Lossless Compression of Semi-Ordered Trees

In this paper, authors are interested in the problem of lossless compression of unlabeled semi-ordered trees. Semi-ordered trees are a class of trees that present an order between some sibling while some other sibling are unordered. They offer a wide possibility of applications especially for the representation of plants architecture. Authors show that these trees present remarkable compression properties covering those of ordered and unordered trees. To illustrate this approach, authors apply these notions to a particular class of semi-ordered trees which is the most studied branching structure particularly for a botanical motivation, namely axial trees.

A Canonical String Encoding for Pure Bigraphs

The bigraph theory, devised by Robin Milner, is a recent mathematical framework for concurrent processes. Its generality is able to subsume many existing process calculi, for example, CCS, CSP, and Petri nets. Further, it provides a uniform proof of bisimilarity, which is a congruence. We present the first canonical string encoding for pure and lean bigraphs by lifting the breadth-first canonical form of rooted unordered trees to a unique representation for bigraphs up to isomorphism (i.e., lean-support equivalence). The encoding’s applicability is limited to atomic alphabets. The time complexity is $$O(n^{2}k\, d \log {d})$$ O ( n 2 k d log d ) , where n is the number of places, d the degree of the place graph and k the maximum arity of a bigraph’s signature. We provide proof of the correctness of our method and also conduct experimental measurements to assess the complexity.

Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

We consider the problem of computing the triplet distance between two rooted unordered trees with n labeled leaves. Introduced by Dobson in 1975, the triplet distance is the number of leaf triples that induce different topologies in the two trees. The current theoretically fastest algorithm is an O( n log n ) algorithm by Brodal et al. (SODA 2013). Recently, Jansson and Rajaby proposed a new algorithm that, while slower in theory, requiring O( n log 3 n ) time, in practice it outperforms the theoretically faster O( n log n ) algorithm. Both algorithms do not scale to external memory. We present two cache oblivious algorithms that combine the best of both worlds. The first algorithm is for the case when the two input trees are binary trees, and the second is a generalized algorithm for two input trees of arbitrary degree. Analyzed in the RAM model, both algorithms require O( n log n ) time, and in the cache oblivious model O( n / B log 2 n / M ) I/Os. Their relative simplicity and the fact that they scale to external memory makes them achieve the best practical performance. We note that these are the first algorithms that scale to external memory, both in theory and in practice, for this problem.

Generating Trees for Comparison

Tree comparisons are used in various areas with various statistical or dissimilarity measures. Given that data in various domains are diverse, and a particular comparison approach could be more appropriate for specific applications, there is a need to evaluate different comparison approaches. As gathering real data is often an extensive task, using generated trees provides a faster evaluation of the proposed solutions. This paper presents three algorithms for generating random trees: parametrized by tree size, shape based on the node distribution and the amount of difference between generated trees. The motivation for the algorithms came from unordered trees that are created from class hierarchies in object-oriented programs. The presented algorithms are evaluated by statistical and dissimilarity measures to observe stability, behavior, and impact on node distribution. The results in the case of dissimilarity measures evaluation show that the algorithms are suitable for tree comparison.