R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. Proc. 27th Symp. on Foundations of Computer Sci., IEEE, 478-491(1986). 31  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm for dense graphs. Nath and Maheshwari [30] provided an algorithm which require O(log 2 n) time using O(n 2 ) processors on the weakest EREW PRAM model. On the powerful CRCW PRAM model, O(log n) time deterministic algorithms exist (e.g. see [40] In particular, Cole and Vishkin [7] attain a nearly optimal processor bound, i.e. O( n m) log (3) n= log n) processors, on the STRONG CRCW PRAM model. Awerbuch and Shiloach [1] provided a PRIORITY CRCW PRAM algorithm requiring O(n m) processors. Johnson and Metaxas [23] provided the first EREW PRAM algorithm requiring o(log ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In IEEE Symposium on Foundations of Computer Science, pages 478--491, 1986.

....results below. A summary is given in Table 1. Our results are obtained using two main techniques (1) using the properties of an arbitrary k wise independent sampler, and (2) re using random bits. 1.1. 1 Parallel MST Connectivity The best deterministic parallel MST and connectivity algorithms [CV86, CV91, CHL99] run in logarithmic time yet they all use superlinear work. There are somewhat simpler logarithmic time linear expected work randomized MST and connectivity algorithms [Gaz91, CKT96, HZ96, PR99, HZ01] but each uses a linear number of random bits. We present a new randomized MST ....

.... Bound Problem Deterministic Bound Best previous This paper Parallel NC O(m) graph O(m (m;n) O(m) random bits O(m) EREW) connectivity [CV91] HZ01] for EREW o(log 3 n) random bits (work) Gaz91] for CRCW Parallel NC O(m) minimum O(m log (3) n) O(m) random bits O(m) EREW) spanning [CV86] PR99] for EREW o(log 3 n) random bits trees (work) CKT96] for CRCW Local O(n log n) O(n log m n n ) O(n log m n n ) sorting (trivial) O(n log m n n ) random bits o(log 1 n) random bits (comparisons) GKKS93] Set O(n log n) O(n log m n n ) O(n log m n n ) maxima (trivial) O(n ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree, and graph problems. In Proc. FOCS'86, pages 478-491, 1986.

....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 particularly important case is when the simulating ....

....computation. Wyllie s algorithm was improved by Vishkin [71] who gave a randomized ranking algorithm for the EREW. This algorithm was later made deterministic [12, 13] made more e#cient but at a cost in total time [40] and still later made faster with the same asymptotic number of operations [14, 6]. 2.4 Randomized Ranking and Deterministic Coin Tossing If our list were represented as a balanced binary tree we could perform prefix computation, or equivalently ranking, as e#ciently as if it were an array. Therefore we can perform ranking by turning the list into a tree. This can be done as ....

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R. Cole and U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to List, Tree and Graph Problems. 27th Symp. Found. Comput. Sci., 1986, 478--491.

....2.2.1 CRCW PRAM Algorithms There are several simple deterministic O(lg n) time CRCW algorithms for this problem, 1] 21] most of these algorithms are however not optimal. A near optimal deterministic O(lg n) time algorithm using (n m)ff(n; m) lg n processors was presented by Cole and Vishkin [8], this algorithm is considered complex. Recently, simpler, near optimal, O(lg n) time deterministic algorithms using (n m)ff(n; m) lg n processor were presented by Hagerup [12] and by Iwama and Kambayashi [16] An optimal O(lg n) time randomised algorithm has been presented by Gazit [10] 2.2.2 ....

R. Cole and U. Vishkin.Approximate and Exact Parallel Scheduling with Applications to List, Tree and Graph Problems. Proc. 27th IEEE FOCS, p478--491, 1986.

....uses O(m n) processors. The bottleneck in all three algorithms is nding the blocks of a graph. Tarjan and Vishkin proposed an algorithm that e ectively reduces the block nding problem to the connected components nding problem [20] If the connected components algorithm of Cole and Vishkin [5] is used, the block nding algorithm runs in O(log n) time on a CRCW PRAM with O( n m) m;n) log n) processors. We call the resulting algorithm the CTV algorithm. Using a more ecient algorithm or an algorithm for a di erent model of parallel computation (for example, see [12] will lead to other ....

....PRAM with O( n m) m;n) log n) processors. Proof. The running time and processor count of the rst step, and of the whole procedure, is dominated by the resources required to nd the spanning tree T of G. This can be done in O(log n) time on a CRCW PRAM with O( m n) m;n) log n) processors [5]. The number of descendants of each vertex in the spanning tree T can be found in O(log n) time on an EREW PRAM with O(n= log n) processors [20] Given this information, v (which must be unique) can be found easily. Finding the connected components of G[D] requires another application of the ....

R. Cole, U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree, and graph problems, in Proc. 27th Annual Symposium on Foundations of Computer Science, (1986), pp. 478-491.

....an elegant O(log n) time sorting algorithm which has virtually settled the problem of sorting on PRAM models; however Flashsort continues to remain the most practical algorithm for networks. The optimal sub logarithmic algorithm for pre x sum stated in lemma 3. 5 was discovered by Cole and Vishkin [CV86]. The rst optimal sub logarithmic time algorithms for General sorting and integer sorting were provided by Rajasekaran and Reif [RR89] The presentation of our general sorting algorithm uses ideas drawn from a lot of the earlier work and was given in Rajasekaran and Reif [RR87] RR87] also ....

R. Cole and U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems, Proc. 27th IEEE Symposium on Foundations of Computer Science, 1986, pp. 478-491.

....O(log n) parallel time with O( m n)ff(m; n) log n) 19 processors on a CRCW PRAM, where ff is the inverse Ackermann function. We first note the following results on optimal and almost optimal parallel algorithms. A List ranking on n elements can be performed optimally in O(log n) on an EREW PRAM [3]. B Connected components and spanning tree of an n node, m edge graph can be found in time O(log n) with O( m n)ff(m; n) log n) on an ARBITRARY CRCW PRAM [3] provided the input is presented as an adjacency list. C Least common ancestors of k pairs of vertices in an n node tree can be found in ....

....on optimal and almost optimal parallel algorithms. A List ranking on n elements can be performed optimally in O(log n) on an EREW PRAM [3] B Connected components and spanning tree of an n node, m edge graph can be found in time O(log n) with O( m n)ff(m; n) log n) on an ARBITRARY CRCW PRAM [3] provided the input is presented as an adjacency list. C Least common ancestors of k pairs of vertices in an n node tree can be found in O(1) time with k processors after O(log n) time preprocessing using O(n= log n) processors on an EREW PRAM using the algorithm in [24] D The Euler tour ....

R. Cole and U. Vishkin, "Approximate and exact parallel scheduling with applications to list, tree, and graph problems," Proc. 27th Symp. Found. Comp. Sci., 1986, pp. 478-491.

....2.1 and 2.3 of algorithm 3 on p(m; n) processors. 2 We note that on an ARBITRARY PRAM the complexity of steps 2.1 and 2. 3 is dominated by that 10 of finding connected components, i.e. they can be performed with almost optimal speedup in time O(log n) using O( m n)ff(m; n) logn) processors ([4]) where ff denotes the inverse Ackermann function. The minimal augmentation step can be done in polylogarithmic time on a linear number of processors as shown in [13] for biconnectivity and 2 edge connectivity. 4 Linear Time Algorithms In this section we adapt algorithm 3 to compute minimal ....

R. Cole AND U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree, and graph problems, Proc. 27th Ann. IEEE Symp. on Foundations of Comp. Sci., 1986, pp.478-491.

....space. Proof. The correctness of the algorithm follows directly from Theorem 1. All the steps can be performed sequentially in O(n) time with straightforward methods. Regarding the parallel complexity, steps 1 and 3 take O(log n) time on a CREW PRAM with n log n processors, using list ranking [5]. Step 2 can be executed by computing a spanning forest of the face sink graph, which takes O(log n) time on a CRCW PRAM with n #(n) log n processors [5] and thus determines the parallel time complexity. 4. Upward planarity and SPQR trees. Let G be a biconnected single source digraph. In this ....

....methods. Regarding the parallel complexity, steps 1 and 3 take O(log n) time on a CREW PRAM with n log n processors, using list ranking [5] Step 2 can be executed by computing a spanning forest of the face sink graph, which takes O(log n) time on a CRCW PRAM with n #(n) log n processors [5], and thus determines the parallel time complexity. 4. Upward planarity and SPQR trees. Let G be a biconnected single source digraph. In this section we give a combinatorial characterization of the upward planarity of G using SPQR trees. 4.1. Basic definitions and main result. A digraph is ....

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree, and graph problems, in Proc. 27th IEEE Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1986, pp. 478--491.

....63 structure of our sequential algorithm, the parallelization of some of the steps required new insights into the problem. Our parallel algorithm can be made to run within the same time bound using O( n m) log log n log n ) processors by using the algorithm for finding connected components in [CV86] and the algorithm for integer sorting in [Hag87] Chapter 4 Smallest Triconnectivity Augmentation: Biconnected Graphs 4.1 Introduction In this chapter, we present a linear time sequential algorithm for finding a smallest augmentation to triconnect a biconnected graph. Our sequential ....

.... also be constructed in O(log n) time using a linear number of processors on a CRCW PRAM [FRT93, Ram93] The algorithm for constructing the 3block tree can be made to run within the same time bound using a sublinear number of processors by using the algorithm for finding connected components in [CV86] and the algorithm for integer sorting in [Hag87] Implied Path in the 3 Block Tree Given two vertices u and v in a biconnected graph G, the implied path between u and v in 3 blk(G) is the path between the two fi vertices that corresponds to the two Tutte components that contain u and v, ....

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R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. 27th Annual IEEE Symp. on Foundations of Comp. Sci., pages 478--491, 1986.

....time on a CRCW PRAM [16] This is the first parallel algorithm for minimum spanning trees that does linear work. In contrast, Karger [13] gives an algorithm running on an EREW PRAM that requires O(log n) time and m= log n n 1 ffl processors for any constant ffl 0. Also, Cole and Vishkin [6] give an algorithm running on a CRCW PRAM that requires O(log n) time on O( n m) log log n= log n) processors. Among remaining open problems, we note especially the following three: 1. Is there a deterministic linear time minimum spanning tree algorithm in the restricted randomaccess model ....

R. Cole and U. Vishkin, "Approximate and exact parallel scheduling with applications to list, tree, and graph problems," Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, Computer Society Press, Los Alamitos, CA, 1986, pp. 478-491.

....We need an extra O(loglogn) factor space for the triangulation procedure. If the input PSLG is triangulated, the number of operations performed is linear in n (follows from the sequential algorithm) and hence by Brent s slow down procedure and using the loadbalancing scheme of Cole and Viskin [9] or the randomized scheme of Miller and Reif [17] the processor bounds can be reduced to O(n logn) without a ecting the asymptotic run time. 2 REMARK : Dadoun and Kirkpatrick [10] show that their algorithm runs in O(logn) expected parallel time. However they do not extend their analysis for the ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. Proc. 27th IEEE Symp. on Foundations of computer Science, pages 511 - 516, 1986.

....directed and undirected series parallel graph recognition and decomposition, which takes time O(log n) with C(m,n) processors; here C(m,n) is the number of processors required to compute connected components of a graph in logarithmic time. The best bound known for this is C(m,n) O(m#(m,n) log n) [5]. We use the stronger concurrent read concurrent write (CRCW) model of parallelism; however any CRCW algorithm can be executed on an EREW machine with a logarithmic loss of time and e#ciency. In our case this results in time bounds of O(log 2 n) matching the previous result; however the number ....

....to ears nested in the ear in which the vertex is interior, and those nested in other ears. Thus, for each graph H i ,wecan compute the adjacency lists of all vertices except the copies of the endpoints of ear E i . But these two remaining adjacency lists can be constructed in a prefix computation [1, 5, 10] simply by scanning all ears properly nested in E i . With these definitions, our problem becomes that of testing, for each H i , whether the non path edges of H i nest, and if so constructing a nesting tree. We must take logarithmic time and a number of operations proportional to the size of H i ....

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R. Cole and U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to Optimal Parallel List Ranking, Info. and Control 70 (1986) 32--53.

....all the partial sums of the form S k = L k i=1 a i for 1 k n. Since these partial sums can be computed in O(n) time sequentially using a straight forward approach, an optimal PT product is O(n) Although an O(log n) time algorithm is not difficult to derive using a binary tree, Cole and Vishkin [24] obtained the following result: Lemma 2.1 In a CRCW (arbitrary) PRAM model, the prefix sum of n elements can be computed optimally in O(log n= log log n) time. 2.1.1 List Ranking A related problem to prefix computation is that of computing the prefix sums when the input is given as a ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. Proc. 27th IEEE Symp. on Foundations of computer Science, pages 511 -- 516, 1986.

....show in Theorem 3 that this algorithm constructs a layout for biconnecteds graph with at most 2n 4 bends, which by Theorem 1 is optimal in the worst case. If the graph G is not biconnected, we first decompose G into its connected and biconnected components. This can be done optimally in parallel [3, 7]. The layout of each component is constructed separately, taking special care of the articulation vertices so that the layouts can later be merged together. The first step of Algorithm GraphLayout can be done in O(logn) time by an EREW PRAM with n= log n processors [13] Steps 2 and 3 are local ....

....rot(P 00 ) 4. 2 Lemma 5 Let P be a symbolic polygon with m vertices. Algorithm SymbolicDecomposition decomposes P into symbolic rectangles in O(log m) time on a CREW PRAM with m= log m processors. Sketch of Proof: Steps 1 and 3 can be done using optimal list ranking and prefix computations [1,3]. Step 2 takes constant time. In Step 4, we use the subroutine described in [2] for finding the next larger of each element of the list rot(i) i = 0; Delta Delta Delta ; m. In a list of numbers, the next larger of a given element x is the first element following x in the list that is larger ....

R. Cole and U. Vishkin, "Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems," Proc. 27th IEEE Symp. on Foundations of Computer Science (1986), 478--491.

....only makes the problem easier and hence adds no additional resources. We note that in practice, we would change Step 7 to include a cycle DeltaC only if it contained some edge that was not already in C. This could be checked in O(log n) time on (m n) log n processors using pointer jumping [4]. Also, we could replace Steps 3, 4, and 5 with the computation of a minimum spanning tree of G with the edges in E(C) weighted with 0 and the rest of the edges weighted with 1. 3.3 Finding a Cover of Size O(m n log n) In this section, we show how to decrease the size of the cycle cover from ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree, and graph problems. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pages 478--491, 1986.

....n ) time on a CRCW PRAM [14] This is the first parallel algorithm for minimum spanning tree that does linear work. In contrast, Karger [11] gives an algorithm running on an EREW PRAM that requires O(log n) time and m= log n n 1 ffl processors for any constant ffl 0. Also, Cole and Viskin [5] give an algorithm running on a CRCW PRAM that requires O(log n) time and O( n m) log log n= log n) processors. Among remaining open problems, we note especially the following three: 1. Is there a deterministic linear time minimum spanning tree algorithm in the restricted random access model 2. ....

R. Cole and U. Vishkin, "Approximate and exact parallel scheduling with applications to list, tree, and graph problems," Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, 1986, pp. 478-491

....(l(v) v) and (r(v) v) the value Gamma1. From the construction of the Euler tour follows that for every vertex v all its ancestors are not marked if and only if the sum of all edges which precede the edge (v; f(v) is equal 3 to 0. Using an optimal O(log n) time algorithm for list ranking (Cole and Vishkin, 1986), we can compute this sum in O(log n) time using n=log n processors on the CREW PRAM. Hence we can find Ceil(h i ) for all candidates h i in O(log n) time with O(n 2 ) total work on the CREW PRAM. 4.4.2 Finding all Candidates Existing in an Optimal Triangulation of the Polygon Below h i For ....

Cole, R. and Vishkin, U. (1986), "Approximate and exact parallel scheduling with applications to list, tree and graph problems," In Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, pages 478--491.

....fact, half as many as there are leaves can be performed in just two steps, without mutual interference. Their scheme is as follows: i) Assume all leaves are numbered from left to right, starting with zero. This numbering is easily computed in O(log n) time on O(n= log n) processors (Cole and Vishkin, 1986). ii) Mark all even numbered leaves. iii) For every junction u such that u.l is a marked leaf, perform contractl(u) iv) For every junction u not involved in the previous step such that u.r is a marked leaf, perform contractr(u) v) Renumber the leaves by halving their numbers. Actually, ....

R. Cole and U. Vishkin (1986). Approximate and exact parallel scheduling with applications to list, tree and graph problems. In 27th IEEE Symposium on Foundations of Computer Science, pages 478--491.

....evaluation, two cases can be distinguished: expressions and circuits. An expression is a formula where every variable and every intermediate result can serve only once as an operand. It can be represented as a tree, and optimal algorithms exist with a EREW S ( n log n ; log n) complexity 1 [1,4,10,15,17], where n is the number of nodes in this tree. Note that this problem is NC 1 complete [14] The evaluation of an arithmetic circuit with operations in a commutative semiring (SR; Theta; 0; 1) can be done by Miller, Ramachandran and Kaltofen s algorithm 2 [18] It has a complexity of ....

....time) will be employed. For this last case, it does not matter which kind of CRCW PRAM is used. 2 In the following, Miller, Ramachandran and Kaltofen will be abbreviated in MRK. the shunt is the equivalent of the rake of Kosaraju and Delcher [15] or of the prune of Cole and Vishkin [4]) A lattice (L; Phi; Omega ) is a set L in which two internal operations, Phi and Omega , are commutative and associative and satisfy the absorption law: 8a; b 2 L; a Phi b) Omega a = a Omega b) Phi a = a. In this paper we shall restrict the work to distributive lattices. Miller and ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In 27 th IEEE Symposium on Foundations of Computer Science, pp 478--491, 1986.

....respectively. The following lemma is immediate from the above analysis: Lemma 3 For an n vertex m edge G, the most vital edge of G w.r.t. minimum spanning tree can be computed in O(t MST (m; n) log n) time using O(pMST (m; n) n) MINIMUM CRCW processors. We use Cole and Vishkin s algorithm [2] for constructing minimum spanning tree on CRCW in Step 0. This gives t MST (m; n) O(log n) and p MST (m; n) O(m log log log n= log n) So we have the following theorem: Theorem 1 Given G with n vertices and m edges, we can compute the most vital edge of G w.r.t. minimum spanning tree in ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In 27th IEEE Symposium on Foundations of Cmputer Science, pages 478--491, 1986.

....Using this prefix sum, each processor sequentially computes log n prefixes of the original input sequence. 2 The above idea of processor improvement was originally used by Brent in his algorithm for expression evaluation, and hence we attribute lemma 2.1 to him. Recently Cole and Vishkin [9] have proved the following Lemma 2.2 Prefix sum computation of n integers (O(log n) bits each) can be performed in O(log n= log log n) time using n log log n= log n CRCW PRAM processors. 2.2 An Assignment Problem Given a set Q = f1; 2; ng of n indices. Each index belongs to exactly one ....

R. COLE AND U. VISHKIN, Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems, Proc. 27th IEEE Symposium on Foundations of Computer Science, 1986, pp. 478-491.

....et al. 1] which sorts n integers in the range 0 : n O(1) in time O(log n= log log n) with n(log log n) 2 = log n processors on an ARBITRARY CRCW PRAM. We need to compute connected components at several places in our algorithm (see below) The most efficient connectivity algorithm ([2]) computes the connected components of a graph with n nodes and m edges represented by adjacency lists in O(log n) time with (m n)ff(m; n) log n processors of an ARBITRARY PRAM; if the graph is represented by an unordered list of its edges, we can construct its adjacency list by sorting the ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree, and graph problems. In Proc. 27th Ann. IEEE Symp. on Foundations of Computer Science, pages 478--491, 1986.

....component problem actually admits a faster algorithm in the sequential context, yet techniques for solving the two problems in parallel are very similar. If concurrent write is allowed, it is relatively simple to solve both problems in O(log n) time using n m processors on the CRCW PRAM [1, 6]. Using randomization, Cole et al. 5] were able to improve the processor bound to (n m) log n, while maintaining O(log n) expected time. For the exclusive write models (i.e. CREW and EREW PRAMs) O(log 2 n) time algorithms for the connected component and MST problems were developed y ....

R. Cole and U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems, FOCS'86, pp. 478-491.

....The following lemma is immediate from the above analysis: Lemma 3 For an n vertex m edge G, the most vital edge of G w.r.t. minimum spanning tree can be computed in O(t MST (m; n) log n) time using O(maxfn; p MST (m; n)g) MINIMUMCRCW processors. We use Cole and Vishkin s algorithm [2] for constructing minimum spanning tree on CRCW in Step 0. This gives t MST (m; n) O(log n) and p MST (m; n) O(m log log log n= log n) The total space used in the algorithm is clearly O(m n) O(m) So we have the following theorem: Theorem 1 Given G with n vertices and m edges, we can ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In 27th IEEE Symposium on Foundations of Cmputer Science, pages 478--491, 1986.

.... In parallel models, the previous results for the MST problem were O(log 2 n) using n 2 = log 2 n CREW PRAM [HCS79, CLC82] or n 2 EREW PRAM processors [NM82] and O(log n) time using n m PRIORITY CRCW PRAM processors [AS87, SV82] or (n m) log log log n= log n STRONG CRCW PRAM processors [CV86] using very elaborate techniques. Other parallel algorithms are reported in [KRS90, KR84, Ben80, SJ81] Recently, CL93] have improved the running time of [JM91] to O(log n log log n) mainly by providing a recursive version of the growth control schedule. It does not appear, however, that this ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. 27th Annual IEEE Symposium on Foundations of Computer Science, pages 478--491, 1986.

....all vertices of graph R 0 according to their indegree using Cole s O(log n) time n processors sorting algorithm [9] By Corollary 3.8, step 2 can be done in O(log n) time with n 2 processors. Steps 3,4,5 can be implemented in O(log n) time, n= log n processors using list ranking technique [2, 10]. In step 5, we use list ranking to propagate value x i to all vertices of graph Q i . 2 9 two path union Let Q be a two path subgraph of G C 0 . Let V 1 be the subset of V (Q) that induces in Q an acyclic tournament with hamiltonian path H = v 1 ; v k . The remaining vertices of Q, ....

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree and graph problems, Proc. 27th Annual IEEE Symposium on Foundations of Computer Science (1986) 478--491.

....its block. The amount of work performed is O(n) and this matches the sequential complexity of the problem. In general, if p n processors are available, this approach can be used to obtain a prefix sum algorithm that runs in O(log p n=p) time by using p blocks of size n=p. Cole and Vishkin [11] describe a slightly faster prefix sum algorithm for O(log n) bit integers which runs in O(log n= log log n) time and performs O(n) work on an Arbitrary CRCW PRAM. The prefix sum algorithm is often used to allocate processors to tasks. Suppose there are n tasks, and that task i requires p i ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. 27th IEEE Symposium on Foundations of Computer Science, pages 478--491, 1986.

....the overall structure of our sequential algorithm, the parallelization of some of the steps required new insights into the problem. Our parallel algorithm can be made to run within the same time bound using a sublinear number of processors by using the algorithm for finding connected components in [3] and the algorithm for integer sorting in [9] ....

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree and graph problems, in Proc. 27th Annual IEEE Symp. on Foundations of Comp. Sci., 1986, pp. 478--491.

....space. Proof: The correctness of the algorithm follows directly from Theorem 1. All the steps can be performed sequentially in O(n) time with straightforward methods. Regarding the parallel complexity, Steps 1 and 3 take O(log n) time on a CREW PRAM with n= log n processors, using list ranking [5]. Step 2 can be executed by computing a spanning forest of the face sink graph, which takes O(log n) time on a CRCWPRAM with n Delta ff(n) log n processors [5] and thus determines the parallel time complexity. 4 Upward Planarity and SPQR Trees Let G be a biconnected single source digraph. In ....

....methods. Regarding the parallel complexity, Steps 1 and 3 take O(log n) time on a CREW PRAM with n= log n processors, using list ranking [5] Step 2 can be executed by computing a spanning forest of the face sink graph, which takes O(log n) time on a CRCWPRAM with n Delta ff(n) log n processors [5], and thus determines the parallel time complexity. 4 Upward Planarity and SPQR Trees Let G be a biconnected single source digraph. In this section we give a combinatorial characterization of the upward planarity of G using SPQR trees. 4.1 Basic Definitions and Main Result A digraph is expanded ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree, and graph problems. In Proc. 27th IEEE Symp. on Foundations of Computer Science, pages 478--491, 1986.

....As we said, the first stage consists of finding the minima of O(n) sets of vertices, each with cardinality O(k) and then to reduce the O(k) resulting pseudotrees of height O(n) to stars. For the first part Brent s technique applies. For the second part we use the optimal list ranking technique of [4]. 2 Acknowledgement. The authors would like to thank Professors Sam Bent and Jim Driscoll for their helpful comments on the presentation of the paper. ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. 27 FOCS, pages 478--491, 1986.

....distance from the root of T and check whether it is an ancestor of all remaining vertices in L v . The distances of all vertices of T to the root can be computed in O(log n) time using O(n= log n) processors using Eulerian cycle techniques [16] combined with an optimal list ranking algorithm [6]. The vertex r of L v with the minimum distance to the root of T can be determined in O(log k) time using k= log k processors. For all vertices in L v simultaneously it can be checked whether r is an ancestor of all remaining vertices in L v can be checked in O(logk) time using O(k= log k) ....

R. Cole, U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems, 27 th Annual Symposium on Foundations of Computer Science (1986), pp. 478-491.

.... Gamma 1) pairs of w(e) and w(e 0 ) g Step 4. Find the max [w(e 0 i ) Gamma w(e i ) for 1 i n Gamma 1. Step 5. Return the tree edge e that corresponds to the above maximum. The parallel run time of Step 1 is O(log n) on (m n) log log log n= log n processors due to Cole and Vishkin [7]. The same holds for Step 2. Step 3 has run time complexity of O(log n) on m processors on a minimum CRCW PRAM (see Sections 3.2 3.4 of [16] Step 4 can be done in O(log n) on n= log n processors [31] Step 5 is trivial. Hence, assuming that m n the algorithm has O(log n) run time on O(m) ....

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree, and graph problems, Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, Computer Society Press, Los Alamitos, CA, 1986, pp. 478-491.

....S = s 0 ; s 1 ; s n Gamma1 ] where s i = a 0 Phi a 1 Phi : Phi a i . When Phi is addition, and the input array consists of n numbers, each of O(log n) bits, the prefix operation can be performed in Theta(log n= log log n) time with n log log n= log n processors on a CRCW PRAM [16]. Compression, in which m marked records out of a total of n records must be compressed to the front of the output array, can easily be reduced to prefix addition, and thus can be performed in the same time bounds. Using only the processors assigned to the marked records, those marked records can ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree, and graph problems. In 27th IEEE Symp. on Foundations of Computer Science, pages 478--491, 1986.

.... The best known parallel algorithm for the CREW PRAM model runs in O(log 2 n) time using n 2 = log 2 n processors [6, 15] For the CRCW PRAM model, in which concurrent writing is permitted, the best known algorithm runs in O(log n) time using slightly more than (n m) log n processors [26, 9, 5]. Simulating this algorithm on the weaker CREW model increases its running time to O(log 2 n) 10, 19, 29] We present here a simple algorithm that runs in O(log 3=2 n) time using n m CREW processors. Finding an o(log 2 n) parallel connectivity algorithm for this model was an open problem ....

....of [15] In the CREW model of parallel computation, concurrent writing to any memory location by more than one processor is not allowed. For the CRCW PRAM model, in which concurrent writing is permitted, the best known algorithm runs in O(log n) time using (n m)ff(n; m) log n processors [26, 9, 5]. There is also a randomized algorithm [11] with the same time complexity. Simulating an algorithm designed for this model on the weaker CREW model increases its running time to O(log 2 n) 10, 19, 29] We present an efficient and simple algorithm that runs in O(log 3=2 n) time using n m ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. 27th Annual IEEE Symposium on Foundations of Computer Science, pages 478--491, 1986.

....HT to search the occurrences of the leaving edge in the cycle history. The first occurrence of the leaving edge in HT can then be selected easily. Selecting the edge (v, w) from the active list L can also be done in O(log n) time on (m n) log n processors using a procedure due to Cole and Vishkin [3]. In the table, the two p(k) entries for step 2 are due to the if then else branches. Note that (m n 1) and (m n) log n are both less than (m n) Step 3, finding the maximum, requires n log n processors and O(log.n) run time (see Valiant [11] The last step is trivial and requires constant ....

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree and graph problems, Proc. 27th Annual Symposium on Foundations of Computer Science, 1986, pp. 478-491.

....processors. However, their result assumes a model in which write con icts are de termined by priority, where the priority of a processor is determined by the weight of the edge assigned to it. Our algorithm assumes a weaker model in which arbitrary processors succeed in writing. Cole and Vishkin [5] have claimed an algorithm running on a CRCW PRAM that requires O(log n) time and O( n m) log log log n= log n) processors. Thus their algorithm is within a triply logarithmic factor of optimal. Their algorithm assumes the same strong model as the algorithm of Awerbuch and Shiloach. Karger [11] ....

R. Cole and U. Vishkin, \Approximate and exact parallel scheduling with applications to list, tree, and graph problems, " Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, 1986, pp. 478-491.

....idea and any appropriate parallel list ranking algorithm. Such a list ranking algorithm is used both indirectly, through the Euler tour technique, and directly. In other words, we solve this class of problems by a logarithmic time optimal parallel reduction into the list ranking problem. Since [CV 86b] provides a logarithmic time optimal parallel list ranking algorithm, the new accelerated centroid decomposition technique also achieves this desired efficiency Recall that [MR 85] gave a deterministic logarithmic time parallel algorithm using a linear number of processors and a randomized ....

....index of this successor. Consider a path that starts from any node and follows the successor relation. We assume that such a path never provides a circuit. Each node v has a weight w(v) The problem: For each node compute the total weight of the nodes following it in its linked list. Recently, [CV 86b] gave a logarithmic time parallel algorithm for this problem which uses an optimal number of processors (n log n) The present paper relies on this algorithm. Note that for all practical purposes we could have used the O(log n log n) time parallel algorithm of [CV 86a] which also uses an ....

[Article contains additional citation context not shown here]

R. Cole and U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree and graph problems, Proc. 27th Symp. on Foundations of Computer Science, 1986, to appear.

....using m n processors, where n and m are, respectively, the number of vertices and edges of the graph. However, their result assumes a model in which write conflicts are resolved by priority, where the priority of a processor is determined by the weight of the edge assigned to it. Cole and Vishkin [6] have claimed an algorithm running on a CRCW PRAM that requires O(log n) time and O( n m) log log log n= log n) processors. Their algorithm assumes the same strong model as the algorithm of Awerbuch and Shiloach. Karger [17] has claimed an algorithm running on an EREW PRAM that requires O(log ....

R. Cole and U. Vishkin, "Approximate and exact parallel scheduling with applications to list, tree, and graph problems, " Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, 1986, pp. 478-491.

....processors. However, their result assumes a model in which write conflicts are determined by priority, where the priority of a processor is determined by the weight of the edge assigned to it. Our algorithm assumes a weaker model in which arbitrary processors succeed in writing. Cole and Vishkin [5] have claimed an algorithm running on a CRCW PRAM that requires O(log n) time and O( n m) log log log n= log n) pro cessors. Thus their algorithm is within a triply logarithmic factor of optimal. Their algorithm assumes the same strong model as the algorithm of Awerbuch and Shiloach. Karger ....

R. Cole and U. Vishkin, "Approximate and exact parallel scheduling with applications to list, tree, and graph problems," Proc. 27th Annual IEEE Symp. on Foundations of Computer Science, 1986, pp. 478-491.

....algorithms for list ranking and (undirected) graph connectivity proved to be central to obtaining optimal algorithms for a considerable number of list, tree and graph problems. First randomized, and later deterministic, optimal parallel algorithms for list ranking were given [Vis84b] CV86b] CV86a] AM88] and [CV89] The deterministic algorithms are based on a deterministic arbitration technique, dubbed deterministic coin tossing [CV86b] Extensions of this technique for sparse graphs and other applications were given [GPS87] CZ90] and [HCD87] Key techniques for parallel algorithms on ....

....as implicitly used in [Win75] for O(log 2 n) time computations. Accelerating centroid decomposition was the motivation for the tree contraction version of [CV88] Two logarithmic time connectivity algorithms were given: 1) a deterministic one which is optimal on all except very sparse graphs [CV86a] 2) a randomized optimal one [Gaz86] For Figure 1, the deterministic algorithms builds on a restricted union find problem, a scheduling problem, dubbed duration unknown task scheduling, and the Euler tour technique, as well as ideas from two previous connectivity algorithms [HCS79] and ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. of the 27th IEEE Annual Symp. on Foundation of Computer Science, pages 478--491, 1986.

....(a) Compute the domain right neighbor of each index i. b) For each element x[i] compute its rank r in the linked list defined by the drn, and let (r) be i. Step (b) can be (trivially) done in O(n) time or in parallel time O(log n) and optimal speedup using a List Ranking algorithm ( AM88] CV86] CV88] CV89] Therefore, our main concern is solving the domain right neighbor problem. For simplicity we assume that m = 2 2 t for some integer t 1. The domain right neighbor, as defined above, is the nearest neighbor from the right. We will also need a definition of the domain left ....

....Lemma 7, the expected number of iterations in (a) is 2. We first show how to check whether S k 5n. Given b(k; j) for all j, the evaluation of the prefix sums M i = 2 P i j=1 b(k; j) 2 (for i = 1; n) and of S k = M n can be done by using the Prefix Sums algorithm of Cole and Vishkin [CV86] CV89] in O(log n= log log n) time and O(n) operations. The evaluation of b(k; j) for each j is done as follows: 1) Sort the n numbers in ff k (x) x 2 Wg into an array C[1: n] 2) Find for each j the rightmost (resp. leftmost) index i 1 (resp. i 2 ) for which C[i 1 ] j (resp. C[i 2 ] ....

R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. In Proc. of the 27th IEEE Annual Symp. on Foundation of Computer Science, pages 478--491, 1986.

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R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems. Proc. 27th Symp. on Foundations of Computer Sci., IEEE, 478-491(1986). 31

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. R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems, Proc. 27th Symp. on Foundations of Comput. 13 Sci., IEEE, 478-491(1986).

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R. Cole and U. Vishkin, Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems, FOCS'86, pp. 478-491.

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. R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems, 27th Symp. on Foundations of Comput. Sci., IEEE, 478-491(1986).

No context found.

. R. Cole and U. Vishkin. Approximate and exact parallel scheduling with applications to list, tree and graph problems, 27th Symp. on Foundations of Comput. Sci., IEEE, 478-491(1986).

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Cole,R., and Vishkin,U., `Approximate and Exact Parallel Scheduling with Applications to List, Tree, and Graph Problems,' Proc. of the IEEE Foundations Of Computer Science, 1986, pp.478-491.

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R. Cole, U. Vishkin, Approximate and exact parallel scheduling with applications to list, tree and graph problems, FOCS 1986, pp. 478--491.

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