Multiple Source Shortest Paths in a Genus g Graph

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1 Mltiple Sorce Shortest Paths in a Gens g Graph Sergio Cabello Erin W. Chambers Abstract We gie an O(g n log n) algorithm to represent the shortest path tree from all the ertices on a single specified face f in a gens g graph. From this representation, any qery distance from a ertex in f can be obtained in O(log n) time. The algorithm ses a kinetic data strctre, where the sorce of the tree iteratiely moes across edges in f. In addition, we gie applications sing these shortest path trees in order to compte the shortest non-contractible cycle and the shortest nonseparating cycle embedded on an orientable -manifold in O(g 3 n log n) time. 1 Introdction A shortest path tree (sp-tree) in a graph is a tree containing shortest paths from some ertex to all other ertices. Sp-trees are a fndamental tool on graphs, and hae applications for flows, distance qeries, connectiity, and many other problems. In the planar graph setting, there are many reslts for mltiple sorce shortest paths. The reslt of primary interest for the prposes of this paper is Klein [9], who addressed the problem of maintaining the sp-tree in a plane graph as the sorce of the tree moes along some face in the graph. He gae an algorithm that represented the sp-tree for all ertices on a common face in the graph in O(n log n) time. Other noteworthy reslts on mltiple sorce shortest paths inclde Frederickson s all-pairs shortest paths representation [6], Lipton and Tarjan s planar separator theorem [11], and Schmidt s O(n log n) algorithm which spports distance qeries for specific sbsets of ertices on a grid [13]. In this paper, we gie an algorithm to maintain the sp-tree as the sorce of the tree moes arond a face on a gens g graph, that is, a graph embedded in an orientable srface of gens g. This reslt extends Department of Mathematics, IMFM, and Department of Mathematics, FMF, Uniersity of Ljbljana, Sloenia, sergio.cabello@fmf.ni-lj.si. Partially spported by the Eropean Commnity Sixth Framework Programme nder a Marie Crie Intra-Eropean Fellowship, and by the Sloenian Research Agency, project J Department of Compter Science, Uniersity of Illinois, erinwolf@ic.ed. Research partially spported by an NSF Gradate Research Fellowship and by NSF grant DMS Klein s reslt on plane graphs, and the rnning time to compte the sp-tree for eery ertex on a gien face becomes O(g n log n). In Klein s algorithm, the sorce of the shortest path tree is moed to a neighbor of a ertex, and then the sptree is pdated edge by edge based on a set of candidate edges which shold now be in the sp-tree. The next edge from the set is chosen based on what Klein refers to as the leafmost nrelaxed edge, which exploits the property that the edges not present in the sp-tree form a tree in the dal graph. A main obstacle to extend this idea to gens g graphs is that the strctre of the edges not present in the sp-tree is more complex, and in particlar the concept of leafmost nrelaxed edge does not carry on. This makes harder to find ot which edges hae to be added and remoed from the sp-tree, as well as in which order these operations shold be done. Or approach consists of looking at the scenario like a kinetic sp-tree, where we maintain the sp-tree as the sorce moes continosly along an edge. Edges to add and remoe from the sp-tree are discrete eents which we compte dring the corse of this moement. (For more backgrond on kinetic data strctres in general, see [7].) As applications, we se or algorithm to find in an embedded graph a shortest non-contractible in O((g + b)g n log n) time, and a shortest non-separating cycle in O(g 3 n log n) time. Here b is the nmber of bondaries of the srface. The best preios reslts for this problem were O(n log n) by Erickson and Har- Peled [5], O(g 3/ n 3/ log n) by Cabello and Mohar [1], and O(g O(g) n log n) by Ktz [10]. Or approach improes them for a large range of ales of g. The main tool we se is the self-adjsting top tree [16]. A reiew of this data strctre along with necessary topological backgrond is proided in Section. Section 3 describe the maintenance of the shortest path tree when the root moes along an edge. First, we present the planar case to illstrate the necessary concepts, and then gie the algorithm on a graph of gens g. Section 4 gies the analysis of the time sed when the root moes along a face. Section 5 gies the algorithms to compte the shortest non-separating and non-contractible cycles on an embedded graph.

2 Backgrond.1 Self-adjsting top trees. For or algorithm, we reqire a dynamic forest data strctre spporting edge insertions and deletions. In addition, each ertex of the forest will maintain a nmber which is its distance to some ertex in the graph. Updates to this distance data mst be spported on both sbtrees and paths within the tree. The data strctre sed here is the self adjsting top tree [16]. Operations sed in or application are join, ct, and expose. We briefly describe these operations. The operations join(, ) and ct(, ) respectiely add an edge between and in different components of the forest and delete the edge between and. The operation expose(, ) makes the path between and in the forest (if it exists) the top-leel path in the data strctre, allowing qick pdates and searches in this path. All of these operations can be spported in O(log n) amortized time.. Graphs. Throghot the paper, we work on a weighted, embedded graph G = (V, E), where V is the ertex set and E is the edge set. The weight of an edge is denoted w(). The dal graph of G, written G, consists of the graph gien by making eery face of G a ertex, and adding edges between adjacent faces. For e E, we se e to denote the edge in the dal graph which goes between the two faces that e borders. As in Klein s paper, we define d T : V IR as the distance in a rooted directed tree T from the sorce of the tree to a gien ertex. (We omit T when the tree is clear from context.) Define the tension of an edge as t( ) = d() w( ) d() (for more examples, see [15]). We say the edge is tense if t( ) > 0. In other words, if is tense, there is a shorter path to which ses a path in the tree to pls the edge. It is a simple exercise to erify if a tree T on G leaes no tense edges in (E \ T ), then T is a sp-tree. For simplicity, we assme that all shortest paths in G are niqe. As pointed ot in [5], this condition can be obtained with high probability: the Isolation Lemma of Mlmley, Vazirani, and Vazirani [1] implies that adding an infinitesimal weight ε r(e) to each edge e, where r(e) is chosen from [1,,..., n ] niformly at random, makes all shortest paths niqe with probability at least 1 1/n..3 Topological backgrond. We briefly present some concepts and definitions from algebraic and combinatorial topology. For more detailed backgrond, see also [14] and [17]. A srface, or -manifold with bondary, is a topological Hasdorff space where each point has a neighborhood homeomorphic to either the plane (if the point is interior) or the closed half-plane (if the point is on the bondary). Srfaces considered here are connected and compact, and are orientable nless otherwise specified. Any connected compact orientable srface is homeomorphic to a sphere with g handles and b open disks remoed. We say g is the gens of the srface and b is the nmber of bondaries. Let M be a srface. A cre or path on M is a continos map p : [0, 1] M. The endpoints are denoted p(0) and p(1). A loop with basepoint x is a path p with x = p(0) = p(1). An arc α is a path whose endpoints are in the bondary. A cycle is a continos map γ : S 1 M, where S 1 is the nit circle. A cre is simple if it is one-to-one. Two cycles are disjoint if the do not intersect, and two paths are disjoint if they intersect only at their endpoints. A homotopy between two paths p and q, with p(0) = q(0) and p(1) = q(1), is a continos map H : [0, 1] [0, 1] M sch that H(0, ) = p, H(1, ) = q, H(, 0) = p(0) = q(0), and H(, 1) = p(1) = q(1). A homotopy between two cycles γ and β is a continos map H : [0, 1] S 1 sch that H(0, ) = γ and H(1, ) = β. A cycle is contractible if it is homotopic to a constant cycle. An arc whose endpoints are in the same bondary component δ is non-contractible if after contracting δ to a point, the obtained loop is non-contractible. A simple cycle or arc is separating if M \ γ has two connected components. For the topological reslts in this paper, we work on the combinatorial srface model. This model has been commonly sed for most related reslts [1], [], [3], [5]. A combinatorial srface is an abstract srface M together with a weighted ndirected graph G embedded on M sch that each open face is a disk. Allowed paths are walks in G, and the length of a path is the sm of the weights of the edges on the path (with mltiplicity). The mltiplicity of a path is the maximm nmber of times that an edge appears in it. The complexity of the combinatorial srface, sally denoted by n, is the nmber of ertices, edges, and faces in the embedding of G. In a combinatorial srface, a cre or cycle in M is simple if it can be infinitesimally pertrbed to a simple cre in M. Any arrangement of cres in the srface can be seen as an embedded graph [3], which basically handles this by increasing the nmber of ertices and edges appropriately to obtain a graph G where the cres are edge-disjoint. This allows s to assme that, in the srface, all the cres considered are generically embedded.

3 3 Updating the sp-tree Sppose that we are gien the sp-tree rooted at some ertex. We wish pdate the tree so that it becomes the shortest path rooted at, a neighbor of. View the sp-tree as a kinetic data strctre, in which the root s slides continosly from to along the edge e =. Any ertex whose shortest path to s goes throgh immediately before reaching s is called a ble ertex. All other ertices are colored red. Note that in the sp-tree rooted at s, the ertices of each color form a sbtree of the sp-tree, since any ertex s shortest path to s mst go throgh either or. The ble ertices are in some sense already in the final sp-tree, since their shortest paths to s do not change as s moes closer to. Eentally, when s arries at, the sp-tree rooted at is obtained. As s slides across e a distance, the distance from s to eery red ertex increases by, and the distance from s to eery ble ertex decreases by. When an edge xy is abot to become tense (i.e. if t( xy) was negatie and jst became = 0), then d(y) = d(x)+w( xy) for some red ertex y and ble ertex x. This means that as s contines to moe along e, the path from s to y that goes throgh x is to be shorter than the preios shortest path. When an edge becomes tense, we hae an eent in the sp-tree. In an eent, the edge immediately preceding y along its shortest path to s shold be deleted, and the edge xy is added to the sp-tree instead. Additionally, y and all the ertices in its sbtree shold be recolored to ble, since the shortest path to s now passes throgh last. The kinetic property we exploit here is that we hae no extraneos eents. Each time an edge becomes tense, we mst do exactly one ct and one join in the sp-tree. Using appropriate data strctres, a rnning time of O(log n) per pdate to the sp-tree is obtained. The main isse becomes detecting what edges come in and ot in the tree. In the algorithm, we se two types of data strctres. The first holds the sp-tree T, and the second maintains a sbgraph of the dal graph (G \ T ). The main difference between the algorithms for the planar case and the gens g case is the dal strctre. In a planar graph, the dals of edges not contained in the sp-tree form the so-called cotree, a spanning tree of the dal graph. Howeer, in a gens g graph, there are O(g) extra edges that rin this cotree strctre and make the algorithm more complicated. In the primal strctre, we hae a directed forest that is a sbgraph of the crrent sp-tree. Each ertex implicitly maintains its distance to s. Note that all of the following operations can be implemented in O(log n) (worst-case) time sing Eler-Tor trees [8]. GetVale() retrns d(). Ct(e ) remoes the specified edge e crrent forest. from the Join(e ) adds the edge e to the forest. (This operation assmes the forest remains acyclic after adding e.) AddSbtree(, x) adds the ale to eery ertex in the sbtree rooted at ertex x. In the dal graph, we need a different set of operations. These differ in the planar and non-planar cases, and will be described in more detail in the appropriate sections. 3.1 Planar graphs. We begin with the algorithm in the planar case. This algorithm is similar to Klein s [9] in that the same tree and cotree decomposition is sed, and the basic idea of relaxing edges is the same. Howeer, we relax edges in or kinetic data strctre one by one as the sorce moes along an edge, instead of changing the root and then relaxing edges iteratiely from a set of tense edges as Klein does. Or algorithm begins with a sp-tree T rooted at a ertex s =, and moes the sorce s of the sp-tree T across a fixed edge e =. In the dal graph, each edge xy dal to an edge xy that is not in T stores t( xy) and t( yx). This set of edges in (G \ T ) forms a spanning tree. We se the following operations, each of which can be performed in O(log n) amortized time sing self adjsting top trees [16]. Ct(e) cts the edge e from the tree, and retrns two new separate trees. Join(e) adds the edge e, joining two trees into a single tree. AddPath(, π) adds ± to the tension of each of the edges in the path π. (The ale + is added to the tension of the edge oriented from red to ble ertex, and is added to the orientation from ble to red.) MaxPath(π) finds the edge in the path π that has maximm tension (where the tension is oriented from a ble ertex to a red ertex). Assme that e is in the sp-tree. Any (primal) edge whose tension is changing mst hae one ble endpoint and one red endpoint, since any edge with monochromatic endpoints has constant tension (its endpoints

4 Figre 1. The thick lines in the graph are the sp-tree rooted and the sorce, shown moing along the edge e. As s moes closer to the target, an edge along the green path in the dal (shown in dotted lines) becomes tense and is inserted. change at the same rate). This means that the set of edges whose tension is changing corresponds to the niqe path in T that goes between the endpoints of e. Call this set of edges green, and let the dal path consisting of all the green edges be π. If the edge e is not in the sp-tree, all ertices are red. At some stage e becomes tense, since the distance from s to along e is going to zero as s slides along e. At this stage, the edge e enters the sp-tree, and the sbtree rooted at is immediately colored ble. Once e is in the sp-tree, the algorithm iterates oer the following steps ntil either eery ertex is ble or s reaches. See Figre 1. We call MaxPath on the two endpoints of e to find the first edge xy that has t( xy) > 0 as slides towards. Let be the amont the root needs to slide for xy to become tense. Next, moe the sorce s a distance along the edge e and pdate the distances in the primal tree and tension in the dal tree. AddPath(π, ) pdates the tensions of the green edges. In the primal tree, AddSbtree(, ) adds to eery distance for ertices in the red sbtree, simlating the root sliding along the edge e by a distance of. Similarly, AddSbtree(, ) pdates the distance from s to the ble ertices. Now we mst actally pdate the sp-tree to reflect the new edge being added. We first Ct(wy) where wy is the last edge in the path in T from s to y. Then Join( xy) reconnects s to y ia the new shorter path. This reconnects the sp-tree, and recolors y and its sbtree ble. To pdate the dal spanning tree, we call Ct(xy ) to remoe the dal edge whose tension is no longer changing, and then Join(wy ) to reconnect the dal tree and pdate the strctre. Each edge change in the tree calls a constant nmber of operations taking O(log n) time. This concldes the description of the algorithm to moe the sorce along a single edge. We smmarize in the following lemma: Figre. As the ble tree expands, the set of green edges cold intersect itself and separate into mltiple components. Here, the ble sbtree (rooted at ) wrapped arond the tors, and the set of green edges is shown as two disconnected cycles in the dal which separate the red sbtree from the ble sbtree. Lemma 3.1. Sp-trees in planar graphs can be represented in sch a way that the sp-tree rooted at can be changed to the sp-tree rooted at a neighbor of in O(k log n) time, where k is the nmber of edges entering or leaing the trees. In this representation, a shortest path distance from the root can be compted in O(log n) time. 3. Graphs of gens g. The algorithm in the primal case does not immediately extend to gens g graphs, since the dal graph (G \ T ) is no longer a tree. Withot this tree strctre, the set of green edges can become more complicated than a path in the dal, and finding tense edges is harder. For example, consider the case when the graph is on the tors. Initially, the ble sbtree is diided from the red sbtree by a cycle of green edges. Howeer, as the ble sbtree grows, it is possible for the ble to meet at other places, and the green bondary splits into different connected components. See Figre. Or algorithm for graphs of gens g is identical to the planar algorithm on the primal data strctre. Howeer, the dal strctre is necessarily more complicated, since we no longer hae a tree in the dal. (Throghot the algorithm, we consider the edge () as part of (G \ T ) to simplify the algorithm.) For each dal edge xy corresponding to an edge xy that is not in T, we maintain the tenseness of xy and yx. Decompose the edges of (G \ T ) () as follows. Iteratiely delete all edges of degree 1. Denote the deleted edges as F, and note that F is a forest in G. After the edges F hae been deleted, we hae a set of paths, called ct paths, that meet at ertices of degree 3. Denote these ct paths as P = {, π,..., π k }, where each π i is a path between two ertices of degree 3. Eler s formla implies that k = O(g) (see [5, Lemma 4.]).

5 The set of green edges, or primal edges whose endpoints are not monochromatic, are again those edges that cold potentially become tense. All the green edges are dal to edges in P. To see this, note that P bonds two topological disks, one containing all the ble ertices and the other containing all the red ertices. For any edge xy F, the edge xy cannot hae endpoints of different colors, since its endpoints are in the same face. Therefore, P mst contain the dal of eery green edge. Moreoer, any path π i P has all the edges dal to either monochromatic edges or to green edges. Each path π i, along with any components of F which are rooted along π i, is pt into a self adjsting top tree T i, where π i is the top leel path. We also maintain two linked lists of pointers representing the crrent bondary of the red and ble faces. For each π i on the bondary of a face, a pointer in the linked list for that face points to the corresponding tree T i. These pointers appear in the linked list in the same order that the paths appear along the bondary. Note that it is possible for a path to bond the same region on both sides; the linked list will simply contain two pointers to the same tree. See Figre 3. The operations on the dal strctre are: 1. Ct(e) and Join(e) are identical to the planar operations.. AddBondary( ) adds ± to the tension of eery edge that borders both the red and the ble faces. (Again, we add + to the orientation going from red to ble, and to the orientation from ble to red.) 3. MinBondary retrns the edge with maximm tenseness among edges that border both faces.. To implement MinBondary, we search both linked lists in O(g) time to find the ct paths that border both regions; this is the candidate list of green edges. We can find the edge in each ct path with highest tension in O(log n) time by qerying the appropriate T i with MinPath. Ths, it takes O(g log n) time to find the edge xy that becomes tense first. To implement AddBondary, we again search both linked lists in order to find the ct paths that contain green edges. These are the only trees that need to be pdated, since edges with monochromatic endpoints hae constant tension. Each of the self adjsting top trees can pdate tensions in O(log n) time sing AddPath, giing a bond of O(g log n) time for pdating all of the ct paths. Now we describe the algorithm. We first call MinBondary to find the first tense edge xy. Let be the amont that s mst moe along e in order for xy to become tense. In the primal sp-tree, we can pdate the distances sing AddSbtree( ) to s sbtree (the red ertices), and AddSbtree( ) to s sbtree (the ble ertices). We then call AddBondary with a ale of. We also need to pdate the sp-tree. This is done sing Ct( wy) to ct y ot of the red tree, where wy is the red edge that connects y to the s in the sp-tree. Then Join( xy) reconnects y and its sbtree to the sorce s. We also hae to pdate the dal strctre. The dal edge yw is now a green edge, since w is still red and y is now ble. The endpoints of wy are either in F or are directly on some path π j, since we know that wy was in the sp-tree. Find the niqe (and possibly empty, if the ertex is on some π j ) path in F connecting each endpoint of wy to the paths P in O(log n) by simply splaying p the self adjsting top tree that they belong to ntil reaching the top leel path. Denote these (possibly empty) paths as π and. We call expose to make π and π the top leel in their respectie trees, and then we combine them into one path π sing Join( wy ). We now insert π into the bondaries of the two faces in or linked lists. In addition, the path π i that xy ct across is no longer a bondary, so we mst pdate the tree for that component and remoe it from the bondary lists. It is also possible that π intersects one or two paths π j ; if so, those trees mst be sbdiided at the appropriate location sing Ct. We pdate the red face s linked list first by ctting (if necessary) the paths that the endpoints of π sbdiided. We then remoe all of the pointers in the linked list that were between those endpoints, since they gae the bondary of the sbtree rooted at y that is now ble and therefore it is no longer a bondary. Note that this atomatically remoes the old path π i from the bondary, since π necessarily either blocks or diides that path. For the ble face s linked list, we mst add in any of the paths π k that were remoed from the red list. We also add in the self adjsting top tree for π, since it now borders the ble face. Altogether, pdating the linked lists can be done in O(g) time, since we traerse each list at most twice while ctting ot and adding in links to the appropriate ct paths. This finishes the description of the algorithm when moing along one edge. We smmarize in the following lemma: Lemma 3.. Sp-trees in graphs on a srface of gens g can be represented in sch a way that the sp-tree rooted at can be changed to the sp-tree rooted at a neighbor

6 π + π π π 4 5 π π + π π π 4 5 π π π + π + π π + π π 4 π π π π 5 π 4 π π π 5 π π + π π + PSfrag π + replacements π π π 5 π 4 π π π π π 4 π 5 π - Figre 3. An example of the algorithm progressing. On the left, the alterations are shown on the srface on the tors, and on the right, they are shown on the polygonal schema obtained by ctting along the initial ct locs of. As new edges become tense, the set of ct paths alters, bt always separates the ble sbtree from the red sbtree. of in O(kg log n) time, where k is the nmber of edges entering or leaing the trees. In this representation, a shortest path distance from the root can be compted in O(log n) time. 4 Moing the root along a face 4.1 Planar graphs. Consider the case when G is a plane graph, and we want to maintain the sp-tree as the root moes along a face f. Klein [9] noted two key facts. First, each edge can enter and leae the tree a constant nmber of times. For graphs with niqe shortest paths, this can be seen ia a relatiely simply crossing argment. (Klein shows this also in general if one maintains a left-most sp-tree.) Also, sing standard techniqes, the graph can be assmed to hae bonded degree, and then one can se persistence [4] to store and search any preios ersions of the sp-tree. (This increases the reqired space to O(n log n).) With these two obserations and Lemma 3.1, the following reslt is obtained. Theorem 4.1 (Klein [9]). Let G be a plane graph with n ertices, and let be f a face of G. With O(n log n) preprocessing time, a shortest path distance from any ertex on f to any other ertex can be fond in O(log n) time. 4. Graphs of gens g. Let G be a graph embedded in a srface of gens g. We are interested on maintaining the sp-tree as the root moes along a face f. We first show a bond on the nmber of times that an edge can enter or leae the sp-tree. Lemma 4.1. As the root of the sp-tree moes along f, each edge enters and leaes the sp-tree O(g) times. Proof: Let e be an edge in G, and let N be the srface obtained by contracting e to a point e and f to a point f. Consider all the shortest paths from ertices in f to any of the endpoints of e. In N, these shortest paths define O(g) distinct homotopy classes of paths with endpoints e and f. This follows from [,

7 Lemma.1], since we can combine pairs of pairwise nonhomotopic paths to get a set of pairwise non-crossing, non-homotopic loops with basepoint f. If we consider all the shortest paths from ertices on f to an endpoint of an edge e, we know that e = will enter or leae the sp-tree when the shortest path from w f to ses e and the shortest path from the neighbor of w on f does not se e. In a homotopy class, the edge e cannot switch from being in and ot of the shortest paths more than constant nmber of times, since a homotopy class of paths forms a planar graph. Note that the homotopy classes cannot interleae, that is, all the shortest paths that hae the same homotopy class in N appear consectiely as we walk along f. Indeed, any two paths in the same homotopy class, together with a piece f of f and e define a topological disk, and it cannot be that a path in a different homotopy class has an endpoint in the piece f. Since there are at most O(g) homotopy classes, each of which occrs as a block, and in each class, an edge comes in and ot O(1) times, the reslt follows. Since we take O(g log n) time each time we pdate the tree, this gies O(g n log n) time total to pdate all the trees oer the corse of the algorithm. Again, we can se standard techniqes to conert to a graph with bonded degree, and se persistence [4] to store and search any preios ersions of the sp-tree. Regarding the space bonds, dring the process, we need O(gn) space to maintain the (dal) strctres. Howeer, for answering distance qeries, we only need to store the primal strctre, and therefore, the final data strctre ses O(gn log n) space. With these two obserations and Lemma 3., we obtain or main reslt. Theorem 4.. Let G be a graph with n ertices embedded in a srface of gens g, and let be f a face of G. With O(g n log n) preprocessing time, a shortest path distance from any ertex on f to any other ertex can be fond in O(log n) time. 5 Compting shortest non-separating and non-contractible cycles In this section we consider the problems of finding a shortest non-separating and a shortest non-contractible cycle in an orientable combinatorial srface M. The se of or techniqe on maintaining shortest path trees is condensed in the following lemma. Lemma 5.1. Let α be a simple cycle or arc in M, and let Cross 1 (α) be the set of cycles that cross α exactly once. A shortest cycle in Cross 1 (α) can be obtained in O(g n log n) time. Proof: Consider the srface obtained by ctting M along α: each ertex in α gies rise to two ertices,, and two bondary arcs or cycles α, α. Let N be the srface obtained by gling topological disks to the bondaries that contain α and α. (If α is an arc, then α and α are contained in a single bondary.) A cycle in M that crosses α once at a point becomes a path in N connecting to. Ths, a shortest cycle that crosses α once at is a shortest path that connects to in N, and ice ersa. Since all the points, with α, belong to a face of N, we can se Theorem 4. to find in O(g n log n) time a closest pair ( 0, 0 ). Compting the shortest path from 0 to 0 gies the reslt. Note that if α is separating, then Cross 1 (α) = becase any cycle crosses α an een nmber of times. We also se that all the cycles in Cross 1 (α) are noncontractible. This is de to the fact that a contractible cycle C is a separating cycle, and any cycle or arc mst cross C an een nmber of times. 5.1 Shortest non-separating cycle. A shortest non-separating cycle in M is also non-separating in the srface obtained by attaching disks to the bondaries. Therefore, we only need to consider srfaces withot bondary. Cabello and Mohar [1] hae shown how to constrct in O(gn log n) time a set S of O(g) simple loops with the property that a shortest cycle in l S Cross 1(l) is a shortest non-separating cycle. Since S consists of O(g) loops, we can apply the preios lemma to Cross 1 (l) for each l in S and take the globally shortest cycle. We smmarize: Theorem 5.1. Let M be an orientable srface, possibly with bondary, of complexity n and gens g. We can find a shortest non-separating cycle in O(g 3 n log n) time. 5. Shortest non-contractible cycle. The main approach is to find a cre whose remoal decreases the gens or nmber of bondaries of the srface, and moreoer, it has the property that there is a shortest non-contractible cycle intersecting it at most once. The main tool to proe the following reslts is an exchange argment. We omit their proof. Lemma 5.. Let M be an orientable srface. (a) Let l x be a shortest non-contractible loop with gien basepoint x M. There is a shortest noncontractible cycle in M that crosses l x at most once.

8 (b) Let δ be a bondary cycle in M and let α be a shortest non-contractible arc with endpoints in δ. There is a shortest non-contractible cycle in M that is homotopic to δ, or that crosses α at most once. (c) Let α be a shortest arc connecting two different specified bondaries of M. There is a shortest noncontractible cycle in M that crosses α at most once. The next reslt smmarizes the time needed to find the cres described in the preios lemma. Lemma 5.3. Let M be an orientable combinatorial srface of complexity n. (a) Gien a basepoint x, we can find in O(n log n) time a shortest non-contractible loop with basepoint x and mltiplicity two. (b) Gien a bondary δ, we can find in O((g+b)n log n) time a shortest cycle homotopic to δ. (c) Gien a bondary δ, we can find in O(n log n) time a shortest non-contractible arc with endpoints in δ, mltiplicity two, and edge-disjoint from δ. (d) If M has two or more bondaries, we can find in O(n log n) time a shortest arc with endpoints in different bondaries, mltiplicity one, and edgedisjoint from all bondaries of M. Proof: (a) See Erickson and Har-Peled [5, Lemma 5.]. (b) Constrct a cross-metric srface N sing a cylinder with a bondary haing 3 edges and another bondary that gets gled to δ. Assign infinite length to each edge not in. Since N has complexity O(n) and consists of three edges, an algorithm of Colin de Verdière and Erickson [3, Theorem 6.1] finds in O((g + b)n log n) time a shortest cycle homotopic to in N. This is the desired cycle in M. (c) Contract δ to a point p and constrct a shortest non-contractible loop with basepoint p as in item (a). This is the desired arc in M. (d) Select a bondary δ of M, contract it to a point p, and constrct in O(n log n) time a shortest path tree from p. Let q be the closest point to p among ertices in the bondary of M. The shortest path from p to q in M is the shortest arc connecting the bondary δ to any other bondary, and it does not se any edge from any bondary. Theorem 5.. Let M be an orientable srface of complexity n, gens g, and b bondaries. We can find a shortest non-separating cycle of M in O((g+b)g n log n) time. Proof: We gie a recrsie algorithm that redces either the gens or the nmber of bondaries of the sbproblems. The recrsion stops when the sbproblem is a topological disk. We distingish three cases; N denotes the srface in the sbproblem and m its complexity. The gens of N is bonded by g. (a) If N is a srface withot bondary, we choose a point x N and find a shortest non-contractible loop l x throgh x. Becase of Lemma 5.(a), there is a shortest non-contractible cycle in N that crosses l x at most once. Consider the srface N = N \l x, and retrn a shortest cycle among: the cycles Cross 1 (l x ) and the shortest non-contractible cycle in N. Note that if l x is separating, then N has two connected components and Cross 1 (l x ) =. From Lemmas 5.1 and 5.3(a), it follows that we spend O(g m log m) time in N, pls the time sed for N. (b) If N has exactly one bondary δ, we find a shortest non-contractible arc α with endpoints in δ. Becase of Lemma 5.(b), there is a shortest noncontractible cycle in N that crosses α at most once, or it is homotopic to δ. Consider the srface N = N \ l x, and retrn a shortest cycle among: the cycles Cross 1 (l x ), a shortest cycle homotopic to δ, and a shortest non-contractible cycle in N. From Lemmas 5.1 and 5.3(b-c), it follows that we spend O(g m log m) time in N, pls the time sed for N. (c) If N has two or more bondaries, we find a shortest arc α connecting two different bondaries. Becase of Lemma 5.(c), there is a shortest noncontractible cycle in N that crosses α at most once. Consider the srface N = N \ l x, and retrn a shortest cycle among: the cycles Cross 1 (l x ) and a shortest non-contractible in N. From Lemmas 5.1 and 5.3(d), it follows that we spend O(g m log m) time in N, pls the time sed for N. This finishes the description of the algorithm. Note that at most O(g + b) recrsie sbproblems are considered: starting from a srface M, we enconter O(b + 1) times cases (a) or (c), ntil we first obtain pieces with one bondary, and then we repeatedly enconter O(g) cases (b-c) that maintain each component with one or two bondaries and decrease their gens. The arcs and loops that are sed for ctting can be obtained from

9 Lemma 5.3, which hae mltiplicity at most two and are edge-disjoint from the bondary. It follows that any srface considered in a sbproblem has at most for copies of an edge of M. Ths, m = O(n), and in each piece we spend O(g n log n) time. References [1] S. Cabello and B. Mohar. Finding shortest nonseparating and non-contractible cycles for topologically embedded graphs. In G. S. Brodal and S Leonardi, editors, Algorithms - ESA 005, 13th Annal Eropean Symposim, Palma de Mallorca, Spain, October 3-6, 005, Proceedings, olme 3669 of LNCS, pages Springer, 005. [] E. W. Chambers, É. Colin de Verdière, J. Erickson, F. Lazars, and K. Whittlesey. Splitting (complicated) srfaces is hard. In SCG 06: Proceedings of the twenty-second annal symposim on Comptational geometry, pages 41 49, New York, NY, USA, 006. ACM Press. [3] É. Colin de Verdière and J. Erickson. Tightening non-simple paths and cycles on srfaces. In SODA 06: Proceedings of the seenteenth annal ACM-SIAM symposim on Discrete algorithm, pages 19 01, New York, NY, USA, 006. ACM Press. [4] J. R. Driscoll, N. Sarnak, D. D. Sleator, and R. E. Tarjan. Making data strctres persistent. In STOC 86: Proceedings of the eighteenth annal ACM symposim on Theory of compting, pages , New York, NY, USA, ACM Press. [5] J. Erickson and S. Har-Peled. Optimally ctting a srface into a disk. Discrete and Compt. Geometry, 31(1):37 59, 004. [6] G. N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Compt., 16(6): , [7] L. Gibas. Kinetic data strctres: A state of the art report, [8] M. R. Henzinger and V. King. Randomized flly dynamic graph algorithms with polylogarithmic time per operation. Jornal of the ACM, 46(4):50 516, [9] P. N. Klein. Mltiple-sorce shortest paths in planar graphs. In SODA 05: Proceedings of the sixteenth annal ACM-SIAM symposim on Discrete algorithms, pages , Philadelphia, PA, USA, 005. Society for Indstrial and Applied Mathematics. [10] M. Ktz. Compting shortest non-triial cycles on orientable srfaces of bonded gens in almost linear time. In SCG 06: Proceedings of the twenty-second annal symposim on Comptational geometry, pages , New York, NY, USA, 006. ACM Press. [11] R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36: , [1] K. Mlmley, U. V. Vazirani, and V. V. Vazirani. Matching is as easy as matrix inersion. Combinatorica, 7: , [13] J. P. Schmidt. All highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings. SIAM J. Compt., 7(4):97 99, [14] J. Stillwell. Classical Topology and Combinatorial Grop Theory. Springer-Verlag, New York, [15] R. E. Tarjan. Data Strctres and Network Algorithms. SIAM, Philadelphia, [16] R. E. Tarjan and R. F. Werneck. Self-adjsting top trees. In SODA 05: Proceedings of the sixteenth annal ACM-SIAM symposim on Discrete algorithms, pages 813 8, Philadelphia, PA, USA, 005. Society for Indstrial and Applied Mathematics. [17] A. Zomorodian. Topology for Compting. Cambridge Uniersity Press, 005.