded graphs. In ths secton, we show that there exst K l -mnor-free graphs, so that no matter how we splt the vertces, the resultng graph wll nclude a mnor whose sze can not be bounded by a functon n l. The key les n the observaton that splttng an apex mght ntroduce arbtrarly large mnors. Ths s a well-known fact n graph mnor theory [8]. For the sake of completeness, we nclude a short proof below. Apces are a fundamental part of mnor-closed graph classes as s demonstrated by the powerful graph-decomposton theorem of Robertson and Seymour [8]. Ths theorem shows, n a sense, that at most a bounded number of apces are allowed n these classes; and ntutvely, splttng an apex wth unbounded degree mght result n an unbounded number of apces and s thus not allowed n general. Proposton.. For every k N, there exsts a K 6 -mnor-free graph G k, so that, f the vertces of G k are splt n any way to acheve a maxmum degree of, the resultng graph G k ncludes a K k-mnor. Proof (Sketch). We defne G k to be a suffcently large planar grd-graph augmented by an apex as follows: consder a sequence S of numbers between and k, so that each possble par of these k numbers s at least once adjacent n S. Let t < k be the length of ths sequence. Choose a set W of t vertces n the grd that are suffcently far away from each other and add an apex v connected to these t vertces. Ths completes the defnton of G k, whch s clearly K 6 -mnor-free. Now, no matter how we splt the vertces of G k, the apex v wll become a path of t vertces, each one connected to exactly one vertex of W. Ths path mposes an order on the vertces n W. We label the vertces n W accordng to ths order usng the sequence S. Let W be the set of vertces n W labeled by ( k). For each, construct a tree T that connects the vertces

5 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) Fg.. A smplfed example for the proof of Proposton.: the apex of the graph n (a) s splt, resultng n the graph (b); the vertces are labeled and connected accordng to the dsjont trees n (c); contractng the thck edges n (b) results n a K -mnor (d). of W n the planar grd. Note that f the grd s suffcently large and the vertces n W are suffcently far away from each other, t s easly possble to choose the trees T to be all dsjont. Let U be the set of edges connectng the vertces n W wth the path resulted from splttng v. Now, f we contract the trees T and the edges n U and delete redundant edges, what remans s a K k -mnor (see Fg. )... Our generalzed recursve dvson algorthm Our modfed algorthm s gven n Algorthm.. Our modfcatons are only n three places but as we already dscussed, they are essental to make the algorthm work for all proper mnor-closed graph classes. In ths subsecton, we dscuss the algorthm and our modfcatons thereof n detal and n the next subsecton, we present ts proof of correctness.

6 678 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) Let the nput graph be G = (V, E). In the frst phase of the algorthm, the nput graph s reduced n sze by buldng mnors. Specfcally, startng wth G = G, a sequence of graphs G, G,..., G I+ s bult, so that for > j, G s a mnor of G j and G I+ s the frst graph n the sequence havng less than n/ log n vertces. To ths end, a sequence of parameters z s used to specfy the sze of the next graph n the sequence as follows: let n be the number of vertces of G. In the orgnal algorthm, the sequence s defned as z = and z + = 7 z/ and has the effect that G s parttoned nto at most n /z connected components, each one havng at most z vertces (usng Frederckson s FndClusters [9]). Each of these components s contracted to construct G +, a mnor of G wth n + n /z vertces. Our frst two changes occur n ths phase of the algorthm. Frst, nstead of usng Frederckson s FndClusters, we make use of the H-Partton procedure of Reed and Wood [], to acheve a smlar effect wthout dependng on the graph havng bounded degree (cf. Secton.). Secondly, we had to change the defnton of the z s to be z = and z + = 4 z/7. Ths s due the fact that n order to prove nequalty () n our case, we need the exponent of z to be nstead of ; but then, n 7 order to ensure that the z s stll grow (extremely fast) towards nfnty, the base of the exponentaton had to be changed from 7 to 4, too. Indeed, 4 s the smallest nteger that can be used, so that the z s grow towards nfnty. Now, usng the H-Partton procedure as n Corollary., we stll have that n + n /z but now, each vertex of G + represents at most c z vertces of G (nstead of the orgnal z ). In the second phase, the algorthm works backwards from G I+ towards G, buldng (r, s)-dvsons and a recursve dvson tree as follows: t starts wth the trval dvson D I+ of G I+ as a sngle regon and ntalzes the recursve dvson tree T wth a sngle node v G. Then, for each G, t consders each regon R of G + and bulds an (r, s)-dvson on t (wth approprate values of r and s defned below); each resultng regon R of R (of G + ) s turned nto a regon R of G by expandng every vertex nto the at most c z vertces t represents n G ; afterwards, a chld v R of v R s added to T. The dvson D s defned to be the decomposton of G by the regons R obtaned ths way. Note that a boundary vertex s expanded n multple regons, creatng multple copes of the edges t expands to; there should be only one copy of these edges and ths may be acheved by assgnng them to one of these regons arbtrarly. It remans to specfy how exactly and wth what parameters the (r, s)-dvson s bult n the teraton above. Ths s the thrd place where our algorthm dffers from the orgnal. The orgnal algorthm uses a modfed verson of Frederckson s (r, s)-dvson algorthm [], called Dvde, as follows: t takes three parameters G, S and r and dvdes the edges of an n- vertex graph G nto at most c ( S / r + n ) regons, each one havng at most r vertces and at most c r r boundary vertces, where c and c are constants; a vertex s consdered as a boundary vertex f () t belongs to more than one regon, or () t belongs to the set S. Internally, the lnear-tme planar O( n)-separator of Lpton and Tarjan [] s used to acheve the desred dvson. We make use of the lnear-tme O(n )-separator of Reed and Wood [] nstead; our Dvde procedure takes four parameters G, S, r, and l and has the propertes specfed n Lemma.4 below; as t can be seen n the lemma, a number of constants and exponents are changed. The parameter l s a constant taken to ndcate the fxed excluded K l - mnor. In the last phase of the algorthm, the edges of each regon R of D are added as chldren of the node v R to the recursve dvson tree T. Ths completes the descrpton of our generalzed algorthm... Correctness of our generalzed recursve dvson algorthm Theorem.. Algorthm. s a lnear-tme algorthm that gven a K l -mnor-free graph G, fnds an (r, f )-recursve dvson of G that satsfes nequalty () for all r exceedng a constant and whose recursve dvson tree has O(n) nodes. The proof of the correctness of the algorthm follows the proof of Henzger et al. [8] very closely. For the sake of completeness, and snce a number of subtle detals and calculatons have to be flled n and replaced at several places, we have ncluded the full proof n ths secton; only the proof of Lemma.4 s left for the Appendx. Ths lemma shows the correctness of the Dvde procedure and s based on the orgnal proof of Frederckson []; Lemmas..7 step-by-step complete the proof of Theorem.. Lemma.4. Replacng the planar separator n Frederckson s Dvde procedure [] wth the separator algorthm of Reed and Wood [] causes the Dvde (G, S, r, l) procedure to work as follows (where G s a graph wth n vertces and excludes K l as a mnor and c and c are constants dependng only on l): t dvdes G nto at most c ( S /r + n ) regons; r each regon has at most r vertces; each regon has at most c r boundary vertces, where the vertces n S also count as boundary; t takes tme O(n log n). Recall that we start wth the graph G = G and repeatedly apply the procedure H-Partton to each G to obtan G +. For each, let n denote the number of vertces of G. Afterwards we work our way back from G I+ to G and obtan a dvson D on each G. Let k denote the number of regons of D. Note that the recursve dvson tree T has depth I +.

7 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) The followng proof has four parts. Frst, we show that each regon of the dvson D has at most O(z ) vertces and at most O(z ) boundary vertces. Second, we show that the number k of regons s O(n /z ). Thrd, we show that Algorthm. takes lnear tme and fnally, we show that the dvson fulflls nequalty (). For notatonal convenence, let z I+ = n I+, so the sngle regon of the dvson D I+ of G I+ has z I+ vertces. Consder teraton I n the second phase of the algorthm. By the correctness of Dvde, the decomposton D R of a regon of D + conssts of regons R of sze at most z. By the correctness of H-Partton, each vertex of G + expands to at most c z vertces of G. Hence, each regon R obtaned from R by expandng ts vertces has sze at most c z. Smlarly, each regon R has at most c z boundary vertces. boundary vertces by the correctness of Dvde, so the correspondng regon R has at most c c z Lemma.. The number k of regons n the dvson D s O(n /z ). Proof. We show by reverse nducton on that k c n /z for all, where and c are constants to be determned. For the base case, we have k I+ =. Consder teraton I n the second phase, and suppose. The regons of D are obtaned by subdvdng the k + regons comprsng the dvson of G +. Snce n + n /z and z + z, we have by the nducton hypothess that k + c n + /z + c n /z. () Each regon R of the dvson of G + has S R c c z+ boundary vertces. Summng over all regons R n D +, we obtan n R = (# of nonboundary vertces + # of boundary vertces) R R n + + R c c z+ n + + c c k + z. + () For each regon R, by correctness of Dvde, the number of subregons nto whch R s dvded s at most c ( S R /z + n R /z ), whch s n turn at most c (c c z+ /z + n R /z ). Summng over all such regons R and usng () and (), we nfer that the total number of subregons s at most R c (c c z+ /z + n R /z ) = c c c k + z c c c k + z+ /z ( ) c n + c c c + /z + c n R /z z + c c c c n + /(z c c c c n /(z z R + c (n + + c c k + z+ )/z ( + c n + /z + c c c z + /z z c n + z + + ) + c n + /z + c c c c n + /(z z ) ) + z+ /z ) + + c n /z + c c c c n /(z z ) + (4) where n the last lne we use the fact that n + n /z. We have obtaned an upper bound on the total number of subregons nto whch the regons of D + are dvded. Each subregon becomes a regon of D. Thus, we have n fact bounded k, the number of regons of D. To complete the nducton step, we show that each of the three terms n (4) s bounded by c n /z. The second term, c n /z, s bounded by c n /z f we choose c c. The thrd term s smaller than the frst term. As for the frst term, recall that z + = 4 z/7. For suffcently large choce of, we can ensure that mples z + c c c /z. Thus, the frst term s also bounded as desred. We conclude that k c n /z, completng the nducton step. We have shown ths nequalty holds for all. As for <, clearly k (z )n /z (z )n /z. Thus, by choosng c to exceed the constant z, we obtan the lemma for every. Lemma.6. The algorthm runs n lnear tme. Proof. The tme requred to form the graphs G, G,..., G I+ s O( n/z ), whch s O(n). For I, the tme to apply Dvde to a regon R of G + wth n R vertces s O(n R log n R ). Each such regon has O(z + ) vertces, so the tme s O(n R log z + ). Summed over all regons R, we get R O(n R log z + ) = O(n + log z + ). The tme to obtan the nduced dvson of G s O(n ).

8 68 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) Thus, the tme to obtan dvsons of all the G s s O(n 7 + log z + ). Snce n + n /z n/z and log z + = O(z ), the sum s O(n). Lemma.7. The recursve dvson obtaned by Algorthm. satsfes nequalty (). Proof. Frst, note that combnng the nequaltes n + n /z, we obtan n n/ j< z j. () Note moreover that each vertex of G expands to at most j< c z j vertces of G. Consder the dvson D of G, and the dvson t nduces on G. The dvson D conssts of O(n /z ) regons, each havng O(z ) vertces and O(z ) boundary vertces. Ths nduces O(n /z ) regons n G, each consstng of O(z j< c z j ) vertces and O(z j< c z j ) boundary vertces. Let r = z j< z j and defne f (r ) = z c z j. j< Then, by (), the nduced dvson of G has O(n/r ) regons each wth O(r c ) vertces and O(f (r )) boundary vertces. Snce c = O( j z j), we get that the number of vertces per regon s O(r ). We have (6) r f (r ) = z z c = z c. (7) Usng the defnton of z, one can verfy that z = θ(log 7 z ) and j< z j = O(log 8 z ). Hence f (r ) = c We also have log r + = log z ( j< z + j z j = c O(log z log 8 log z ). (8) ) z j = O(log(z + log8 7 z + )) = O(log z + ) = O(z ) (9) + and consequently j= log r 7 j = O(z ). For a suffcently large constant, we have for all, ( ) + 8 f (r ) log r + log r j 8 c O(log z log z )O(z )O(z ) j= = 8 c 7 O(z log z ) z = r c f (r ), () snce the z s grow much faster than any exponental functon havng a constant n the base; specfcally, we can see below that z g log z for any constant g f s larger than a constant: log z log g + log log z 7 log 4z log g + log log 4 z 7 7 z g 7 + g log z + g, () 7 for some constants g, g, and g. And the last nequalty s true f s large enough, snce z nequalty () s fulflled for all r exceedng the constant r. grows much faster than. So,

9 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) Fg.. (a) A gven graph wth regons ndcated by dfferent lne-styles and shaded boundary vertces; (b) the transformed graph, such that every vertex has degree at most ; the number of boundary vertces of each regon has exactly doubled... Establshng the degree requrement After havng computed a recursve dvson, we stll have to transform the graph to have maxmum degree ; otherwse, Theorem. can not be appled, see Secton.4. We can acheve ths, usng our recursve dvson, by the followng lemma. Note that accordng to Proposton.. the resultng graph mght not be K l -mnor-free but t wll stll serve our purpose of fndng shortest paths n lnear tme, snce t s now accompaned by a recursve dvson satsfyng nequalty (). Lemma.8. Let G be an edge-weghted drected graph excludng a fxed mnor and let T be a recursve dvson tree representng an (r, f )-recursve dvson of G. Then one can replace every vertex of G wth a zero-weght cycle to obtan a graph G and at the same tme modfy T nto a tree T, so that G has n-/outdegree at most and T represents an (r, f )-recursve dvson of G. Ths modfcaton takes lnear tme. Proof. Recall that the leaves of T represent the edges of G and that nternal nodes of T correspond to regons of G, namely, the regon nduced by all the leaves n the subtree of that node. We modfy G and T at the same tme. Frst, for every vertex v of G wth degree d(v) (the sum of the ndegree and outdegree), we add new vertces v,..., v d(v) to G. We do an n-order traversal of T and for every leaf of T representng an edge e = vw of G, we do the followng: let e be the th edge of v and the jth edge of w that we encounter. We change the endponts of e to be the vertces v and w j and add two new zero-weght edges v v + and w j w j+ as sblngs of e to T (f = d(v), we use v d(v) v nstead; same for w). Ths way, every vertex v of G s replaced by a zero-weght cycle (v,..., v d(v) ) (see Fg. ). The orgnal vertces of G wll become solated and can be removed. We call the resultng graph G and the modfed recursve dvson tree T. Note that snce T has sze O(n), ths procedure takes only lnear tme. Also note that we only added new leaves to T and thus, the nternal nodes of T and T correspond one-to-one to each other. Now consder an nternal node q of T. It represents a regon R of G and corresponds to a node q of T, representng a regon R of G. R has r O() vertces and O(f (r)) boundary-vertces. The number of edges of R s at most three tmes as large as n R and the number of vertces s proportonal to the number of edges of R. But R s a subgraph of G, excludes the same fxed mnor and thus, the number of ts edges s lnear n the number of ts vertces. Hence, R stll has r O() vertces and edges. Also, snce R s represented by the subtree rooted at q, ts edges were traversed n order whle buldng T and G. So, every vertex v n R s replaced by a path v, v +,..., v j wth j d(v) n R. Thus, f v s a boundary vertex of R, then nstead, we have v and v j as boundary vertces of R. So R has at most twce as many boundary vertces as R,.e. stll O(f (r)) (see Fg. ). So, T represents an (r, f )-recursve dvson of G. Proof of Theorem.. Note that up to the choce of the start- and endvertex nsde the zero-weght cycles of G, shortest paths n G and G correspond one-to-one to each other. G fulflls all the requrements of Theorem. and combnng ths wth Theorem., and Lemma.8, we obtan our man theorem, namely, Theorem.. 4. Stener tree approxmaton We show how to mplement Mehlhorn s -approxmaton algorthm for the Stener tree problem [] n lnear tme on proper mnor-closed graph classes usng the result above and the observaton that Mehlhorn s dstance network s a mnor of the nput graph. Frst, we brefly revew Mehlhorn s algorthm and then we present our mplementaton. 4.. Overvew of Mehlhorn s algorthm Gven an n-vertex graph G = (V, E) wth nonnegatve edge-weghts and a vertex-subset K of termnals, one can determne a Stener tree of K n G as follows: frst, buld Mehlhorn s dstance network N = (K, D E D ), a specal graph defned on the set of termnals, n whch every edge corresponds to a path n G. To calculate N D, we frst have to partton the graph nto Vorono regons wth respect to the set of termnals K. Every vertex of the graph belongs to the Vorono regon of ts closest termnal (f a vertex happens to have the same dstance to more than one termnal, t should belong to the Vorono regon of the termnal wth the smallest ndex). Vorono regons n graphs can be calculated easly usng a shortest-paths

10 68 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) computaton: add a super-source s to the graph and connect t to every termnal wth a drected zero-weght edge; fnd the shortest paths from s to every vertex and then remove s from the resultng shortest-paths tree. The tree falls apart nto K connected components, each havng a termnal as ther root. These components correspond exactly to the Vorono regons of the termnals. Usng Djkstra s algorthm, one obtans a runnng tme of O(n log n + m) for general graphs. In the dstance network N D, there exsts an edge between two termnals u and v f and only f there exsts an edge between two vertces x and y n G, so that x belongs to the Vorono regon of u and y belongs to the Vorono regon of v. The weght of such an edge s the length of the shortest such paths connectng u and v. Once the Vorono regons of G wth respect to K are determned, N D can be constructed n lnear tme usng bucket sort. After the dstance network N D s determned, one can fnd ts mnmum spannng tree and replace every edge wth the correspondng path n G. Mehlhorn shows that the resultng graph s ndeed a tree and ts weght s at most ( K ) tmes the weght of the mnmum Stener tree of K n G. The mplementaton he offers runs n tme O(n log n + m) for general graphs. 4.. A lnear-tme mplementaton for proper mnor-closed classes Theorem 4.. There s a lnear-tme algorthm that calculates a -approxmaton for the Stener mnmum tree problem n any proper mnor-closed class of graphs. We frst show how to fnd the Vorono regons n lnear tme. In graphs excludng a fxed mnor K l, we observe that the graph wth an added super-source wll exclude K l+ ; so, Theorem. apples and shortest paths can be calculated n lnear tme. Alternatvely, usng a smlar method as n Secton., one can frst fnd a recursve dvson of G and then add the super-source and ts edges to G and to the recursve dvson. Ths could result n much better constants n the runnng tme of the algorthm, especally for planar graphs. We get Lemma 4.. For a graph G excludng a fxed mnor and havng nonnegatve edge-weghts and a gven set of termnals K n G, the Vorono regons of G wth respect to K can be determned n lnear tme. Corollary 4.. In a proper mnor-closed class of graphs, the dstance network N D can be calculated n lnear tme for any gven set of termnals n a gven graph from the class. The next step of Mehlhorn s algorthm s to calculate the mnmum spannng tree of N D. But notce that N D s obtaned by contractng the Vorono regons of the graph (whch are connected) and removng loops and parallel edges,.e. Observaton 4.4. For a gven graph G and a set of termnals, the dstance network N D s a mnor of G. Thus, N D belongs to the same proper class of mnor-closed graphs as G and one can apply the lnear-tme mnmum spannng tree algorthm of Mares [4]. When we are dealng wth planar graphs, the algorthm of Cherton and Tarjan can be used []. As mentoned before, the last step of Mehlhorn s algorthm s to replace the edges of N D wth the correspondng paths from G and ths can clearly be done n lnear tme. Hence, Theorem 4. s proven. Acknowledgement The frst author was supported by the Deutsche Forschungsgemenschaft (DFG), grant MU48/-. Appendx. Proof of Lemma.4 Lemma A.. Replacng the planar separator n Frederckson s Dvde procedure [] wth the separator algorthm of Reed and Wood [] causes the Dvde (G, S, r, l) procedure to work as follows (where G s a graph wth n vertces and excludes K l as a mnor and c and c are constants dependng only on l): t dvdes G nto at most c ( S /r + n ) regons; r each regon has at most r vertces; each regon has at most c r boundary vertces, where the vertces n S also count as boundary; t takes tme O(n log n). Proof. In the followng, when we refer to boundary vertces, we mean vertces that belong to more than one regon or vertces that belong to the set S. The Dvde procedure works as follows: assgn weght to each vertex of G and fnd a n O(n )-separator n G; recursvely apply the separaton algorthm to each regon wth more than r vertces. Now each regon has at most r vertces. Whle there s a regon wth more than c r boundary vertces, do the followng: f such a regon has n boundary vertces, assgn weght n to each of them, assgn weght zero to the other vertces of that regon and apply the

11 S. Tazar, M. Müller-Hannemann / Dscrete Appled Mathematcs 7 (9) separator theorem. In the end, all regons wll have the desred propertes and the algorthm takes tme O(n log n). It remans to show the bound on the number of the regons. Consder the dvson before the regons are further splt to enforce the requrement on the number of boundary vertces (.e. just when we have acheved that each regon has sze at most r). Let V B be the set of vertces that are ncluded n more than one regon. For a vertex v V B, let b(v) be one less than the number of regons that contan v n the dvson. Let B(n, r) be the total of b(v) over all vertces v V B. Thus, B(n, r) s the sum of the number of vertces v V B weghted by the count b(v). From the separaton theorem n [], we have the followng recurrence: B(n, r) d n + B(αn, r) + B(( α)n, r) for n > r, B(n, r) = for n r (A.) where d s a constant and α. We clam that B(n, r) d n r d n for n r, (A.) wth some constants d and d. The clam can be shown by nducton: As the base of the nducton, we consder the cases r n r. Note that snce after splttng a regon, each subregon stll has at least one-thrd of the total vertces, t s suffcent to only consder graphs wth at least r/ vertces. By choosng d d, we have d n r d n n = d n d n d n = B(n, r). (A.) For the nductve step,.e. for n > r, we have r B(n, r) d n αn + d d α n ( α)n + d d ( α) n = d n r d n r r r + n (d d α d ( α) ) d n f we choose d d α + d ( α) d. Ths can be acheved by settng d = d d. α +( α) In partcular, we have shown so far that B(n, r) = O(n/r ). The sum of the number of vertces n each regon s n + B(n, r) = n + O(n/r ) and each regon has Θ(r) vertces, so the number of regons we have so far s Θ(n/r). Let t be the number of regons wth boundary vertces (recall that n our defnton, the set of boundary vertces s V B S). We have t = v VB(b(v) + ) + S \ V B < B(n, r) + S = O(n/r + S ). (A.) Let s() be an upper bound on the number of splts that have to be appled to a graph wth at most r vertces and boundary vertces, untl each of ts regons has at most c r boundary vertces, for a constant c to be determned. We have that (A.4) s() s(α + d r ) + s(( α) + d r ) + for > c r s() = for c r (A.6) where α. We clam that s() d c r d d for c r c for some constant d. We prove our clam by nducton. Lke n the prevous nducton, for the base case we may assume c r c r. By choosng d = and c = 8d, we have d c r d d c 4d 8d = = s(). (A.8) (A.7)

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