These, and closely related, roblems has been extensively studied for the sequential [1, 16] and the shared memory (PRAM) arallel [5, 6, 12, 13, 14, 15

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1 Coarse Grained Parallel Algorithms for Detecting Convex Biartite Grahs Extended Abstract E. Caceres y A. Chan z F. Dehne x G. Prencie { Abstract In this aer, we resent arallel algorithms for the coarse grained multicomuter (CGM) and bulk synchronous arallel comuter (BSP) for solving two well known grah roblems: (1) determining whether a grah G is biartite, and (2) determining whether a biartite grah G is convex. Our algorithms require O(log ) ando(log 2 ) communication rounds, resectively, and linear sequential work er round on a CGM with rocessors and N= local memory er rocessor, N=jGj. The algorithms assume that N for some xed >0, which is true for all commercially available multirocessors. Our results imly BSP algorithms with O(log ) ando(log 2 ) suerstes, resectively, O(g log() N ) communication time, and O(log() N ) local comutation time. Our algorithm for determining whether a biartite grah is convex includes a novel, coarse grained arallel, version of the PQ tree data structure introduced by Booth and Lueker. Hence, our algorithm also solves, with the same time comlexity as indicated above, the roblem of testing the consecutive-ones roerty for(0 1) matrices as well as the chordal grah recognition roblem. These, in turn, have numerous alications in grah theory, DNA sequence assembly, database theory, and other areas. 1 Introduction In this aer, we study the roblem of detecting biartite grahs and convex biartite grahs. That is, given an arbitrary grah G, determine whether G is a biartite grah and, given a biartite grah G, determine whether G is a convex biartite grah. Biartite and convex biartite grahs are formally dened as follows. Denition 1 Agrah G =(V E) is a biartite grah if V can be artitioned into two sets A and B such that A \ B =, A [ B = V and E ((A B) [ (B A)). Abiartite grah G is also denoted as G =(A B E). Denition 2 Abiartite grah G =(A B E) isaconvex biartite grah if there exists an ordering (b1 b2 b jbj ) of B such that, for all a 2 A and 1 i<jjbj, if (a b i ) 2 E and (a b j ) 2 E then (a b k ) 2 E for all i k j. Research artially suorted by the Natural Sciences and Engineering Research Council of Canada and by PRONEX-FINEP, Brazil. y Det. de Comutacao e Estatistica, UFMS, Camo Grande, Brasil, edson@dct.ufms.br z School of Comuter Science, Carleton University, Ottawa, Canada K1S 5B6, achan@scs.carleton.ca x School of Comuter Science, Carleton University, Ottawa, Canada K1S 5B6, dehne@scs.carleton.ca { Diartimento di Informatica, Corso Italia 40, Pisa, Italy, rencie@di.unii.it 1

2 These, and closely related, roblems has been extensively studied for the sequential [1, 16] and the shared memory (PRAM) arallel [5, 6, 12, 13, 14, 15] domain. Unfortunately, theoretical results from PRAM algorithms do not necessarily match the seedus observed on real arallel machines. In this aer, we resent arallel algorithms that are more ractical in that the assumtions and cost model used reects better the reality of commercially available multirocessors. More recisely, we will use a version of the BSP model, referred to as the coarse grained multicomuter (CGM) model. In comarison to the BSP model, the CGM [7, 8, 9, 10] allows only bulk messages in order to minimize message overhead costs. A CGM is comrised of a set of rocessors P1 ::: P with O(N=) local memory er rocessor and an arbitrary communication network (or shared memory). All algorithms consist of alternating local comutation and global communication rounds. Each communication round consists of routing a single h-relation with h = O(N=), i.e. each rocessor sends O(N=) data and receives O(N=) data. We require that all information sent from a given rocessor to another rocessor in one communication round is acked into one long message, thereby minimizing the message overhead. A CGM comutation/communication round corresonds to a BSP suerste with communication cost g N (lus the above \acking requirement"). Finding an otimal algorithm in the coarse grained multicomuter model is equivalent tominimizing the number of communication rounds as well as the total local comutation time. The CGM model has the advantage of roducing results which corresond much better to the actual erformance of imlementations on commercially available arallel machines. In addition to minimizing communication and comutation volume, it also minimizes imortant other costs like message overheads and rocessor synchronization. In this aer, we resent arallel CGM algorithms for detecting biartite grahs and convex biartite grahs. The algorithms require O(log ) and O(log 2 ) communication rounds, resectively, and linear sequential work er round. They assume that the local memory er rocessor, N=, is larger than for some xed > 0. This assumtion is true for all commercially available multirocessors. Our results imly BSP algorithms with O(log ) suerstes, O(g log() N ) communication time, and O(log() N ) local comutation time. The algorithm for detecting biartite grahs is fairly simle and is essentially a combination of tools develoed in [3]. The larger art of this aer deals with the roblem of detecting convex biartite grahs. This is clearly a much harder roblem. It has been extensively studied in the literature and is closely linked to the consecutive ones roblem for (0 1)-matrices as well as chordal grah recognition [1, 5, 6, 12, 13, 14, 15, 16]. Our algorithm for determining whether a biartite grah is convex includes a novel, coarse grained arallel, version of the PQ tree data structure introduced by Booth and Lueker [1]. Hence, our algorithm also solves, with the same time comlexity as indicated above, the roblem of testing the consecutive-ones roerty for(0 1)-matrices as well as the chordal grah recognition roblem. These, in turn, have numerous alications in grah theory, DNA sequence assembly, database theory, and other areas. [1, 5, 6, 12, 13, 14, 15, 16] 2 Detecting Biartite Grahs In this section, we resent a fairly simle CGM algorithm for detecting biartite grahs. It is essentially a combination of tools develoed in [3]. Algorithm 1 Detection of Biartite Grahs 2

3 Inut: A Grah G =(V E) with vertex set V and edge set E, jgj = N, stored on a CGM with rocessors and O(N=) memory er rocessor N= for some xed >0. V and E are arbitrarily distributed over the memories of the CGM. Outut: A Boolean indicating whether G is a biartite grah and, if it is, a artition of V into two disjoint set A and B such that E ((A B) [ (B A)). (1) Comute a sanning forest of G [3]. (2) For each tree in the forest, select one arbitrary node as the root. Aly the CGM Euler Tour algorithm in [3] to determine the distance between each node and the root of its tree. Classify the nodes into two grous: the nodes with an odd numbered distance to the root, and the nodes with an even numbered distance to the root. (3) Each rocessor examines the edges stored in its local memory. If any such edge has two vertices that belong to the same grou, the result for that rocessor is\failure" otherwise, the result is \success". (4) By alying CGM sort [11] to all \failure"/\success" values, it is determined whether there was any rocessor with a \failure" result. If there was any \failure", the grah G is not biartite. Otherwise, G is abiartite grah, and the two grous of vertices identied in Ste 2 are the sets A and B. Theorem 1 Algorithm 1 detects whether G = (V E), jgj = N, is a biartite grah and, if so, artitions E into sets A and B such that E ((A B) [ (B A)) in O(log ) communication rounds and O( N ) local comutation er round on a CGM with rocessors and O( N ) memory er rocessor, N for some xed >0. Proof. Omitted due to age restrictions consult [4] for details. 2 3 Detecting Convex Biartite Grahs We now turn our attention to the roblem of testing whether a given biartite grah is a convex biartite grah. The sequential solution, resented by Booth and Lueker [1], introduced a data structure called PQ-tree. Our coarse grained arallel solution will include a coarse grained arallel version of the PQ-tree. We will rst review Booth and Lueker's PQ-tree denition. 3.1 PQ-Trees A PQ-tree [1] is a tree data structure that reresents a class of ermissible ermutations over a universal set. Denition 3 A tree T is a PQ-tree if every internal node of T can be classied as either a P-node or a Q-node. A P-node is an internal node that has at least 2 children, and the children can be ermuted in any order. A Q-node is an internal node that has at least 3 children, and the children can only be ermuted in two ways: the original order or the reverse order. The leaves of the PQ-tree are elements of a universal set S = fa1 ::: a n g, usually called the ground set. The order of the ground set in the PQ-tree, from left to right, is called its frontier. The frontier of a PQ-tree is clearly a ermutation of the ground set. Given a PQ-tree T and using only ermissible ermutations of its internal nodes, we can generate a number of ermutations of S. We will denote with L(T ) the set of all these ermissible ermutations. A PQ-tree T 0 is equivalent to T if T 0 can be transformed into T using only ermissible ermutations of the internal nodes (if L(T 0 ) and L(T )have the same elements). 3

4 Given a set A S, we say that 2 L(T ) satises A if all elements of A aear consecutively in. The main oeration on a PQ-tree T is called reduce: given a reduction set A = fa1 ::: A k g of subsets of S and a PQ-tree T, we want obtain a PQ-tree T 0, if it exists, such that each ermutation in L(T 0 ) satises every A i,1 i k. Let m = k i=1 ja ij and N = n + m. In order to store T and A, we require a coarse grained multicomuter with rocessors and N= local memory er rocessor. Two articular PQ-trees are the universal and the emty tree: the rst one has only one internal node (the root of T ) and that internal node is a P-node the second one (also called a null PQ-tree) is used to reresent an imossible reduction, that is when it is imossible to reduce a PQ-tree with resect to a given reduction set. 3.2 Multile Disjoint Reduce Oerations on a PQ-Tree (MDReduce) In this section, we will resent a coarse grained arallel algorithm for the secial case of erforming multile disjoint reductions on a PQ-tree. We will then use this solution to develo the general algorithm in the subsequent section. More recisely, given a PQ-tree T we will rst study how to erform the reduce oeration for a set A = fa1 ::: A k g of subset of the universal set S where A1 ::: A k are disjoint. We shall refer to our algorithm as Algorithm MDReduce. For ease of discussion, each set A i is assigned a unique color, and we color the leaves of the PQ-tree accordingly. As shown in [13], MDReduce can be artitioned into the following hases: (1) re-rocess the PQ-Tree (Algorithm 2) (2) rocess all the P-nodes (Algorithm 3) (3) rocess all the Q-nodes (Algorithm 4) (4) if T is not a null tree, each rocessor examines its nodes and relabel each orientable node (dened later) to become an R-node (5) ost-rocess the PQ-Tree (Algorithm 5). In the following subsections of Section 3.2, we resent coarse grained arallel algorithms for each hase. The PQ-tree denitions used are from [1, 13, 14, 15] Pre-Processing the PQ-Tree The re-rocessing hase extends the coloring of the leaves to a coloring of all nodes of the PQ-tree T. For an internal node v of T, we say that a color is comlete at v if all the leaves with that color are descendants of v. We say a color is incomlete at v if some, but not all, of the leaves of that color are descendants of v. We say that a color covers v if all the leaves below v are of that color, and that v is uncovered if no color covers v. Let LCA(c) be the lowest common ancestor of all leaves with color c. Let COLORS(v) denote the set of colors assigned to leaves that are descendents of v. Let INC(v) be the set of colors which are incomlete at v. Then INC(v) = COLORS(v) - fc: LCA(c) is a descendent ofvg. Algorithm 2 Pre-Processing the PQ-Tree Inut: The original PQ-tree T. Outut: The original PQ-tree T in which each node is assigned a "coloring", or, if failure occurs, a null tree. (1) Aly the coarse grained arallel Lowest Common Ancestor (LCA) algorithm [3]. (2) Exand T intoabinary tree B, as described in Klein [13]. (3) Perform tree contraction on B[3]. For eachnodev b in B, let v be the node in T from which v b is created. Let w 1 and w 2 be the children of v b. The oeration for the tree contraction is INC(v b ) = INC(w 1 ) [ INC(w 2 ) - fc: LCA(c) is a descendent of v g. If at any oint the size of INC is more than two, sto and return a null tree. 4

5 (4) Let c v be a new color unique to node v. Each rocessor, for all its nodes, v, calculates (v) = <c 1 c 2 > as follows: If two colors are incomlete at v, then c 1 and c 2 are these colors. If only one color c is incomlete at v but c does not cover v, then c 1 = c and c 2 = c v. If one color c is incomlete at v and covers v, then c 1 = c 2 = c. If no color is incomlete at v, thenc 1 = c 2 = c v. Lemma 1 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algorithm 2 can be comleted ino(log ) communication rounds with O( N ) local comutation er round. Proof. Omitted due to age restrictions consult [4] for details Processing P-Nodes Anodev in a PQ-tree is orientable if it is a Q-node and the two colors in its (v) =<c1 c2 > are dierent, i.e. c1 6= c2. ( c if c 2 INC(v) For a color c, dene h v (c) = c v if c =2 INC(v) For a PQ-tree T where w1 and w k are the leftmost and rightmost elements, resectively, of the frontier fr T (v), let l T = h v ((w1)) and r T = h v ((w k )). If lr T [v] =<l T [v] r T [v] > then we use the following notation: <a b>< a 0 b 0 > if fa bg = fa 0 b 0 g. Algorithm 3 Processing P-Nodes Inut: The PQ-Tree outut from Algorithm 2. Outut: The original PQ-tree T in which all the P-nodes have been rocessed, or, if failure occurs, a null tree. (1) If the inut PQ-tree T is a null tree, return T. (2) Each rocessor setsvariable FAILURE to F ALSE (3) Each rocessor, for each P-node v, reorder the children of v such that for eachcolor c all children covered by c are consecutive. (4) Each rocessor, for each P-node v and each color c, if there are at least two children covered by c (and at least one child not covered by c) then insert a new P-node w c between these c-covered children and v. (5) Each rocessor, for each P-node v, constructs an auxiliary grah G v whose nodes are the children of v and where for each color c there is an edge between children v i and v j at which c is incomlete if v i or v j is covered by c, or there is no child covered by c. If any node has more than 2 neighbors, set FAILURE to TRUE to indicate a failure condition. (6) Perform a multi-broadcast of the variable FAILURE. If any of the broadcast values is TRUE, return a null tree. (7) Each rocessor uses list-ranking to identify the connected comonents of each G v and veries that each of these connected comonents is a simle ath. If any of these comonents is a cycle, set FAILURE to TRUE to indicate a failure condition. We call these aths color chains. (8) Perform a multi-broadcast of the variable FAILURE. If any of the broadcast values is TRUE, return a null tree. (9) Each rocessor, for each color chain containing at least 2 nodes, chooses one of the 2 orientations of arbitrarily. Reorder the children of v so that the nodes of are consecutive, and insert a new Q-node between these nodes of v. (10) Each rocessor, for each P-node v, let S = fv i : v i is a child of v, andinc(v i )= g. If every child of v is in S, then return. Otherwise, reorder the children of v to make S consecutive, insert a new P-node v 0 between v and the subset S (if jsj > 1), and rename v to be a Q-node. Lemma 2 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algorithm 3 can be comleted ino(log ) communication rounds with O( N ) local comutation er round. 5

6 Proof. Omitted due to age restrictions consult [4] for details Processing Q-Nodes For each Q-node v, we dene an orientation LR(v) which iseither(v i ) or (v i ) R. Note that if (v i ) =< c1 c2 > than (v i ) R =< c2 c1 > For < a b >< a 0 b 0 > and a 6= b, we dene < a b > swa < a 0 b 0 > equals TRUE if < a b >=< b 0 a 0 >, F ALSE if <a b>=< a 0 b 0 >. ForaQ-nodev, fli is dened as the oeration which re-orders all its children in reverse order. Algorithm 4 Processing Q-Nodes Inut: The PQ-tree outut from Algorithm 3. Outut: The original PQ-tree T in which all the Q-nodes have been rocessed, or, if failure occurs, a null tree. (1) If the inut PQ-tree T is a null tree, return T. (2) Each rocessor setsvariable FAILURE to F ALSE (3) Each rocessor, for each Q-node v and children be v 1 ::: v s, assign to each LR[v i ] either (v i ) or (v i ) R such that every color in the sequence LR[v 1 ] ::: LR[v s ] occurs consecutively, and such that h v (< L[v 1 ] R[v s >]) (v). If this is imossible, set FAILURE to TRUE to indicate a failure condition, otherwise, set LR[v] to h v (< L[v 1 ] R[v s ] >). (4) Perform a multi-broadcast of the variable FAILURE. If any of the broadcast values is TRUE, return a null tree. (5) Each rocessorforeachnodev: if v is orientable, then set OP P [v] to LR[v] swa LR[v], otherwise, set OP P [v] to F ALSE. L L (6) Each rocessor for each node v: set REV [v] to OP P [u] (Note: u is an ancestor of v denotes "exclusive-or"). (7) For eachorientable node v, ifrev [v] is TRUE, then i v. Lemma 3 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algorithm 4 can be comleted ino(log ) communication rounds with O( N ) local comutation er round. Proof. Omitted due to age restrictions consult [4] for details Post-Processing the PQ-Tree Algorithm 5 Post-Processing the PQ-Tree Inut: The PQ-tree outut from Algorithm 4, with all R-nodes renamed. Outut: Result of Algoithm MDReduce. (1) If T is a null tree, return. (2) Each rocessor temorarily cuts the links of its Q-nodes to their arents. (3) Each rocessor erforms ointer juming for all its nodes that are children of R-nodes to determine their lowest Q-node ancestor. (4) Each rocessor restores the links cut in Ste 2. (5) Each rocessor eliminate its R-nodes by setting the arents of their children to their lowest Q-node ancestors. Lemma 4 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algoritm 5 can be comleted using in O(log ) communication rounds with O( N ) local comutation er round. Proof. Omitted due to age restrictions consult [4] for details. 2 6

7 3.2.5 Analysis Theorem 2 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algorithm MDReduce erforms a multile disjoint reduce for a PQ-tree T in O(log ) communication rounds with O( N ) local comutation er round. Proof. Omitted due to age restrictions consult [4] for details Multile General Reduce Oerations on a PQ-Tree (MReduce) Using the coarse grained arallel MDreduce algorithm resented in the revious section, we will now develo coarse grained arallel algorithm for the general MReduce oeration: given a PQ-tree T over the ground set S with n elements, erform the reduce oeration for an arbitrary reduction sets A = fa1 ::: A k g Our CGM algorithm for the general MReduce oeration consists of two hases. In the rst hase, we execute 3log times an algorithm similar to the one roosed by Klein for the PRAM [13, 14, 15]. We call this oeration Klein-like-Mreduce(T fa1 ::: A k g 0). Our contribution here is the coarse grained arallel imlementation of the various stes which is, in some cases, non trivial. After the rst hase, we have reduced the roblem to one in which we are left with a set of smaller PQ-trees over ground sets whose size is at most n=. Hence, each tree can be stored in the local memory of one rocessor. However, we can not guarantee that all the reduction sets of these PQ-trees do also t in the local memory of one rocessor. In the second art of our algorithm, we use a merging strategy to comlete the algorithm. We will refer to this hase as the Merging Phase. An illustration of our general algorithm strategy is given in Figure First Phase: Klein-like-MReduce For a node v of a PQ-tree, leaves T (v) denotes the set of endant leaves of v, i.e. leaves of T having v as ancestor. Let lca T (A) denote the least common ancestor in T of the leaves belonging to A. Suose that v = lca T (A) has children v1 :::v s in order. We say A is contiguous in T if either (1) v is a Q-node, and for some consecutive subsequence v ::: v q of the children of v, A = S iq leaves T (v i ), or (2) v is a P-node or a leaf, and A = leaves T (v). Suose that E is contiguous in T. T je denotes the subtree consisting of lca T (E) and those children of lca T (E) whose descendents are in E (it is still a PQ-tree whose ground set is E). For a set A, dene ( A A i je = i \ E if A i \ E 6= E if A i \ E = E Let? E denote lca T (E). T=E denotes the subtree of T obtained by omitting all the roer descendents of lca T (E) that are ancestors of elements of E (it is still a PQ-tree whose ground set is S ; E [f? E g). For a set A, dene ( A A i =E = i ; E [f? E g if A i E A i ; E otherwise Algorithm 6 Klein-like-Mreduce(T fa 1 ::: A k g i)[13, 14,15]: 7

8 (1) If i = 3 log, return. (2) Purge the collection of inut sets A i of emty sets. If no sets remain, return. (3) Let n be the size of the ground set of T. If n 4, carry out the reduction one by one. If the size of the inut is smaller than the size of the local memory of the rocessors, than solve the roblem sequentially using the Booth and Lueker's algorithm. (4) Otherwise, let A be the family of (nonemty) sets A i. Let S consist of the sets A i such that ja i j n=2. We call such sets "small". Let L be the remaining, "large", sets in A. Find the connected comonents of the intersection grah of A, nd a sanning forest of the intersection grah of S, and nd the intersection \L of the large sets. (5) Proceed according to one of the following cases: (a) The intersection grah of A is disconnected. In this case, let C 1 ::: C r be the connected comonents of A. For i =1 ::: r,lete i be the union of sets in the connected comonent C i. Call MDreduce to reduce T with resect to the disjoint sets E 1 ::: E r. Next, for each i =1 ::: r in arallel, recursively call Klein-like-Mreduce(T je i C i i+1). (b) The union of sets in some connected comonent of S has cardinality at least n=4. In this case, from the small sets making u this large connected comonent, select a subset whose union has cardinality between n=4 and 3n=4. Let E be this union, and call subreduce(t E fa 1 ::: A k g i). (c) The cardinality of the intersection of the large sets is at most 3n=4. In this case, from the large sets choose a subset whose intersection has cardinality between n=4 and 3n=4. Let E be this intersection, and call subreduce(t E fa 1 ::: A k g i). (d) The other case do not hold. In this case, let E be the intersection of the large sets, and call subreduce(t E fa 1 ::: A k g i). In the full version of this aer [4], we show how to imlement the above on a coarse grained multicomuter with rocessors and O( n ) storage er rocessor in O(log ) communication rounds. The non trivial arts are Ste 4, Ste 5b, the comutation of E, T=E, and T je, aswell as the subreduce oeration. The latter involves another oeration called Glue. Due to age restrictions, we can not resent this art of our result in the extended abstract. Instead, we give one examle whichshows the coarse grained arallel comutation of the set E in Ste 5(b) of Algorithm 6. Algorithm 7 Comutation of E. Inut: The set S and the sanning forest of its intersection grah. (1) In order to nd a connected comonent C in the sanning forest of S, such that the union of its sets has cardinality at least n=4, order all the comonents according to the labeling given by the coarse grained arallel sanning forest algorithm [3]. (2) Sort each comonent with resect to the values of its elements and mark as "valid" only one element er distinct value. (3) Sort again with resect to the comonents' labels. Comute the cardinality of the union of the elements of each comonent (that is the size of each comonent), with a rex-sum comutation, counting only the "valid" elements. (Hence, we do not count twice the elements with same values and comute correctly the cardinality of the union.) (4) If a rocessor nds a comonent whose size is n=4, then it broadcasts the label of this comonent. Otherwise it broadcast a "not-found" message. (5) If everybody sent "not-found", go to ste 4(c) of Mreduce algorithm. Otherwise, among all the labels received in the revious ste, choose as C the comonent with the smallest label. (6) For each of the sets comrising C, comute the distance in the sanning tree (from the root) using the coarse grained arallel Euler-tour technique [3]. (7) Sort the sets according to distance, and let B 1 ::: B s be the sorted sequence. Sort each sets with resect to the values of its elements and mark as "valid" only one element er distinct value. Sort 8

9 the sets again, according to distance, and let ^{ be the minimum i such that j S i j=1 B jjn=4. (^{ can be found with a rex-sum comutation on the "valid" elements.) Broadcast ^{. (8) Mark all "valid" elements in B 1 ::: B^{ as elements of E Second Phase: The Merging Phase Consider the tree R of recursive calls in Klein-like-Mreduce. We observe that, after l =3log 4=3 levels of R (when the rst art of our algorithm stos), the sizes of the ground sets associated with the nodes in R at level l are at most n=. This is due to the fact that the descendants of a node u in R that are 3 levels below u are smaller than u by aroximately a factor 3=4. More recisely, ifn(u) denotes the size of the ground set of T (u) (the subtree rooted at u) then, for every node w three levels below u, n(u) 3n(w)=4 +1[13, 14, 15]. Hence, each PQ-tree obtained at the end of the rst hase ts comletely into the local memory of one rocessor. Unfortunately, the same argument does not hold for the reduction sets. Recall that m = k i=1 ja ij. Let u be an internal node of R, A u1 ::: A uj its reduction sets, and m u = j i=1 ja u i j. Since the sizes of the reduction sets of the children of u deend strictly on the A ui and on how they intersect with the set E comuted for u, it is ossible that the A ui are slit in an unbalanced way. That is, we canhave j i=1 ja u i jej = O(m u ) and j i=1 ja u i =Ej = O(1) (or vice versa). If this continues u to level 3 log of R, it is ossible that for a recursive call associated with a node v at level l, f i=1 ja vij > m=. Therefore, while the ground set of T (v), and hence T (v), can t in one rocessor, the reduction sets could ossibly not. Thus, at this oint of the comutation, we can not simly use the sequential algorithm of Booth and Lueker [1] for comleting the reduction. Our idea for solving this roblem is the following. Let us consider a node v at level l in R that has m u >m=. Since, at any level of recursion, the sum of the sizes of all reduction sets is at most 2m, we can create v coies of T (v), with v = b mv c. We observe that m= m v v2l v = v2l b m v m= c v2l m= m v2lm v 2m =2: m Hence, we require at most two coies er rocessor. The reduction roblem of each node v at level l of R will be solved by the v rocessors that have coies of T (v). The next ste is the distribution of the reduction sets associated to v among these v rocessors. Each of these v rocessors can solve locally the roblem of reducing T (v) with resect to the reduction sets that it has stored, using Booth and Lueker's algorithm [1]. For each rocessor, let T 0 (v) refer to this reduced tree. Now, we need to merge these v trees, T 0 (v). More recisely, we need to comute a PQ-tree b T (v) such that L( b T (v)) = L(T (v)), where T (v) is the PQ-tree that we would have obtained by reducing T (v) directly with resect its reduction sets. For the construction of b T (v), we merge the T 0 (v) trees in a binary tree fashion as deicted in gure 1. Algorithm 8 Merging Phase Inut: h PQ-trees T (i), withjt (i)j n= and i jt (i)j n, and their reduction sets. Outut: The T (i) reduced with resect their reduction sets. (1) Let m i be the sum of the sizes of the reduction sets of T i. Make i = b mi m= c coies of each T (i). Distribute the reduction sets of each T i between the rocessors that have the coies of T (i). (2) Each rocessor executes the sequential algorithm [1] for its PQ-trees with the reduction sets that it has stored. Let T 0 (i) refer to the trees obtained. 9

10 (3) The i rocessors associated with each T (i) merge the T 0 (v) trees in a binary tree fashion, as deicted in Figure 1. The details are discussed below. We will now discuss the details of Ste 3. The following Theorem 3 shows that the merge oeration in Ste 3 of Algorithm 8 reduces to a tree intersection oeration [13, 14, 15]. Theorem 3 Let T be a PQ-tree over the ground set S and let T 0 be a coy of T. Let T and T 0 betheresult of the reduction of T with resect to fa1 ::: A r g and of T 0 with resect to fb1 ::: B t g, resectively. Let T be the PQ-tree obtained by reducing T with resect to fa1 ::: A r B1 ::: B t g. Then, 2 L(T ), 2 L(T ) \ L(T 0 ): Proof. L(T ) is the intersection of the sets of all orderings that satisfy A1 ::: A r, B1 ::: B t, and L(T ). L(T ) is always the same, indeendently of the order in which we reduce T. If 2 L(T ), then must belong to the intersection between the set of all orderings that satisfy A1 ::: A r and L(T) and it must also belong to the intersection between the set of all orderings that satisfy B1 ::: B t and L(T), that is 2 L(T ), 2 L(T 0 ) and 2 L(T ). Hence belongs to L(T ) \ L(T 0 ). The reverse can be shown analogously. 2 Set E is called segregated in T if E = leaves(lca T (E)). If E is contiguous in T,we can modify T, obtaining T 0, so that E is segregated in T 0 and L(T )=L(T 0 ). Namely, ife is not already segregated, then v = lca T (E) is a Q-node (from the denition of contiguous). Insert an R-node z between the children v ::: v q and v. In the resulting tree T 0, z = lcat T 0(E) and leaves(z) = S q i= leaves(v i)=e. Moreover, it follows from the denition of the R-node that L(T 0 )=L(T ). For a given tree T of n nodes, a node v of T with s children determines a searation of T into s +1 subtrees: T1 :::T s, the subtree of T rooted at the children of v, and T0, the subtree obtained from T by deleting T1 ::: T s. We say v is a good searator of T if each subtree T0 T1 ::: T s has no more than n=2 nodes. Every tree with at least 2 nodes has a good searator, that can be found by comuting size(v), that is the number of descendants of v, for each nodev of T. (This can be easily comuted using the Euler tour technique [3]. We recall the following PRAM algorithm from [13, 14, 15]: Algorithm 9 Intersect(T,T') Inut: 2 PQ-trees T T 0 over the same ground set. (1) If T has only one node, return. (2) Find a good searator v of T with children v 1 ::: v s. Set A = leaves T (v) and A i = leaves T (v i ) for i =1 ::: s. Let T 0 = T=A and T i = T ja i for i =1 ::: s. (3) Reduce T 0 with resect to A and then with resect to the disjoint sets A 1 ::: A s. If v is a Q-node, also reduce T 0 with resect to the disjoint sets A 1 [A 2 A 3 [A 4 :::and with resect to the disjoint sets A 2 [ A 3 A 4 [ A 5 ::: (4) Segregate T 0 with resect to A, and let T 0 0 = T 0 =A. (5) For i =1 ::: s, segregate T 0 with resect to A i, and let Ti 0 = T 0 ja i. (6) Identify a node v with lca T 0(A). The corresonding subtrees T i and Ti 0 have identical ground set. Recursively intersect the corresonding subtrees. We will now outline a coarse grained arallel imlementation of Algorithm 9. We observe that, for a coarse grained arallel imlementation, the recursion deth reduces to O(log ) because at that level, subroblems t into a single rocessor and can be handled sequentially. Finding a good searator (Ste 2) can be done using the Euler tour technique. Since the 10

11 two PQ-trees to intersect t in the local memory of one rocessor, this comutation can be done sequentially in O(n) time and O(1) communication rounds. The same bounds hold for the comutation of T0 and T i. The reduction of T 0 required in Ste 3 can be done using Booth and Lueker's sequential algorithm. Since T 0 is on the same ground set as T, jt 0 j 2n. Therefore, we can erform this ste in O(1) communication rounds and O(n) time. With a similar argument, segregating T 0 with resect to A and all the A i (Ste 4), can be erformed sequentially in linear time. The same bounds hold for comuting T 0 0 and all T 0 i Summary Theorem 4 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, Algorithm MReduce erforms a reduce oeration for a PQ-tree T in O(log 2 ) communication rounds with O( N ) local comutation er round. 3.4 Convex Biartite Grahs Recall the denition of convex biartite grahs (Denition 2). Given a biartite grah G = (A B E) with A = fa1 a2 a k g and B = fb1 b2 b n g. Let A = fa1 ::: A k g where A i = fb 2 B :(a i b) 2 Eg, and let T be a PQ-tree over the ground set B consisting of a root with children b1 b2 b n. The roblem of determining whether G is convex and, if this is the case, comuting the correct ordering of the elements in B is equivalent to the MReduce oeration on T with resect to A. Theorem 5 On a coarse grained multicomuter with rocessors and O( N ) storage er rocessor, the roblem of determining whether G is convex (and comuting the correct ordering of the elements in B) can be solved in O(log 2 ) communication rounds with O( N ) local comutation er round. References [1] K. S. Booth and G. S. Lueker, \Testing for the Consecutive Ones Proerty, Interval Grahs, and Grah Planarity Using PQ-Tree Algorithms," in Journal of Comuter and System Sciences, Vol. 13, , [2] P. Bose, A. Chan, F. Dehne, and M. Latzel, \Coarse grained arallel maximum matching in convex biartite grahs," in Proc. 13th International Parallel Processing Symosium (IPPS'99), [3] E. Caceres, F. Dehne, A. Ferreira, P. Flocchini, I. Rieing, A. Roncato, N. Santoro, and S. W. Song, \Ecient arallel grah algorithms for coarse grained multicomuters and BSP," in Proc. 24th International Colloquium on Automata, Languages and Programming (ICALP'97), Bologna, Italy, 1997, Sringer Verlag Lecture Notes in Comuter Science, Vol. 1256, [4] E. Caceres, A. Chan, F. Dehne, G. Prencie, \Coarse grained arallel algorithms for detecting convex biartite grahs," Technical Reort, School of Comuter Science, Carleton University, Ottawa, Canada K1S 5B6. 11

12 [5] L.Chen and Y.Yesha, \Parallel recognition of the consecutive ones roerty with alications," J. Algorithms, vol. 12, no. 3, [6] Lin Chen, \Grah isomorhism and identication matrices: arallel algorithms," IEEE Trans. on Parallel and Distr. Systems, Vol. 7, No. 3, March 1996, [7] F. Dehne (Ed.), \Coarse grained arallel algorithms," Secial Issue of Algorithmica, Vol. 24, No. 3/4, 1999, [8] F. Dehne, A. Fabri, and A. Rau-Chalin, \Scalable Parallel Geometric Algorithms for Coarse Grained Multicomuters," in Proc. ACM 9th Annual Comutational Geometry, ages 298{307, [9] F. Dehne, A. Fabri, and C. Kenyon, \Scalable and Architecture Indeendent Parallel Geometric Algorithms with High Probability Otimal Time," in Proc. 6th IEEE Symosium on Parallel and Distributed Processing, ages 586{593, [10] F. Dehne, X. Deng, P. Dymond, A. Fabri, and A.A. Kokhar, \A randomized arallel 3D convex hull algorithm for coarse grained multicomuters," in Proc. ACM Symosium on Parallel Algorithms and Architectures (SPAA'95),. 27{33, [11] M.T. Goodrich, \Communication ecient arallel sorting," ACM Symosium on Theory of Comuting (STOC), [12] X. He and Y.Yeshua, "Parallel recognition and decomosition of two termninal series arallel grahs," Information and Comutation, vol. 75, , 1987 [13] P.N. Klein, Ecient Parallel Algorithms for Planar, Chordal, and Interval Grahs PhD. Thesis, MIT, [14] P. Klein. \Ecient Parallel Algorithms for Chordal Grahs". Proc. 29th Sym. Found. of Com. Sci., FOCS 1989,. 150{161. [15] P. Klein. \Parallel Algorithms for Chordal Grahs". In Synthesis of arallel algorithms, J. H. Reif (editor). Morgan Kaufmann Publishers, 1993,. 341{407. [16] A.C. Tucker, "Matrix characterization of circular-arc grahs," Pacic J. Mathematics, vol. 39,no. 2, , [17] L. Valiant, \A bridging model for arallel comutation," Communications of the ACM, Vol. 33, No. 8, August

13 The Recursion Tree R log v h Each T(i) at level log in R is coied α times i Processor 1... α 1 α α 1+ α 2... T(1) T(1) T(1) T(1) T(2) T(2) T(2) T(2) T(h) Each rocessor runs Booth and Lueker locally Processor 1... α1 α α 1+ α 2... T'(1) T'(1) T'(1) T'(1) T'(2) T'(2) T'(2) T'(2) T'(h) Figure 1: Illustration of the Main Algorithm. 13