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1 Raning and Unraning Algorithms for -ary Trees in Gray Code Order Fateme Ashari-Ghomi, Najme Khorasani, and Abbas Nowzari-Dalini Abstract In this paper, we present two new raning and unraning algorithms for -ary trees represented by x-sequences in Gray code order These algorithms are based on a gray code generation algorithm developed by Ahrabian et al In mentioned paper, a recursive bactracing generation algorithm for x-sequences corresponding to -ary trees in Gray code was presented This generation algorithm is based on Vajnovszi s algorithm for generating binary trees in Gray code ordering Up to our nowledge no raning and unraning algorithms were given for x-sequences in this ordering we present raning and unraning algorithms with O(n 2 ) time complexity for x-sequences in this Gray code ordering Keywords -ary Tree Generation, Raning, Unraning, Gray Code I INTRODUCTION TRees are one of the most important structures in computer science Trees have many applications such as database generation, decision table programming, analysis algorithms, string matching [], switching theory, and even in the theoretical VLSI circuit design [26] Trees are also widely used data structures for maintaining data [9] and as auxiliary structures for compressing data [0] In addition, the exhaustive generation of all trees of a certain type is often useful; for example, a list of all trees with a given number of nodes n, may be used to test and analyze of algorithm complexity, and prove the correctness of an algorithm [] As such, the problem of generating binary trees [], [4], [7], [2], [22], [30], -ary trees [2], [5], [6], [7], [8], [3], [8], [23], [28], and other types of trees[20], [25], has been thoroughly investigated and many papers have been published in the literature for generating trees In most of these algorithms, trees are encoded as integer sequences and then these sequences are generated with a certain order, and consequently their corresponding trees are also generated in a specific order The most well-nown orderings on trees are A-order and B-order [22], [33], and the orderings on the sequences are lexicographical [33] and Gray code [5], [7] Typically, an important issue for algorithms to enumerate various classes of objects is that they must be run in a constant time for each generation Therefore so many algorithms have been proposed to generate -ary trees in constant time in the worst case These algorithms generate the sequences corresponding to trees in Gray code order [4], [5], [6], [24], [32] Fateme Ashari-Ghomi, Najme Khorasani and Abbas Nowzari-Dalini are with the Department of computer Science, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran addresses: (ftmashari@utacir, nhorasani@utacir, nowzari@utacir) Beside the generation algorithm for -ary trees, raning and unraning algorithms are also important in the concept of generation [23], [28], [3], [33] Given a specific order on the set of -ary trees, the ran of a tree (or corresponding sequence) is its position in the exhaustive generated list, and the raning algorithm computes the ran of a given tree (or corresponding sequence) in this order The reverse operation of the raning is called unraning, which generates the tree (or sequence) corresponding to a given ran Usually tree generation papers present raning and unraning algorithms for tree generation algorithm Raning and unraning algorithms also have many applications For example, in traditional tree compression algorithm for encoding the tree to code sequence and decoding the code sequence bac to a tree, the raning and unraning algorithms can be used As mentioned, in tree generation algorithms, trees are represented as sequences and then these sequences are generated Two well-nown representations of -ary trees with n nodes are x-sequences (or 0- sequences) and z-sequences presented by Zas [33] He presented a generation algorithm for z- sequences encoding in lexicographical ordering such that the corresponding trees are generated in B-order, and generates one sequence from a given sequence in O(n) (n is the length of z-sequence) He also presented two raning and unraning algorithms for this generation in O(n) Later, some other -ary tree generation algorithms were presented on this encoding [7], [24], [32] Vajonvszi presented a new Gray code for the x-sequence corresponding to binary trees with n nodes [27] A constant average time algorithm for generating this Gray code is also given Ahrabian et al introduced another algorithm for generating -ary trees represented by x-sequence in Gray code order [3], which is based on Vajnovszi s paper [27] This algorithm is a bactracing generation algorithm Up to now, no raning and unraning algorithms are presented for x- sequences in Gray code ordering given by Vajonvszi [27] and Ahrabian et al [3] In this paper, we present raning and unraning algorithms for x-sequences corresponding to -ary trees with n nodes generated in Gray code based on the Ahrabian et al [3] generation algorithm The time complexity of raning algorithm is O(n 2 ) and unraning algorithm is also O(n 2 ) The rest of this paper is organized as follows Section II introduces the definitions and notations that are used subsequently In Section III, raning and unraning algorithms are discussed Finally, some concluding remars are offered in Section IV 020

2 II PRELIMINARIES In graph theory, a tree is a connected and undirected graph without any cycle A rooted tree is a tree with a distinct node which is called root Rooted trees are directed trees and the direction is from the root to the offspring A rooted tree where the children of each node have a designated order is called ordered tree A -regular ordered tree is an ordered tree in which each node has either children (internal node) or no children (external node or leaf), and is called -ary tree An n- node -ary tree has n internal nodes and ( )n+ external nodes or leaves, and the total number of -ary trees with n internal nodes is denoted by C n, and is nown to have the value C n, ( )n + ( n n A -ary tree T can also be defined recursively as being either an external (leaf) or an internal node together with a sequence T,T 2,,T of -ary trees, which T i is defined as the ith subtree of T [] Let us introduce basic notations used through this paper and a definition of -ary trees by means of choice functions of indexed families of sets (from [9], [2], [3]) Let Γ{,,m} and I {,,l} be two sets, and l, m, then < Γ i > i I where Γ i Γ, show an family of sets indexed by I Any mapping function f which chooses one element from each set Γ,, Γ l is called a choice function of the indexed family < Γ i > i I With supplementary restrictions, various classes of combinatorial objects can be modeled by choice functions [9], [2], [3] If I {,,l} and Γ i {0, }, then any choice function χ < x i > i I, that belongs to the indexed family < Γ i > i I, is called binary choice function of this family [2] If l n for a given and n, each binary choice function which x + +x i i/, for i n, is called binary choice function with -dominating properties There exist bijections between set of choice functions χ and sets of -ary trees with n internal nodes in widely used representations All -dominating binary choice functions, with l n and the number of x + + x i n, are bitstring representations of all -ary trees of the set Γ This bitstring representation is called x-sequence [33] By -dominating definition, in each subsequence {x j } i ( i n) the accumulated number of s is at least i/ Clearly, each sequence has n s and ( )n 0 s For any given choice functions δ < d,,d l > and γ < g,,g l >, we say that δ and γ are in Gray code order, if they differ by a constant number of changes [29] The x-sequence can be obtained directly from -ary trees Given a regular -ary tree with n internal nodes, we label each internal node with and each external node with 0 Reading tree labels in pre-order (recursively, visit first root and then all the subtrees from left to right), we get a bitstring with n s and ( )n + 0 s As the last visited node is an external node, we omit the corresponding 0 [33] For example, the x-sequence corresponding to the tree presented in Fig is x Now, let be a set of all -dominating binary choice functions with the number of x + +x i n and m l n, ) in other words the set of all bitstrings with a -dominating property with m 0 s and n s such that m ( )n 0 In particular case where m ( )n, is the set of all x-sequences of length n As it is mentioned, we call a list of bitstrings in Gray code list if every two successive bitstrings in the list differ by an interchange of two bits Also, let α and β be two bitstrings then we denote αβ the concatenation of α and β By these definitions, the following lemma shows the recursion property of the set, which helps us in the generation schema [3] Lemma For any >0, and m, n 0, we have 0 m if n 0, if m ( )n, 0S m,n if m>( )n >0 For example, Table I shows all bitstrings S 6,2 3 0S 5,2 3 S 6, 3 With regard to the recursion property of the set, we can easily count the number of elements in this set Let be the number of bitstrings in According to the above lemma, can be obtained as follows Corollary For any >0, and m, n 0, we have Fig if n 0, if m ( )n, S m,n + if m>( )n > A 3-ary tree with 4 internal nodes TABLE I ALL BITSTRINGS IN S 6,2 3 0S 5,2 3 S 6, 3 0S 5,2 3 S 6,

3 Corollary 2 For any >0, n 0, and m ( )n 0, we have m + ( )n m + ( m + n With regard to the definition of, it should noted that S ( )n,n is the set of bitstrings corresponding to -ary trees with n internal nodes Therefore, we can have the following theorem Theorem In particular case of Corollary 2 where m ( )n, ( ) S ( )n,n n ( )n + n Obviously S ( )n,n C n, (the number of all -ary trees with n nodes) With regard to the above discussion and using Lemma the generation -ary trees algorithm in Gray code order was presented in [3] As an example, a list of 22 x- sequences corresponding to 4-ary trees with n 3nodes in this Gray code order is shown in Table II III RANKING AND UNRANKING ALGORITHM The ran of a -ary tree with respect to some ordering is the number of previously generated trees in that ordering The reverse operation of the raning is unraning As mentioned, for the x-sequences in Gray code, no raning and unraning algorithms are presented in the literature in Gray code ordering given by Vajonvszi [27] and Ahrabian et al [3] In this section, the raning and unraning algorithms for these generation and ordering are presented For raning and unraning algorithms we need for given, m, and n, that are defined and computed earlier We assume that these values are computed and stored in arrays S[m][n] for a fixed Also, for computing the ran of a given bitstring or unraning a given ran, we need three other variables prepos, curpos, and dir In the raning and unraning algorithm, when the ith of a bitstring is investigated, the variable curpos holds the position of this (ith ) and the variable prepos holds the position of previous ((i )th ) The variable dir holds the direction of the ith in a bitstring Direction of the ith in a bitstring is down (represented by 0) if the ith has been shifted one place to the left in the next bitstring in Gray code order, and if the ith has been shifted one place to the right, the direction is up (represented by ) For example, consider two bitstrings and , for the first bitstring the direction of the second is down, because in the next bitstring, the second has been shifted to left But for two bitstrings and , the direction of second in the first bitstring is up, because this has been shifted to right Since the first bit in the input bistring is always equal to and does not shift, we can assume that the direction of the first bit is equal to 0 n ) To determine the direction of the ith for 2 i n, we can use the following relation: If direction of the (i )th is equal to 0 then direction of the ith is equal to { 0 if (i )th is in its last position, if (i )th is not in its last position If direction of the (i )th is equal to then direction of the ith is equal to { if (i )th is in its last position, 0 if (i )th is not in its last position In this relation, if the position of the ith is equal to (i ) +, then it is in the last position As mentioned, a x-sequence corresponding to a -ary tree with n nodes, has n s and m ( )n 0 s In the Gray code order, for generating a bitsring from the previous bitsring, each should be shifted from its position to next position Therefore, each has a shift direction and can be computed using the above relation If the position of the ith in the jth bitsring is p and its direction is up (dir is equal to ) and this is a candidate for shifting, then its position in the (j +)th bitsring is p + Otherwise, if the direction of the ith in the jth bitsring is down then its position in the (j +)th bitsring is p For a given bitsring, ith have a last and a first position The last position of the ith is shown by p last and is equal to (i ) + and the first position of it is shown by p first, and is equal to the position of the (i )th plus A is shiftable if it is not in its last or first position, in other words, a is shiftable if its shift direction is up and it is not in its last position, or if its shift direction is down and it is not in its first position In order to compute the ran of a bitsring, we count the number of generated bitsring before this bitsring in the Gray code order This can be done, using by the shiftable s For this purpose, the bitsring should be scaned from left to right and a shiftable should be found As mentioned, the first is never shifted and is always in position, and shift direction of ith is determined by shift direction of (i )th Assume that we investigate the ith in the jth bitsring, we have two cases which are explained as follows In the first case, the shift direction of the ith in the jth bitsring is equal to down (dir 0) So, the first possible position is p last for this, where is equal to (i ) +, and the last possible position is p first for this, where is equal to the position of the (i )th plus In all the consecutive bitsrings that are generated by shifting of the ith, the possible positions for this can be the positions p last, p last, p last 2,, p first Therefore, if the position of the ith is p then the positions of this in previous bitsrings are p last, p last,, p+2, p+ The number of such bitsrings should be obtained for raning At first, we count the number of bitsrings which the ith of them is in position p last The positions of the first to (i )th are fixed and we have to count the number of 0 s and s which their positions are after the position of ith The number of s that their positions are after the ith in the jth bitsring is equal to n j n i Now 022

4 TABLE II THE 22 x-sequences OF LENGTH 2 FOR n 3AND 4 ran bitstring ran bitstring let q be the number of 0 s which their positions are before the ith in the jth bitsring, so the number of 0 s which their positions are after p is equal to m j m q (it is obvious that q p i) Therefore, the number of bitsrings that the ith of them is in position p last is equal to S m j,n j With the same way, we can count the number of bitsrings which the ith of them are in the positions p last,, p +2, p + Finally, the number of previous bitsrings before bitsring j that the ith of them are in positions p last, p last,, p +2, p + is equal to: p last jp+ S m j,n j In the second case, the shift direction of the ith in the jth bitsring is equal to up (dir ) So the first possible position for this is p first In all the consecutive bitsrings which are generated by shifting of the ith, the possible positions for this can be p first, p first +, p first +2,, p last Therefore, if the position of the ith is p then the positions of this in previous bitsrings are p first, p first +,, p 2, p The number of such bitsrings should be obtained for raning This number can be computed similar to the first case and we have: p S m j,n j, jp first where the values of m j and n j are equal to m q and n i, respectively (similar to the first case) Now, the ran of a given bitsring can be computed as: n p u S m j,n j, i2 jp l where p l and p u are lower and upper bound for positions of the ith With regards to the above discussion, the raning algorithm is given in Algorithm This algorithm taes n,, and bitstring x corresponding to the -ary tree with n nodes as input parameters, and returns the ran of x in variable r Considering the above discussion, the time complexity of the raning algorithm is O(n 2 ) The unraning algorithm essentially reverses the steps carried out in computing the ran The unraning algorithm given in Algorithm 2 taes r,, and n as input, and returns the x- sequence corresponding to the -ary tree with n nodes that has ran r Foragivenr as a ran, the corresponding bitstring x x x n is computed by the following operations Initially x is set to and all x i s (2 i n) are set to zero, the position of the first is set to, and the direction of this is set to 0 The bitstring x has to have n s, therefore the correct position of the ith (2 i n) is computed in a loop In the iteration i (2 i n) of this loop, at first, the direction of the ith (dir) is computed using by the direction and position of the (i )th, after that, the position of the ith (curp os) is computed using s[n ( ) curp os + i][n i] The values of s[n ( ) curp os + i][n i] determine the possible number of shifts which ith can have Finally, the value of this position is set to in bitstring x Considering the above discussion, the time complexity of the unraning algorithm is O(n 2 ) IV CONCLUSION we have introduced two new raning and unraning algorithms for n-node -ary trees represented by x-sequences In these algorithms, it is assumed that the order of x-sequences corresponding to the list of all -ary trees with n nodes is Gray code order The time complexity of both raning and unraning algorithms is O(n 2 ) REFERENCES [] H Ahrabian and A Nowzari-Dalini, On the generation of binary trees from (0-) codes, Intern J Comput Math 69 (998), [2] H Ahrabian and A Nowzari-Dalini, Generation of t-ary trees with Ballot sequences, Intern J Comput Math 80 (2003), [3] H Ahrabian and A Nowzari-Dalini, Gray code algorithm for listing -ary trees, Studies in Information and Control 3 (2004), [4] H Ahrabian and A Nowzari-Dalini, Parallel generation of binary trees in A-order, Parallel Comput 3 (2005), [5] H Ahrabian and A Nowzari-Dalini, Parallel Generation of t-ary trees in A-order, Comput J 50 (2007), [6] A Ahmadi-Adl, A Nowzari-Dalini, and H Ahrabian, Raning and unraning algorithms for loopless generation of t-ary trees, Logic Journal of IGPL 9 (20), [7] MC Er, Efficient generation of -ary trees in natural order, Comput J 35 (992),

5 Algorithm The raning algorithm Procedure Raning(x : bitstring; n, : integer) begin r ; curp os ; dir 0; for i 2to n do begin prep os curp os; curp os curp os +; while (x[curp os]! ) curp os curp os +; if (((dir 0) &&(prep os (i 2) +)) ((dir ) &&(prep os! (i 2) +)))then begin dir 0; for j (i ) + downto curp os + do r r + s[( ) n j + i][n i]; else begin dir ; for j prep os + to curp os do r r + s[( ) n j + i][n i]; return r ; Algorithm 2 The unraning algorithm Procedure UnRaning(r, n, : integer) begin x[] ; for i 2to n do x[i] 0; prep os ; dir 0; for i 2to n do begin if (((dir 0) &&(prep os (i 2) +)) ((dir ) &&(prep os! (i 2) +)))then begin dir 0; curp os (i ) +; while r>s[n ( ) curp os + i][n i] do begin r r s[n ( ) curp os + i][n i]; curp os curp os ; else begin dir ; curp os prep os +; while r>s[n ( ) curp os + i][n i] do begin r r s[n ( ) curp os + i][n i]; curp os curp os +; x[curp os] ; prep os curp os; return x ; 024

6 [8] S Heubach, N Li, and T Mansour, Staircase tilings and -Catalan structures, Discrete Math 308 (2008), [9] A Kapralsi, New methods for the generation of permutations, combinations and other combinatorial objects in parallel, J Parallel Distrib Comput 7 (993), [0] J Katajainen and E Mainen, Tree compression and optimization with application, Inter J Found Comput Sci (990), [] DE Knuth, The Art of Computer Programming, Vol : Fundamental Algorithms, Addision Wesly, Reading, 973 [2] Z Koosinsi, On the generation of permutations through decomposition of symmetric group into cosets, Bit 30 (990), [3] Z Koosinsi, Parallel enumeration of tary trees in ASC SIMD model, International Journal of Computer Science and Networ Security (20), [4] JF Korsh, Loopless generation of -ary tree sequences, Inform Process Lett 52 (994), [5] JF Korsh and P LaFollette, Loopless generation of Gray codes for -ary trees, Inform Process Lett 70 (999), 7 [6] J F Korsh and P LaFollette, A loopless Gray code for rooted trees, ACM Transactions on Algorithms 2 (2006), [7] J Lucas, D Roelants van Baronaigien, and F Rusey, On rotations and the generation of binary trees, J of Algorithms 5 (993), [8] K Manes, A Sapounais, I Tasoulas, and P Tsiouras, Recursive generation of -ary trees, Journal of Integer Sequences 2 (2009), 8 [9] O O Olugbenga, E F Adebiyi, S Fatumo and A Dawodu, PQ trees, consecutive ones problem and applications, International Journal of Natural and Applied Sciences 4 (2008), [20] J M Pallo, A simple algorithm for generating neuronal dritic trees, Comput Meth Prog Bio 33 (990), [2] J M Pallo, Rotational tree structures on binary trees and triangulations, Acta Cybernetica 7 (2006), [22] J M Pallo and R Racca, A note on generating binary trees in A-order and B-order, Intern J Comput Math 8 (985), [23] F Rusey, Generating t-ary trees lexicographically, SIAM J Comput 7 (978), [24] D Roelants Van Baronaigien, A loopless Gray code algorithm for listing -ary trees, J Algorithms 35 (2000), [25] E Seyedi-Tabari, H Ahrabian, and A Nowzari-Dalini, A new algorithm for generation of different types of RNA, Intern J Comput Math 87 (200), [26] T Uehara and WM Cleemput, Optical layout of cmos functional arrays, IEEE Trans Comput 7 (98), [27] V Vajnovszi, Constant time algorithm for generating binary tree Gray codes, Studies in Informatics and Control 5 (996), 5 2 [28] V Vajnovszi and J M Pallo, Raning and unraning of -ary trees with 4-4 letter alphabet, Journal of Information and Optimization Science 8 (997), [29] V Vajnovszi and R Vernay, Restricted compositions and permutations: From old to new Gray codes, Inform Process Lett (20), [30] R Wu, J Chang, and Y Wang, A linear time algorithm for binary tree sequences transformation using left-arm and right-arm rotations, Theor Comput Sci 335 (2006), [3] R Wu, J Chang, and C Chang, Raning and unraning of non-regular trees with a prescribed branching sequence, Math Comput Model, 53 (20), [32] L Xiang, K Ushijima, and C Tang, Efficient loopless generation of Gray codes for -ary trees, Inform Process Lett 76 (2000), [33] S Zas, Lexicographic generation of ordered trees, Theor Comput Sci 0 (980),

An Efficient Ranking Algorithm of t-ary Trees in Gray-code Order

The 9th Workshop on Combinatorial Mathematics and Computation Theory An Efficient Ranking Algorithm of t-ary Trees in Gray-code Order Ro Yu Wu Jou Ming Chang, An Hang Chen Chun Liang Liu Department of