Breadth-Depth Search is P-complete. Raymond Greenlaw. University of New Hampshire. address: Abstract

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1 Breadth-Depth Search is P-complete Raymond Greenlaw Department of Computer Science University of New ampshire Durham, New ampshire address: greenlawcs.unh.edu Abstract The parallel complexity of a search strategy that combines attributes of both breadth-rst search and depth-rst search is studied. The search called breadth-depth search was dened by orowitz and Sahni. The search technique has applications in branch-and-bound strategies. Kindervater and Lenstra posed the complexity of this type of search strategy as an open problem. We resolve their question by showing that a natural decision problem based on breadth-depth search is P-complete. Specically, we prove that if given a graph G = (V; E) either directed or undirected, a start vertex s 2 V, and two designated vertices u and v in V, then the problem of deciding whether u is visited before v by a breadth-depth search originating from s is P-complete. The search can be based either on vertex numbers or xed ordered adjacency lists. Our reductions dier for directed/undirected graphs and depending on whether vertex numbers/xed ordered adjacency lists are used. These results indicate breadth-depth search is highly sequential in nature and probably will not adapt to a fast parallel solution, unless N C equals P. Keywords: Breadth-depth search, breadth-rst search, depth-rst search, parallel algorithms, P-completeness. 1 Introduction Graph searching techniques like breadth-rst search (BFS) and depth-rst search (DFS) are fundamental procedures in many algorithms. Applying parallel computers to speedup graph searching has become an especially important problem as the size of the search space for many problems has grown very large. We study a search strategy called breadth-depth search (BDS) that combines features of both BFS and DFS. orowitz and Sahni dened BDS and demonstrated its applications to branch-and-bound strategies [8]. Kindervater and Lenstra posed as an open question the complexity of this type of search [9, page 182]. In this paper we classify the complexity of BDS by showing that a natural decision problem based on BDS is P-complete. Our results indicate BDS is probably inherently sequential. Table 1 summarizes research that examines the complexities of various search methods. Reif showed that computing an ordered DFS was P-complete [10]. This result led researchers to believe This research was partially funded by National Science Foundation Grants CCR and CCR

2 DFS was highly sequential. Fast parallel algorithms were found for certain restricted version of DFS for example, restricted to planar graphs [11] or to directed acyclic graphs [3, 4, 6]. Aggarwal and Anderson showed that by using randomization one could obtain a fast parallel algorithm for unordered DFS in undirected graphs [1]. Aggarwal, Anderson, and Kao showed a similar result for directed graphs [2]. It is well-known that computing BFS level numbers is in N C. We showed a natural decision problem based on an implementation of BFS using stacks was P-complete [6]. Comparing the results in this paper with those in Table 1 seems to indicate BDS is more like DFS than BFS. Table 1: A summary of results about search strategies. Search Method Complexity Reference(s) depth-rst search in planar graphs N C [11] ordered depth-rst search P-complete [10] ordered depth-rst search in DAGs N C [3, 4, 6] ordered breadth-rst search N C [3, 4, 6] stack breadth-rst search P-complete [6] unordered depth-rst search RN C [1, 2] breadth-depth search P-complete [5, 6] The remainder of this paper is organized as follows. Section 2 contains preliminary material. In Section 3 we formally dene BDS and present an algorithm for BDS. Reductions are presented in Section 4 proving decision problems based on the BDS algorithm are P-complete. Section 5 contains conclusions. 2 Preliminaries P-completeness theory has been an active research area over the last few years. An introduction to the theory as well as a comprehensive list of P-complete problems is given in [7]. We assume the reader is familiar with the basic principles of P-completeness. The theory has an important practical signicance because problems that are P-complete probably do not adapt to fast (log k n for input size n and some constant k) parallel solutions using a polynomial number of processors. That is, it is unlikely that they are in the complexity class N C. For evidence supporting this statement, the reader is referred to [7]. There are two main problems that we use to prove dierent variants of BDS are P-complete. The rst is the NOR Circuit Value Problem (NOR CVP). The problem is known to be P-complete (see [7]) and is dened below. Denition 2.1 NOR Circuit Value Problem (NOR CVP) Given: An encoding of a Boolean circuit constructed only of NOR gates, plus inputs x 1 ; : : :; x n. The gates of the circuit are numbered in topological order. The circuit is restricted to have fan-out one for circuit inputs and fan-out two for NOR gates. All inputs are restricted to being TRUE. Problem: Does on inputs x 1 ; : : :; x n output 1 2

3 The second problem we use in the reductions is the ordered DFS problem. problem below [10]. We dene the Denition 2.2 Ordered Depth-First Search (ODFS) Given: A graph G = (V; E) with xed ordered adjacency lists, a start vertex s, and two designated vertices u and v. Problem: Does vertex u get visited before vertex v in the ordered DFS of G induced by the xed ordered adjacency lists The search produced in Denition 2.2 is sometimes called a greedy DFS. Reif showed this problem was P-complete by a reduction from a variant of NOR CVP. Theorem 2.1 [10] The ordered depth-rst search problem is P-complete under log space reducibility for both directed and undirected graphs. The order depth-rst search problem remains P-complete if the search order is based on vertex numberings instead of xed ordered adjacency lists. For clarity, we formally dene this version of the problem as well. Denition 2.3 Vertex Ordered Depth-First Search (Vertex ODFS) Given: A graph G = (V; E) with a numbering on the vertices, a start vertex s, and two designated vertices u and v. Problem: Does vertex u get visited before vertex v in the ordered DFS of G induced by the vertex numbering of V It is not obvious that this version of the problem is P-complete. The following result proved in [5] shows that it is. Theorem 2.2 [5] The vertex ordered depth-rst search problem is P-complete under log space reducibility for both directed and undirected graphs. 3 Breadth-Depth Search The algorithm shown below depicts a BDS that is based on xed ordered adjacency lists. The algorithm is easily modied to work based on a vertex numbering of its input graph. The search order produced by the algorithm is dierent from a BFS in that it does not proceed level by level in visiting nodes. It is dierent from DFS in that the algorithm has a \breadth" component. Figures 3 through 6 provide examples of BDS. Breadth-Depth Search Algorithm Input: A graph G = (V; E) represented by xed ordered adjacency lists and a node s. Output: Starting from s all nodes visited are labeled with the level in which they occur and are numbered consecutively in the order in which they were visited within their level. Comment: next(v) { returns the next unlabeled vertex on v's adjacency list; unlabeled(v) { returns the total number of unvisited vertices adjacent to v; level[v] { contains v's level number; order[v] { contains v's number within a given level. begin 3

4 level[s] 0; order[s] 1; count 1; push s on stack S; while stack S 6= null do v pop S; T unlabeled(v); for i 1 to T do u next(v); order[u] count; count count + 1; level[u] level[v] + 1; push u on stack S; endfor endwhile end. We say a node has been explored if all of its adjacent vertices have been visited. The BDS algorithm operates by exploring nodes. The start vertex s is the rst node explored. The next node explored is the last one visited from s the vertex most recently popped o the stack. This process continues so that a single node at each level is fully explored each time a node on that level is popped o the stack. This diers from DFS where only a single vertex is visited from the current node. The current node in DFS is not explored unless it is at the end of a path in the DFS tree and even then it may not be fully explored next. BDS diers from BFS because in general only the nodes adjacent to a single vertex per level are explored initially. In BFS an entire level is visited before proceeding to the next level. In the next section we show that several natural decision problems based on BDS are log space reducible (see [7] for a denition) to either NOR CVP, ODFS, or vertex ODFS. Clearly, BDS can be solved in polynomial time (by running the BDS algorithm), this combined with the reductions proves the problems are P-complete. 4 Breadth-Depth Search is P-complete Our goal in this section is to prove that several decision problems based on breadth-depth search are P-complete. We dene the breadth-depth search problem below. Denition 4.1 Breadth-Depth Search Problem (BDS) Given: A graph G = (V; E) with a numbering on the vertices in V, a start vertex s, and two designated vertices u and v. Problem: Does vertex u get visited before vertex v in the BDS of G induced by the vertex numbering of V We present three groups of reductions in this section that show BDS is P-complete for either directed or undirected graphs. The reductions are based on NOR CVP, vertex ODFS, and a variant of BDS that is based on xed ordered adjacency lists. The third group of reductions was suggested by Richard Andersion (Personal communication, Richard Anderson, 1988). The rst result we present for BDS based on vertex numberings implies the theorem for xed ordered adjacency lists. Since the alternative reduction is quite simple and interesting, we present it as well. 4

5 This reduction does not work for BDS when the search is based on vertex numbering (Personal communication, Amarjit Singh, 1992). The table below summarizes the results that are proved in this section. Table 2: A summary of results concerning the variants of BDS. Search Based on Type of Graph Reduction From Theorem vertex numbers directed NOR CVP 4.1 vertex numbers undirected NOR CVP 4.1 vertex numbers directed vertex ODFS directed graph 4.2 vertex numbers undirected vertex ODFS undirected graph 4.2 xed ordered adjacency lists directed ODFS directed graph 4.3 xed ordered adjacency lists undirected ODFS undirected graph 4.3 Theorem 4.1 The breadth-depth search problem, as stated in Denition 4.1, is P-complete under log space reducibility for either directed or undirected graphs. Proof: As noted previously, the BDS algorithm can be used to solve the BDS problem in P for either directed or undirected graphs. We prove the directed case rst. The reduction is from NOR CVP. Consider an instance of NOR CVP. Let 1; : : :; m denote input numbers and m + 1; : : :; N denote NOR gate numbers. The idea behind the reduction is to construct from an instance of BDS (G = (V; E), s, u, and v) such that the search order used to explore G allows us to determine the value of the output gate of. We construct gadgets to represent NOR gates. These gadgets are linked together. By adding a SPECIAL node to the end of the string of gadgets, the search of G is such that the value of a particular gate in is TRUE if and only if a specied vertex S(i) in each gadget is visited before the SPECIAL node. We take u to be S(output gate) and v to be SPECIAL. G is described below. The circuit inputs are represented in G as shown in Figure 1. Input i connected to gate number j is labeled IN(j; i). For input i there is an associated vertex labeled ENTER(i). A single chain of inputs does not suce for this reduction because the search has a breadth component as well as a depth component. ENTER(i) 1 R ENTER(i + 1) 3 IN(j; 2 i) Figure 1: Input i's contribution to G where input i is connected to gate j. Figure 2 depicts the contribution to G from each gate. The numbers inside the nodes indicate their relative numberings within the gadget and are fully specied below. We assume, for this 5

6 numbering only, that i 1 is less than i 2. ENTER(m) is connected to ENTER(m + 1). EXIT(N) is connected to the vertex SPECIAL. For the sake of clarity, we have depicted the gadget so that EXIT(i 1) is dierent from ENTER(i). The two nodes could be combined. EXIT(i 1) $ D(i) ENTER(i) 4-1 IN(i,i 1 ) 2 IN(i,i 2 ) 3 S(i) 5 IN(j 1,i) 6 IN(j 2,i) 7 T(i) EXIT(i) R 8 9 % Figure 2: Contribution to G from NOR gate i with inputs i 1 and i 2, and outputs j 1 and j 2. We specify a numbering of the vertices in G such that the search order of G can be used to evaluate the output gate of. Number the nodes D(m + 1); : : :; D(N) with values 1; : : :; (N m), respectively. The gadgets corresponding to the input nodes are numbered next. Number ENTER(i) and IN(j; i) with values N m + 2i 1 and N m + 2i, respectively for 1 i m. The remaining vertices that have not received numbers ENTER(i), S(i), IN(j 1 ; i), IN(j 2 ; i), T(i), and EXIT(i) are numbered N + m + 6i; : : :; N + m + 6i + 5, respectively. The SPECIAL vertex is numbered 7N + m + 6. We take s in the instance of BDS to be node ENTER(1), u to be S(output gate), and v to be SPECIAL. The reduction is easily seen to be log space. We claim that evaluates to TRUE if and only if u is visited before v in the breadth-depth search of G originating from s. The lemma below shows the reduction is correct by proving a slightly stronger result. Lemma 4.1 Vertex S(i) in the gadget for gate i is visited in G by the breadth-depth search algorithm before vertex SPECIAL if and only if gate i evaluates to TRUE in. 6

7 Proof: We assume input i is connected to gate j i. From node ENTER(1) it is easy to see that the breadth-depth search algorithm will initially visit the following sequence of nodes: ENTER(1), IN(j 1 ; 1), : : :, ENTER(m), and IN(j m ; m). The IN() nodes are all unexplored and pushed on the stack, whereas, the ENTER() nodes are explored. IN() nodes that have been visited represent TRUE values and those that have not represent FALSE values. We assume for the induction hypothesis that for all i 0, where m < i 0 i N, one of the following two cases hold. 1. If g i 0 is TRUE, the search proceeded through gadget i 0 visiting ENTER(i 0 ), D(i 0 ), IN(i 0,i 0 1 ), IN(i0,i 0 2 ), S(i0 ), IN(j 1,i 0 ), IN(j 2,i 0 ), T(i 0 ), and EXIT(i 0 ). 2. If g i 0 is FALSE, the search proceeded through gadget i 0 visiting ENTER(i 0 ), D(i 0 ), (possibly IN(i 0,i 0 1 )), T(i0 ), and EXIT(i 0 ). Because of the vertex numbering and the search order, no \outputs" of a gadget are fully explored when the gadget is visited. Suppose gate g i = g i1 NOR g i2 and both inputs to g i are FALSE. By the induction hypothesis IN(i; i 1 ) and IN(i; i 2 ) have not been previously visited, so the nodes of gadget i are visited in the order as specied in CASE 1 above. If either input to g i is TRUE then by the induction hypothesis either IN(i; i 1 ) or IN(i; i 2 ) has already been visited. Upon reaching the one that has been visited, the search backtracks and the nodes of the gadget are visited as in CASE 2 above. For values of i falling into CASE 1 (CASE 2), S(i) gets visited before (after) SPECIAL. This completes the proof of Lemma 4.1. Lemma 4.1 proves the reduction for the directed case is correct. To prove the theorem in the undirected case, we make a simple modication to the gadget Figure 2 depicts. This allows us to use essentially the same proof of correctness. A new node is inserted between IN(i; i 2 ) and S(i), and S(i)'s name is changed to F(i). The new node is labeled S(i). The relative numbering of the vertices within the gadgets remains the same. F(i)'s number is between S(i)'s and IN(j 1 ; i)'s. Again, we have two cases to consider for the induction hypothesis to demonstrate the correctness of the reduction. Making the same assumptions about i 0, our induction hypothesis for this case is as follows: 1. If g i 0 is TRUE, the search proceeded through gadget i 0 visiting ENTER(i), D(i), IN(i,i 1 ), IN(i,i 2 ), S(i), F(i), IN(j 1,i), IN(j 2,i), T(i), and EXIT(i). 2. If g i 0 is FALSE, the search proceeded through gadget i 0 visiting ENTER(i), D(i), (possibly IN(i,i 1 )), T(i), F(i), and EXIT(i). 7

8 Reasoning proceeds analogously to the directed case, so an undirected version of Lemma 4.1 holds. This proves the theorem for the undirected case and completes the proof of Theorem 4.1. The next theorem shows vertex ODFS (see Denition 2.3) is log space reducible to BDS providing an alternative P-completeness proof. The reduction illustrates an interesting relationship between ODFS and BDS. Theorem 4.2 Vertex ordered depth-rst search is log space reducible to breadth-depth search. Proof: We rst present the proof for the directed case. The undirected case requires a separate reduction. Given an instance of vertex ODFS G = (V; E) with a numbering on the vertices in V and three vertices s, u, and v; we construct a new instance G 0 = (V [ V 0 ; E 0 ), s, u, and v of BDS such that u is visited before v in the instance of vertex ODFS if and only if u is visited before v in the corresponding instance of BDS. Without loss of generality, we will assume G is connected. We modify G to prove the theorem. R R R R R R (a) R R (b) R 10 6 Figure 3: An example illustrating the reduction performed in Theorem 4.2 for the directed case. The original graph with its vertex numbering is shown in part (a). Part (a) also shows the vertex ODFS. The search order is 1, 2, 4, 5, 6, and 3; whereas for the BDS shown in part (b) the search order is 1, 8, 9, 2, 16, 4, 29, 30, 5, 6, 3, 22, and 24. Thus, vertices 1{6 are visited in the same relative order. The idea behind the construction of G 0 from G is to insert a new vertex on every directed edge of G. Thus, there are jej new vertices inserted in G 0. Since each edge in E is split, there are 2jEj edges in E 0. The new edges are both directed in the same direction as the edge they split was. The numbering of the new nodes is specied below. Let b be a node in the original graph having neighbors b 1 ; : : :; b k. Suppose the b i 's are numbered in increasing order as listed. That is, number(b 1 ) < number(b 2 ) < < number(b k ). Let c 1 ; : : :; c k 8

9 be the nodes added by the reduction and splitting the edges (b; b 1 ); : : :; (b; b k ), respectively. For example, the directed edges involving c 1 are (b; c 1 ) and (c 1 ; b 1 ). The c i 's are numbered so that number(c k ) < < number(c 1 ). That is, the c i 's numbers are such that the c i 's are ordered just the reverse of the b i 's. More specically, c i receives number b jv j + b k+1i. An example illustrating the reduction is presented in Figure 3. Because of the \reversal" of the search order by BDS after a node has been fully explored, the b i 's are visited in the same relative order by the BDS algorithm as they are by the vertex ODFS algorithm. We prove this fact formally below. We show u is visited before v in an instance of ODFS if and only if u is visited before v in an instance of BDS. In fact, the following lemma shows a slightly stronger result holds. Lemma 4.2 For all w 2 V, w gets visited in the same relative order among the vertices in V by the vertex ordered depth-rst search algorithm as by the vertex number based breadth-depth search algorithm. Proof: The proof is by induction. For the base case, the start vertex s gets visited rst by each search algorithm. Suppose w k is the k-th vertex visited by each algorithm among the vertices in V. Let w k+1 be the next node visited by the vertex ODFS algorithm. We show w k+1 is also the next vertex in V visited by the BDS algorithm. We divide the argument into two cases. The rst case is that where w k+1 is a neighbor of w k. Since w k+1 is the rst vertex visited from w k, w k+1 must be the lowest numbered neighbor of w k that has not been visited yet. Suppose d 1 ; d 2 ; : : :; d r are all the neighbors of w k in G and that they are listed in increasing order of their numbers. In G 0 there are vertices d 0 1 ; d0 2 ; : : :; d0 r splitting the edges formed between w k and its neighbors in G. In the BDS of G 0, these were all visited in the order from d 0 r to d 0 immediately after 1 w k was visited. The \reversal" of the search order by the BDS algorithm means the search continues in G 0 from d 0 up to 1 d0 r, so the next vertex in V to be visited corresponds to the lowest numbered unvisited vertex that is the sole neighbor of one of the d 0 i 's. This is the vertex w k+1 that is visited in G from w k. The second case to consider is one in which w k+1 is not a neighbor of w k. That is, all neighbors of w k have been visited by the ODFS. In this case w k+1 is a neighbor of one of the vertices visited before w k in G. By employing reasoning similar to the previous case at the point where the search backed up to, it is not hard to see that w k+1 will be the next vertex in V visited by the BDS algorithm in G 0. This completes the proof of the lemma. Lemma 4.2 shows u is visited before v in an instance of ODFS if and only if u is visited before v in an instance of BDS. It is easy to verify that the reduction is a log space reduction. A dierent reduction is required for undirected graphs. We reduce the problem to vertex ODFS for undirected graphs (which is P-complete by Theorem 2.2). We only sketch the reduction. The idea is to insert two vertices on each undirected edge. For each vertex w in the original graph G, its new neighbors are used to \reverse" the search out of w. A numbering scheme similar to the one used for the directed case can be used here. Figure 4 illustrates the reduction. The theorem proved below is an immediate corollary of Theorem 4.1. We present an alternative proof suggested by Richard Anderson. Theorem 4.3 The breadth-depth search problem whose search order is based on xed ordered adjacency lists is P-complete under log space reducibility for either directed or undirected graphs. 9

10 (a) (b) Figure 4: An example illustrating the reduction performed in Theorem 4.2 for the undirected case. The original graph with its vertex numbering is shown in part (a). Note, in the vertex ODFS in part (a) the search order is 1, 2, 4, 3, 6, and 5; whereas for the BDS shown in part (b) the search order is 1, 8, 9, 16, 2, 13, 30, 4, 26, 27, 29, 22, 3, 19, 24, 40, 6, 39, 34, and 5. Thus, the vertices 1{6 are visited in the same relative order. Proof sketch: The reduction is from the ODFS problem (see Denition 2.2). We sketch the reduction for the undirected case rst. Given an instance of ODFS G = (V; E) and three vertices s, u, and v; we construct a new instance G 0 = (V [ V 0 ; E 0 ), s, u, and v of BDS such that u is visited before v in the instance of ODFS if and only if u is visited before v in the corresponding instance of the BDS problem. Without loss of generality, we assume G is connected. The idea behind the construction of G 0 from G is to insert a new vertex between every pair of connected nodes in G. Thus, there are jej new vertices inserted in G 0. Since each edge in E is split, there are 2jEj edges in E 0. The ordering of the new nodes on the adjacency lists is specied below. Let b be a node in the original graph having neighbors b 1 ; : : :; b k. Suppose the b i 's are ordered on b's adjacency list as shown. Let c 1 ; : : :; c k be the nodes added by the reduction and splitting the edges (b; b 1 ); : : :; (b; b k ), respectively. The c i 's are ordered on the adjacency lists just the reverse of the b i 's, i.e. from c k down to c 1. Let m 1 (m 2 ) be the minimum (maximum) vertex number of c i 's two neighbors. We number the c i 's by taking jv j m 1 + m 2 for c i. Because of the \reversal" of the search order by BDS after all the neighbors of a node have been visited, the b i 's are visited in the same relative order by the BDS algorithm as they are by the ordered DFS algorithm. It is easy to verify that the reduction can be done in log space. Since the BDS algorithm runs in polynomial time, the proof of the theorem is complete for undirected graphs. (An example of the reduction is presented in Figure 5 and is described further below.) A similar reduction works for directed graphs since ordered DFS is P-complete for directed graphs as well (see Theorem 2.1). (An example of the reduction is presented in Figure 6 and is described further below.) 10

11 The reduction in Theorem 4.3 for the undirected case is illustrated in Figure 5. That is, given the adjacency lists specied below the numbers outside the nodes indicate the search order for ODFS in part (a) and BDS in part (b). The adjacency lists for part (a) are 1: 2, 3; 2: 1, 4; 3: 1, 4, 6; 4: 2, 3, 5, 6; 5: 4; 6: 3, 4; and for part (b) are 1: 9, 8; 2: 16, 8; 3: 24, 22, 9; 4: 30, 29, 22, 16; 5: 29; 6: 30, 24; 8: 1, 2; 9: 1, 3; 16: 2, 4; 22: 3, 4; 24: 3, 6; 29: 4, 5; 30: 4, (a) (b) Figure 5: An example illustrating the reduction performed in Theorem 4.3 for the undirected case. The adjacency list orderings are specied in the text. Note, in the ODFS in part (a) the search order is 1, 2, 4, 3, 6, and 5; whereas for the BDS shown in part (b) the search order is 1, 9, 8, 2, 16, 4, 30, 29, 22, 3, 24, 6, and 5. Thus, the vertices 1{6 are visited in the same relative order. The reduction in Theorem 4.3 for the directed case is illustrated in Figure 6. That is, given the adjacency lists specied below the numbers outside the nodes indicate the search order for ODFS in part (a) and BDS in part (b). We use the same convention for numbering the c i 's as we did in the undirected case, where c i 's two \neighbors" are the two nodes having edges involving c i. The adjacency lists for part (a) are 1: 2, 3; 2: 4; 3: 4, 6; 4: 5, 6; 5: ; 6: ; and for part (b) are 1: 9, 8; 2: 16; 3: 24, 22; 4: 30, 29; 5: ; 6: ; 8: 2; 9: 3; 16: 4; 22: 4; 24: 6; 29: 5; and 30: 6. Note that vertices 1{6 are visited in the same relative order by both ODFS and BDS. 5 Conclusions Our proofs indicate the BDS problem is inherently sequential. Therefore, it is unlikely that the ordered version of the BDS algorithm can be parallelized eciently. It is possible that an alternative approach to BDS could lead to a fast parallel algorithm. Such algorithms have been devised for DFS [1, 2]. It would be interesting to show that a non-greedy version of BDS is in RN C or N C. 11

12 R R 3 4 R (a) R R R R R (b) R 10 6 Figure 6: An example illustrating the reduction performed in Theorem 4.3 for the directed case. The adjacency list orderings are specied in the text. Note, in the ODFS in part (a) the search order is 1, 2, 4, 5, 6, and 3; whereas for the BDS shown in part (b) the search order is 1, 9, 8, 2, 16, 4, 30, 29, 5, 6, 3, 24, and 22. Thus, the vertices 1{6 are visited in the same relative order. 6 Acknowledgements I would like to thank Richard Anderson for suggesting the problem be reduced to the ordered DFS problem, Amarjit Singh for correspondence on this problem, and the referees for several helpful suggestions. References [1] A. Aggarwal and R. J. Anderson. A random NC algorithm for depth rst search. Combinatorica, 8(1):1{12, [2] A. Aggarwal, R. J. Anderson, and M. Kao. Parallel depth-rst search in general directed graphs. SIAM Journal on Computing, 19(2):397{409, [3] P. de la Torre and C. Kruskal. Fast parallel algorithms for all sources lexicographic search and path nding problems. Technical Report CS-TR 2283, University of Maryland, [4] P. de la Torre and C. Kruskal. Fast and ecient parallel algorithms for single source lexicographic depth-rst search, breadth-rst search and topological-rst search. In International Conference on Parallel Processing, volume III, pages 286{287,

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