F.Ruskey and T.C.Hu, "Generating binary trees lexicographically, " SIAM Journal on Computing, Vol. 6, no. 4, 1977, pp.745-758.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to the binary vector of length n as decoding (these mappings are also called ranking and unranking ) We assume that coding stands for the pair of mappings (encoding, decoding) The problem under consideration, when N is the smallest odd integer satisfying (1) was addressed by many authors ([3], 4] 5] and other papers) However, implementation of the known coding algorithms seems to be dicult if n is big enough. We develop the approach, proposed for solving the data transmission problem in asynchronous mode [6] and include a small redundancy in the representation, which allows us ....

Ruskey, F., and Hu, T. C. (1977) Generating binary trees lexicographically. SIAM J. Comput., 6, 745-758.

....equal probability of being generated. For small queries a simple, and in efficient, bruteforce filter method is feasible, but for bigger queries a more elaborate method has to be used. In efficient generation. A straight forward method is to generate binary trees at random, like proposed in [RH77] and permute the relations to the leaves. Since this method generates all valid and invalid plans we need to check if the resulting plan is either valid or 14 Number of relations All plans Valid plans Fraction 5 40320 576 0.014 10 6:4x10 15 1:3x10 11 2:1x10 Gamma5 15 3:0x10 29 7:6x10 ....

....the numbers 1 through 5 to the trees of Qg in which e appears at level 1; numbers 6 through 10 to those in which e is at level 2; 11 through 15 to those in which e is at 3; and finally 16 through 18 to those trees in which e is at level 4. Our unranking procedure is based on those presented in [RH77, Li86] Theorem 2. Association trees of a given acyclic query graph G on n relations can be unranked in polynomial time. Since trees are numbered, and we can reconstruct efficiently any of them given its number, the next corollary follows. Corollary of theorem 2. Assuming a source of random ....

F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM journal of Computation, 6(4):745--758, December 1977.

....the numbers 1 through 5 to the trees of Qg in which e appears at level 1; numbers 6 through 10 to those in which e is at level 2; 11 through 15 to those in which e is at 3; and finally 16 through 18 to those trees in which e is at level 4. Our unranking procedure is based on those presented in [RH77, Li86]. Theorem 2. Association trees of a given acyclic query graph G on n relations can be unranked in polynomial time. Since trees are numbered, and we can reconstruct efficiently any of them given its number, the next corollary follows. Corollary of theorem 2. Assuming a source of random numbers ....

F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM journal of Computation, 6(4):745--758, December 1977.

....on which relations can be joined together, and counting them does not reduce, in general, to the enumeration of familiar classes of trees e.g. binary trees, trees representing equivalent expressions on an associative operator, etc. A variety of techniques are used to enumerate graphs and trees [Knu68, HP73, RH77, VF90]. The scheme we use is similar to that used, for example, in [GLW82] in the sense that an auxiliary structure serves to guide the counting and ranking of elements of the space, instead of applying a closed formula. Previous work has identified restricted classes of queries for which valid ....

F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM journal of Computation, 6(4):745--758, December 1977.

....constructing the corresponding plan. Once an unranking mechanism is available, uniform sampling of elements in the space reduces to random generation of numbers in the range 0, N 1. None of the known ranking and unranking techniques for tree structures apply to the current problem [10, 2], as the space of alternatives considered by industrial query optimizers is not restricted to an abstract combinatorial problem, such as join reordering. Multiple execution algorithms, index utilization, reordering of grouping operators, special purpose physical operators, and heuristics to ....

R. Ruskey and T. C. Hu. Generating Binary Tree Lexicographically. SIAM Journal of Computation, 6(4):745--758, December 1977.

....of t. That is, Pi(T ) foe 2 S n : t(oe) Tg. The Pi(T ) s are the equivalence classes of S n . Note that some authors have considered 1 to 1 mappings between the symmetric group and the space of binary trees in a way that gives a method for ordering and ranking trees. See, for instance, Ruskey and Hu (1977) and Trojanowski (1978) By contrast, here we are considering the set of all permutations which can be identified with a particular tree. The move to root (MTR) operation is defined as a series of simple exchanges between nodes. A simple exchange (SE) for a requested record j is as follows: i) ....

Ruskey, F., and Hu, T. C. (1977). Generating binary trees lexicographically. SIAM J. Comp., 6 745--758.

....trees is precise, but we also have to transform the messages to codewords and vise versa. This problem is discussed below. There are enumerative coding procedures that map the message to a binary vector having a xed number of ones [1] and similar methods of enumerating the preorder codewords [2], 4] If the number of messages is not very big, these methods can be e ectively used in practice. Otherwise, organization of the operations with the binomial coecients included in their description seems to be a hard problem. The statement of Proposition 2 allows us to overcome the diculties in ....

F. Ruskey and T. C. Hu, \Generating binary trees lexicographically," SIAM Journal on Computing, vol. 6, no. 4, pp. 745-758, 1977.

....scheme for every other member of the family. However, the resulting lists may not look like Gray codes, since bijections need not preserve minimal changes between elements. The problem of generating all binary trees with a given number of nodes was considered in several early papers, including [RH77], Zak80] and [Zer85] However, Gray codes in the Catalan family were first considered in [PR85] where binary trees were represented as strings of balanced parentheses. It was shown in [PR85] that strings of balanced parentheses could be listed so that consecutive strings differ only by the ....

F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM Journal on Computing, 6(4):745--758, 1977.

....in general, to the enumeration of familiar classes of trees e.g. binary trees, trees representing equivalent C. Galindo Legaria was supported by an ERCIM postdoctoral fellowship. expressions on an associative operator, etc. A variety of techniques are used to enumerate graphs and trees [Knu68, HP73, RH77, GLW82, VF90], but none of them seems to apply directly to our problem. Previous work has identified restricted classes of queries for which valid operator trees map one to one to permutations or to unlabeled binary trees the first class known as star queries, and the second as chain queries, see for ....

.... using Lemma 6 we find fi fi fiT Gj fabg fi fi fi b = 0; 1] fi fi fiT Gj fabcg fi fi fi c = 0; 1; 1] fi fi fiT Gj fabcdg fi fi fi d = 0; 2; 2; 1] and jT G j e = 0; 5; 5; 3; 1] The computation in the above example is isomorphic to the one used to count unlabeled binary trees in [RH77]. This is the case for chain queries i.e. those with nodes fv 1 ; vn g and edges f(v 1 ; v 2 ) v 2 ; v 3 ) v n Gamma1 ; vn )g. Then, as shown in [RH77] the closed form for jT G j is 1=n Delta 2n Gamma 2 n Gamma 1 , for a chain query of n nodes. Unfortunately, ....

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F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM journal of Computation, 6(4):745--758, December 1977.

....the numbers 1 through 5 to the trees of Qg in which e appears at level 1; numbers 6 through 10 to those in which e is at level 2; 11 through 15 to those in which e is at 3; and finally 16 through 18 to those trees in which e is at level 4. Our unranking procedure is based on those presented in [RH77, Li86]. Theorem 2. Association trees of a given acyclic query graph G on n relations can be unranked in polynomial time. Since trees are numbered, and we can reconstruct efficiently any of them given its number, the next theorem follows. 3 A loose upper bound on theorems 1,2 and 3 is O(n 3 ) ....

F. Ruskey and T. C. Hu. Generating binary trees lexicographically. SIAM journal of Computation, 6(4):745--758, December 1977.

....of a given word. This is in fact an application of another algorithm by Duval [3] that computes, in linear time, the factorization of a word into Lyndon words. The algorithm for systematic generation of Lyndon words is similar, in structure, to algorithms for systematic generation of trees [11, 14] or of other combinatorial objects [12] For these objects, known algorithms have constant average running time. We show that the same holds for Duval s algorithm : the average cost is given by q 1 q Gamma 1 1 2q (q 2 Gamma 1)n O 1 n 2 : 1) We even give an evaluation of ....

F. Ruskey and T.C. Hu. Generating binary trees lexicographically. SIAM J. Comput., 6:745--758, 1977.

....2,k 1) S(n, k 1) S(n 2,k 1) 2 the electronic journal of combinatorics 5 (1998) #R44 19 Define T (n, k) to be the number of well formed parentheses strings of length 2n, which begin with exactly k left parentheses. The following recurrence relation for T (n, k)isproveninHuandRuskey[9]. T (n, k) # # # # # T (n, 2) if k =1 T (n, k 1) T (n 1,k 1) if 1 k n 1ifk = n. We use T (n, k) to demonstrate a relationship between S(n)andC(n) the nth Catalan number. The Catalan numbers count the total number of well formed parentheses strings of length 2n. Two equations for ....

T.C. Hu and F. Ruskey, "Generating Binary Trees Lexicographically," SIAM J. Computing, 6 (1977) 745-758.

....5. A constant average time algorithm is presented in this case as well. 1 Introduction Binary Trees are of fundamental importance in computer science. In recent years there has been some interest in algorithms that generate all binary trees with a fixed number of nodes (for example, Ruskey and Hu [17], Proskurowski [10] Zaks [20] Pallo [9] Zerling [21] or restricted classes of binary trees (for example, Lee, Lee, and Wong [6] Li [7] Usually the trees are represented as integer sequences and those sequences are then generated. A natural question is whether the sequences can be generated ....

....section. A reason for considering a Gray code for T(n; k) is that a Gray code for T(n 1; 1) can be trivially transformed into a Gray code for T(n) by ignoring the 10 prefix of every bitstring in T(n 1; 1) Let T (n; k) be the number of bitstrings in T(n; k) It was shown in Ruskey and Hu [17] that the numbers T (n; k) satisfy the following recurrence relation. This recurrence, perhaps in a different form, certainly occurred earlier in connection with the study of Ballot problems, 14] T (n; k) 8 : T (n; 2) if k = 1 T (n; k 1) T (n Gamma 1; k Gamma 1) if 1 k n ....

F. Ruskey and T.C. Hu. Generating binary trees lexicographically. SIAM J. Computing, 6:745--758, 1977.

.... 2; i) by Lemma 3.6) k 2 X i=1 S(n 2; i) S(n 2; k 1) S(n; k 1) S(n 2; k 1) 2 Define T(n; k) to be the number of well formed parentheses strings of length 2n, which begin with exactly k left parentheses. The following recurrence relation for T(n; k) is proven in Ruskey [9]. T(n; k) 8 : T(n; 2) if k = 1 T(n; k 1) T(n 1; k 1) if 1 k n 1 if k = n. We use T(n; k) to demonstrate a relationship between S(n) and C(n) the nth Catalan number. The Catalan numbers count the total number of well formed parentheses strings of length 2n. Two equations for ....

T.C. Hu and F. Ruskey, "Generating Binary Trees Lexicographically," SIAM J. Computing, 6 (1977) 745-758.

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F.Ruskey and T.C.Hu, "Generating binary trees lexicographically, " SIAM Journal on Computing, Vol. 6, no. 4, 1977, pp.745-758.

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Ruskey, F. and Hu, T. C. (1977) Generating binary trees lexicographically. SIAM J. Comput., 6, 745--758.

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