Reingold, E. M., Nievergelt, J. & Deo, N. (1977), Combinatorial Algorithms: Theory and Practice, Prentice-Hall.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....: The parallel fan out algorithm given in [2, 15] Substitution :The elimination tree based forward and back substitution algorithms given in [14] Mapping of columns onto processors is an important issue. For the bidirectional scheme, we have used the block wrap around mapping using gray code [25] whereas for the regular algorithm we have used the subtree to processor mapping [8] based on elimination tree. For the purpose of simulation we used three test matrices, described in table 1, from the Harwell Boeing Collection. Due to memory constraints, the maximum dimension of the test matrix ....

E.M.Reingold, J.Nievergelt, and N.Deo, Combinatorial Algorithms : Theory and Practice, Prentice Hall, Englewood Clis, NJ, 1977. 21

....in 1992, under the guidance of Barry Crabtree. Chris Voudouris is employed by the EPSRC funded project, ref GR H75275. I. Introduction Due to their combinatorial explosion nature, many real life constraint optimization problems are hard to solve using complete methods such as branch bound [17, 14, 21, 23]. One way to contain the combinatorial explosion problem is to sacrifice completeness. Some of the best known methods which use this strategy are local search methods, the basic form of which often referred to as hill climbing. The problem is seen as an optimization problem according to an ....

E. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial algorithms: theory and practice, Englewood Cliffs, N.J., Prentice hall, 1977.

....: The parallel fan in algorithm given in [1] Substitution :The elimination tree based forward and back substitution algorithms given in [14] Mapping of columns onto processors is an important issue. For the bidirectional scheme, we have used the block wrap around mapping using gray code [23] whereas for the regular algorithm we have used the subtree to processor mapping [9] based on elimination tree. The parameters that were varied were the grid size k(16 and 32) the number of processors p(1 to 1024) the number of b vectors for which solution vector x was obtained, and the C=E ....

E.M.Reingold, J.Nievergelt, and N.Deo, Combinatorial Algorithms : Theory and Practice, Prentice Hall, Englewood Clis, NJ, 1977.

....algorithm [3] does not have the interchange property but runs in constant amortized time. We present an algorithm that has both properties. The problem of developing a constant amortized time algorithm for generating bag permutations is given as an exercise by Reingold, Nievergelt, and Deo [9]. However, the solutions manual [4] presents Hu and Tien s algorithm [5] and references two other algorithms, all of which do not run in constant amortized time. We consider bags over the elements 0; 1; t. We represent a bag by a sequence n = hn 0 ; n t i, where each n i is the ....

.... Ko [6] 10 4 Concluding Remarks When all n i equal 1, Versions A and B produce the same list of #n permutations, and this list appears to be different than any of those produced by the permutation generation algorithms surveyed in Sedgewick [12] or Lipski [7] The proof technique used in [1] [9] to show the interchange property is different than that used here. Their proof is inductive and is based on the starting permutation being 0 and the ending permutation being 10 n1 Gamma1 . For t 1 the ending permutations are not so easy to specify. For example, 303131122233 and ....

E.M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, 1977.

....explored again. When an extension of a position is found in the transposition table, the subtree generated from this extension does not need to be explored again, and we can instead join the extension and the position in the transposition table (this is also known as folding or fusing positions) [1, 23]. In the triangle puzzle, because the number of joined positions is large, the size of the search tree is greatly reduced by using a transposition table. For problem sizes 5, 6, and 7, 66 , 83 , and 90 of all positions explored are joined, respectively. Because joining occurs, what we refer to as ....

....there is a time space tradeoff. We decided to dynamically compute reflections of positions because physical memory per node on Alewife is low (5MB maximum usable memory) and there is no virtual memory yet. The triangle puzzle is similar to other tree search problems, such as the N queens problem [9, 12, 23, 27, 28], the Hi Q puzzle [23] and chess [18] Many of the search techniques, such as exploiting symmetry to reduce the search space and using a transposition table, arise in solving these problems. 535353 Conclusions We presented parallel shared memory and message passing implementations that solve ....

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E. Reingold, J. Nievergelt, N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice Hall, 1977.

....T of diameter 4 has a T code, and that no tree T of diameter 3 has a T code. Mathematical Reviews Subject Number: 05C45. 1 Introduction The utility of the ubiquitous binary reflected Gray code is undisputed. See, for example, the books of Nijenhuis and Wilf [5] Reingold, Nievergelt, and Deo [6], and Wilf [8] For certain applications, however, other Gray codes are desired. Many other Gray codes have been proposed, both for specific values of n and general constructions. For example, Goddyn, Lawrence, and Nemeth [3] motivated by an issue in the design of photon detectors, study the ....

Reingold, Nievergelt, and Deo, Combinatorial Algorithms: Theory and Practice, Prentice Hall, 1977.

....these strongly connected components. The first initialization task performed by the preprocessor module is preparing the strongly connected components of the graph for retiming, as described below. First it identifies all of the nontrivial strongly connected components (SCC s) of the input graph [9]. Next, using the Algorithm IE in [8] an interval or equality constraint is generated for each edge. Later, using these constraints, the solutions manager module will generate alternative retiming solutions, as needed. The second initialization task is the computation of the lower bounds on the ....

E. M. Reingold, J.Nievergelt, N. Deo, "Combinatorial Algorithms: Theory and Practice", Englewood Cliffs, New Jersey: Prectice-Hall Inc., 1977.

....circuits are designed to satisfy global zero clock skew. A great deal of effort has been applied to the design of clock distribution networks which maintain global zero clock skew across the circuit [5] B. Circuit model The work described in this paper is based upon a connected undirected graph [9, 10] model of a synchronous circuit. Formally, the graph H C of a circuit C with r registers and p local data paths is the six tuple H C IKJ V L C M G E L C M G A L C M G h L C M l G h L C M u G h L C M d N , where V L C M IPO v 1 G Q Q Q v r R , E L C M ....

....of cycles from the set. Note that a skew basis must not contain a cycle. If it did, the basis skews would be linearly dependent. Alternatively, linear independence in a kernel is guaranteed by choosing the cycles such that each cycle contains a unique edge from H . From graph theory [9 11], a spanning tree of the circuit graph H defines a basis of exactly n b I r 5 1 edges an example is indicated by the thicker edges shown in Fig. 3. The n c p 5 n b edges outside the basis are called chords any choice of basis naturally yields a kernel of exactly n c cycles, each ....

E. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice. Prentice-Hall, 1977.

....of their level numbers using fourth color pair y. breakcycles(root, y) end algorithm Figure 7: An eight coloring algorithm. number of a cycle C i is equal to the number of top nodes of other cycles dominating the top node of C i and this can be found in linear time using depth first search [5]. Another simple search of the colored tree, in which any left or right side node can be chosen as the one to be recolored with the fourth color pair y, can be used to implement breakcycles. Therefore, eightcolortree takes time linear in the size of the given quasi binary tree (or the size of the ....

E. M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms : Theory and Practice. Prentice-Hall, 1977.

.... 1 pieces. Thus, P k i=0 v i = m dk. Next we count all possible combinations for integers v 0 ; v k which sum up to m dk. This gives the number of compositions of m dk into k 1 parts and is the number of clone families having k clones of length d in the knapsack array of length m. By [51] (p. 190) the number of compositions is (m dk) k 1) 1 (k 1) 1 = m kd k k : So, the number of all clone families is bm=dc X k=0 m kd k k (11) bm=dc X k=0 (m kd k) m kd k 1) m kd 1) k bm=dc X k=0 (m kd 1) m kd 1) m kd ....

....procedure. The composition algorithm has three input parameters: the length of the whole time interval m, number k of clones we want to compose into a plan and the set of allowed lengths S = fc 0 ; c q g of clones (q 2 N) This algorithm is 51 based on a composition algorithm given in [51]. Few minor modi cations, however, are needed. The idea is to compose integer m into 2k 1 integers stored in array s and hence, to interpret each composition as a set of clones containing exactly k separate clones. We achieve this by restricting and interpreting the 2k 1 integers in s in the ....

Edward M. Reingold, Jurg Nievergelt, and Narsingh Deo. Combinatorial Algorithms: Theory and Practice. Prentice Hall, 1977.

....fast and constant time. Since the intersection heuristic is predicated on the ability to find set cardinalities, an efficient means of calculating the number of elements in a set is required. We use the card routine shown in Figure 10.2. This uses one of the bit counting techniques discussed in [102], and takes O(jssj) time. Although [102] also gives routines which are constant time, in our case jssj is always small certainly less than k, and usually less than t and card is very efficient. The complete routine makes extensive use of dynamic data structures, in the form of linked lists. ....

....heuristic is predicated on the ability to find set cardinalities, an efficient means of calculating the number of elements in a set is required. We use the card routine shown in Figure 10.2. This uses one of the bit counting techniques discussed in [102] and takes O(jssj) time. Although [102] also gives routines which are constant time, in our case jssj is always small certainly less than k, and usually less than t and card is very efficient. The complete routine makes extensive use of dynamic data structures, in the form of linked lists. While this allows the programme to cope ....

Edward M. Reingold, Jurg Nievergelt, and Narsingh Deo. Combinatorial Algorithms: theory and practice. Prentice-Hall, 1977.

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E. M. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, NJ, 1977.

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Reingold, E. M., Nievergelt, J. & Deo, N. (1977), Combinatorial Algorithms: Theory and Practice, Prentice-Hall.

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E. M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms - Theory and Practice. Prentice-Hall, 1977.

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E.M. Reingold, J. Neivergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice Hall, Englewood Cli#s, N.J., (1977)

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E.M. Reingold, J. Neivergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice Hall, Englewood Cli#s, N.J., (1977)

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E.M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, New Jersey, 1977.

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E.M. Reingold, J. Neivergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice Hall, Englewood Clis, N.J., (1977)

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Reingold E., Nievergelt J., Deo N.: Combinatorial Algorithms: Theory and Practice, Prentice Hall, New Jersey, 1977

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E. M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, Inc., Englewood Clis, New Jersey, 1977.

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E.M.Reingold, J.Nievergelt, and N.Deo, Combinatorial Algorithms : Theory and Practice, Prentice Hall, Englewood Clis, NJ, 1977.

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E. M. Reingold, J. Nievergelt, and N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, Inc., Englewood Clis, New Jersey, 1977.

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E. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, NJ, 1977.

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E. M. Reingold, J. Nievergeld, and N. Deo. Combinatorial Algorithms: Theory and Practice. Prentice-Hall, 1976. 13

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Reingold, E.M., Nievergelt, J. , and Deo, N. Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1977.

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