CERIAS Tech Report EFFICIENT PARALLEL ALGORITHMS FOR PLANAR st-graphs. by Mikhail J. Atallah, Danny Z. Chen, and Ovidiu Daescu

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1 CERIAS Tech Report EFFICIENT PARALLEL ALGORITHMS FOR PLANAR t-graphs by Mikhail J. Atallah, Danny Z. Chen, and Ovidiu Daecu Center for Education and Reearch in Information Aurance and Security, Purdue Univerity, Wet Lafayette, IN 47907

2 Algorithmica (2003) 35: DOI: / Algorithmica 2002 Springer-Verlag New York Inc. Efficient Parallel Algorithm for Planar t-graph 1 Mikhail J. Atallah, 2 Danny Z. Chen, 3 and Ovidiu Daecu 4 Abtract. Planar t-graph find application in a number of area. In thi paper we preent efficient parallel algorithm for olving everal fundamental problem on planar t-graph. The problem we conider include all-pair hortet path in weighted planar t-graph, ingle-ource hortet path in weighted planar layered digraph (which can be reduced to ingle-ource hortet path in certain pecial planar t-graph), and depthfirt earch in planar t-graph. Our parallel hortet path technique exploit the pecific geometric and graphic tructure of planar t-graph, and involve cheme for partitioning planar t-graph into ubgraph in a way that enure that the reulting path length matrice have a monotonicity property [1], [2]. The parallel algorithm we obtain are a coniderable improvement over the previouly bet known olution (when they are applied to thee t-graph problem), and are in fact relatively imple. The parallel computational model we ue are the CREW PRAM and EREW PRAM. Key Word. Algorithm, EREW PRAM, Merging, Multi-election, Partitioning, Sorting, Parallel computing. 1. Introduction. An n-vertex planar t-graph G = (V, E) i a planar directed acyclic graph with exactly one ource vertex and exactly one ink vertex t, uch that G can be embedded in the plane with both and t on the boundary of the external face of the embedding of G. Planar t-graph find application in a number of area, including partial order, computational geometry (e.g., path planning, planar point location, and viibility), graph theory (e.g., graph drawing, planar graph embedding, and graph planarization), VLSI deign (e.g., floorplanning and layout compaction), and motion planning (ee [32] for reference). Serie-parallel graph are a pecial cae of planar t-graph. In thi paper we preent efficient determinitic parallel algorithm for olving everal fundamental problem on planar t-graph. In particular, we conider the following problem: All-pair hortet path in weighted planar t-graph (whoe edge are aociated with nonnegative weight). Single-ource hortet path in weighted planar layered digraph. Note that hortet path in planar layered digraph can be reduced to hortet path in certain pecial planar t-graph. Depth-firt earch in planar t-graph. 1 The firt author gratefully acknowledge the upport of the COAST Project at Purdue Univerity and it ponor, in particular Hewlett Packard, DARPA, and the National Security Agency. The reearch of the econd and third author wa upported in part by the National Science Foundation under the NSF Grant CCR Department of Computer Science, Purdue Univerity, Wet Lafayette, IN 47907, USA. mja@c.purdue.edu. 3 Department of Computer Science and Engineering, Univerity of Notre Dame, Notre Dame, IN 46556, USA. dchen@ce.nd.edu. 4 Department of Computer Science, Univerity of Texa at Dalla, Richardon, TX 75083, USA. daecu@utdalla.edu. Received June 29, Communicated by H. W. Leong and H. Imai. Online publication December 9, 2002.

3 Efficient Parallel Algorithm for Planar t-graph 195 We henceforth aume all graph have n vertice and have edge whoe weight are nonnegative. The parallel computational model we ue are the CREW PRAM and EREW PRAM [14]. For convenience, all our algorithm are decribed uing two parallel complexity bound: time and work (the work i the time proceor product of an algorithm). The proceor bound of all our parallel algorithm can be eaily derived by uing Brent theorem [4]. The problem of computing hortet path in graph i fundamental in computer cience, and ha application in olving many cientific and engineering problem. Sequentially, the problem i quite well tudied and efficient algorithm for variou verion of graph hortet path problem have been given (e.g., ee [6] and [8]). However, deigning efficient parallel algorithm for computing hortet path in graph ha remained an eluive tak depite coniderable effort. Several intereting parallel graph algorithm for all-pair hortet path are known. Han et al. [11] preented an O(log 2 n) time, o(n 3 ) work EREW PRAM algorithm for all-pair hortet path in general directed graph. Cohen [5] gave an O(log 4 n) time, O(n 2 log n) work CREW PRAM algorithm for all-pair hortet path in planar directed graph. An O(log n) time, O(n 2 ) work EREW PRAM algorithm for all-pair hortet path in erie-parallel digraph can be obtained from the reult of [33] (by uing the parallel tree contraction technique [14]). Note that a erie-parallel digraph i a directed acyclic graph with exactly one ource and exactly one ink, uch that the graph can be contructed by erie and parallel compoition. Serie-parallel digraph are a pecial cae of planar t-graph. There are alo parallel graph algorithm for ingle-ource hortet path. Pan and Reif [25], [26] developed an O(log 3 n) time, O(n 1.5 ) work algorithm for ingle-ource hortet path in planar undirected graph, and thi reult ha been generalized to planar directed graph [5], [11]. Klein and Subramanian [21] gave a linear-proceor, polylogtime algorithm for ingle-ource hortet path in planar directed graph, but the exponent in their polylogarithmic running time i rather large. An O(log n) time, O(n) work EREW PRAM algorithm for ingle-ource hortet path in erie-parallel digraph wa preented in [33]. There are a few parallel algorithm for computing a hortet path between one pair of vertice in certain graph. Aggarwal and Park [1] and Apotolico et al. [2] obtained an O(log 2 n) time, O(n log n) work CREW PRAM algorithm for finding a ource-toink hortet path in planar directed acyclic grid graph (ee Figure 1(a) for an example of uch graph). An intereting generalization i the reult of Sairam et al. [28], who conidered planar layered directed acyclic graph (called planar layered digraph). Note that the directed grid graph conidered in [1] and [2] are in fact a pecial cae of planar t-graph, and can be tranformed to planar layered digraph (ee Figure 1(b) for an example). The CREW PRAM algorithm in [28] find a ource-to-ink hortet path in planar layered digraph in O(log 2 n) time and O(n log n) work, baed on an elegant one-way eparator contruction. It wa alo hown in [28] that hortet path in planar layered digraph can be reduced to hortet path in ome pecial planar t-graph. The problem of computing a depth-firt earch tree in a graph i fundamental in graph theory, and i eaily olved in linear time equentially [6]. In parallel, performing ordered depth-firt earch that viit vertice of a graph in a given order i known to be P-complete

4 196 M. J. Atallah, D. Z. Chen, and O. Daecu t t (a) (b) Fig. 1. (a) A planar directed grid graph. (b) A correponding planar layered digraph. [27]. There are parallel algorithm for depth-firt earch in variou pecial graph, uch a chordal graph [20], erie-parallel graph [12], planar undirected graph [10], [13], [15], [19], [29], [30], and planar directed graph [16] [18]. In particular, Kao and Klein [18] gave an O(log 10 n) time, O(n log 10 n) work algorithm for depth-firt earch in general planar directed graph, and Kao [17] preented an O(log 5 n) time, O(n log 4 n) work algorithm for depth-firt earch in trongly connected planar directed graph. In thi paper we preent the following efficient parallel olution: An O(log 2 n) time, O(n 2 ) work CREW PRAM algorithm for all-pair hortet path in planar t-graph. The technique of thi algorithm, when applied to computing allpair hortet path in the planar directed acyclic grid graph of [1] and [2], yield an O(log 2 n) time, O(n 2 ) work CREW PRAM algorithm. An O(log 2 n) time, O(n log n) work CREW PRAM algorithm for ingle-ource hortet path in planar layered digraph. Thi generalize the ource-to-ink hortet path reult of [28]. An O(log n) time, O(n) work EREW PRAM algorithm for depth-firt earch in planar t-graph. In comparion, our parallel algorithm are a coniderable improvement over the previouly bet known reult (which are often for more general clae of graph) when they are applied to thee t-graph problem, and in fact are relatively imple. Further, note that our parallel depth-firt earch and all-pair hortet path algorithm are optimal, and the work bound of our parallel ingle-ource hortet path algorithm i within only a log n factor of the time bound of it correponding optimal equential algorithm. All our parallel algorithm exploit the pecific geometric and graphic tructure of planar t-graph. Our parallel hortet path technique involve two graph partitioning cheme: the trict one-way eparator for planar t-graph (which make ue of the viibility repreentation of planar t-graph in [32]), and the one-way eparator for planar layered digraph (which are a light modification of the one ued in [28]). Thee cheme partition planar t-graph or planar layered digraph into ubgraph in a way that enure that the reulting path length matrice have a monotonicity property. Thi monotonicity property enable one to perform matrix multiplication in a very efficient manner. In particular, we get around the following difficulty that arie in all our hortet path computation. Oberve that for the four vertice a, b, c, and d that are all on the

5 Efficient Parallel Algorithm for Planar t-graph 197 t G d t d G c a h b c a b (a) (b) Fig. 2. (a) Path that mut cro each other. (b) Path that need not cro each other. boundary of the external face of an embedded planar t-graph G a hown in Figure 2(a), the hortet a-to-d path in G mut cro the hortet b-to-c path in G at a vertex h (becaue G i a planar t-graph). Such a croing property i a key to obtaining matrice with the monotonicity property. However, when ome of the vertice are not on the boundary of the external face of G (a hown in Figure 2(b)), hortet path in G between thee vertice need not cro each other. In all our hortet path computation (epecially for the all-pair cae), we mut ue appropriate graph partitioning cheme for achieving path length matrice that have the monotonicity property. For our parallel hortet path algorithm, we only decribe the verion for computing the length of the hortet path. The length verion of the algorithm can be eaily modified to generate actual hortet path tree a well a the hortet path length. The ret of the paper i organized a follow. Section 2 review ome ueful definition and preliminary reult. Section 3 give our parallel algorithm for all-pair hortet path in planar t-graph. Section 4 preent our parallel algorithm for ingle-ource hortet path in planar layered digraph. Section 5 dicue our parallel depth-firt earch algorithm on planar t-graph. 2. Preliminarie. Let G = (V, E) be an n-vertex input graph (either a planar t-graph or a planar layered digraph), with edge whoe weight are nonnegative. A in [32], we aume that the input graph repreentation i embedded, that i, for each vertex v of G, the cyclic ordering of v neighboring vertice (both incoming and outgoing) in the embedding i given in tandard form (a a doubly linked edge lit). Without lo of generality, we aume that the embedding of G i uch that all it edge are directed upward (e.g., by rotating the embedding of the graph in Figure 1 by π/4). Planar layered digraph are alo called planar proper hierarchie [7], [34]. The definition of a planar layered digraph i reviewed a follow. DEFINITION 1. A planar layered digraph i a planar directed acyclic graph G = (V, E) that allow an embedding in the plane, called k-line embedding, that ha the following propertie: (i) The vertice of G are partitioned into k ubet called layer.

6 198 M. J. Atallah, D. Z. Chen, and O. Daecu t layer 6 layer 5 layer 5 layer 4 layer 4 layer 3 layer 3 layer 2 layer 2 layer 1 (a) layer 1 (b) layer 0 Fig. 3. (a) A five-layer planar layered digraph. (b) The correponding planar layered t-graph. (ii) The k layer of G are conecutively embedded in k parallel line, with layer i on the ith line. (iii) Every edge of G correpond to a directed traight line egment from a vertex of layer i to a vertex of layer i + 1, for ome i with 1 i < k. (iv) No two edge of G cro each other in the embedding. A planar layered digraph may have multiple ource and ink (e.g., ee Figure 3(a)). Note that planar layered digraph are a generalization of planar directed acyclic grid graph in the ene that uch a grid graph can be eaily tranformed to a planar layered digraph (e.g., ee Figure 1). It ha been hown in [28] that given a k-line embedding of an n-vertex planar layered digraph G, it i poible to tranform G to an (n + 2)-vertex planar layered t-graph G with a (k + 2)-line embedding, uch that hortet path problem in G are equivalent to hortet path problem in G (thi i done by adding to G a new ource and a new ink t, and adding edge with a weight of + ). Furthermore, thi tranformation can be done in O(log n) time and O(n) work on the EREW PRAM. An example of uch a planar layered t-graph for the planar layered digraph in Figure 3(a) i given in Figure 3(b) (with the dahed edge being the newly added edge with a weight of + ). Hence, from now on, we aume that the given planar layered digraph i an n-vertex planar layered t-graph with a k-line embedding. We need to review ome ueful tructure of planar t-graph, baed on the development of Tamaia and Preparata [31]. Let F be the et of face of a planar t-graph G = (V, E). DEFINITION 2. Four function left( ), right( ), low( ), and high( ) are defined on the et V E F, a follow: 1. For a vertex v V, left(v) (rep., right(v)) i the face in F that i to the left (rep., right) of v and eparate the incoming edge of v from it outgoing edge; low(v) = high(v) = v. 2. For an edge e E from vertex u to vertex v, left(e) (rep., right(e)) i the face in F that i immediately to the left (rep., right) of e; low(e) = u, and high(e) = v.

7 Efficient Parallel Algorithm for Planar t-graph For a face f F, left( f ) = right( f ) = f ; low( f ) (rep., high( f )) i the common tarting (rep., ending) vertex of the two bounding path of f. The external face f of G i conceptually partitioned into two face: the left and right external face of G, with the left (rep., right) external face being to the left (rep., right) of the left (rep., right) bounding path of f. DEFINITION 3. The dual graph G = (V, E ) of a planar t-graph G = (V, E) i the directed graph obtained a follow: V = F {, t }, where (rep., t ) i the left (rep., right) external face of G. For every edge e E, there i an edge (left(e), right(e)) E from left(e) to right(e). It i eay to ee that G i alo a planar t-graph with ource and ink t. DEFINITION 4. Two partial order, denoted by and, are defined on V E F of G a follow: for any x, y V E F, it i aid that x i below y (denoted by x y) if there i a path from high(x) to low(y) in G, and that x i to the left of y (denoted by x y) if there i a path from right(x) to left(y) in the dual graph G of G. Exactly one of the following hold for any x, y V E F: x y, y x, x y, or y x [31]. DEFINITION 5. Two total order, denoted by < L and < R, are defined on V E F of G a follow: for any x, y V E F, x < L y iff x y or x y; x < R y iff x y or y x. The equence of all element in V E F orted according to < L (rep., < R ) i called the left equence (rep., right equence) ofg. The two total order < L and < R are very ueful. For example, one can make ue of the fact that there i a path from a vertex v to another vertex w in a planar t-graph G iff v precede w in both the left and right equence of G [31]. Tamaia and Vitter [32] gave an O(log n) time, O(n) work EREW PRAM algorithm for computing the two total order < L and < R. Without lo of generality, we aume that the ource vertex for our ingle-ource hortet path and depth-firt earch computation i the ource of the input planar t-graph G. When the ource vertex for thee problem i v, one can reduce uch a problem to one on a (poibly maller) planar t-graph whoe ource i v, ino(log n) time and O(n) work on the EREW PRAM (by uing the parallel tranitive cloure reult of [32]). A number of reult on computing variou eparator for planar graph are known (e.g., ee [9], [19], [22] [24]). The graph partitioning cheme of our parallel hortet path algorithm are baed on two type of eparator pecifically for planar t-graph. Thee planar t-graph eparator have ome pecial propertie. One type of the eparator, called one-way eparator, wa introduced by Sairam et al. [28]. The following definition review and generalize the idea of one-way eparator. DEFINITION 6. Let S be a ubet of vertice of a directed graph G = (V, E ) uch that removing S and the edge adjacent to any vertex of S from G partition V S

8 200 M. J. Atallah, D. Z. Chen, and O. Daecu into ubet V 1, V 2,..., V h, with h > 1. For any ubet W of V, let G (W ) be the ubgraph of G whoe vertice are the vertice of W and whoe edge are the edge of G connecting vertice of W. S i called a one-way eparator of G if for any directed path P in G, one of the following hold for every V i : (1) P G (V i S) i empty, (2) P G (V i S) i a connected ubpath of P, and (3) P G (V i ) i a connected ubpath of P. S i called a trict one-way eparator of G if S i a one-way eparator of G and if for any directed path P in G, P G (S) i either empty or a connected ubpath of P. Intuitively, for a one-way eparator S of the graph G, a directed path in G can cro the ubgraph G (S) at mot once, but may enter and leave G (S) multiple time (without croing it). In comparion, for a trict one-way eparator S, a directed path in G can enter and leave G (S) at mot once. Computationally, thee two type of eparator have quite different conequence. For a region R of a graph G that i bounded by trict oneway eparator, all directed path (and hence hortet path) between any two vertice of R never go outide R (although thee path can touch the boundary vertice of R ). Therefore, the hortet path in G between any two vertice of R can be computed imply from the region R of G. However, for a region R of a graph G that i bounded by one-way eparator, a directed path P (and hence a hortet path) that i between two vertice of R and that ha an end vertex on the boundary of R need not tay inide the boundary of R uch a path may pa through vertice of G outide R (ee Figure 4 for an example). Therefore, the hortet path between two vertice of R and with an end vertex on the boundary of R that are computed only from the region R may not be true hortet path in G. Thi make the hortet path computation uing one-way eparator omewhat more difficult. A in [1], [2], and [28], our algorithm involve multiplying pecial kind of matrice (matrice with the monotonicity, or Monge, property), in the (max, +) cloed emiring, i.e., (M M )(i, j) = max k {M (i, k) + M (k, j)}. Although the ituation depicted in Figure 2(b) implie that the tructure that give rie to uch matrice i not alway available, the fact that we can deal with the ituation in Figure 2(a) will be ueful. For two dijoint vertex et A and B on the boundary of a region R of G, let matrix M AB contain the length of the hortet path that tart in A and end in B (by convention, thee path go through the vertice of G on their way from A to B). t G" P R" One-way eparator Fig. 4. A hortet path P between two vertice of R may go outide R.

9 Efficient Parallel Algorithm for Planar t-graph 201 DEFINITION 7. Let X and Y be two dijoint vertex et on the boundary of a region R of G, each totally ordered in ome way (i.e., uing one of the order in Definition 5), and uch that the row (rep., column) of the matrix M XY are a in the ordering for X (rep., Y ). Then matrix M XY i Monge iff for any two ucceive vertice p, p in X and two ucceive vertice q, q in Y,wehaveM XY (p, q) + M XY (p, q ) M XY (p, q ) + M XY (p, q). The following well-known lemma [2], [1] will alo be ueful. LEMMA 1. Aume that matrice M XY and M YZ are Monge, with X =c 1 Y c 2 Z for ome poitive contant c 1 and c 2. Then M XY M YZ can be computed in O(log Y ) time and O( X Z ) work in the CREW PRAM model. 3. All-Pair Shortet Path in Planar t-graph. Thi ection preent our O(log 2 n) time, O(n 2 ) work CREW PRAM algorithm for computing all-pair hortet path in planar t-graph. Our parallel algorithm ue a divide-and-conquer approach. A key to thi divide-and-conquer algorithm i a partitioning cheme that make ue of trict one-way eparator. We dicu firt our graph partitioning cheme, and then our divideand-conquer algorithm Strict One-Way Separator. The trict one-way eparator we ue are baed on a geometric tructure that i decribed by the following viibility repreentation for planar t-graph. DEFINITION 8. A viibility repreentation VR(G) for a planar t-graph G i a mapping of G in the plane a follow: every vertex u of G i mapped to a horizontal line egment VR h (u), and every directed edge (u,w) of G i mapped to a vertical line egment VR v ((u,w)) whoe lower (rep., upper) endpoint i on the horizontal egment VR h (u) (rep., VR h (w)) uch that VR v ((u,w)) doe not interect any other horizontal egment of VR(G) (ee Figure 5 for an example). It ha been hown in [32] that a viibility repreentation VR(G) of a planar tgraph G can be computed in O(log n) time and O(n) work on the EREW PRAM, a t b (a) c b a t (b) c Fig. 5. (a) A planar t-graph. (b) A viibility repreentation for the graph of (a).

10 202 M. J. Atallah, D. Z. Chen, and O. Daecu uch that all the y-coordinate of the horizontal egment of VR(G) are ditinct integer. Actually, the equence of the horizontal egment of VR(G) in the increaing order of their y-coordinate correpond to a topologically orted equence of the vertice of G [32]. Without lo of generality, we aume that uch a viibility repreentation VR(G) in the plane i already available for the planar t-graph G we ue. A trict one-way eparator S G for an n-vertex planar t-graph G i defined a follow: Let L(S G ) be a horizontal line in the plane uch that n/2 horizontal egment of VR(G) are trictly above (rep., trictly below) L(S G ). Then L(S G ) interect the interior of a number of vertical egment of VR(G).IfL(S G ) interect the vertical egment VR v (e) for an edge e of G, then we aume that L(S G ) interect VR v (e) at an interior point which correpond to an artificial-vertex av(e) of G. Thee artificial-vertice are treated like ordinary vertice of G in the following way: If av(e) i an artificial-vertex on the original edge e = (u,v)of G, then it i a if the edge (u,v)i replaced by edge (u, av(e)) and (av(e), v) in G. However, artificial-vertice have no repreentation in VR(G), and hence they do not affect our graph partitioning cheme (i.e., we compute trict one-way eparator baed only on the original information of the graph G). (A i to be hown later, artificial-vertice are ueful in our hortet path computation.) Let S G be the equence of uch artificial-vertice created by the line L(S G ), uch that S G i in the increaing order of the x-coordinate of the interection point between L(S G ) and the correponding vertical egment of VR(G). LEMMA 2. The orted equence S G of all artificial-vertice created by the horizontal line L(S G ) on VR(G) i a trict one-way eparator for the planar t-graph G, uch that S G partition G into two region (for which the number of original vertice differ by at mot one). Furthermore, for any two conecutive artificial-vertice av(e ) and av(e ) in S G, their correponding edge e and e are on the boundary of the ame face f of G, with e (rep., e ) being on the left (rep., right) bounding path of f. PROOF. The equence S G i a trict one-way eparator becaue of the following: (1) L(S G ) cut VR(G) into two part, (2) any directed path from an artificial-vertex av(e) in S G can only go trictly upward in VR(G), and (3) each artificial-vertex i in the interior of an original edge of G. By it definition, S G clearly partition G into two region, each with the ame number of original vertice. The fact on any two conecutive artificial-vertice av(e ) and av(e ) in S G follow from the way in which a horizontal line interect the tructure of VR(G). Actually, it i not hard to ee that any directed path in G interect at mot one artificial-vertex of S G (i.e., no directed path in G pae through two ditinct artificialvertice of S G ). From the viibility repreentation VR(G) of G, it i eay to compute the trict one-way eparator S G of G, ino(log n) time and O(n) work on the EREW PRAM. With uch trict one-way eparator, we have the following graph partitioning cheme for the planar t-graph G: 1. Ue the trict one-way eparator S G to partition G into two region G 1 and G 2 (with G 1 below G 2 ). Note that G 1 and G 2 each contain n/2 original vertice of G, but

11 Efficient Parallel Algorithm for Planar t-graph 203 S G and the edge of G that contain an artificial-vertex of S G are excluded from G 1 and G Recurively partition each of G 1 and G 2, until every region contain exactly one vertex of G. One can ee that every region R of G generated by the above partitioning cheme i bounded by two trict one-way eparator, one on a horizontal line bounding R from above and the other on another horizontal line bounding R from below. Further, every directed path (and hence hortet path) between two vertice of R tay inide R (ince R i bounded by trict one-way eparator). We ue a partition tree T G to capture the above partitioning proce on the graph G. The root of T G i aociated with G and with the trict one-way eparator S G of G. The left (rep., right) child of the root of T G i the root of the partition tree T G1 (rep., T G2 ) for the lower (rep., upper) region G 1 (rep., G 2 )ofg. Note that the leave of T G, in the left-to-right order, correpond to the horizontal egment of VR(G) in the increaing order of their y-coordinate. Clearly, the number of level of the tree T G i O(log n). Since computing the trict one-way eparator S G of G take O(log n) time and O(n) work, T G can be obtained in altogether O(log 2 n) time and O(n log n) work on the EREW PRAM. LEMMA 3. Let z be a node of the tree T G that i aociated with a ubgraph R of G and with the trict one-way eparator S R of R. Let R contain r original vertice of G. Then the ize of S R can be a big a O(r) and a mall a zero. PROOF. Since R i an r-vertex planar graph, S R can certainly be O(r) (i.e., the horizontal line L(S R ) cut O(r) edge of R in the viibility repreentation VR(G) of G). On the other hand, it i alo poible that S R = 0 ince there may be no edge connecting the two region into which S R partition R. We maintain T G explicitly for our hortet path computation. Every node of T G tore certain information: the root root(t G ) of T G tore G and S G, the left child of root(t G ) tore G 1 and S G1, etc. Some hortet path information i alo tored at the node of T G (more on thi later) All-Pair Shortet Path Computation. We are now ready to dicu the parallel computation for all-pair hortet path in the planar t-graph G. Note that the partitioning cheme in Section 3.1 put exactly one artificial-vertex av(e) on every original edge e of G. Hence G ha O(n) artificial-vertice. We hall be computing the O(n 2 ) all-pair hortet path between the O(n) original and artificial vertice of G. To implify the computation, we need to reduce the planar t-graph G (with weighted edge) to another planar t-graph with weighted vertice: (1) for every original vertex v of G, let the weight of v be zero, (2) for every artificial-vertex av(e) of G, let the weight of av(e) be the weight of the correponding original edge e of G, and (3) let the weight of all edge of G be zero. Thi reduction i eay to perform. We till let G denote the reulting planar t-graph.

12 204 M. J. Atallah, D. Z. Chen, and O. Daecu The two lemma below form the bai for achieving efficiency in our hortet path computation. LEMMA 4. Let S R be the trict one-way eparator for a region R of G. For any vertex v of G, if there are two directed path in G, one between v and an artificial-vertex av(e) of S R and the other between v and another artificial-vertex av(e ) of S R (with av(e) preceding av(e ) in S R ), then there i a directed path in G between v and every artificial-vertex of S R that i between av(e) and av(e ). PROOF. Without lo of generality, aume that the two directed path are from the vertex v to the artificial-vertice of S R (if any). Let P 1 (rep., P 2 ) be a directed path from v to the ink t of G paing through av(e) (rep., av(e )). Then every artificial-vertex av(e ) of S R that i between av(e) and av(e ) i inide the region of G that i bounded by P 1 and P 2. Since there i a directed path P 3 from the ource of G to av(e ) in G, P 3 mut interect a vertex on either P 1 or P 2. Thi implie that there i a directed path in G from v to av(e ). Lemma 4 help u decide for which pair of artificial-vertice of G (on variou trict one-way eparator generated by our graph partitioning cheme in Section 3.1) we need to compute hortet path. Note that a query on whether there i a directed path from a vertex v to another vertex w in G can be anwered in O(1) work after an O(log n) time, O(n) work EREW PRAM preproceing [32]. Let SP R (u,v)denote a hortet path inide a graph region R from a vertex u to another vertex v. Let Length(P) denote the length of a path P. LEMMA 5. Let S and S be two trict one-way eparator tored in the node of the partition tree T G, uch that the horizontal line L(S ) containing S i bellow the horizontal line L(S ) containing S. Let a and b be two artificial-vertice of S with a preceding bins, and let c and d be two artificial-vertice of S with c preceding d in S. Then the hortet path SP G (a, d) interect the hortet path SP G (b, c) at a vertex of G (ee Figure 6). Further, Length(SP G (a, c)) + Length(SP G (b, d)) Length(SP G (a, d)) + Length(SP G (b, c)). PROOF. To prove that the hortet path SP G (a, d) and SP G (b, c) interect each other at a vertex of G, let P(, a) (rep., P(d, t)) be a directed path in G from the ource of G to a (rep., from d to the ink t of G). Then the vertice b and c are on oppoite ide of the upward directed path Q = P(, a) SP G (a, d) P(d, t). Hence the hortet path SP G (b, c) mut cro the path Q. Since SP G (b, c) goe upward in the viibility repreentation VR(G) of G, uch a croing cannot be below the line L(S ) and cannot be above the line L(S ). Therefore SP G (a, d) and SP G (b, c) interect each other (ay) at a vertex h of G. To how that Length(SP G (a, c)) + Length(SP G (b, d)) Length(SP G (a, d)) + Length(SP G (b, c)), we only need to point out that Length(SP G (a, c)) Length(SP G (a, h)) + Length(SP G (h, c)) and that Length(SP G (b, d)) Length(SP G (b, h)) + Length (SP G (h, d)) (ee Figure 6).

13 Efficient Parallel Algorithm for Planar t-graph 205 t P(d,t) c S" d L(S") SP G (a,c) h SP G (a,d) SP G (b,c) a S P(,a) SP G (b,d) b L(S ) Fig. 6. Illutrating Lemma 5. Lemma 5 enure that the length matrice for hortet path between artificial-vertice on any two trict one-way eparator generated by our graph partitioning cheme in Section 3.1 have the monotonicity property [1], [2]. Therefore, we can ue the efficient parallel algorithm for monotone matrix multiplication in [1] and [2] to compute hortet path in planar t-graph. The main tep of our all-pair hortet path algorithm are a follow: 1. A bottom-up tage on the partition tree T G. In thi tage, for every region R in T G,we compute and tore the hortet path length from the artificial-vertice on the lower boundary of R to the artificial-vertice on the upper boundary of R. 2. A top-down tage on the partition tree T G. In thi tage, for each level k in T G and for each two region R and R on level k, uch that R i below R, we compute the hortet path length from the artificial-vertice on the upper boundary of R to the artificial-vertice on the lower boundary of R. Thu, at the end of thi tage, we have computed the all-pair hortet path length between the artificial vertice in G. 3. An update tage, to obtain the hortet path length between original vertice of G. The computation of the firt tage proceed upward level by level on T G, tarting from the leave of T G. Suppoe that the computation reache a node z of T G at the current level. Let z be aociated with a graph region R of G and with the trict one-way eparator S R. Then S R i the common boundary of the two region aociated with the two children of z in T G. The computation at the left child of z ha obtained and tored hortet path length from the artificial-vertice on the lower boundary of R to the artificial-vertice on S R, and the computation at the right child of z ha obtained and tored hortet path length from the artificial-vertice on S R to the artificial-vertice on the upper boundary of R. The computation at z then obtain and tore the hortet path length from the artificial-vertice on the lower boundary of R to the artificial-vertice on the upper boundary of R, uing the hortet path information tored at the two children of z. Thi hortet path computation i baed on Lemma 4 and 5.

14 206 M. J. Atallah, D. Z. Chen, and O. Daecu After the firt tage reache the root of T G, the econd tage take over. The computation of the econd tage proceed downward level by level on T G. Let u and v be any two node of T G at the current level, uch that u (rep., v) i aociated with a region R (rep., R )ofg and with the trict one-way eparator S R (rep., S R ), and uch that R i below R. Then the hortet path length from the artificial-vertice on the upper boundary of R to the artificial-vertice on the lower boundary of R are computed. Thi hortet path computation get help from the hortet path information already computed and tored at the parent node u of u and at the parent node v of v, and i alo baed on Lemma 4 and 5. For a node x T G, let S l (x) (rep., S r (x)) be the one-way eparator bounding the region R x, aociated with x, from below (rep., above). For two one-way eparator S 1 and S 2, let M[S 1, S 2 ] be the matrix of hortet path length from the artificial vertice on S 1 to the artificial vertice on S 2. There are four poible cae to conider for every pair of node u and v at the current level, depending on whether u (rep., v) i the left or right child of it parent node: Cae 1: u i the right child of u and v i the left child of v. length were already computed at the previou level. Then the hortet path Cae 2: u i the right child of u and v i the right child of v. Then the hortet path matrix M[S r (u), S l (v)] i computed a a monotone matrix multiplication between the hortet path matrix M[S r (u ), S l (v )], tored at u (computed earlier in the econd tage), and the hortet path matrix M[S l (x), S r (x)], tored at the left child x of v (computed in the firt tage). Cae 3: u i the left child of u and v i the left child of v. Then the hortet path matrix M[S r (u), S l (v)] i computed a a monotone matrix multiplication between the hortet path matrix M[S l (x), S r (x)], tored at the right child x of u (computed in the firt tage) and the hortet path matrix M[S r (u ), S l (v )], tored at u (computed earlier in the econd tage). Cae 4: u i the left child of u and v i the right child of v. Then the hortet path matrix M[S r (u), S l (v)] i computed a two monotone matrix multiplication. The firt multiplication, between the hortet path matrix M[S l (x), S r (x)], tored at the right child x of u (computed in the firt tage) and the hortet path matrix M[S r (u ), S l (v )], tored at u (computed earlier in the econd tage), compute the hortet path matrix M[S l (x), S l (v )]. The econd multiplication, between the hortet path matrix M[S l (x), S l (v )] and the hortet path matrix M[S l (y), S r (y)], tored at the left child y of v (computed in the firt tage), give the hortet path matrix M[S r (u), S l (v)]. To obtain the hortet path length between the original vertice of G, we only need to oberve that the hortet path length between two artificial vertice av(e) and av(e ), on edge e and e repectively, i alo the hortet path length between the original vertice z and z, where z i the lower vertex of av(e) and z i the upper vertex of av(e ) in VR(G). Our parallel all-pair hortet path algorithm take altogether O(log 2 n) time and O(n 2 ) work on the CREW PRAM. To obtain thi reult, we perform an amortized analyi over all the O(log n) level in T G, in order to bound the total work of our algorithm. Since the work in the top-down tage dominate the work in the bottom-up tage, we only need

15 Efficient Parallel Algorithm for Planar t-graph 207 to analyze the work in the top-down tage. Recall that the total number of artificial vertice (between which we compute hortet path length) i O(n) and the work to compute the hortet path length between the boundary vertice of two region R and R, uch that R (rep., R ) ha k (rep., k ) boundary vertice, i O(k k ). Then the work in thi tage (over all the O(log n) level in T G )io( n i, j=1,i j k ik j ), where n i=1 k i = O(n) and n j=1 k j = O(n). Since O( n i, j=1,i j k ik j ) = O(( n i=1 k i)( n j=1 k j)) = O(n 2 ),it follow that the work in the top-down tage i O(n 2 ). Once the all-pair path length are available, the actual ingle-ource hortet path tree SPT u, for every ource vertex u G, i eay to generate in O(log n) time and O(n) work. Thi computation i baed on the fact that, for every vertex v with Length(SP G (u,v)) + (thu, v SPT u ), the equality Length(SP G (u,v)) = Length(SP G (u,w)) + Length(w, v) hold for at leat one vertex w, uch that (w, v) i an incoming edge of the vertex v. Note that our parallel all-pair hortet path algorithm on planar t-graph, when applied to computing all-pair hortet path in the planar directed acyclic grid graph conidered in [1] and [2], yield an O(log 2 n) time, O(n 2 ) work CREW PRAM olution. 4. Single-Source Shortet Path in Planar Layered t-graph. Thi ection preent our O(log 2 n) time, O(nlog n) work CREW PRAM algorithm for computing ingleource hortet path in planar layered t-graph. A in the all-pair algorithm in Section 3, our parallel ingle-ource algorithm ue a divide-and-conquer approach. A key to thi divide-and-conquer algorithm i a partitioning cheme that make ue of one-way eparator (rather than the trict one-way eparator in Section 3). One might wonder why we ue one-way eparator intead of trict one-way eparator to compute in parallel ingle-ource hortet path in planar layered t-graph. Recall from Lemma 3 that the ize of a trict one-way eparator S R for a ubgraph R of a planar t-graph can be proportional to the ize of R. Thi lead to a parallel hortet path algorithm whoe work bound i at leat quadratic. Such a parallel algorithm would certainly be too expenive for the ingle-ource problem on planar layered t-graph, whoe equential time bound i only linear. The one-way eparator we ue in thi ection are a modified verion of thoe ued in [28]. The ize of uch a one-way eparator for a graph region i proportional to the quare-root of the ize of the region. Hence, thee one-way eparator are more likely to yield an efficient parallel ingle-ource algorithm. On the other hand, a dicued in Section 2, one-way eparator do not enure a much locality a trict one-way eparator for hortet path computation (i.e., true hortet path may go outide a region bounded by one-way eparator). Therefore, hortet path computation baed on one-way eparator mut be done with much more care in order to achieve the deired efficiency. LEMMA 6 [28]. uch that Every n-vertex planar layered t-graph G ha a one-way eparator X (1) X c n for ome poitive contant c, and

16 208 M. J. Atallah, D. Z. Chen, and O. Daecu (2) X partition G X into at mot four region, uch that the number of vertice of each region i no bigger than 2n/3. Furthermore, uch a eparator can be computed in O(log n) time and O(n) work on the EREW PRAM if a k-line embedding of G i given. PROOF. See Theorem 2 and 3 in [28]. We modify the algorithm for computing a one-way eparator in [28] to generate a lightly different one-way eparator a hown in the next lemma. LEMMA 7. Given a k-line embedding of an n-vertex planar layered t-graph G, a oneway eparator X of G can be obtained in O(log n) time and O(n) work on the EREW PRAM, uch that (1 ) X c n for ome poitive contant c, and (2 ) X partition G X into exactly two region A and B, uch that neither A nor B ha more than 2n/3 vertice. PROOF. We firt review briefly the propertie of the one-way eparator ued in [28] (the reader i referred to [28] for more detail). The one-way eparator X computed in Theorem 3 of [28] conit of two layer L i and L l of the graph G, and poibly a pecial directed path p in the portion M of G between L i and L l (ee Figure 7). The two layer L i and L l together partition G into at mot three region A, B, and M. If M ha le than 2n/3 vertice, then no path p i needed, and X = L i L l.ifm ha at leat 2n/3 vertice, then a path p in M i computed a follow: Tranform M into a planar layered t-graph (by mainly adding a ource and a ink to M); let m be the middle vertex of the vertice of M in the left equence of M; obtain a path p by going up from m and following the leftmot outgoing edge until L l i reached, and by going down from m and following the rightmot incoming edge until L i i reached. The path p partition M into two region C and D, each with the ame number of vertice (ee t A D layer l C m p M B layer i Fig. 7. Illutrating the one-way eparator.

17 Efficient Parallel Algorithm for Planar t-graph 209 Figure 7). Let X = L i L l p. X o obtained i a one-way eparator with the following propertie: None of A and B ha more than 2n/3 vertice, and none of C and D ha more than n/2 vertice. X = L i L l p c n for ome poitive contant c. We modify the algorithm in [28] o that it compute a one-way eparator X that partition G into two region A and B. Our algorithm conit of the following tep: (a) Ue layer L i and L l to partition G into A, B, and M a in Figure 7. (b) If either A or B ha at leat n/3 vertice (ay, A ha n/3 vertice), then let A = A and B = M B. (c) Otherwie (none of A and B ha n/3 vertice), ue the path p to partition M into C and D, and let A = A D and B = B C. The parallel complexity bound of our algorithm are jut the ame a thoe in Lemma 6. We now claim that (i) the common boundary X between A and B i a one-way eparator with X c n, and (ii) none of A and B ha more than 2n/3 vertice. Proof of the Claim. It i clear that X eparate G X into two region A and B. Claim (i) follow from the proof of Theorem 2 of [28], becaue X L i L l p. We now prove claim (ii). Note that A and B are obtained from tep (b) or (c). In tep (b), uppoe A ha n/3 vertice; then claim (ii) eaily follow becaue A = A and A ha 2n/3 vertice. In tep (c) (none of A and B ha n/3 vertice), let 2n CD be the number of vertice in M and let n A (rep., n B ) be the number of vertice in A (rep., B). Then (n A +n CD )+(n B +n CD ) n. Aume that A ha more than 2n/3 vertice (the proof for B i imilar), that i, n A +n CD > 2n/3. Then, ince none of A and B ha n/3 vertice, (n A + n CD ) + (n B + n CD )>n, which i a contradiction. Thu, claim (ii) follow. Thi complete the proof of thi lemma. Note that although the algorithm in [28] find a one-way eparator that partition the graph G into at mot four region (each of which ha no more than 2n/3 vertice), every uch region may have a few a O(1) vertice. In contrat, we compute a oneway eparator that partition the graph G into exactly two region, whoe ize are nicely bounded from above and from below. Thi implifie the ubequent hortet path computation. Baed on our one-way eparator, we obtain a graph partitioning cheme. A in Section 3, we maintain explicitly a partition tree T G that capture the partitioning proce on the graph G. The main tep of our ingle-ource hortet path algorithm are a follow: 1. A bottom-up tage on the partition tree T G. In thi tage, when the level-by-level computation reache a node of T G that i aociated with a graph region R of G, we do the following: (a) Compute the all-pair hortet path length between all boundary vertice of the region R that do not go outide R. (b) Compute the all-pair hortet path length between all boundary vertice of the region R which may go through a neighboring region of R.

18 210 M. J. Atallah, D. Z. Chen, and O. Daecu The computation in thi tage i baed on the parallel algorithm in [1] and [2] for multiplying monotone matrice of ize O(m) O(m). 2. A top-down tage on the partition tree T G. In thi tage, when the level-by-level computation reache a node of T G that i aociated with a graph region R of G, the hortet path from the ource of G to all boundary vertice of R are computed. Thi computation i baed on the parallel algorithm of Atallah and Koaraju [3] for earching monotone matrice of ize O(m) O(m). Note that the hortet path algorithm in [28] alo ue a bottom-up procedure (but not a top-down one) for computing a ource-to-ink hortet path in G. Although the bottom-up procedure in [28] i baed on one-way eparator (which may allow true hortet path to go outide a region bounded by uch eparator), it i able to compute correctly the ource-to-ink hortet path in G. The reaon for thi i that uch a bottomup procedure can correctly compute hortet path between boundary vertice of a graph region provided that the hortet path do not go outide that region. Oberve that the ource-to-ink hortet path in G i eventually computed between two boundary vertice of a graph region that i G itelf. Our ingle-ource algorithm alo hinge on computing hortet path between boundary vertice of the graph region that are generated by our graph partitioning cheme baed on one-way eparator. However, in the ingle-ource cae, we mut compute the true hortet path from to all vertice of G. Conider a region R generated by our partitioning cheme uch that R i a proper ubregion of G. In order to compute the true hortet path from to all the boundary vertice of R, we need to compute correctly and efficiently the true hortet path between the boundary vertice of R. However, ome of thee hortet path may go outide R (even though thee path are till inide the larget region G). See Figure 8 for an example. Our ingle-ource algorithm mut handle thi difficulty carefully. Our idea for reolving thi difficulty i to divide the candidate hortet path into everal group, uch that the computation on each group of path can be done by uing the monotone matrix earching or multiplication algorithm in [1] [3]. The correct hortet path are then elected among their candidate path. t R R R' m p P R L l R" MR L i R Fig. 8. A hortet path P R goe outide R through R.

19 Efficient Parallel Algorithm for Planar t-graph 211 Let M[R, R] denote the matrix of hortet path length between the boundary vertice of a region R in T G and let M[, R] denote the matrix of hortet path length from to the boundary vertice of R. We compute the hortet path length between boundary vertice in the bottom-up tage. Suppoe that the bottom-up computation reache a node u of T G that i aociated with a graph region R. Let X R be the one-way eparator aociated with R, and let L i and L l be the two horizontal layer of X R (ee Figure 8). The difficulty with our one-way eparator arie when X R = L i L l p, that i, when the ubgraph region M R of R between L i and L l ha more than 2r/3 vertice, where r i the number of vertice in R. To ee why thi i the cae, let v and w be the left and right children of u in T G, aociated with region R and R, repectively. Then the hortet path between the vertice on the boundary of R (rep., R ) which tay inide R (rep., R ) are already computed at v (rep., w). However, ome hortet path from vertice on the boundary of R to vertice on the directed path p of X R may go through R (e.g., the path P R in Figure 8). To compute thee hortet path, we do the following: 1. From the hortet path matrix tored at R, elect the ubmatrix of hortet path between pair of vertice on p. 2. Ue the hortet path between vertice on p to compute the correct hortet path (which may go through R ) between the vertice on the boundary of R and the vertice on p, and update the hortet path information at v. Thi computation can be done a a monotone matrix multiplication between M[R, p] and M[p, p]. 3. Chooe the better path length for M[R, R ] from thoe taying inide R and from thoe going through R. Finally, the hortet path between vertice on R are computed. Thi hortet path computation can be done a a monotone matrix multiplication between M[R, R ] and M[R, R ]. After the firt tage reache the root of T G, the econd tage take over. The computation of the econd tage proceed downward level by level on T G, at each level computing the hortet path length from to the boundary vertice of each region on that level. The computation at a node u T G that i aociated with a graph region R ue the parallel algorithm in [3] to compute M[, R]. Let M[, R ] be the hortet path length matrix for the region R, aociated with the parent of u in T G. From M[, R ], we only need the ubmatrix of hortet path length from to the boundary vertice of R that are alo boundary vertice of R. It i traightforward to extract thi ubmatrix once M[, R ] i available. We refer to thi ubmatrix a M [, R ]. Note that if M[R, R] ianm m matrix, then M [, R ]ia1 m matrix. Then M[, R] can be computed in O(log m) time, with O(m) EREW PRAM proceor [3], by multiplying the matrix M [, R ] with the matrix M[R, R]. We now analyze the work performed by our algorithm: 1. For the bottom-up phae. In thi phae, for each region R on every level k in T G, we compute the hortet path length between boundary vertice of R. At level k, there are 2 k region and the number of boundary vertice (which are vertice on ome one-way eparator) for each region i O( n/2 k/2 ). Then the work at level k i O(n).

20 212 M. J. Atallah, D. Z. Chen, and O. Daecu Summing over all O(log n) level in T G, the work of our algorithm in the bottom-up phae i O(n log n). 2. For the top-down phae. In thi phae, for each region R on every level k in T G,we compute the hortet path length from to the boundary vertice of R. The work at level k i O(2 k n/2 k/2 log( n/2 k/2 )). Summing over all O(log n) level in T G, the work of our algorithm in the top-down phae i bounded from above by O(n log n). Thu, our parallel ingle-ource hortet path algorithm take altogether O(log 2 n) time and O(n log n) work on the CREW PRAM. 5. Depth-Firt Search in Planar t-graph. Mot known parallel depth-firt earch algorithm on planar graph are hinged on ome kind of eparator reult [10], [13], [15] [19], [29], [30]. However, the tak of computing in parallel ueful eparator for planar graph i quite nontrivial [9], [19], [24]. In contrat, our optimal parallel depthfirt earch algorithm on planar t-graph doe not rely on any eparator reult and i actually very imple. Let be the root of the depth-firt earch tree on G. Without lo of generality, we aume that i the ource of G (other cae can be handled eaily). Suppoe that, for every vertex v of the planar t-graph G = (V, E), the cyclic ordering of v outgoing edge i given in the clockwie direction, tarting from it leftmot outgoing edge. Let incoming lm (v) be the leftmot incoming edge of each vertex v of G (incoming lm () i empty for the ource of G). Let INCOMING lm (V ) ={incoming lm (v) v V }. LEMMA 8. The ubgraph T = (V, INCOMING lm (V )) of the planar t-graph G = (V, E) i a depth-firt earch tree of G rooted at the ource of G. PROOF. We firt prove that T i a panning tree of G rooted at the ource of G, and then that it i a depth-firt earch tree of G. A previouly, we aume that the embedding of G i uch that all edge are oriented upward. From the contruction of T, each vertex u ha an unique parent parent(u) (the end vertex of incoming lm (u) which i different from u). Then, to prove that T i a panning tree for G rooted at, we only need to how that for every vertex u, there i a unique directed path from to u in T. To prove thi, for every vertex u G, we contruct a path P a follow: (1) Let P be initially empty. (2) Add to P the edge incoming lm (u) (which i an edge in T ). (3) Repeat the previou tep, with u replaced by the other end vertex of incoming lm (u), until a vertex v i reached uch that v ha no incoming edge. Such a vertex v mut exit and it can only be (ince only incoming lm () i empty). Thi complete the proof that T i a panning tree of G rooted at. Let T DFS be the depth-firt earch tree of G rooted at, contructed by firt viiting the leftmot edge of a vertex u of G, and aume that there i a vertex w G uch that the path P from to w in T i different from the path P from to w in T DFS. Then there exit two vertice u (poibly ) and v (poibly w) uch that u,v P, u,v P, and the path P uv from u to v in T i vertex dijoint from the path P uv from u to v in T DFS. Let u (rep., u ) be the vertex ucceeding u in P (rep., P ), and let v (rep., v ) be the vertex preceding v in P (rep., P ) (ee Figure 9). We denote the directed edge