An external memory data structure for shortest path queries

Discrete Applied Mathematics 126 (2003) 55 – 82 An external memory data structure for shortest path queries
David Hutchinsona , Anil Maheshwarib;1 , Norbert Zehb; ∗ b School a Department of Computer Science, Duke University, Durham, NC, USA of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ont., Canada K1S 5B6 Received 22 September 1999; received in revised form 28 November 2000; accepted 7 September 2001 Abstract We present results related to satisfying shortest path queries on a planar graph stored in external memory. Let N denote the number of vertices in the graph and sort(N ) denote the number of input=output (I=O) operations required to sort an array of length N : (1) We describe a blocking for rooted trees to support bottom-up traversals of these trees in O(K=B) I=Os, where K is the length of the traversed path. The space required to store the tree is O(N=B) blocks, where N is the number of vertices of the tree and B is the block size. √ (2) We give an algorithm for computing a 23 -separator of size O( N ) for a given embedded planar graph. Our algorithm takes O(sort(N )) I=Os, provided that a breadth- rst spanning tree is given. (3) We give an algorithm for triangulating embedded planar graphs in O(sort(N )) I=Os. We use these results to construct a data structure for answering shortest path queries on planar 3=2 graphs. The data structure uses O(N √ =B) blocks of external memory and allows for a shortest path query to be answered in O(( N + K)=DB) I=Os, where K is the number of vertices on the reported path and D is the number of parallel disks. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: External-memory algorithms; Graph algorithms; Shortest paths; Planar graphs
A preliminary version appeared in [14]. Corresponding author. E-mail addresses: [email protected] (D. Hutchinson), [email protected] (A. Maheshwari), [email protected] (N. Zeh). 1 Research supported by NSERC and NCE GEOIDE. ∗ 0166-218X/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 2 1 8 X ( 0 2 ) 0 0 2 1 7 - 2 56 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 1. Introduction Answering shortest path queries in graphs is an important and intensively studied problem. It has applications in communication systems, transportation problems, scheduling, computation of network ows, and geographic information systems (GIS). Typically, an underlying geometric structure is represented by a combinatorial structure, which is often a weighted planar graph. The motivation to study external memory shortest path problems arose in our GIS research and in particular, with an implementation of the results in [16] for shortest path problems in triangular irregular networks. In this application the given graph represents a planar map, i.e., it is planar and embedded. Quite commonly it is too large to t into the internal memory of even a large supercomputer. In this case, and in many other large applications, the computation is forced to wait while large quantities of data are transferred between relatively slow external (disk-based) memory and fast internal memory. Thus, the classical internal memory approaches to answering shortest path queries in a planar graph (e.g. [6,8–10,15]) may not work eciently when the data sets are too large. 1.1. Model of computation Unfortunately, the I=O-bottleneck is becoming more signi cant as parallel computing gains popularity and CPU speeds increase, since disk speeds are not keeping pace [22]. Thus, it is important to take the number of input=output (I=O) operations performed by an algorithm into consideration when estimating its eciency. This issue is captured in the parallel disk model (PDM) [24], as well as a number of other external memory models [5,25]. We adopt the PDM as our model of computation for this paper due to its simplicity, and the fact that we consider only a single processor. In the PDM, an external memory, consisting of D disks, is attached to a machine with an internal memory capable of holding M data items. Each of the disks is divided into blocks of B consecutive data items. Up to D blocks, at most one per disk, can be transferred between internal and external memory in a single I=O operation. The complexity of an algorithm is the number of I=O operations it performs. 1.2. Previous results Shortest path problems can be divided into three general categories: (1) computing a shortest path between two given vertices of a graph, (2) computing shortest paths between a given source vertex and all other vertices of a graph (single source shortest paths (SSSP) problem), and (3) computing the shortest paths between all pairs of vertices in a graph (all pairs shortest paths (APSP) problem). Previous results in the RAM model: In the sequential RAM model, much work has been done on shortest path problems. Dijkstra’s algorithm [6], when implemented using Fibonacci heaps [11], is the best-known algorithm for the SSSP-problem for general graphs (with nonnegative edge weights). It runs in O(|E| + |V | log |V |) time, where |E| and |V | are the numbers of edges and vertices in the graph, respectively. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 57 The APSP-problem can be solved by applying Dijkstra’s algorithm to all vertices of the graph, which results in an O(|V E| + |V |2 log |V |) running time. For planar
graphs, an O(N log N )-algorithm for the SSSP-problem and an O(N 2 )-algorithm for the APSP-problem, where N = |V |, are given in [9]. A linear-time SSSP-algorithm for planar graphs is presented in [15]. An alternate approach is to preprocess the given graph for online shortest path √ queries. For graphs for which an O( N )-separator theorem holds (e.g., planar graphs), an O(S)-space data structure (N 6 S 6 N 2 ) that answers distance queries in O(N 2 =S) time is presented in [8]. The corresponding shortest path can be reported in time proportional to the length of the reported path. (For planar graphs slightly better bounds are given.) It is known that every tree or outerplanar graph has a 23 -separator of size O(1). √ In [17] it is shown that every planar graph has a 23 -separator of size O( N ), and a linear-time algorithm for nding such a separator is given. Other results include separator algorithms for graphs of bounded genus [1] and for computing edgeseparators [7]. Previous results in the PRAM model: A PRAM algorithm for computing a 23 √ separator of size O( N ) for a planar graph is presented in [12]. The algorithm runs in O(log2 N ) time and uses O(N 1+ ) processors, where  ¿ 0 is a constant. In [13] a PRAM algorithm is given that computes a planar separator in O(log N ) time using O(N=log N ) processors, provided that a breadth- rst spanning tree (BFS-tree) of the graph is given. Previous results in external memory: In the PDM, sorting, permuting, and scanning an array of size N take sort(N ) = ((N=DB) log M=B N=B); perm(N ) = (min{N; sort (N )}), and scan(N ) = (N=DB) I=Os [23,24]. For a comprehensive survey on external memory algorithms, refer to [23]. The only external-memory shortest path algorithm known to us is the SSSP-algorithm in [4], which takes O(|V |=D+(|E|=DB) logM=B |E|=B) I=Os with high probability, on a random graph with random weights. We do not know of previous work on computing separators in external memory; but one can use the PRAM-simulation results in [3] together with the results of [12,13] cited above. Unfortunately, the PRAM simulation introduces O(sort(N )) I=Os for every PRAM step, and so the resulting I=O complexity is not attractive for this problem. 1.3. Our results The main results of this paper are: (1) A blocking to store a rooted tree T in external memory so that a path of length K towards the root can be traversed in at most ⌈K=DB⌉ + 3 I=Os, for 0 ¡  ¡ 1. √ If  ¿ 3−2 5 , the blocking uses at most (2 + 2=(1 − ))|T |=B + D blocks of external √ storage. For  ¡ 3−2 5 , a slight modi cation of the approach reduces the amount of storage to (1 + 1=(1 − 2))|T |=B + D blocks. For xed , the tree occupies optimal O(|T |=B) blocks of external storage and traversing a path takes optimal O(K=DB) I=Os. Using the best previous result [20], the tree would use the same 58 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 amount of space within a constant factor, but traversing a path of length K would take O(K=logd (DB)) I=Os, where d is the maximal degree of the vertices in the tree (see Section 3). √ (2) An external memory algorithm which computes a separator consisting of O( N ) vertices for an embedded planar graph in O(sort(N )) I=Os, provided that a BFStree of the graph is given. Our algorithm is based on the classical planar separator algorithm in [17]. The main challenge in designing an external memory algorithm for this problem is to determine a good separator corresponding to a fundamental cycle (see Section 4). (3) An external memory algorithm which triangulates an embedded planar graph in O(sort(N )) I=Os (see Section 5). (4) An external memory data structure for answering shortest path queries online. Results 1–3, above, are the main techniques that we use to construct this data structure. Our data structure uses O(N 3=2 =B) blocks of √ external memory √ and answers online distance and shortest path queries in O( N =DB) and O(( N + K)=DB) I=Os, respectively, where K is the number of vertices on the reported path (see Section 6). Our separator and triangulation algorithms may be of independent interest, since graph separators are used in the design of ecient divide-and-conquer graph algorithms and many graph algorithms assume triangulated input graphs. 2. Preliminaries 2.1. De nitions An undirected graph (or graph for short) G = (V; E) is a pair of sets V and E, where V is called the vertex set and E is called the edge set of G. Each edge in E is an unordered pair {v; w} of vertices v and w in V . Unless stated otherwise, we use |G| to denote the cardinality of V . In a directed graph, every edge is an ordered pair (v; w). In this case, we call v the source vertex and w the target vertex of edge (v; w). A graph G is planar if it can be drawn in the plane so that no two edges intersect, except possibly at their endpoints. Such a drawing de nes for each vertex v of G, an order of the edges incident to v clockwise around v. We call G embedded if we are given this order for every vertex of G. By Euler’s formula, |E| 6 3|V | − 6 for planar graphs. A path from a vertex v to a vertex w in G is a list p = v = v0 ; v1 ; : : : ; vk = w of vertices, where {vi ; vi+1 } ∈ E for 0 6 i ¡ k. The length of path p is the number k of edges in the path. We call p a cycle if v0 = vk . Paths and cycles are de ned analogously for directed graphs. A directed acyclic graph (DAG) is a directed graph that does not contain cycles of length ¿ 0. A graph G is connected if there is a path between any two vertices in G. A subgraph G ′ = (V ′ ; E ′ ) of G is a graph with V ′ ⊆ V and E ′ ⊆ E. Given a subset X ⊆ V , we denote by G[X ] = (X; E[X ]) the subgraph of G induced by X , where E[X ] = {{v; w} ∈ E: {v; w} ⊆ X }. The graph G–X is de ned D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 59 as G[V \ X ]. The connected components of G are the maximal connected subgraphs of G. A tree with N vertices is a connected graph with N − 1 edges. A rooted tree is a tree with a distinguished root vertex. The level or depth of a vertex v in a rooted tree is the number of edges on the path from v to the root. For an edge {v; w} in a tree T we say that v is w’s parent and w is v’s child if w’s depth in T is greater than v’s. A vertex v is an ancestor of a vertex w, and w is a descendant of v, if v is w’s parent or it is an ancestor of w’s parent. A common ancestor of two vertices v and w is a vertex u which is an ancestor of v and w. The lowest common ancestor (lca)(v; w) of v and w is the common ancestor of v and w at maximum depth among all common ancestors of v and w. A preorder numbering of a rooted tree T with N vertices is an assignment of numbers 0 through N − 1 to the vertices of T such that every vertex has a preorder number less than any of its descendants and the preorder numbers of each vertex and all its descendants are contiguous. Given an ordering of the children of each node, a lexicographical numbering of T is a preorder numbering of T such that the preorder numbers of the children of any node, sorted by the given order, are increasing. An independent set I ⊆ V in a graph G is a set of vertices such that for every vertex v ∈ I and every edge {v; w} ∈ E, w ∈ I . That is, no two vertices in I are adjacent. A k-coloring of a graph G is an assignment f : V → {1; : : : ; k} of colors to the vertices of G such that for any edge {v; w} ∈ E, f(v) = f(w). A spanning tree of a graph G =(V; E) is a tree T =(V; F), where F ⊆ E. A BFS-tree is a rooted spanning tree T of G such that for any edge {v; w} in G, the levels of v and w in T di er by at most 1. Let c : E → R+ be an assignment of non-negative costs to the edges of G. The cost k−1 p of a path p = v0 ; : : : ; vk is de ned as p = i=0 c({vi ; vi+1 }). A shortest path (v; w) is a path of minimal cost from v to w. Let w :  V → R+ be an assignment of non-negative weights to the vertices of G such that v∈V w(v) 6 1. The weight w(H ) of a subgraph H of G is the sum of the weights of the vertices in H . An -separator, 0 ¡  ¡ 1, of G is a subset S of V such that none of the connected components of G − S has weight exceeding . For a given DAG G = (V; E), a topological ordering is a total order O ⊆ V × V such that for every edge (v; w) ∈ E, (v; w) ∈ O. We will describe results on paths in a rooted tree which originate at an arbitrary node of the tree and proceed to the root. We will refer to such paths as bottom-up paths. For a given block size B, a blocking of agraph G = (V; E) is a decomposition of q V into vertex sets B1 ; : : : ; Bq such that V = i=1 Bi and |Bi | 6 B. The vertex sets Bi do not have to be disjoint. Each such vertex set Bi corresponds to a block in external memory where its vertices are stored. 2.2. External memory techniques First, we introduce some useful algorithmic techniques in external memory which we will use in our algorithms. These include external memory stacks and queues, list ranking, time-forward processing and their applications to processing trees. 60 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 We have already observed that scanning an array of size N takes O(scan(N )) I=Os using at least D blocks of internal memory to hold the D blocks currently being scanned. Using 2D blocks of internal memory, a series of N stack operations can be executed in O(scan(N )) I=Os. These 2D blocks hold the top-most elements on the stack. Initially, we hold the whole stack in internal memory. As soon as there are 2DB elements on the stack, and we want to push another element on the stack, we swap the DB bottom-most elements to external memory. This takes one I=O. Now it takes at least DB stack operations (DB pushes to ll the internal bu er again or DB pops to have no elements left in internal memory) before the next I=O becomes necessary. Thus, we perform at most one I=O per DB stack operations, and N stack operations take O(N=DB)=O(scan(N )) I=Os. Similarly, to implement a queue, we need 2D blocks of internal memory, D to bu er the inputs at the end of the queue and D to bu er the outputs at the head of the queue. The list-ranking problem is the following: Given a singly linked list L and a pointer to its head, compute for every node of L its distance to the tail of L. A common variant is to assign weights to the nodes of L and to compute for every node the weight of the sublist of L starting at this node. In [3], a recursive list-ranking procedure was developed. If the list has size at most M , the problem can be solved in internal memory. Reading the input and writing the output take O(scan(N )) I=Os in this case. If the list contains more than M nodes the procedure 3-colors the list and removes the largest monochromatic set of vertices. For every removed vertex x, the weight of x’s predecessor y is updated to w(y) = w(y) + w(x). After recursively applying the same technique to the remaining sublist, which has size at most 32 N , the removed vertices are reintegrated into the list and their ranks computed. In [3] it is shown that the 3-coloring of the list and the removal and reintegration of the independent set can be done in O(sort(N )) I=Os. Thus, this procedure takes T (N ) 6 T ( 23 N )+O(sort(N ))=O(sort(N )) I=Os. In [26] it is shown that the 3-coloring technique can be extended to rooted trees, which allows the application of the same framework to the problem of computing the level of every node in a rooted tree. Alternatively, one can use the Euler-tour technique together with the list-ranking technique to compute these levels. The Euler-tour technique can be described as follows: Given a rooted tree T , we replace every edge {v; w} by two directed edges (v; w) and (w; v). For every vertex v, ′ let e0 ; : : : ; ek−1 be the incoming edges of v and e0′ ; : : : ; ek−1 be the outgoing edges of ′ ′ v, where ei is directed opposite to ei . We de ne edge e(i+1) mod k to be the successor ′ of edge ei , for 0 6 i ¡ k. At the root of T we de ne edge ek−1 to have no successor. This de nes a traversal of the tree, starting at the root and traversing every edge of the tree exactly twice, once per direction. To compute the levels of all vertices in T , for instance, we assign weight 1 to all edges directed from parents to children and weight −1 to all edges directed from children to parents. Then we can apply the list-ranking technique to compute the levels of the endpoints of all edges. If we choose the order of the edges e0 ; : : : ; ek−1 carefully, we can use the Euler-tour technique to compute a lexicographical numbering of a given rooted tree. Recall that in this case we are given a left to right ordering of the children of every vertex. We construct the Euler tour by choosing for every vertex v edge e0 to be the edge connecting v to its parent. The remaining edges e1 ; : : : ; ek−1 are the edges connecting v to its D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 61 children, sorted from left to right. It is easy to see that the Euler tour thus constructed traverses the tree T lexicographically. To compute the lexicographical numbers of all vertices, we assign weight 1 to edges directed from parents to children and weight 0 to all remaining edges. After applying list ranking to this list, the lexicographical number of a vertex v is the rank of the edge with weight 1 and target vertex v. An important ingredient of the 3-coloring technique in [3] as well as most of the algorithms presented in this paper is time-forward processing, which was introduced in [3]. This technique is useful for processing DAGs. We view the DAG as a circuit and allow sending a constant amount of information along every edge. Every node can use the information sent along its incoming edges to compute a function of these values and then send a constant amount of information along each of its outgoing edges. The technique presented for this problem in [3] has two constraints: (1) the fan-in of the vertices in the DAG has to be bounded by some constant d and (2) the ratio m = M=B has to be large enough. These constraints have been removed in [2] using the following elegant solution: Given a DAG G, sort the vertex set topologically, which de nes a total order on this set. Then evaluate the nodes in their order of appearance, thereby ensuring that all nodes u with an outgoing edge (u; v) have been evaluated before v. Every such node u inserts the information it wants to send to v into a priority queue, giving it priority v. Node v performs din (v) DELETEMIN operations to retrieve its inputs, where din (v) is the fan-in of v. It is easy to verify that at the time when v is evaluated, the din (v) smallest elements in the priority queue have indeed priority v. After sorting the vertex and edge sets of the DAG, this technique performs 2|E| priority queue operations, which take O(sort(|E|)) I=Os [2]. Thus, this technique takes O(sort(|V | + |E|)) I=Os. 3. Blocking rooted trees In this section, we consider the following problem: Given a rooted tree T , store it in external memory so that for any query vertex v ∈ T , the path from v to the root of T can be reported I=O-eciently. One can use redundancy to reduce the number of blocks that have to be read to report the path. However, this increases the space requirements. The following theorem gives a trade-o between the space requirements of the blocking and the I=O-eciency of the tree traversal. Theorem 1. Given a rooted tree T of size N and a constant ; 0 ¡  ¡ 1; we can store T on D parallel disks so that traversing any bottom-up path of length K in T takes at most ⌈K=DB⌉+3 √I=Os. The amount of storage used is at most (2+2=(1−))N=B +D √ blocks for  ¿ 3−2 5 and at most (1 + 1=(1 − 2))N=B + D blocks for  ¡ 3−2 5 . Proof. This follows from Lemmas 2 and 3 below. Intuitively, our approach is as follows. We cut T into layers of height DB. This divides every bottom-up path of length K in T into subpaths of length DB; each subpath stays in a particular layer. We ensure that each such subpath is stored in 62 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 1. A tree is cut into layers of height DB. (Here, DB = 2.) The tree is cut along the dashed lines. The resulting subtrees Ti are shaded. at most D blocks, and each of the D blocks is stored on a di erent disk. Thus each subpath can be traversed at the cost of a single I=O operation. This gives us the desired I=O-bound because any bottom-up path of length K crosses at most ⌈K=DB⌉+1 layers and is thus divided into at most ⌈K=DB⌉ + 1 subpaths. More precisely, let h represent the height of T , and let h′ = DB be the height of the layers to be created (we assume that h′ is an integer). We cut T into layers L0 ; : : : ; L⌈h=h′ ⌉−1 , where layer Li is the subgraph of T induced by the vertices on levels ih′ through (i+1)h′ −1 (see Fig. 1). Each layer is a forest of rooted trees, whose heights are at most h′ . Suppose that there are r such trees, T1 ; : : : ; Tr , taken over all layers. We decompose each tree Ti , 1 6 i 6 r, into possibly overlapping subtrees Ti; 0 ; : : : ; Ti; s having the following properties: Property 1. |Ti; j | 6 DB; for all 0 6 j 6 s. Property 2. s j=0 |Ti; j | 6 (1 + 1=(1 − ))|Ti |. Property 3. For every leaf l of Ti ; there is a subtree Ti; j containing the whole path from l to the root of Ti . Lemma 2. Given a rooted tree Ti of height at most DB; 0 ¡  ¡ 1; we can decompose Ti into subtrees Ti; 0 ; : : : ; Ti; s having Properties 1–3. Proof. If |Ti | 6 DB; we “decompose” Ti into one subtree Ti; 0 = Ti . Then Properties 1–3 trivially hold. So assume that |Ti | ¿ DB. Given a preorder numbering of the vertices of Ti , we denote every vertex by its preorder number. Given a parameter 0 ¡ t ¡ DB to be speci ed later, let s=⌈|Ti |=t⌉−1. We de ne vertex sets V0 ; : : : ; Vs , where Vj ={jt; : : : ; (j +1)t −1} for 0 6 j 6 s (see Fig. 2(a)). The vertex set Vs may contain less than t vertices. The tree Ti; j = Ti (Vj ) is the subtree of Ti consisting of all vertices in Vj and their ancestors in Ti (see Fig. 2(b)). We claim that these subtrees Ti; j have Properties 1–3, if we choose t appropriately. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 63 Fig. 2. (a) A rooted tree Ti with its vertices labelled with their preorder numbers. Assuming that t = 8, V0 is the set of black vertices, V1 is the set of gray vertices, and V2 is the set of white vertices. (b) The subtrees Ti (V0 ); Ti (V1 ), and Ti (V2 ) from left to right. Property 3 is guaranteed because for every leaf l, the subtree Ti; j , where l ∈ Vj , contains the whole path from l to the root of Ti . To prove Property 1, let Aj be the set of vertices in Ti; j which are not in Vj . Every such vertex x is an ancestor of some vertex y ¿ jt. That is, x ¡ y. As x ∈ Vj and the vertices in Vj are numbered contiguously, x ¡ jt. Vertex y is in the subtree of Ti rooted at x and x ¡ jt 6 y. Thus, vertex jt must be in this subtree as well because the vertices in the subtree rooted at x are numbered contiguously. Hence, every vertex in Aj is an ancestor of vertex jt. This implies that |Aj | 6 DB because jt has at most one ancestor at each of the at most DB levels in Ti . Hence, |Ti; j | = |Aj | + |Vj | 6 DB + t. Choosing t = DB − DB, we guarantee that every subtree Ti; j has size at most DB. Now Property 2 holds because s  j=0 |Ti; j | 6 ⌈|Ti |=(DB−DB)⌉−1  DB   |Ti | + 1 DB DB − DB 1 |Ti | + DB = 1−   1 ¡ + 1 |Ti |: 1− 6 (1) j=0 (2) (3) (4) The step from lines (3) to (4) uses the fact that |Ti | ¿ DB. Lemma 3. If a rooted tree T of size N is decomposed into subtrees Ti; j such that Properties 1–3 hold; T can be stored on D parallel disks so that any bottom-up path of length K in T can be traversed in at most ⌈K=DB⌉ + 3 I=Os; for 0 ¡  ¡ 1. For √ 3− 5  ¿ 2 ; the amount of storage used is at most (2 + 2=(1 − ))N=B + D blocks. For √  ¡ 3−2 5 ; the tree occupies at most (1 + 1=(1 − 2))N=B + D blocks. Proof. We consider D disks of block size B as one large disk divided into superblocks of size DB. Thus; by Property 1; each subtree Ti; j ts into a single superblock and can be read in a single I=O. 64 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 A bottom-up path p of length K in T crosses at most k = ⌈K=DB⌉ + 1 layers. This divides p into k maximal subpaths such that none of them crosses more than one layer. Each such subpath p′ is a leaf-to-root path in some subtree Ti , or a subpath thereof. Thus, by Property 3, there exists a subtree Ti; j that contains the whole of the subpath p′ . That is, each subpath p′ can be accessed in a single I=O. Therefore, the traversal of p takes at most ⌈K=DB⌉ + 1 I=Os. r By Property 2, all subtrees Ti; j together use at most (1 + 1=(1 − )) i=0 |Ti | = (1 + 1=(1 − ))N space. Initially, we store every subtree Ti; j in a separate superblock, which may leave many superblocks sparsely populated. As long as there are at least two superblocks that are at most half full we keep merging pairs of superblocks. Finally, if we are left with a single half full superblock we try to nd a superblock that can be merged with it. If we nd one, we merge the two superblocks. Otherwise, the half-full superblock together with any other superblock contains more than DB vertices. All other superblocks are at least half full. Thus, on average, each superblock is at least half full, and we use ⌈(2 − 2=(1 − ))N=DB⌉ superblocks, i.e., D⌈(2 − 2=(1 − ))N=DB⌉ 6 (2 + 2=(1 − ))N=B√+ D blocks to store all subtrees Ti; j . For  ¡ 3−2 5 ≈ 0:38; 1 + 1=(1 − 2) ¡ 2 + 2=(1 − ). In this case, we apply the following strategy. We cut the given tree T into layers of height 2DB. We guarantee that the vertices of every subtree Ti; j are stored contiguously, but not necessarily in a single block. As every tree Ti; j has size at most DB, it is spread over at most two blocks. Thus, two I=Os suce to traverse any leaf-to-root path in a subtree Ti; j , and traversing a path of length K takes at most 2(⌈K=2DB⌉ + 1) 6 ⌈K=DB⌉ + 3 I=Os. On the other hand, we guarantee that all superblocks except for the last one are full. Thus, we use ⌈(1 − 1=(1 − 2))N=DB⌉ superblocks, i.e., D⌈(1 − 1=(1 − 2))N=DB⌉ 6 (1 + 1=(1 − 2))N=B + D blocks of external memory. 4. Separating embedded planar graphs We now present an external-memory algorithm for separating embedded planar graphs. Our algorithm is based on the classical linear-time separator algorithm of [17]. √ It computes a 32 -separator S of size O( N ) for a given embedded planar graph G in O(sort(N )) I=Os, provided that a BFS-tree of the graph is given. 2 The input to our algorithm is an embedded planar graph G and a spanning forest F of G. Every tree in F is a BFS-tree of the respective connected component. (In the remainder of this section, we call F a BFS-forest.) The graph G is represented by its vertex set V and its edge set E. To represent the embedding, let the edges incident to each vertex v be numbered in counterclockwise order around v, starting at an arbitrary edge as the rst edge. This de nes two numbers nv (e) and nw (e), for every edge e = {v; w}. Let these numbers be stored with e. The spanning forest F is given implicitly by marking every edge of G as “tree edge” or “non-tree edge” and storing with each vertex v ∈ V , the name of its parent in F. 2 The currently best-known algorithm for computing a BFS-tree [19] takes O(|V | + |E|=|V |sort(|V |)) I=Os. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 65 4.1. Framework of the algorithm First, we compute the connected components of the given graph G. If there is a component whose weight is ¿ 23 we compute a separator S of that component. Then we compute the connected components of G–S, which gives the desired partition of G into subgraphs of weight at most 23 each. The connected components can be computed in O(sort(N )) I=Os [3]. In the next subsection, we describe how to compute the separator S using O(sort(N )) I=Os, leading to the following theorem. Theorem 4. Given an embedded planar√graph √ G with N vertices and a BFS-forest F of G; a 23 -separator of size at most 2 2 N for G can be computed in O(sort(N )) I=Os. 4.2. Separating connected planar graphs In this section, we present an external-memory algorithm for computing a 23 -separator √ of size O( N ) for a connected embedded planar graph G of size N and weight at least 32 , provided that a BFS-tree T of G is given. We assume that G is triangulated. If it is not, it can be triangulated in O(sort(N )) I=Os using the algorithm in Section 5. 3 Also, we assume that no vertex has weight exceeding 13 because otherwise S = {v}, where w(v) ¿ 31 , is trivially a 23 -separator of G. The algorithm (Algorithm 1) is based on the following observation: In a BFS-tree T of a given graph G, non-tree edges connect vertices on the same level or on consecutive levels. Thus, the removal of all vertices of any level in T disconnects the subgraph of G corresponding to the upper part of T from the subgraph corresponding to the lower part (see Fig. 3(a)). We will nd two levels l0 and l2 that divide the graph into three parts G1 ; G2 , and G3 , where w(G1 ) 6 23 , w(G3 ) 6 32 , and the number of vertices on √ √ levels l0 and l2 is at most 2 2 N − 2(l2 − l0 − 1) (Step 2; see Fig. 3(b)). Thus, G1 and G3 already have the desired weight, and we can a ord to add up to two vertices per level between l0 and l2 to the separator, in order to cut G2 into pieces whose individual weights are at most 32 (Step 3; see Fig. 3(c)). 3 Note, however, that the graph has to be triangulated before computing the BFS-tree. Otherwise, T might not be a BFS-tree of the triangulation anymore. SEPARATECONNECTED(G; T ): 1: 2: 3: 4: Label every vertex in T with its level in T . Compute levels l0 and l2 cutting G into subgraphs G1 , G2 , and G3 such √ √ that w(G1 ) 6 32 ; w(G3 ) 6 32 and L(l0 ) + L(l2 ) + 2(l2 − l0 − 1) 6 2 2 N . Find a 23 -separator S ′ of size at most 2(l2 − l0 − 1) for G2 . Remove the vertices on levels l0 and l2 and in S ′ and all edges incident to these vertices from G and compute the connected components of G − S. Algorithm 1: Separating a connected embedded planar graph. 66 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 3. Illustrating the three major steps of the separator algorithm. √ √ Lemma 5. Algorithm 1 computes a 32 -separator S of size at most 2 2 N for an embedded connected planar graph G with N vertices in O(sort(N )) I=Os; provided that a BFS-tree of G is given. Proof. First; let us assume that Step 3 computes the desired separator of G2 in O(sort (N )) I=Os. Then the major diculty in Algorithm 1 is nding the two levels l0 and l2 in Step 2. For a given level l; let L(l) be the number of vertices on level l and W (l) be the weight  of the vertices on level l. We rst compute  the level l1 closest to the root such that l6l1 W (l) ¿ 12 (see Fig. 3(a)). Let K = l6l1 L(l). Then we compute √ levels l√0 6 l1 and l2 ¿ l1 such that L(l0 ) + 2(l1 − l0 ) 6 2 K and L(l2 ) + 2(l2 − l1 − 1) 6 2 N − K (see Fig. 3(b)). The existence of level l1 is obvious. The existence of levels l0 and l2 has been shown in [17]. It is easy to see that levels l0 and l2 have the desired properties. Now we turn to the correctness of Step 3. In this step, we shrink levels 0 through l0 to a single root vertex r of weight 0. Next, we remove levels l2 and below, and retriangulate the resulting graph, obtaining a triangulation G ′ (see Fig. 3(c)). Then we use the techniques of Section 4.3 to compute a fundamental cycle C of G ′ which is a 32 -separator of G ′ . Graph G2 is a subgraph of G ′ . Thus, G2 ∩ C is a 32 -separator of G2 . The fundamental cycle can have length at most 2(l2 − l0 ) − 1. If the length is indeed 2(l2 − l0 ) − 1, C contains the root vertex r, which is not part of G2 . Thus, S ′ = G2 ∩ C has size at most 2(l2 − l0 − 1), as desired. When shrinking levels 0 through l0 to a single vertex we have to be a bit careful because we have to maintain the embedding of the graph. To do this we number the vertices of T lexicographically and sort the vertices v1 ; : : : ; vk on level l0 + 1 by increasing numbers (i.e., from left to right). Let wi be the parent of vi on level l0 . Then, we replace edge {vi ; wi } by edge {r; vi } and assign nvi ({r; vi }) = nvi ({vi ; wi }) and nr ({r; vi }) = i. This places edges {r; vi } counterclockwise around r and guarantees that edge {r; vi } is embedded between the appropriate edges incident to vi . We construct a BFS-tree T ′ of G ′ by adding the edges {r; vi } to the remains of T in G ′ . It is easily veri ed that T ′ is a spanning tree of G ′ . It is not so easy to see that T ′ is a BFS-tree of G ′ , which is crucial for simplifying Algorithm 2 in Section 4.3, which computes the desired separator for G ′ and thus G2 . Consider the embedding of G in the plane (see Fig. 4(a)). Then, we de ne Rl0 to be the union of triangles that have at least one vertex at level l0 or above in T (the outer face in Fig. 4(a)). Analogously, we de ne Rl2 to be the union of triangles that D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 67 Fig. 4. (a) An embedding of a given graph G. Regions Rl0 ; Rl2 , and R′ are the white, light gray, and dark gray parts of the plane, respectively. The additional curves in region Rl0 are level l0 + 1 vertices. (b) In Step 3 of Algorithm 1, we replace the graph in region Rl0 by a single root vertex r and connect it to all vertices at level l0 + 1 (dotted curves). We also remove the interior of region Rl2 . All non-triangular faces like face f have to be triangulated. have at least one vertex at level l2 or below in T (the light gray interior faces in Fig. 4(a)). The embedding of G2 then lies in R′ = R2 \ (Rl0 ∪ Rl2 ) (the dark gray regions in Fig. 4(a)). The boundary of R′ is a set of edge-disjoint cycles, and the interior of R′ is triangulated. Also, note that no vertex at level at most l0 can be on the boundary between Rl0 and R′ because all incident triangles are in Rl0 . On the other hand, for a vertex at level at least l0 + 2 to be on that boundary it has to be on the boundary of a triangle that has a vertex at level at most l0 on its boundary, which is also impossible. Thus, the boundary between Rl0 and R′ is formed by level l0 + 1 vertices only. (Note that some level l0 + 1 vertices are interior to Rl0 (the additional curves in region Rl0 in Fig. 4(a)).) Analogously, the boundary between Rl2 and R′ is formed by level l2 − 1 vertices only. When shrinking the vertices above and including level l0 to a single root vertex, we can think of this as removing the parts of G in Rl0 and placing the new root vertex r into Rl0 (see Fig. 4(b)). The tree edges connecting r to level l0 + 1 vertices are embedded in Rl0 . As we did not alter the triangulation of R′ (i.e., G2 ), the only faces of G ′ where the triangulation algorithm adds diagonals are in Rl0 or Rl2 . As all vertices on the boundary of Rl0 are already connected to r, the triangulation algorithm only adds edges connecting two level l0 + 1 or two level l2 − 1 vertices, thereby not destroying the BFS-property of T ′ with respect to G ′ . Now, we analyze the complexity of Algorithm 1. Step 1 takes O(sort(N )) I=Os (see Section 2.2). In Step 2, we sort the vertices by their levels and scan it until the total weight of the vertices seen so far exceeds 21 . We continue the scan until we reach the end of a level. While scanning we maintain the count K of visited vertices. We scan the vertex list backward, starting at l1 , and count the number of vertices on each √ visited level. When we nd a level l0 with L(l0 ) + 2(l1 − l0 ) K we stop. In the same way we nd l2 scanning forward, starting at the level following l1 . As we sort and scan a constant number of times, Step 2 takes O(sort(N )) I=Os. Removing all levels 68 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 5. A non-tree edge e and the corresponding fundamental cycle C(e) shown in bold. R1 (e) is the region inside the cycle and R2 (e) is the region outside the cycle. below and including l2 takes O(sort(N )) I=Os: We sort the vertex set and rst sort the edges by their rst endpoints; in a single scan we mark all edges that have their rst endpoint on level l2 or below; after sorting the edges by their second endpoints, we scan the edge list again to mark all edges with their second endpoints on level l2 or below; nally, it takes a single scan to remove all marked vertices and edges. Shrinking levels 0 through l0 to a single root vertex takes O(sort(N )) I=Os: O(sort(N )) I=Os to number the vertices lexicographically, O(sort(N )) I=Os to sort the vertices on level l0 + 1 and another scan to replace edges {vi ; wi } by edges {r; vi }. By Theorem 7, we can retriangulate the resulting graph in O(sort(N )) I=Os. The removal of the vertices and edges in G1 can be done in a similar way as for G3 . The rest of Step 3 takes O(sort(N )) I=Os by Lemma 6. Steps 2 and 3 have marked the vertices on levels l0 and l2 and in S ′ as separator vertices. Then we use the same technique as for removing G3 to remove all separator vertices and incident edges in O(sort(N )) I=Os. Computing the connected components of G − S takes O(sort(N )) I=Os [3]. 4.3. Finding a small simple cycle separator Let G ′ be the triangulation as constructed at the beginning of Step 3 of Algorithm 1 and T ′ be the BFS-tree of G ′ constructed from T . Every non-tree edge e = {v; w} in G ′ de nes a fundamental cycle C(e) consisting of e itself and the two paths in the tree T ′ from the vertices v and w to the lowest common ancestor u of v and w (see Fig. 5). (Note that in a BFS-tree, u is distinct from both v and w.) Given an embedding of G ′ , any fundamental cycle C(e) separates G ′ into two subgraphs R1 (e) and R2 (e), one induced by the vertices embedded inside C(e) and the other induced by those embedded outside. In [17] it is shown that there is a non-tree edge e in G ′ such that R1 (e) and R2 (e) have weights at most 23 each, provided that no vertex has weight exceeding 31 and G ′ is triangulated. Moreover, for any non-tree edge e, the number of vertices on the fundamental cycle C(e) is at most 2height(T ′ ) − 1 = 2(l2 − l0 ) − 1. Next, we show how to nd such an edge and the corresponding fundamental cycle I=O-eciently. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 69 We compute labels for the non-tree edges of G ′ so that we can compute the weights w(R1 (e)) and w(R2 (e)) of regions R1 (e) and R2 (e) only using the labels stored with edge e. Then we can nd the edge e whose corresponding fundamental cycle is a 2 ′ 3 -separator in a single scan over the edge set of G . Consider Fig. 5. Given a lexicographical numbering of the vertices in T ′ , denote the number assigned to vertex x by nl (x). For a given edge e = {v; w}, we denote by v the endpoint with smaller lexicographical number, i.e., nl (v) ¡ nl (w). Then the vertices in the white and striped subtrees and on the path from u to w are exactly the vertices with lexicographical  numbers between nl (v) and nl (w). We compute for every vertex x a label l (x) = nl (y)6nl (x) w(y). Then the weight of the vertices in the white and striped subtrees and on the path from u to w is l (w) − l (v). Given for every vertex x the weight (x) of the vertices on the path from x, inclusive, to the root r of T ′ , the weight of the vertices on the path from u to w, including w but not u, is (w) − (u), and the weight of the white and striped subtrees is l (w) − l (v) + (w) − (u). It remains to add the weight of the cross-hatched trees and to subtract the weight of the white trees to obtain the weight w(R1 (e)) of region R1 (e). For every edge e, we compute a label (e). If e = {v; w} is a tree edge and v is further away from the root of T ′ than w, then (e) is the weight of the subtree of T ′ rooted at v. If e is a non-tree edge, (e) = 0. Let e0 ; : : : ; ek be the list of edges incident to a vertex x, sorted counterclockwise around x and so that e0 = {x; p(x)}, i where p(x) denotes x’s parent in T ′ . Then we de ne x (e0 ) = 0 and x (ei ) = j=1 (ej ) for i ¿ 0. Now the weight of the white subtrees in Fig. 5 is v (e), and the weight of the cross-hatched subtrees is w (e). Thus, we can compute the weight of region R1 (e) as w(R1 (e)) = l (w) − l (v) + (w) − (u) + w (e) − v (e): (5) It is easy to see that the weight of region R2 (e) is w(R2 (e)) = w(G ′ ) − w(R1 (e)) − (v) − (w) + 2(u) − w(u) (6) (the weight of the whole graph minus the weight of the interior region minus the weight of the fundamental cycle). Assuming that all these labels are stored with e, we can scan the edge set of G ′ and compute for every visited non-tree edge the weights of R1 (e) and R2 (e) using Eqs. (5) and (6). It remains to show how to compute these labels. We provide the details in the proof of the following lemma. CYCLESEPARATOR(G ′ ; T ′ ): 1: Compute the vertex and edge labels required to compute the weights of regions R1 (e) and R2 (e), for every non-tree edge e. a: Label every vertex x in G ′ with a tuple A(x) = (W (x); nl (x); (x); l (x)), 70 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 where • W (x) is the weight of the subtree of T ′ rooted at x, • nl (x) is x’s lexicographical number in T ′ , • (x) is  the total weight of all ancestors of x in T ′ , inclusive, and • l (x) = nl (y)6nl (x) w(y) (“weighted lexicographical number of x”). b: For every edge e = {v; w}, (1) Compute the lowest common ancestor u of v and w and copy the tuples A(u), A(v), and A(w) to e, and (2) Compute a label (e) de ned as (e) =  0 if e is a non-tree edge W (v) if e is a tree edge and w = p(v): c: For every vertex x let e0 ; : : : ; ed be the set of edges incident to x sorted counterclockwise around x and so that e0 = {x; p(x)}. Compute labels x (ei ) =  0 if i = 0 x (ei−1 ) + (ei ) if i ¿ 0: 2: Scan the edge list of G ′ and compute for every non-tree edge e, the weights of R1 (e) and R2 (e) using equations (5) and (6). Choose a non-tree edge e such that w(R1 (e)) 6 23 and w(R2 (e)) 6 23 . 3: Report the fundamental cycle C(e). Algorithm 2: Finding a simple cycle separator in a triangulation. Lemma 6. Given a triangulated graph G ′ with N vertices and a BFS-tree T ′ of G ′ ; Algorithm 2 takes O(sort(N )) I=Os to compute a 23 -simple cycle separator of size at most 2height(T ′ ) − 1 for G ′ . Proof. The labelling of vertices in Step 1a takes O(sort(N )) I=Os: The subtree weights W (v) and the weighted levels (v) of the vertices can be computed using time-forward processing in the tree. To compute the subtree weights; we process the tree bottom-up. To compute the weighted levels; we process the tree top-down. In Section 2.2; we have described how to compute a lexicographical numbering of T ′ in O(sort(N )) I=Os. To compute l (x) for all vertices x; we sort the vertex set of T ′ by increasing numbers nl (x) and then scan this sorted list to compute labels l (x) as the pre x sums over the weights w(x). As there are at most 2N − 5 non-tree edges in T ′ , we have to compute O(N ) lowest common ancestors in T ′ in Step 1b, which takes O(sort(N )) I=Os [3]. To copy the labels of u, v, and w to edge e ={v; w}, we rst sort the vertex set by increasing vertex names. Then we sort the edge set three times, once by each of the values of u, v, and w. After each sort, we scan the vertex and edge sets to copy the tuples A(u), A(v), D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 71 and A(w), respectively, to the edges. Computing labels (e) then takes an additional scan over the edge set of T ′ . To compute labels x (e) in Step 1c, we create a list L1 containing two tuples (v; nv (e); p(v); w; (e)) and (w; nw (e); p(w); v; (e)) for every edge e = {v; w}. We sort this list lexicographically, so that (the tuples corresponding to) the edges incident to vertex x are stored consecutively, sorted counterclockwise around x. Then we scan L1 and compute a list L2 of triples (v; w; v (e)) and (v; w; w (e)). Note that in L1 , the edge (x; p(x)) is not necessarily stored as the rst edge in the sublist of edges incident to x causing us to skip some edges at the beginning of the sublist until we nd edge (x; p(x)) in the list. The skipped edges have to be appended at the end of the sublist. We can use a queue to do this. After sorting L2 as well as the edge list of T ′ , it takes a single scan of the two sorted lists to copy the labels v (e) and w (e) to all edges e. Step 2 searches for a non-tree edge whose corresponding fundamental cycle is a 2 -separator. As already observed, this search takes a single scan over the edge list of 3 G ′ , using the labels computed in Step 1 to compute the weights w(R1 (e)) and w(R2 (e)) of the interior and exterior regions. Step 3 is based on the following observation: Given a lexicographical numbering nl (x) of the vertices of T ′ , this numbering is also a preorder numbering. Given a vertex v with preorder number x, let the subtree rooted at v have size m. Then the vertices in this subtree have preorder numbers x through x + m − 1. This implies that the ancestor of a vertex v at a given level l is the vertex u such that nl (u) = max{nl (u′ ): l(u′ ) = l ∧ nl (u′ ) 6 nl (v)}, where l(x) denotes the level of vertex x. Using this observation we sort the vertices by increasing levels in T ′ and in each level by increasing lexicographical numbers and scan this sorted list of vertices backward, nding the ancestors of v and w at every level until we come to a level where v and w have the same ancestor, u. Thus, Step 3 also takes O(sort(N )) I=Os. 5. Triangulating embedded planar graphs In this section, we present an O(sort(N ))-algorithm to triangulate a connected embedded planar graph G = (V; E). We assume the same representation of G and its embedding as in the previous section. Our algorithm consists of two phases. First, we identify the faces of G. We represent each face f by a list of vertices on its boundary, sorted clockwise around the face. In the second phase we use this information to triangulate the faces of G. We show the following theorem. Theorem 7. Given an embedded planar graph G; it can be triangulated in O(sort(N )) I=Os. Proof. This follows from Lemmas 8 and 11. 5.1. Identifying faces As mentioned above, we represent each face f by the list of vertices on its boundary, sorted clockwise around the face. Denote this list by Ff . Let F be the concatenation of 72 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 6. (a) The directed graph D (dotted arrows) corresponding to a given graph G (solid lines). Note that vertex v appears twice on the boundary of face f, and vertex w appears twice on the boundary of the outer face. (b) The directed graph Ĝ (white vertices and dotted arrows) corresponding to D. the lists Ff for all faces f of G. The goal of the rst step is to compute F. The idea of this step is to replace every edge {v; w} of G by two directed edges (v; w) and (w; v) and decompose the resulting directed graph, D, into directed cycles, each representing the clockwise traversal of a face of G (see Fig. 6(a)). (“Clockwise” means that we walk along the boundary of the face with the boundary to our left. Thus, a clockwise traversal of the outer face corresponds to walking counterclockwise along the outer boundary of the graph. This somewhat confusing situation is resolved if we imagine the graph to be embedded on a sphere because then all faces are interior.) Removing one edge from each of these cycles gives us a set of paths. The vertices on such a path appear in the same order on the path as clockwise around the face represented by the path. (Note that the same vertex may appear more than once on the boundary of the same face, if it is a cutpoint (see Fig. 6(a)).) Considering the set of paths as a set of lists, we can rank the lists. This gives us the orders of the vertices around all faces, i.e., the lists Ff . The problem with the directed cycles in D is that they are not vertex-disjoint. Hence, we cannot apply standard graph external-memory algorithms to extract these cycles. The following modi cation solves this problem: Instead of building D, we build a graph Ĝ for G (see Fig. 6(b)), which is closely related to the representation of G by a doubly-connected edge list [21]. Ĝ contains a vertex v(v; w) for every edge (v; w) in D and an edge between two vertices v(u; v) and v(v; w) if the corresponding edges (u; v) and (v; w) are consecutive in some cycle of D representing a face of G. Graph Ĝ consists of vertex-disjoint cycles, each representing a face of G. We compute the connected components of Ĝ, thereby identifying the cycles of G, and apply the same transformations to Ĝ that we wanted to apply to D. Step 1 of Algorithm 3 gives the details of the construction of Ĝ. In Step 2, we use Ĝ to compute F. Lemma 8. Algorithm 3 takes O(sort(N )) I=Os to constructs the list F for a given graph G. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 73 Proof. Correctness: Two vertices in Ĝ that are consecutive on a cycle of Ĝ represent two edges that are consecutive on the boundary of a face of G in clockwise order. Thus; these two edges are consecutive around a vertex of G in counterclockwise order. The correctness of Step 1 follows from this observation. After removing an edge from every cycle in Ĝ, the connected components of Ĝ are paths. Considering these paths as lists, we can rank these lists. Sorting the vertices in V̂ by component labels and decreasing ranks (Step 2d) arranges the vertices representing the same face of G consecutively and sorts them clockwise around the face. Given a sublist V̂ f of V̂ representing a face f of G, the vertices of V̂ f represent the edges of D clockwise around the face. IDENTIFYFACES(G): 1: Construct a directed graph Ĝ = (V̂ ; Ê): a: For each edge e = {v; w} ∈ E, add two vertices, v(v; w) and v(w; v) to V̂ . b: For each vertex v ∈ V , let {v; w0 }; : : : ; {v; wk−1 } be the edges incident to v, in counterclockwise order. c: Add directed edges (v(wi ;v) ; v(v; w(i+1) mod k ) ), 0 6 i ¡ k, to Ê. 2: Construct F: a: Compute the connected components of Ĝ (considering Ĝ as undirected), labelling every vertex v̂ with a label c(v̂) identifying its connected component. b: Remove one edge from each connected component of Ĝ. c: Rank the resulting lists. d: Sort V̂ by component labels c(v̂) and decreasing ranks. e: Scan V̂ and write for each vertex v(v; w) , a copy of v with face label f(v)= c(v(v; w) ) to F. Algorithm 3: Identifying the faces of G. We can think of Step 2e as scanning this list of edges in D and writing the sorted sequence of source vertices of these edges to disk. This produces the desired list Ff , and the application of Step 2e to the whole of V̂ produces the concatenation F of all such lists Ff . Note that the lists Ff are distinguished in F, as two vertices in F have the same label f(v) if and only if they belong to the same list Ff . Complexity: Step 1a requires a single scan over the edge list E of G. For step 1b, recall that we are given the embedding as two labels nv (e) and nw (e) for every edge e = {v; w}. We replace e by two triples (v; nv (e); w) and (w; nw (e); v), and sort the resulting list of triples lexicographically. In the resulting list, all (triples representing) edges incident to a vertex v are stored consecutively, sorted counterclockwise around the vertex. Thus, Step 1c requires a single scan over this sorted list of triples, and Step 1 takes O(sort(N )) I=Os. Steps 2a and 2c take O(sort(N )) I=Os [3] as do Steps 2d and 2e. The details of Step 2b are as follows: Note that for every vertex v̂, there is exactly one edge (v̂; ŵ) in Ĝ. 74 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Thus, we sort V̂ by vertex names and Ê by the names of the source vertices, thereby placing edge (v̂; ŵ) at the same position in Ê as v̂’s position in V̂ . Scanning V̂ and Ê simultaneously, we assign component labels c((v̂; ŵ)) = c(v̂) to all edges (v̂; ŵ) in Ê. Then we sort Ê by these component labels. Finally we scan Ê again and remove every edge whose component label is di erent from the component label of the previous edge. Also, we remove the rst edge in Ê. As this procedure requires sorting and scanning V̂ and Ê twice, the complexity of Step 2 is O(sort(N )) I=Os. 5.2. Triangulating faces We triangulate each face f in four steps (see Algorithm 4). In Step 1, we reduce f to a simple face f̂. That is, no vertex appears more than once in a clockwise traversal of f̂’s boundary. Accordingly, we reduce the list Ff to Ff̂ . In Step 2, we triangulate f̂. We guarantee that there are no parallel edges in f̂. But we may add parallel edges to di erent faces (see Fig. 8(a) for an example). Let e1 ; : : : ; ek be the set of edges with endpoints v and w. In Step 3, we select one of these edges, say e1 , and mark edges e2 ; : : : ; ek as con icting. Each of these edges is said to be in con ict with e1 . In Step 4, we retriangulate all faces so that con icts are resolved and a nal triangulation is obtained. The following lemma states the correctness of Step 1. Lemma 9. For each face f of G; the face f̂ computed by Step 1 of Algorithm 4 is simple. The parts of f that are not in f̂ are triangulated. Moreover; Step 1 does not introduce parallel edges. Proof. We mark exactly one copy of every vertex on f’s boundary. For the rst vertex; we append a second marked copy to the end of Ff only to close the cycle. This copy is removed at the end of Step 1. We remove an unmarked vertex v by adding an edge between its predecessor u and successor w on the current boundary of the face; thereby cutting triangle (u; v; w) o the boundary of f. As we remove all unmarked vertices this way; the resulting face f̂ is simple and the parts that have been removed from f to produce f̂ are triangulated. Next, we show that Step 1 does not add parallel edges to the same face f. Assume for the sake of contradiction that we have added two edges with TRIANGULATEFACES(G; F): 1: Make all faces of G simple: For each face f, (a) mark the rst appearance of each vertex v in Ff , (b) append a marked copy of the rst vertex in Ff to the end of Ff , (c) scan Ff backward and remove each unmarked vertex v from f and Ff by adding a diagonal between its predecessor and successor in the current list, and (d) remove the last vertex from list Ff . Call the resulting list Ff̂ . D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 75 2: Triangulate the simple faces: Let Ff̂ = v0 ; : : : ; vk . Then add “temporary diagonals” {v0 ; vi }, 2 6 i 6 k − 1, to f̂. 3: Mark con icting diagonals: Sort E lexicographically, representing edge {v; w} by an ordered pair (v; w), v ¡ w, and so that edge {v; w} is stored before any “temporary diagonal” {v; w}. Scan E and mark all occurrences except the rst of each edge as con icting. Restore the original order of all edges and “temporary diagonals”. 4: Retriangulate con icting faces: a: For each face f̂, let Df̂ = {v0 ; v2 }; : : : ; {v0 ; vk−1 } be the list of “temporary diagonals”. b: Scan Df̂ until we nd the rst con icting diagonal {v0 ; vi }. c: Replace the diagonals {v0 ; vi }; : : : ; {v0 ; vk−1 } by new diagonals {vi−1 ; vi+1 }; : : : ; {vi−1 ; vk }. Algorithm 4: Triangulating the faces of G. endpoints v and w to f (see Fig. 7(a)). We can add two such edges e1 and e2 only if one of the endpoints, say w, appears at least twice in a clockwise traversal of f’s boundary. Then, however, there is at least one vertex x that has all its occurrences between two consecutive occurrences of w because e1 and e2 form a closed curve. That is, the marked copy of x also appears between these two occurrences of w. Adding e2 to f would remove this marked copy from f’s boundary; but we never remove marked vertices. Thus, e2 cannot exist. Now assume that we add two edges e1 and e2 with endpoints v and w to di erent faces f1 and f2 . By adding e1 we remove a vertex u from f1 ’s boundary (see Fig. 7(b)). As this copy is unmarked, there has to be another, marked, copy of u. Consider the region R1 enclosed by the curve between the marked copy of u and the removed copy of u, following the boundary of f1 . In the same way we de ne a region R2 enclosed by the curve between the removed copy of u and the marked copy of u. Fig. 7. Illustrating the proof of Lemma 9. 76 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 8. (a) A simple face f̂ with a con icting diagonal edge d = {v0 ; vi }. Diagonal d con icts with d′ and divides f̂ into two parts f1 and f2 . One of them, f1 , is con ict-free. Vertex vi−1 is the third vertex of the triangle in f1 that has d on its boundary. (b) The con ict-free triangulation of f̂. Any face other than f1 that has v on its boundary must be in R1 . Any face other than f1 that has w on its boundary must be in R2 . However, R1 and R2 are disjoint. Thus, we cannot add a diagonal {v; w} to any face other than f1 . Step 2 triangulates all simple faces f̂, possibly adding parallel edges to di erent faces. Consider all edges e1 ; : : : ; ek with endpoints v and w. We have to remove at least k − 1 of them. Also, if edge {v; w} was already in G, we have to keep this edge and remove all diagonals that we have added later. That is, the edge {v; w} is the edge with which all diagonals {v; w} are in con ict, and we have to label all diagonals as con icting while labelling edge {v; w} as non-con icting. If edge {v; w} was not in G, we can choose an arbitrary diagonal {v; w} with which all other diagonals with the same endpoints are in con ict. This strategy is realized in Step 3. The following lemma states the correctness of Step 4, thereby nishing the correctness proof for Algorithm 4. Lemma 10. Step 4 makes all faces f̂ con ict-free; i.e.; the triangulation obtained after Step 4 does not contain parallel edges. Proof. Let d={v0 ; vi } be the edge found in Step 4b (see Fig. 8(a)). Then d cuts f̂ into two halves, f1 and f2 . All diagonals {v0 ; vj }, j ¡ i are in f1 ; all diagonals {v0 ; vj }, j ¿ i are in f2 . That is, f1 does not contain con icting diagonals. Vertex vi−1 is the third vertex of the triangle in f1 that has d on its boundary. Step 4c removes d and all edges in f2 and retriangulates f2 with diagonals that have vi−1 as one endpoint. (Intuitively it forms a star at the vertex vi−1 ; see Fig. 8(b).) Let d′ be the edge that d is in con ict with. Then d and d′ form a closed curve. Vertex vi−1 is outside this curve and all boundary vertices of f2 excluding the endpoints of d are inside this curve. As we keep d′ , no diagonal, except for the new diagonals in f̂, can intersect this curve. Thus, the new diagonals in f̂ are non-con icting. The “old” diagonals in f̂ were in f1 and thus, by the choice of d and f1 , non-con icting. Hence, f̂ does not contain con icting diagonals. D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 77 Lemma 11. Given the list F as computed by Algorithm 3; Algorithm 4 triangulates the given graph G in O(sort(N )) I=Os. Proof. We have already shown the correctness of Algorithm 4. In Step 1; we rst sort every list Ff by vertex names. Then it takes a single scan to mark the rst occurrences of all vertices in Ff . Using another sort we restore the original order of the vertices in Ff . The rest of Step 1 requires scanning F; writing the marked vertices to Ff̂ ; in their order of appearance; and keeping the last visited marked vertex in main memory in order to add the next diagonal. Thus; Step 1 takes O(sort(N )) I=Os. Step 2 requires a single scan over the list F; as modi ed by Step 1. Assuming that all edges in G before the execution of Step 2 were labelled as “edges”; we label all edges added in Step 2 as “diagonals”. Then Step 3 requires sorting the list of edges and diagonals lexicographically and scanning this sorted list to label con icting diagonals. Note; however; that Step 4 requires the diagonals to be stored in the same order as added in Step 2. Thus; before sorting in Step 3; we label every edge with its current position in E. At the end of Step 3; we sort the edges by these position labels to restore the previous order of the edges. Of course; Step 3 still takes O(sort(N )) I=Os. Step 4 takes a single scan over E. Thus; the whole algorithm takes O(sort(N )) I=Os. A problem that we have ignored so far is embedding the new diagonals. Next, we describe how to augment Algorithm 4 in order to maintain an embedding of G under the edge insertions performed by the algorithm. To do this, we have to modify the representation of the embedding slightly. Initially, we assumed that the edges e1 ; : : : ; ek incident to a vertex v are assigned labels nv (ei ) = i clockwise around v. During the triangulation process we allow labels nv (e) that are multiples of 1=N . Note that this does not cause precision problems because we can represent every label nv (e) as an integer N · nv (e) using at most 2 log N bits (while nv (e) uses log N bits). Let v0 ; e0 ; v1 ; e1 ; : : : ; vk−1 ; ek−1 be the list of vertices and edges visited in a clockwise traversal of the boundary of a face f (i.e., Ff = v0 ; : : : ; vk−1 ). During the construction of Ff , we can easily assign labels n1 (vi ) and n2 (vi ) to the vertices, where n1 (vi ) = nvi (e(i−1) mod k ) and n2 (vi ) = nvi (ei ). When we add a diagonal d = {vi ; vj }, i ¡ j, we update the labels of vi and vj to n2 (vi )=n2 (vi )−1=N and n1 (vj )=n1 (vj )+1=N and embed d assigning nvi (d) = n2 (vi ) and nvj (d) = n1 (vj ). Assuming that n1 (vi ) ¡ n2 (vi ) − 1=N or n1 (vi ) ¿ n2 (vi ) and n2 (vj ) ¿ n1 (vj ) + 1=N or n2 (vj ) 6 n1 (vj ), this embeds d between e(i−1) mod k and ei at vi ’s end and e(j−1) mod k and ej at vj ’s end. It remains to show that this assumption is always satis ed. We maintain the following invariant for every face f: Let v0 ; e0 ; : : : ; vk−1 ; ek−1 be the boundary description of f as given above. Then for every vertex vi , either n1 (vi )+ (k − 3)=N ¡ n2 (vi ) or n1 (vi ) ¿ n2 (vi ). Initially, this is true because all labels nv (e) are integers and k 6 N . Adding diagonal d to f as above cuts f into two faces f1 and f2 . For all vertices vl , l ∈ {i; j}, the labels do not change; but the sizes of f1 and f2 are less than the size of f. Thus, for all these vertices the invariant holds. We show that the invariant holds for vi . A similar argument can be applied for vj . Let f1 be the face with vertices vj ; : : : ; vi on its boundary and f2 be the face with vertices vi and vj on the 78 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 Fig. 9. A given planar graph G (a) and its separator tree ST (G) (b). Note that any path between vertices v and w, which are stored in the white and gray subtrees of ST (G), respectively, must contain a black or horizontally striped separator vertex. These separators are stored at common ancestors of (v) and (w). boundary. Let the size of fi be ki 6 k −1. If n2 (vi ) 6 n1 (vi ), then this is also true after subtracting 1=N from n2 (vi ). Otherwise, n2 (vi )−n1 (vi ) ¿ (k −3)=N −1=N ¿ (k1 −3)=N . Thus, the invariant holds for all vertices on f1 ’s boundary. In f2 we do not have any room left to add diagonals incident to vi or vj . However, Steps 1 and 4 of Algorithm 4 scan along the boundaries of the faces and keep cutting o triangles. Choosing the indices i and j in the above description so that f2 is the triangle that we cut o , we never add another diagonal to f2 . (Note that it would be just as easy to maintain the embedding in Step 2; but we need not even do this because diagonals are added only temporarily in Step 2, and nal diagonals are added in Step 4.) 6. The shortest path data structure In this section, we incorporate the main ideas of the internal memory shortest path data structure in [8] and show how to use them together with the external-memory techniques developed in this paper to design an ecient external-memory shortest path data structure. The data structure in √ [8] uses O(S) space and answers distance queries on graphs with separators of size O( N ) in O(N 2 =S) time, where 1 6 S 6 N 2 . Reporting the corresponding shortest path takes O(K) time, where K is the number of vertices on the path. The basic structure used to obtain the above trade-o √is an O(N 3=2 ) size internal memory data structure that answers distance queries in O( N ) time. Our 3=2 external-memory data structure is fully blocked. That √ is, it uses O(N =B) blocks of external memory and answers distance queries in O( N =DB) I=Os. The corresponding shortest path can be reported in O(K=DB) I=Os. Given a planar graph G (see Fig. 9(a)), we compute a separator tree ST (G) for G (see Fig. 9(b)). This tree is de ned recursively: We compute a 32 -separator S()
of size O( |G|) for G. Let G1 ; : : : ; Gk be the connected components of G–S(). Then we store S() at the root  of ST (G) and recursively build separator trees ST (G1 ); : : : ; ST (Gk ), whose roots become the children of . Thus, every vertex v of G is stored at exactly one node (v) ∈ ST (G). For two vertices v and w, let D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 79 (v; w)=lca( (v); (w)). For a vertex ∈ ST (G), we de ne S − ( ) (resp. S + ( )) as the sets of vertices stored at and its ancestors (resp. descendants). For convenience, we denote the sets S( (v)) and S( (v; w)) by S(v) and S(v; w), respectively. Sets S − (v), S + (v), S − (v; w) and S + (v; w) are de ned analogously. In [8], it is shown that any path between v and w in G must contain at least one vertex in S − (v; w). For a given graph H , we denote the distance between two vertices v and w in H by dH (v; w). Then d(v; w) = minx∈S − (v; w) {dG[S + (x)] (x; v) + dG[S + (x)] (x; w)}. If we represent all sets S − (v) and S − (v; w) as lists sorted by increasing depth in ST (G), then S − (v; w) is the longest common pre x of S − (v) and S − (w). Let D(v) be a list of distances, where the ith entry in D(v) is the distance dG[S + (x)] (v; x) between v and the ith entry x in S − (v). Given the lists S − (v), S − (w), D(v), and D(w), we can compute d(v; w) by scanning these four lists. We scan S − (v) and S − (w) in “lock-step” fashion and test whether xv = xw , where xv and xw are the current entries visited in the scans of the two lists. If xv = xw , then we proceed in the scans of D(v) and D(w) to compute dG[S + (xv )] (xv ; v) + dG[S + (xv )] (xv ; w) and compare it to the previously found minimum. If xv = xw , we have reached the end of S − (v; w) and stop. For every separator vertex x, our data structure contains a shortest path tree SPT (x) with root x. This tree represents the shortest paths between x and all vertices in G[S + (x)]. That is, every vertex v of G[S + (x)] is represented by a node in SPT (x), and the shortest path from v to x in G[S + (x)] corresponds to the path from v to x in SPT (x). Given the distance d(v; w) between two vertices v and w, there must be some vertex xmin ∈ S − (v; w) such that d(v; w) = dG[S + (xmin )] (xmin ; v) + dG[S + (xmin )] (xmin ; w). The shortest path (v; w) from v to w is the concatenation of the shortest paths from v to xmin and from xmin to w. Given SPT (xmin ), we can traverse the paths from v and w to the root xmin and concatenate the traversed paths to obtain (v; w). To traverse these two paths in the tree, we have to nd the two (external) memory locations holding the two nodes representing v and w in SPT (xmin ). We construct lists P(v) for all vertices v of G holding pointers to the representatives of v in all shortest path trees. Let x be the separator vertex stored at the ith position in S − (v). Then the ith position of P(v) holds a pointer to the node representing v in SPT (x). That is, if xmin is stored at position i in S − (v) and S − (w), the ith positions of P(v) and P(w) hold the addresses of the representatives of v and w in SPT (xmin ), giving us enough information to start traversing and reporting (v; w). The
size of the separator S( ) stored at every vertex in the separator tree ST (G) is O( |G[S + ( )]|). From one level in ST (G) to the next the sizes of the subgraphs G[S + ( )] associated with the vertices decrease by a factor of at least 32 . Thus, the sizes of the graphs G[S + ( )] on a root-to-leaf path in ST (G) are bounded from above by a geometrically decreasing sequence, and the sizes of the separators S( ) stored at these √ vertices form a geometrically decreasing sequence as well. Hence, there√are only O( N ) separator vertices on any such path and each list S − (v) has size O( N ). As we√have to scan lists S − (v), S − (w), D(v), and D(w) to compute d(v; w), this takes O( N =DB) I=Os. It takes two more I=Os to access the pointers in P(v) and P(w). Assuming that all shortest path trees have been blocked as described in Section 3, traversing the paths from v and w to xmin in SPT (xmin ) takes O(K=DB) I=Os, where K is the number of vertices on the shortest path (v; w). 80 D. Hutchinson et al. / Discrete Applied Mathematics 126 (2003) 55 – 82 It remains to show that the data structure√can be stored in O(N 3=2 =B) blocks. As every list S − (v), D(v), and P(v) has size O( N ) and there are 3N such lists, all lists require O(N 3=2 ) space and can thus be stored in O(N 3=2 =B) blocks. There is exactly one shortest path tree for every separator vertex. Consider tree SPT (x) and a node v in this tree. Then this node corresponds to the entry for x in list S − (v). That is, there is a one-to-one correspondence between entries in the lists S − (v) and shortest path tree nodes. Thus, the total size of all shortest path trees is O(N 3=2 ) as well. As blocking the shortest path trees increases the space requirements by only a constant factor, the shortest path trees can also be stored in O(N 3=2 =B) blocks of external memory. We have shown the following theorem. Theorem 12. Given a planar graph G with N vertices; one can construct√a data structure that answers distance queries between two vertices in G in O( N =DB) I=Os. The corresponding shortest path can be reported in O(K=DB) I=Os; where K is the length of the reported path. The data structure occupies O(N 3=2 =B) blocks of external memory. 7. Conclusions The I=O-ecient construction of our shortest path data structure is still a problem, as there are no algorithms for BFS, embedding, and the single source shortest path problem that perform I=O-eciently on planar graphs. The separator algorithm in Section 4 tries to address the problem of computing the separators required to build the separator tree I=O-eciently. Also, in [26] an O(sort(N )) algorithm for transforming a given rooted tree of size N into the blocked form described in Section 3 is given. A shortcoming of our separator algorithm is that it needs a BFS-tree. Most separator algorithms rely on BFS, but BFS and depth- rst search (DFS) seem to be hard problems in external memory. Thus, it is an important open problem to develop an I=O-ecient separator algorithm that does not need BFS or DFS. The existence of an I=O-ecient planar embedding algorithm is also open. Recently, Maheshwari and Zeh [18] presented O(sort(N )) algorithms for outerplanarity testing, computing an outerplanar embedding, BFS, DFS, and computing a 2 3 -separator of a given outerplanar graph. It is an important question whether there are other interesting classes of graphs with similar I=O-complexities for these problems. They also showed (perm(N )) lower bounds for computing an outerplanar embedding, and BFS and DFS in outerplanar graphs. As outerplanar graphs are also planar, the lower bounds for BFS and DFS also apply to planar graphs. A similar technique as in [18] can be used to show that planar embedding has an (perm(N )) lower bound. 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