Zaks, S.: Lexicographic generation of ordered trees. Theoretical Computer Science 10 (1980) 63--82  Home/Search   Document Not in Database   Summary   Related Articles   Check

....conditions: 1. m= n 1)t 1# 2. a i a j , for i j and i, j 2 I#and3.a i 2f1, i 1)t 1g,fori2 I # are satisfied then anychoice function i = a i i2I , that belongs to the indexed family A i i2I , is called increasing choice function with restricted growth of this family (z sequence [31]) In the above mappings we deal in fact with indexed sets T i = f1, i 1)t 1gaeA i . If A i = f0,1g and I =f1, tng then any choice function = a i i2I , that belongs to the indexed family A i i2I , is called binary choice function of this family (x sequence [31] All binary ....

....(z sequence [31] In the above mappings we deal in fact with indexed sets T i = f1, i 1)t 1gaeA i . If A i = f0,1g and I =f1, tng then any choice function = a i i2I , that belongs to the indexed family A i i2I , is called binary choice function of this family (x sequence [31]) All binary choice functions, with the number of a 1 : a i i=t, for 1 i tn, are bitstring representations of all t ary trees of the set A. Simple transformations convert choice functions into choice functions i and . Let us introduce now lexicographic order on the set of all choice ....

Zaks, S.: Lexicographic generation of ordered trees. Theoretical Computer Science 10 (1980) 63--82

....e.g. 1] and : 1 q 1 4 xv (1 2 u) 2 1 q 1 4 xv (1 2 u) 2 1 . Every secondary structure w can be reconstructed from the semi Dyck word (w) by inserting unary symbols j at the appropriate positions. Thus, since we can identify binary trees and semi Dyck words (see e.g. [22]) it becomes possible to consider secondary structures based on these generating functions by inserting linear lists. For j a string of zero or more symbols j and j a string of at least one symbol j, the cases to be distinguished for this procedure are: j (j )j ( v ....

S. Zaks, Lexicographic Generation of Ordered Trees, Theoretical Computer Science 10 (1980), 63-82.

.... . Then for each secondary structure w, w) is a semi Dyck word, i.e. a word over the alphabet f( g which ful lls the conditions (1) and (2) of De nition 4. The following oneto one correspondence of semi Dyck words of length 2n and ordered binary trees with n nodes is well known (see e.g. [27]) u)v ( n u v : To put the meaning of this symbolic notation into words, the rst bracket in the semi Dyck word together with its corresponding closing bracket represent the root of the tree, the subword u (resp. v) represents the left (resp. right) subtree (and vice versa) Note, that ....

S. Zaks, Lexicographic Generation of Ordered Trees, Theoretical Computer Science 10 (1980), 63-82.

....(i 1) mod 2 ] b i 1 2 c j Gamma1 , where u 2 T , x 2 T and y = ae fi if x = if x = fi . 3.6 Ranking and Unranking Extended Ordered Binary Trees (Zaks) Now, we consider extended ordered binary trees. In an extended ordered binary tree, each degree of a node is either 0 or 2. In [Za80] an extended ordered binary tree is coded as follows: The internal nodes (resp. leaves) are labelled by 1 (resp. 0) Then the tree is traversed in preorder and all labels except the last one, which is a 0, are concatenated. This string corresponds to a Dyckword of length 2n with one type of ....

..... 111000 110100 110010 101100 101010 123 OE lex 124 OE lex 125 OE lex 134 OE lex 135 Figure 11: All trees in E3 arranged according to the lexicographical order OE lex . In contrast to [Za80] we present a ranking algorithm that reads the word from left to right and an unranking algorithm that writes the word from left to right. The algorithms make use of the one to one correspondence between En and D 1 2n . The ranking algorithm computes the rank of a word e 2 En by the computation ....

S. Zaks, Lexicographic Generation Of Ordered Trees, Theoretical Computer Science 10, 1980, 63-82.

.... 391, 394] G [152] G schemes [152] Galois [74, 79] game [76, 426] games [357, 183] gamma [84] General [96, 128, 138, 194, 406, 267] general purpose [406] Generalization [178, 67] Generalized [74, 378, 39, 211, 133, 82] generated [367, 216, 160] generating [171] generation [6]. generators [455] geometric [410] Ginsburg [178] Ginsburg Rice [178] given [361] global [371] Godel [143, 314] grammar [139, 162, 164, 3, 335, 228] grammar which [162] grammars [139, 330, 149, 154, 392, 141, 247, 436, 206, 350, 184, 125, 254, 393] grammatical [99] Graph [26, ....

.... 247, 25, 189, 306, 178, 423, 54, 20, 351, 230, 224, 77, 382, 433, 277, 161, 125, 312, 319, 419, 259, 244, 378, 444] large [407] largest [262, 320] lattice [283] least [129, 52] lemma [87] lemmas [419] length [38] less [218] letter [436] letters [433] level [66, 335] Lexicographic [6]. lies [426] like [157, 440, 14, 19, 371] limited [109, 57, 244] line [241, 278, 273] Linear [401, 78, 444, 21, 400, 420, 230, 276, 195, 354, 171, 265, 234, 188] linear history [400] Linear time [78] lists [211] little [213] LL [51, 250, 254] LL iteration [250] LLP [125] LLR ....

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S. Zaks. Lexicographic generation of ordered trees. Theoretical Computer Science, 10(1):63--82, January 1980. CODEN TCSCDI. ISSN 0304-3975. Schnorr:1980:BNC

....in an analogous way. Remark 2: It is well known that there are several one to one correspondences between the normal Dyck language and other combinatorial objects. For example, the shape of an ordered tree or an extended ordered binary tree can be coded in the normal Dyck language (for example [Za80]) but the labels at the nodes or the edges can not be coded in the normal Dyck language. In [FS96, p. 266] was mentioned a coding of ordered trees with labelled nodes by another generalization of the Dyck language. With the generalization discussed here, we can also code the labels at the edges. ....

S. Zaks, Lexicographic Generation of Ordered Trees, Theoretical Computer Science 10, 1980, 63-82. 14

....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 interchange ....

S. Zaks. Lexicographic generation of ordered trees. Theoretical Computer Science, 10:63-- 82, 1980.

.... 391, 394] G [152] G schemes [152] Galois [74, 79] game [76, 426] games [357, 183] gamma [84] General [96, 128, 138, 194, 406, 267] general purpose [406] Generalization [178, 67] Generalized [74, 378, 39, 211, 133, 82] generated [367, 216, 160] generating [171] generation [6]. generators [455] geometric [410] Ginsburg [178] Ginsburg Rice [178] given [361] global [371] Godel [143, 314] grammar [139, 162, 164, 3, 335, 228] grammar which [162] grammars [139, 330, 149, 154, 392, 141, 247, 436, 206, 350, 184, 125, 254, 393] grammatical [99] Graph [26, ....

.... 247, 25, 189, 306, 178, 423, 54, 20, 351, 230, 224, 77, 382, 433, 277, 161, 125, 312, 319, 419, 259, 244, 378, 444] large [407] largest [262, 320] lattice [283] least [129, 52] lemma [87] lemmas [419] length [38] less [218] letter [436] letters [433] level [66, 335] Lexicographic [6]. lies [426] like [157, 440, 14, 19, 371] limited [109, 57, 244] line [241, 278, 273] Linear [401, 78, 444, 21, 400, 420, 230, 276, 195, 354, 171, 265, 234, 188] linear history [400] Linear time [78] lists [211] little [213] LL [51, 250, 254] LL iteration [250] LLP [125] LLR ....

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S. Zaks. Lexicographic generation of ordered trees. Theoretical Computer Science, 10(1):63--82, January 1980. CODEN TCSCDI. ISSN 0304-3975. Schnorr:1980:BNC

....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 so that successive sequences ....

....the internal nodes, and zeros representing the leaves in the preorder traversal (the zero corresponding to the last leaf being implicit) This is perhaps the most natural of all sequence representations of binary trees. Algorithms for generating binary trees in this representation may be found in [20], 11] and elsewhere. The elements of T(n) can also be interpreted as well formed parentheses strings where a one corresponds to a left parenthesis and a zero corresponds to a right parenthesis. Notice that they also have a symmetric suffix property that the number of 0 s is at least as great as ....

S. Zaks. Lexicographic generation of ordered trees. Theoretical Computer Science, 10:63--82, 1980.

....of this algorithm when the real costs of these operations are taken into account. 1353 1 Introduction During the last decade, there has been a great deal of work on algorithms for selecting trees. In particular, there are now efficient algorithms for listing rooted trees of a given size [3] [18], listing all free trees of a given size [12] 4] 8] 16] choosing uniformly at random a free tree having a given number of vertices [14] ranking and unranking rooted and free trees [15] 17] Several works concern the random generation of binary trees [5] 9] J.L.R emy [13] designed a ....

S.Zaks, Lexicographic generation of ordered trees, T.C.S., 10, 63-82, 1980.

....corresponding to a binary tree with n nodes is of length 2n, and equation (4) does not contain digits greater than n, we have Theorem 2. 1 The method of Arnold and Sleep generates balanced strings of parentheses in linear time using integers no larger than O(n 2 ) In Zaks coding method [17] we first label each intenal node by 1 and each leaf by 0. The labels are then read in preorder to obtain the code word. The Zaks sequence of the sample tree shown in Figure 2 is 11001110010010. Zaks sequences have dominating property, i.e. in each proper prefix the number of ones is not smaller ....

....word. The Zaks sequence of the sample tree shown in Figure 2 is 11001110010010. Zaks sequences have dominating property, i.e. in each proper prefix the number of ones is not smaller than the number of zeros. Zaks sequences on length 2n are in 1 1 correspondence with binary trees with n nodes [17]. Zaks sequences can be produced by the grammar with productions S 1SS; S 0: Given a binary tree, the corresponding string of balanced parentheses and the corresponding Zaks sequence are generated such that the strings of 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 Figure 2: A sample tree with Zaks ....

Zaks, S., Lexicographic generation of ordered trees. Theoret. Comput. Sci. 10 (1980), 63--82.

....in this model that match the performance of their CREW PRAM counterparts. Typically, for the same running time, the MMB uses more processors. Encoding the shape of an ordered tree is a basic step in a number of algorithms in integrated circuit design, automated theorem proving, and game playing [15]. The common characteristic of these applications is that the information stored at the nodes is irrelevant, as one is only interested in detecting whether two ordered trees have the same shape . As it turns out, if we ignore the contents of the nodes of an n node tree T , then the remaining ....

....irrelevant, as one is only interested in detecting whether two ordered trees have the same shape . As it turns out, if we ignore the contents of the nodes of an n node tree T , then the remaining shape information can be uniquely captured by a string of 2n bits, referred to as the encoding of T [11, 15]. Conversely, given a string of 2n bits, a number of practical applications ask to recover the unique n node ordered tree (if any) corresponding to this encoding. The main goal of this work is to fathom the suitability of the MMB architecture for some tree related computations. Our contribution ....

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S. Zaks, Lexicographic generation of ordered trees, Theoretical Computer Science, 10 (1980), 63--82.

....Rao [7] The coding system related to G 2 makes difference only between internal nodes and leaves. The former are labeled with a s and the latter with b s. Again, the labels are read in preorder. The code words obtained are often called Zaks sequences, since they are introduced by Zaks in [8]. In what follows we study G 1 and G 2 and other context free grammars generating valid binary tree code words. Xiang, Tang and Ushijima [5] have proved several theorems concerning the languages L(G 1 ) and L(G 2 ) generated by G 1 and G 2 . We show that these results follow directly from the fact ....

....w 0 is a suffix of a word in L(G 2 ) we have b(w 0 ) Gamma a(w 0 ) 1: Usually this result is given in the form Proposition 3.3 If w 0 is a proper prefix of a word in L(G 2 ) we have a(w 0 ) Gamma b(w 0 ) 0: Proposition 3. 3 is known as the dominating property of Zaks sequences [8]. The rest of this section is devoted to some new results concerning G 1 and G 2 and the languages generated by them. Proposition 3.4 Each word w, w 6= d, in L(G 1 ) can be written in the form w = xyz, y 6= such that all words xy i z, i = 0; 1; are in L(G 1 ) Proof If w has a ....

Zaks, S. (1980) Lexicographic generation of ordered trees. Theoret. Comput. Sci., 10, 63--82.

....algorithms for context free languages generated by arbitrary unambiguous context free grammars. The present paper concerns a somewhat similar but more difficult problem of enumerating regular and context free languages in lexicographic order. The widely studied problem of coding binary trees [3, 7] can be considered as a subproblem of our present problem. For example, in Zaks coding method [7] we label the nodes and the leaves of a binary tree by 1 and 0, respectively. By traversing the tree in preorder we obtain a code word consisting of n (the number of nodes) 1 s and n 1 0 s. The same ....

....The present paper concerns a somewhat similar but more difficult problem of enumerating regular and context free languages in lexicographic order. The widely studied problem of coding binary trees [3, 7] can be considered as a subproblem of our present problem. For example, in Zaks coding method [7] we label the nodes and the leaves of a binary tree by 1 and 0, respectively. By traversing the tree in preorder we obtain a code word consisting of n (the number of nodes) 1 s and n 1 0 s. The same set of words is obtained by considering the context free language generated by productions S 1SS ....

S. Zaks, Lexicographic generation of ordered trees. Theor. Comput. Sci. 10 (1980) 63--82.

....one representation is often used to generate the other. One natural investigation has been that of generating (i.e. listing) all parenthesis strings in some specified order; e.g. lexicographic order. Algorithms for listing parenthesis strings in lexicographic order may be found in Zaks [10] and Makinen [6] Gray code algorithms for generating parentheses strings may be found in Ruskey and Proskurowski [9] and Lucas, Roelants van Baronaigien, and Ruskey [5] A parallel algorithm for generating well formed parentheses is given by Akl and Stojmenovic [1] Listings of well formed ....

S. Zaks, "Lexicographic Generation of Ordered Trees," Theoretical Computer Science, 10 (1980) 63-82.

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Zaks, S.: Lexicographic generation of ordered trees. Theoretical Computer Science 10 (1980) 63--82

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