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A new encoding for labeled trees employing a stack and a queue
 Bulletin of the Institute of Combinatorics and its Applications
, 2002
"... A novel algorithm for encoding finite labeled trees is proposed in this paper. The algorithm establishes a onetoone mapping between trees of order n and (n – 2)tuples of the node labels. This encoding is similar to those proposed by Prüfer [8] and Neville [6], except that our encoding scheme has ..."
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Cited by 11 (3 self)
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A novel algorithm for encoding finite labeled trees is proposed in this paper. The algorithm establishes a onetoone mapping between trees of order n and (n – 2)tuples of the node labels. This encoding is similar to those proposed by Prüfer [8] and Neville [6], except that our encoding scheme has additional properties (not present in earlier codes), which allows us to efficiently determine the center(s), radius, and diameter directly from the code, without having to explicitly construct the tree. We also present conditions for modifying the proposed code in order to obtain trees at distance one from the original. A queue is used during encoding while a stack is used during decoding. Both encoding and decoding require O(n) time.
RandomTree Diameter and the DiameterConstrained MST
 CONGRESSUS NUMERANTIUM
, 2000
"... A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tre ..."
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Cited by 10 (1 self)
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A minimum spanning tree (MST) with a small diameter is required in numerous practical situations. It is needed, for example, in distributed mutual exclusion algorithms in order to minimize the number of messages communicated among processors per critical section. Understanding the behavior of tree diameter is useful, for example, in determining an upper bound on the expected number of links between two arbitrary documents on the World Wide Web. The DiameterConstrained MST (DCMST) problem can be stated as follows: given an undirected, edgeweighted graph G with n nodes and a positive integer k, find a spanning tree with the smallest weight among all spanning trees of G which contain no path with more than k edges. This problem is known to be NPcomplete, for all values of k; 4 k #n  2). In this paper, we investigate the behavior of the diameter of MST in randomlyweighted complete graphs (in ErdősRnyi sense) and explore heuristics for the DCMST problem. For the case when the diameter bound k is smallindependent of n, we present a onetimetreeconstruction (OTTC) algorithm. It constructs a DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to the tree at each stage of the tree construction. This algorithm is fast and easily parallelizable. We also present a second algorithm that outperforms OTTC for larger values of k. It starts by generating an unconstrained MST and iteratively refines it by replacing edges, one by one, in the middle of long paths in the spanning tree until there is no path left with more than k edges. As expected, the performance of this heuristic is determined by the diameter of the unconstrained MST in the given graph. We discuss convergence, relative merits, and implementation of t...
Prüferlike codes for labeled trees
 Congressus Numerantium
, 2001
"... In 1918 Prüfer showed a onetoone correspondence between nnode labeled trees and (n – 2)tuples of node labels. The proof employed a tree code, computed by iteratively deleting the leaf with the smallest label and recording its neighbor. Since then other tree codes have been proposed, based on dif ..."
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Cited by 10 (1 self)
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In 1918 Prüfer showed a onetoone correspondence between nnode labeled trees and (n – 2)tuples of node labels. The proof employed a tree code, computed by iteratively deleting the leaf with the smallest label and recording its neighbor. Since then other tree codes have been proposed, based on different node deletion sequences. These codes have different properties, interesting and useful in graph theory and computer science. In this paper we survey and classify these Prüferlike codes as well as new codes based on the parameter values of the proposed generic treeencoding algorithm and study their properties. 1.
Computing A DiameterConstrained Minimum Spanning Tree
, 2001
"... In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path wi ..."
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Cited by 9 (0 self)
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In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameterconstrained minimum spanning tree (DCMST) of a given undirected, edgeweighted graph, G, is the smallestweight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NPcomplete for all values of k; 4 k (n  2), except when all edgeweights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree datastructure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomialtime algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter4 tree is also used for evaluating the quality of o...
A Unified Approach to Coding Labeled Trees
 IN PROCEEDINGS OF THE 6TH LATIN AMERICAN SYMPOSIUM ON THEORETICAL INFORMATICS (LATIN ’04), LNCS 2976
, 2004
"... We consider the problem of coding labeled trees by means of strings of node labels and we present a unified approach based on a reduction of both coding and decoding to integer (radix) sorting. Applying this approach to four wellknown codes introduced by Prufer [18], Neville [17], and Deo and M ..."
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Cited by 8 (4 self)
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We consider the problem of coding labeled trees by means of strings of node labels and we present a unified approach based on a reduction of both coding and decoding to integer (radix) sorting. Applying this approach to four wellknown codes introduced by Prufer [18], Neville [17], and Deo and Micikevicius [5], we close some open problems. With respect to
Parallel algorithms for computing Prüferlike codes of labeled trees
, 2001
"... In 1918 Prüfer showed a onetoone correspondence between labeled trees on n nodes and sequences of (n2) node labels, obtained by iteratively deleting the leaf with the smallest label in the current tree and appending the label of the adjacent node to the code. In 1953 Neville proposed two additi ..."
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Cited by 4 (1 self)
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In 1918 Prüfer showed a onetoone correspondence between labeled trees on n nodes and sequences of (n2) node labels, obtained by iteratively deleting the leaf with the smallest label in the current tree and appending the label of the adjacent node to the code. In 1953 Neville proposed two additional methods for encoding a labeled tree of order n into a sequence of (n2) labels. Recently, yet another treecode was proposed by the authors. As a generalization, we define a treecode to be Prüferlike, if it is a sequence of (n2) node labels computed by iteratively deleting the leaves of the tree in some deterministic order. The four aforementioned Prüferlike codes can be computed sequentially in O(n) time. Two EREW PRAM O(log n)time algorithms for computing the Prüfer code have been proposed in the literature. Neither of these algorithms is workoptimal, since they require O(n) processors. In this paper we propose three different EREW PRAM O(log n)time algorithms for computing the treecodes proposed by Neville, and by the authors. The algorithm for Neville's third encoding requires O(n / log n) processors and therefore is workoptimal. To the best of our knowledge, no other workoptimal algorithm for computing Prüferlike treecode has been published.
On the Expected Number of LevelI Nodes in a Random Labeled Tree
, 2004
"... A queuebased Pruferlike code is used to determine the expected number of leveli nodes in a random labeled tree on n nodes. Level1 nodes are the leaves of a given tree and leveli nodes are leaves after all nodes in levels 1 through (i1) have been deleted. More precisely, we study the expect ..."
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Cited by 2 (1 self)
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A queuebased Pruferlike code is used to determine the expected number of leveli nodes in a random labeled tree on n nodes. Level1 nodes are the leaves of a given tree and leveli nodes are leaves after all nodes in levels 1 through (i1) have been deleted. More precisely, we study the expected fraction f(i) of n nodes that are in levels 1 through i. Tight bounds on f(i) are obtained and used to estimate the expected radius of a random labeled tree.
Bijective Linear Time Coding and Decoding for kTrees
"... The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. ktrees are one of the most natural and interesting generalizations of trees and there is considerab ..."
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The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. ktrees are one of the most natural and interesting generalizations of trees and there is considerable interest in developing efficient tools to manipulate this class of graphs, since many NPComplete problems have been shown to be polynomially solvable on ktrees and partial ktrees. In 1970 Rényi and Rényi generalized the Prüfer code, the first bijective code for trees, to a subset of labeled ktrees. Subsequently, non redundant codes that realize bijection between ktrees (or Rényi ktrees) and a well defined set of strings were produced. In this paper we introduce a new bijective code for labeled ktrees which, to the best of our knowledge, produces the first coding and decoding algorithms running in linear time with respect to the size of the ktree.
Parallel Algorithms for Encoding and Decoding Blob Code
"... Abstract. A bijective code is a method for associating labeled ntrees to (n − 2)strings of node labels in such a way that different trees yield different strings and vice versa. For all known bijective codes, optimal sequential encoding and decoding algorithms are presented in literature, while pa ..."
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Abstract. A bijective code is a method for associating labeled ntrees to (n − 2)strings of node labels in such a way that different trees yield different strings and vice versa. For all known bijective codes, optimal sequential encoding and decoding algorithms are presented in literature, while parallel algorithms are investigated only for some of these codes. In this paper we focus our attention on the Blob code: a code particularly considered in the field of Genetic Algorithms. To the best of our knowledge, here we present the first parallel encoding and decoding algorithms for this code. The encoding algorithm implementation is optimal on an EREW PRAM, while the decoding algorithm requires O(log n) time and O(n) processors on CREW PRAM. 1