Bijective Linear Time Coding and Decoding for k-Trees

Abstract

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.

Learning Bounded Tree-Width Bayesian Networks via Sampling

“…Caminiti et al [5] proposed to establish a one-to-one correspondence between a k-tree and what is called Dandelion codes.…”

“…A code (Q, S) ∈ A n,k is a pair where Q ⊆ N is a set of integers of size k and S is a 2 × (n−k −2) matrix of integers drawn from N ∪ { }, where is an arbitrary number not in N (see [5] for details). Dandelion codes can be sampled uniformly at random by a trivial lineartime algorithm that uniformly chooses k elements out of N to build Q, and then uniformly samples n−k−2 pairs of integers in N ∪{ }.…”

“…Following the proposal distribution, we use the rejection sampling algorithm (Algorithm 1) to generate a sample of Dandelion codes, and then employ the implementation of [5] to decode it into a k-tree.…”

“…We design an approximate approach based on sampling k-trees, which are the maximal graphs of tree-width k. The sampling method is based on a fast bijection between k-trees and Dandelion codes [5]. We design a sampling scheme, called Distance Preferable Sampling (DPS), in order to effectively cover the space of k-trees using limited samples, in which we give a larger probability for a sample in the unexplored area of the space, based on the existing samples.…”

Efficient learning of Bayesian networks with bounded tree-width

“…We design an approximate approach based on sampling k-trees, which are the maximal graphs of tree-width k. The sampling method is based on a fast bijection between k-trees and Dandelion codes [9]. We design a sampling scheme, called distance preferable sampling (DPS), in order to effectively cover the space of k-trees using limited samples, in which we give a larger probability for a sample in the unexplored area of the space, based on the existing samples.…”