Decomposition algorithms for the tree edit distance problem

We study the behavior of dynamic programming methods for the tree edit distance problem, such as [P. Klein, Computing the edit-distance between unrooted ordered trees, in: Proceedings of 6th European Symposium on Algorithms, 1998, p. 91–102; K. Zhang, D. Shasha, SIAM J. Comput. 18 (6) (1989) 1245–1262]. We show that those two algorithms may be described as decomposition strategies. We introduce the general framework of cover strategies, and we provide an exact characterization of the complexity of cover strategies. This analysis allows us to define a new tree edit distance algorithm, that is optimal for cover strategies.     

Pattern avoidance in binary trees

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns.     

Efficient Algorithms for Approximate String Matching with Swaps

Most research on the edit distance problem and thek-differences problem considered the set of edit operations consisting of changes, insertions, and deletions. In this paper we include theswapoperation that interchanges two adjacent characters into the set of allowable edit operations, and we present anO(t min(m, n))-time algorithm for the extended edit distance problem, wheretis the edit distance between the given strings, and anO(kn)-time algorithm for the extendedk-differences problem. That is, we add swaps into the set of edit operations without increasing the time complexities of previous algorithms that consider only changes, insertions, and deletions for the edit distance andk-differences problems.     

Tree-edges deletion problems with bounded diameter obstruction sets

We study the following problem: given a tree G and a finite set of trees H, find a subset O of the edges of G such that G-O does not contain a subtree isomorphic to a tree from H, and O has minimum cardinality. We give sharp boundaries on the tractability of this problem: the problem is polynomial when all the trees in H have diameter at most 5, while it is NP-hard when all the trees in H have diameter at most 6. We also show that the problem is polynomial when every tree in H has at most one vertex with degree more than 2, while it is NP-hard when the trees in H can have two such vertices.The polynomial-time algorithms use a variation of a known technique for solving graph problems. While the standard technique is based on defining an equivalence relation on graphs, we define a quasiorder. This new variation might be useful for giving more efficient algorithm for other graph problems.     

On spanning tree problems with multiple objectives

We investigate two versions of multiple objective minimum spanning tree problems defined on a network with vectorial weights. First, we want to minimize the maximum ofQ linear objective functions taken over the set of all spanning trees (max-linear spanning tree problem, ML-ST). Secondly, we look for efficient spanning trees (multi-criteria spanning tree problem, MC-ST).Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance between two consecutive solutions is less than a given accuracy.Partially supported by Deutsche Forschungsgemeinschaft and HCº Contract no. ERBCHRXCT 930087.Partially supported by Alexander von Humboldt-Stiftung.     

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.     

On agreement forests

Tree rearrangement operations are widely used to measure the dissimilarity between phylogenetic trees with identical leaf sets. The tree bisection and reconnection (tbr) distance for unrooted trees can be equivalently defined in terms of agreement forests. For both the tbr distance and the less general subtree prune and regraft (spr) distance, we use such forests to derive new upper and lower bounds on the maximal possible distance between two trees with n leaves.     

A bijection between ordered trees and bicoloured ordered trees

A bijection is introduced between ordered trees and bicoloured ordered trees, which maps leaves in an ordered tree to odd height vertices in the related tree. Consequently, a bijection between two sets of bridges specified with four parameters is derived.     

A dynamic edit distance table

In this paper we consider the incremental/decremental version of the edit distance problem: given a solution to the edit distance between two strings A and B, find a solution to the edit distance between A and B′ where B′=aB (incremental) or bB′=B (decremental). As a solution for the edit distance between A and B, we define the difference representation of the D-table, which leads to a simple and intuitive algorithm for the incremental/decremental edit distance problem.     

Partial γ-positivity for quasi-Stirling permutations of multisets

We prove that the enumerative polynomials of quasi-Stirling permutations of multisets with respect to the statistics of plateaux, descents and ascents are partial γ-positive, thereby confirming a recent conjecture posed by Lin, Ma and Zhang. This is accomplished by proving the partial γ-positivity of the enumerative polynomials of certain ordered labeled trees, which are in bijection with quasi-Stirling permutations of multisets. As an application, we provide an alternative proof of the partial γ-positivity of the enumerative polynomials on Stirling permutations of multisets.     

Optimal mappings with minimum number of connected components in tree-to-tree comparison problems

In this paper we consider the problem of finding a minimum cost mapping between two unordered trees which induces a graph with a minimum number of connected components. The proposed algorithm is based on the generalization of an algorithm for computing an edit distance between trees, and it solves the stated problem in sequential time precisely in O(|T₁|×|T₂|×(degT₁+degT₂)×log₂(degT₁+degT₂)).     

Comparison of alignment free string distances for complete genome phylogeny

In this paper, we compare the accuracy of four string distances on complete genomes to reconstruct phylogenies using simulated and real biological data. These distances are based on common words shared by raw genomic sequences and do not require preliminary processing steps such as gene identification or sequence alignment. Moreover, they are computable in linear time. The first distance is based on Maximum Significant Matches (MSM). The second is computed from the frequencies of all the words of length k (KW). The third distance is based on the Average length of maximum Common Substrings at any position (ACS). The last one is based on the Ziv–Lempel compression algorithm (ZL). We describe a simulation process of evolution to generate a set of sequences having evolved according to a random tree topology T. This process allows both base substitution and fragment insertion/deletion, including horizontal transfers. The distances between the generated sequences are computed using the four formulas and the corresponding trees T′ are reconstructed using Neighbor-Joining. T and T′ are compared according to topological criteria. These comparisons show that the MSM distance outperforms the others whatever the parameters used to generate sequences. Finally, we test the MSM and KW distances on real biological data (i.e. prokaryotic complete genomes) and we compare the NJ trees to a Maximum Likelihood 16S + 23S RNA tree. We show that the MSM distance provides accurate results to study intra-phylum relationships, much better than those given by KW.     

Revisiting dynamic programming for finding optimal subtrees in trees

In this paper we revisit an existing dynamic programming algorithm for finding optimal subtrees in edge weighted trees. This algorithm was sketched by Maffioli in a technical report in 1991. First, we adapt this algorithm for the application to trees that can have both node and edge weights. Second, we extend the algorithm such that it does not only deliver the values of optimal trees, but also the trees themselves. Finally, we use our extended algorithm for developing heuristics for the k-cardinality tree problem in undirected graphs G with node and edge weights. This NP-hard problem consists of finding in the given graph a tree with exactly k edges such that the sum of the node and the edge weights is minimal. In order to show the usefulness of our heuristics we conduct an extensive computational analysis that concerns most of the existing problem instances. Our results show that with growing problem size the proposed heuristics reach the performance of state-of-the-art metaheuristics. Therefore, this study can be seen as a cautious note on the scaling of metaheuristics.     

The distance-domination numbers of trees

The P-center problem is to locate P centers in a graph G so that the maximum distance between centers and non-centers is minimized. A related problem is to determine the maximum number of vertices that can be “covered” (within a distance of α) by a vertex set of cardinality P in G. In this paper we describe an O(n³P) algorithm which solves the maximum coverage problem on trees. We also apply the same idea to solve the P-median problem on trees.     

Efficient algorithms for wavelength assignment on trees of rings

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm which solves the WA problem on a tree of rings with an arbitrary (node) degree using at most 3L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3L upper bound is tight since there are instances of the WA problem that require 3L wavelengths even on a tree of rings with degree four. We also give a 3L and 2-approximation (resp. 2.5-approximation) algorithm for the WA problem on a tree of rings with degree at most six (resp. eight). Previous results include: 4L (resp. 3L) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).     

Uniformly continuous maps between ends of $${\mathbb{R}}$$-trees

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper Lipschitz maps) between the trees.     

Underestimating the cost of a soft constraint is dangerous: revisiting the edit-distance based soft regular constraint

Many real-life problems are over-constrained, so that no solution satisfying all their constraints exists. Soft constraints, with costs denoting how much the constraints are violated, are used to solve these problems. We use the edit-distance based SoftRegular constraint as an example to show that a propagation algorithm that sometimes underestimates the cost may guide the search to incorrect (non-optimal) solutions to an over-constrained problem. To compute correctly the cost for the edit-distance based SoftRegular constraint, we present a quadratic-time propagation algorithm based on dynamic programming and a proof of its correctness. We also give an improved propagation algorithm using an idea of computing the edit distance between two strings, which may also be applied to other constraints with propagators based on dynamic programming. The asymptotic time complexity of our improved propagator is always at least as good as the one of our quadratic-time propagator, but significantly better when the edit distance is small. Our propagators achieve domain consistency on the problem variables and bounds consistency on the cost variable. Our method can also be adapted for the violation measure of the edit-distance based Regular constraint for constraint-based local search.     

Constructing a 3-tree for a 3-connected matroid

In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M.