G.L. Miller and J. Reif, "Parallel Tree Contraction and its Application," Proc. 26th Ann. IEEE Symp. on Foundations of Computer Sci., (1985), pp. 478--489.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are not allowed to produce a node with more than two children. Brent [Bre74] was the first to show that a logarithmic number of such stages is sufficient, and he applied this result to the restructuring of algebraic expression trees, producing trees of logarithmic depth. Subsequent work [MR85, GR86, GMT88, KD88, ADKP89] concentrated on the efficient parallel computation of contraction sequences, and eventually resulted in work optimal logarithmic time tree contraction algorithms on the EREW PRAM. We consider the tree contraction problem on the boolean hypercube and on similar networks. In this model, computing ....

....call of the algorithm. 2. The remaining tree may still be quite large, but, if so, has a structure that allows a significant further reduction by eliminating the leaves and contracting chains of nodes with a single child each to one edge, an operation similar to the compact operation in [MR85]. C 1 C2 C3 C4 compact T T Figure 1: The compact operation applied to T 3. To contract the remaining tree, we emulate the PRAM algorithm of [KD88] We first compute the communication structure of this algorithm when executed on the remaining tree. After rearranging the nodes of the tree ....

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G. Miller and J. Reif. Parallel tree contraction and its application. In Proceedings of the 26st Annual Symposium on Foundations of Computer Science, pages 478--489, 1985.

....constant at each leaf and an operator at each internal vertex. AEE involves computing the value of the expression at the root of the tree. Hence, AEE is a direct application of the well studied tree contraction technique, a systematic way of shrinking a tree into a single vertex. Miller and Reif [16] designed an exclusive read exclusive write (EREW) PRAM algorithm for evaluating any arithmetic expression of size n, which runs in O(logn) time using O(n) processors (with O(nlogn) work) Subsequently Cole and Vishkin [6] and Gibbons and Rytter [8] independently developed O(logn) time ....

G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In Proc. 26th Ann. IEEE Symp. Foundations of Computer Science (FOCS), pages 478--489, Portland, OR, October 1985. IEEE Press.

....two theorems give the main idea. Every binary tree T = V, E) has a tree decomposition Xi[i I ; T = I, F) with treewidth 3, and the depth of T is at most 2[log( VD] and T is a binary tree. Proof. Our result is based upon the method of parallel tree contraction of Miller and R2if [11]. We will obtain a series of (rooted) trees T = T0 = V0, E0) T1 = V1,E1) T2 = V2, E2) Tr = V, E, with [V[ 1. To each v 14 we assign a set (v,i) V representing the set of vertices that are contracted to v . Define t0(v, 0) v . Each Ti l is obtained from Ti by applying the ....

....of nodes vl, vk is a chain if Vj l is the only child of vj, and vk has exactly one child and that child is not a leaf. Now, in each maximal chain, identify vi and V l for j odd and 1 j k. Let ivi be the new node. We take O(iv, i . 1) O( j, i) 0( 1 , i) Miller and P,2if [11] showed that after [log n] simultaneous applications of RAKE and COMPRESS, T is reduced to a single vertex. So it follows that r [ 1og hi. Each t0(v, i) represents the set of vertices that are contracted to v in i contractions. Note that each t0(v, i) induces a connected subtree oft and that ....

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G. Miller and J. Reif. Parallel tree contraction and its application. In Proc. of the 6th Annual IEEE Syrup. on the Foundations of Comp. Science, pages 478-489, 1985.

....in linear time. Our results In this paper, we give the rst ecient parallel algorithms for a host of chordal graph problems. Our deterministic algorithms take O(log n) time and use only n m processors of a CRCW P RAM for n node, m edge graphs. Moreover, using randomized techniques [19] [33], 38] we can achieve the same time bound with only (n m) log n processors. Thus our algorithms are nearly optimal in their use of parallelism, in contrast to the previous parallel algorithms that required about n processors to achieve the same time bound. The chordal graph problems we solve ....

G. L. Miller and J. H. Reif, \Parallel tree contraction and its application," 26th FOCS (1985), pp. 478-489.

....of a parallel version of Earley s algorithm is described. ffl In [Kle85, KR88] recognizers for deterministic context free languages are described. ffl In [BOV85] an optimal parallel algorithm is described that can transform any arithmetical expression into its corresponding syntax tree. ffl In [MR85] a parallel tree contraction algorithm is described. ffl In [GR86] a parallel pebble game is given. ffl In [Ryt85b, Ryt85a, Ryt87] several recognizers and parsers for ambiguous and unambiguous grammars in CNF are given. On these foundations [Vre90b, VH91] were written, on which this thesis is ....

....by using the round Robin principle such that the algorithm runs with Theta(n ) processors in O(log n) time for some k. 6.4 Binary Dag Contraction 6.4. 1 Introduction In this section we present a parallel contraction algorithm for binary dags similar to existing tree contraction algorithms [ADKP87, CV88b, DNP86, GMT88, KD88, MR85]. These algorithms make use of several well known techniques like Brent s scheduling [Bre74] finding an Euler tour, and list ranking [AM88, AM91, CV88a] Contraction algorithms can be used as the backbone of an expression computing algorithm. In [BOV85] an optimal parallel algorithm is described ....

G.L. Miller and J.H. Reif. Parallel tree contraction and its application. In Proc. 26 Ann. IEEE Symposium on the Foundations of Computer Science, pages 478--489, 1985.

.... Altogether there are at most O(dj Rj) requests whp that are spread over p buffers, thus, because of the random node distribution, each buffer gets O(log n dj Rj=p) O(log n) requests whp (Chernoff bounds, j Rj n=r, p = maxf g) The requests are placed by randomized dart throwing [18]. If each processor is responsible for the placement of a group of O(log requests (which may go to different buffers) Step 4 takes O(log n) time whp. The dart throwing progress is regularly monitored. In the unlikely case of stagnation (buffers are chosen too small) the buffer sizes are ....

G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In 26th

....the goal is to construct a universal formula evaluator algorithm. Such an algorithm takes as input a description of a formula, with all of its inputs speci ed, and produces as output the value of the formula. Parallel algorithms for this problem have been proposed by Gupta [6] Miller and Reif [9]; Buss [2] Buss, Cook, Gupta, and Ramachandran [3] and Kosaraju and Delcher [8] These yield NC algorithms for the problem that also produce, for any given formula of size S, a circuit of depth O(log S) When these circuits are expressed as formulas, the sizes are S ) for various 2. ....

....formulas of the same depth. We conclude with an example illustrating that polynomial size blowup can arise from this approach, even if one is restricted to division free formulas. In particular, we shall exhibit a formula of size n such that when the formula evaluation algorithm of Miller and Reif [9] is applied to it, the resulting formula is of size n 1 ) for a xed 0. For each n, de ne the formula Fn (x 1 ; x 2 ; x 2n 1 ) as Fn (x 1 ; x 2 ; x 2n 1 ) x 1 x 2 ) x 3 ) x 4 ) x 2n ) x 2n 1 : Clearly, depth(Fn (x 1 ; x 2n 1 ) 2 n and jF n ....

G. L. Miller and J. Reif, Parallel tree contraction and its application, Proc. 26th Ann. IEEE Symp. on Foundations of Computer Sci., (1985), pp. 478-489.

....out the main features of their interesting technique based upon an efficient evaluation of expression trees. Even if the described algorithm runs on the CRCW PRAM model, we might exploit the techniques known in the literature for the evaluation of expression trees on the weakest EREW PRAM model [27, 29], and thus implement this algorithm also on this model without loosing the efficiency. Jung and Mehlhorn [24] solve the single vertex insertion problem by evaluating an expression associated with the original MST, where an operator is assigned to every vertex of the tree. In such a way, while the ....

....k ) processors. 14 (b) For every new vertex z i , form one tree from its copies of components by making z i the new root. The maximum cost edge on the path from every vertex to the root can be computed work optimally and in logarithmic time by applying the parallel tree contraction technique [29]. 3. The edge (z i ; v) 2 G b is labeled by the maximum cost edge on the i path of G z connecting z i to v. Such an edge has been already computed in Step (2.b) since v is a vertex in the tree built for z i . Since we are considering i paths departing from each new vertex and entering the ....

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G. L. Miller and J. H. Reif. Parallel tree contraction and its applications. In IEEE Symposium on Foundations of Computer Science, pages 478--489, 1985.

.... biconnectivity algorithm of [30] There are no direct results on computing triconnected components I O efficiently, although one may apply the PRAM simulation of [9] to the triconnectivity algorithm of [13] A number of PRAM algorithms for planarity testing and planar embedding have been proposed [18, 19, 26, 28]. In [19] the first such algorithm using a linear number of processors was presented; the algorithm runs in O(log 2 2 N) time. The algorithm of [28] runs in O(log 2 N) time using O(C(N) processors, where C(N) is the number of processors required to compute the connected components of a graph ....

G. Miller and J. Reif. Parallel tree contraction and its applications. In Proceedings of the 26th IEEE Annual Symposium on Foundations of Computer Science, pages 478--489, 1985. 35

....respect to two di erent elimination orders. Each step i depicts the trees present in of algorithm el2dt after having processed variable (i) 18 A B C D E F G H I J K L M N O I K MN L J O A C E G I K M B H N L D J F O ABCD EFGH COMPRESS RAKE Figure 17: Demonstrating the contract operation of [24]. Algorithm bal dt bal dt(T ) for each internal node N in T , label(N) empty dtree for each leaf node N in T , label(N) dtree N op compose R nal node resulting from successive applications of contract to T return label(R) Figure 18: Pseudocode for balancing a dtree. soundness is ....

....Balancing Dtrees We now present an algorithm for balancing a dtree while increasing its width by no more than a constant factor. The algorithm is similar to el2dt except that the composition process is not driven by an elimination order. Instead, it is driven by applying the contract operation of [24] to the given dtree. We need to explain this operation rst. contract is an operation which is applied to a tree. It simply absorbs some of the tree nodes into their neighbors, therefore, producing a smaller tree. To absorb node N 1 into node N 2 is to make the neighbors of N 1 into neighbors of ....

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In Proc. 26th IEEE Symp. on Foundations of Computer Science, pages 478-489, Portland, OR, 1985.

....and O( n p ) local memory per processor, n p p ffl (ffl 0) using O(log p) communication rounds and O( n p ) local computation per round. 4. 3 Tree Contraction and Expression Tree Evaluation We observe that the classical tree contraction and expression tree evaluation algorithm of [27] can be easily implemented on a CGM to run in O(log p)communication rounds. Recall that the tree contraction algorithm of [27] applies an alternating sequence of log n rake and compress operations to contract a tree T into a single node. On a CGM, one can simply apply log p rake and compress ....

....computation per round. 4.3 Tree Contraction and Expression Tree Evaluation We observe that the classical tree contraction and expression tree evaluation algorithm of [27] can be easily implemented on a CGM to run in O(log p)communication rounds. Recall that the tree contraction algorithm of [27] applies an alternating sequence of log n rake and compress operations to contract a tree T into a single node. On a CGM, one can simply apply log p rake and compress operations, which require O(log p) rounds, and compresses the tree into a smaller tree T 0 of size O(n=p) The tree T 0 can ....

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G.L. Miller, J.H. Reif, "Parallel tree contraction and its application," Proc. IEEE Symposium on Foundations of Computer Science, 1985, pp. 478--489.

....this problem is called mixed domination problem [1] In this paper we present new parallel NC algorithm to find smallest mixed dominating set in trees. The algorithm is based on tree compression techniques that have been traditionally used to evaluate linear arithmetic expressions in parallel [2], 4] 5] In this respect the paper generalizes the application of tree compression techniques to solve combinatorial problems in trees. The model of parallel computation used is the CRCW P RAM (Concurrent Read Concurrent Write Parallel RAM) where more than one processor can concurrently read ....

....of the algorithm is not attractive for parallel implementation. In this paper we propose a parallel algorithm to solve the mixed domination problem in trees. Our algorithm uses parallel tree compression techniques that have been traditionally used for evaluating arithmetic expressions in parallel [2], 4] 5] In this sense this paper extends the scope of tree compression techniques to solve combinatorial problems. The model for parallel computation is CRCW P RAM; where more than one processor can read from or write into same memory location during one memory cycle. In case of writing ....

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G. E. Miller, and J.H. Reif, Parallel Tree Contraction and its Applications, Proc. 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp 478-489.

....algorithm may have the same bounds as a deterministic algorithm, but may be much simpler. Also, randomized algorithms may be discovered before any equally e#cient deterministic solution. Examples include algorithms for list ranking and for tree contraction, discovered first in randomized versions [52, 71], and later made deterministic [6, 14, 15, 25] 1.3 Simulations Among PRAM Models An algorithm designed for a weak PRAM model can clearly be used in a stronger model. But we would like to use any PRAM algorithms on any model, so it is important to simulate stronger models by weaker ones. A ....

....functions can be computed by processing each leaf, removing the leaves and repeating until the tree becomes empty. However, this takes time proportional to the depth of the tree, which may be too large. In an improvement to this method, alternately called centroid decomposition or tree contraction [64, 47, 52], one alternates this removal of leaves with the removal of vertices having only one child (sometimes with the further restriction that the parent of the vertex also only has one child) Cole and Vishkin [15] and Gazit et al. 25] give centroid decomposition algorithms that take logarithmic time ....

G.L. Miller and J.H. Reif, Parallel Tree Contraction and its Application. 26th Symp. Found. Comput. Sci., 1985, 478--489.

....B to which MATCH is applied is such that its complement B contains a quadratic number (jV (B)j 2 ) of edges even if the original graph G is sparse. This is our reason for not using any of the known algorithms for constructing a maximal matching (see [1] 10] 14] 15] 16] 18] 19] [20]) The subroutine MATCH runs in O(log n) time on an EREW PRAM with O(n m) processors. For graphs with a dense compliment, MATCH constructs a matching of size jV (B)j) which is maximum up to a constant. On the other hand, it is not necessarily maximal. 4 M. GOLDBERG and T. SPENCER Our algorithm ....

....(1) The processor time product for our algorithm is O( n m) log 4 n) We would like to reduce the total amount of work that our algorithm does by reducing the number of processors it requires while not increasing its running time. We note that the technique developed by Miller and Reif in [20] does not seem to apply to this algorithm. 2) There is a trivial sequential algorithm that colors a given graph G in at most 1 colors, where is the maximal degree of a vertex in G. Using the standard reduction of VC to MIS, we get an NC algorithm for 1 coloring which is run on O(n 2 ....

G. L. Miller and J. H. Reif, Parallel tree contraction and its applications, in Proc. 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, pp. 478-489.

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Miller,G.L., Reif,J.H., `Parallel Tree Contraction and its Applications,' Proc. IEEE Symposium on Foundations Of Computer Science, 1985, pp.478-489.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its applications. In 26th Symposium on Foundations of Computer Science, pages 478--489, IEEE, Portland, Oregon, 1985.

....for reducing a planar DAG to a constant size and then expanding it back. This paradigm is developed from a property of planar directed graphs we refer to as the Poincare index formula. Using this new paradigm we then overlay our application in a fashion similar to parallel tree contraction [MR85, MR89]. We also discuss some of the changes needed to extend the reduction procedure to work for general planar digraphs. Using the strongly connected components algorithm of Kao [Kao93] we can compute multiple source reachability for general planar digraphs in O(log time using O(n) processors. This ....

Gary L. Miller and John H. Reif. Parallel tree contraction and its application. In 26th Symposium on Foundations of Computer Science, pages 478--489, Portland, Oregon, October 1985. IEEE.

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G.L. Miller and J. Reif, "Parallel Tree Contraction and its Application," Proc. 26th Ann. IEEE Symp. on Foundations of Computer Sci., (1985), pp. 478--489.

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G. L. Miller and J. H. Reif, Parallel Tree Contraction and Its Application, Proc. IEEE Symposium on Foundations of Computer Science, 1985, pp. 478--489.

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G.L. Miller, J.H. Reif, "Parallel tree contraction and its application," IEEE Symp. on Foundations of Computer Science, 1985, pp. 478--489.

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Gary L. Miller and John H. Reif. Parallel tree contraction and its application. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pages 487--489, 1985.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pages 487--489, 1985.

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G. L. Miller and J. Reif. Parallel tree contraction and its applications. In Proc. of 26th IEEE Symposium on Foundations of Computer Science, pages 478--489, 1985.

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G. L. Miller and J. H. Reif, Parallel Tree Contraction and Its Application, Proc. IEEE Symposium on Foundations of Computer Science, 1985, pp. 478--489.

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G.L. Miller, J.H. Reif, "Parallel tree contraction and its application," IEEE Symp. on Foundations of Computer Science, 1985, pp. 478--489.

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G. L. Miller and J. H. Reif, Parallel Tree Contraction and Its Application, Proc. IEEE Symposium on Foundations of Computer Science, 1985, pp. 478--489.

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G.L. Miller, J.H. Reif, "Parallel tree contraction and its application," IEEE Symp. on Foundations of Computer Science, 1985, pp. 478--489.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. Proc. 1985 IEEE FOCS, 478-489.

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. G. L. Miller and J. H. Reif. Parallel tree contraction and its application. Proc. 26th Symp. on Foundations of Computer Science, IEEE, 291-298(1985).

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. Proc. 26th Symp. on Foundations of Computer Science, IEEE, 478-489(1985).

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. Proc. 26th Symp. on Foundations of Computer Science, IEEE, 478-489(1985). 32

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. G. L. Miller, J. H. Reif. Parallel tree contraction and its application. Proc. 1985 IEEE Foundations of Computer Science, 478-489.

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Miller, G. L., Reif, J. H. Parallel tree contraction and its application. 26th Symp. on Foundations of Computer Sci., IEEE, 1985, pp. 478-489.

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. G. L. Miller, J. H. Reif. Parallel tree contraction and its application, 26th Symp. on Foundations of Computer Sci., IEEE 478-489(1985).

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G. L. Miller, J. H. Reif, "Parallel tree contraction and its application," in Proc. 26th Symp. on Foundations Comput. Sci., IEEE, 1985, 478-489.

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. G. L. Miller, J. H. Reif. Parallel tree contraction and its application, 26th Symp. on Foundations of Computer Sci., IEEE 478-489(1985).

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Miller, G., Reif, J., Parallel tree contraction and its applications, Proc. 26th Symposium on Foundations of Computer Science (1985), pp. 478--489.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pages 487--489, 1985.

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Gary L. Miller and John Reif. Parallel tree contraction and its application. In Proceedings of the 26 Symposium on Foundations of Computer Science, IEEE, pages 478--489, 1985.

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G. L. Miller and J. H. Reif, Parallel tree contraction and its application, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1985, pp. 478--489.

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Miller, G., Reif, J., Parallel tree contraction and its applications, Proc. 26th Symposium on Foundations of Computer Science (1985), pp. 478--489.

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Gary L. Miller and John H. Reif. Parallel tree contraction and its application. In Proceedings Symposium on Foundations of Computer Science, pages 478--489, October 1985.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, pages 487--489, 1985.

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Miller, G., Reif, J., Parallel tree contraction and its applications, Proc. 26th Symposium on Foundations of Computer Science (1985), pp. 478--489.

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G. L. Miller and J. H. Reif, Parallel tree contraction and its application, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1985, pp. 478--489. 19

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G. L. Miller and J. H. Reif, Parallel Tree Contraction and Its Application, Proc. IEEE Symposium on Foundations of Computer Science, 1985, pp. 478--489.

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G. L. Miller and J. H. Reif. Parallel tree contraction and its application. In 26th Annual Symposium on Foundations of Computer Science, pages 478--489, Portland, OR, October 1985. IEEE Computer Sociery Press.

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G. L. Miller and J. F. Reif. Parallel tree contraction and its applications. In Proceedings of the 26th IEEE FOCS, pages 478--489, 1985.

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G.L. Miller, J.H. Reif, "Parallel tree contraction and its application," IEEE Symp. on Foundations of Computer Science, 1985, pp. 478--489.

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G. L. Miller and J. F. Reif. Parallel tree contraction and its applications. In Proceedings of the 26th IEEE FOCS, pages 478--489, 1985.

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