A Unified Approach to Coding Labeled Trees

Caminiti, Saverio; Finocchi, Irene; Petreschi, Rossella

doi:10.1007/978-3-540-24698-5_38

Cited by 12 publications

(22 citation statements)

References 8 publications

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“…After the Algorithm A terminates, c = [2,4,0,1,3,3] which is the Prufer code of T . The algorithms for coding and decoding Prufer codes of a labeled tree in the literatures require time usually.…”

Section: End Of Algorithmmentioning

confidence: 99%

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An Optimal Algorithm for Prufer Codes

Wang¹,

Wang

Wu³

2009

JSEA

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Section: End Of Algorithmmentioning

confidence: 99%

“…Follows the decoding algorithm on line 6 the next node to be added is 2. c [1]=4 is the other node label of the edge to be added. The edge {2,4} is added and d [4] is decreased by 1 on line 5 of the decoding algorithm.…”

Section: Examplesmentioning

confidence: 99%

An Optimal Algorithm for Prufer Codes

Wang¹,

Wang

Wu³

2009

JSEA

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“…To encode a tree as its corresponding Prüfer string, we iteratively delete the leaf vertex with the smallest label and write down its unique neighbour, until just a single edge remains. For example, the unique Prüfer string corresponding to the tree T ∈ T 15 shown in Figure 1 below is P = (12,6,15,15,6,6,3,11,1,11,1,3,15) ∈ P 15 . (In this example, the vertex deletions occur in the following order: 2, 4, 5, 7, 8, 9, 6, 10, 12, 13, 11, 1, 3.)…”

Section: The Prüfer Code Encoding Algorithm (From Tree To Prüfer String)mentioning

confidence: 99%

Developing New Locality Results for the Prüfer Code using a Remarkable Linear-Time Decoding Algorithm

Paulden¹,

Smith²

2007

Electron. J. Combin.

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The Prüfer Code is a bijection between the $n^{n-2}$ trees on the vertex set $[1,n]$ and the $n^{n-2}$ strings in the set $[1,n]^{n-2}$ (known as Prüfer strings of order $n$). Efficient linear-time algorithms for decoding (i.e., converting string to tree) and encoding (i.e., converting tree to string) are well-known. In this paper, we examine an improved decoding algorithm (due to Cho et al.) that scans the elements of the Prüfer string in reverse order, rather than in the usual forward direction. We show that the algorithm runs in linear time without requiring additional data strutures or sorting routines, and is an 'online' algorithm — every time a new string element is read, the algorithm can correctly output an additional tree edge without any knowledge of the future composition of the string. This new decoding algorithm allows us to derive results concerning the 'locality' properties of the Prüfer Code (i.e., the effect of making small changes to a Prüfer string on the structure of the corresponding tree). First, we show that mutating the $\mu$th element of a Prüfer string (of any order) causes at most $\mu + 1$ edge-changes in the corresponding tree. We also show that randomly mutating the first element of a random Prüfer string of order $n$ causes two edge-changes in the corresponding tree with probability $2(n-3)/n(n-1)$, and one edge-change otherwise. Then, based on computer-aided enumerations, we make three conjectures concerning the locality properties of the Prüfer Code, including a formula for the probability that a random mutation to the $\mu$th element of a random Prüfer string of order $n$ causes exactly one edge-change in the corresponding tree. We show that if this formula is correct, then the probability that a random mutation to a random Prüfer string of order $n$ causes exactly one edge-change in the corresponding tree is asymptotically equal to one-third, as $n$ tends to infinity.

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“…Several different bijective codes that realize associations between labeled trees and strings of labels have been introduced, see for example [9,13,15,[23][24][25][26]. From an algorithmic point of view, the problem has been investigated thoroughly and optimal coding and decoding algorithms for these codes are known [4,7,9,13].…”

Section: Introductionmentioning

confidence: 99%

Bijective Linear Time Coding and Decoding for k-Trees

2008

Self Cite

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Abstract. 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. k-trees 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 NP-Complete problems have been shown to be polynomially solvable on k-trees and partial k-trees. In 1970 Rényi and Rényi generalized the Prüfer code, the first bijective code for trees, to a subset of labeled k-trees. Subsequently, non redundant codes that realize bijection between k-trees (or Rényi k-trees) and a well defined set of strings were produced. In this paper we introduce a new bijective code for labeled k-trees 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 k-tree.

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A Unified Approach to Coding Labeled Trees

Cited by 12 publications

References 8 publications

An Optimal Algorithm for Prufer Codes

An Optimal Algorithm for Prufer Codes

Developing New Locality Results for the Prüfer Code using a Remarkable Linear-Time Decoding Algorithm

Bijective Linear Time Coding and Decoding for k-Trees

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