54 research outputs found
Analogies between the crossing number and the tangle crossing number
Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with leaves decreases the tangle crossing number by at most , and this
is sharp. Additionally, if is the maximum tangle crossing number of
a tanglegram with leaves, we prove
. Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in time. This lower bound may be tight, even for
tanglegrams with tangle crossing number .Comment: 13 pages, 6 figure
Drawing Binary Tanglegrams: An Experimental Evaluation
A binary tanglegram is a pair of binary trees whose leaf sets are in
one-to-one correspondence; matching leaves are connected by inter-tree edges.
For applications, for example in phylogenetics or software engineering, it is
required that the individual trees are drawn crossing-free. A natural
optimization problem, denoted tanglegram layout problem, is thus to minimize
the number of crossings between inter-tree edges.
The tanglegram layout problem is NP-hard and is currently considered both in
application domains and theory. In this paper we present an experimental
comparison of a recursive algorithm of Buchin et al., our variant of their
algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer
quadratic program that yields optimal solutions.Comment: see
http://www.siam.org/proceedings/alenex/2009/alx09_011_nollenburgm.pd
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
On the enumeration of tanglegrams and tangled chains
Tanglegrams are a special class of graphs appearing in applications
concerning cospeciation and coevolution in biology and computer science. They
are formed by identifying the leaves of two rooted binary trees. We give an
explicit formula to count the number of distinct binary rooted tanglegrams with
matched vertices, along with a simple asymptotic formula and an algorithm
for choosing a tanglegram uniformly at random. The enumeration formula is then
extended to count the number of tangled chains of binary trees of any length.
This includes a new formula for the number of binary trees with leaves. We
also give a conjecture for the expected number of cherries in a large randomly
chosen binary tree and an extension of this conjecture to other types of trees
Visualizing Co-Phylogenetic Reconciliations
We introduce a hybrid metaphor for the visualization of the reconciliations
of co-phylogenetic trees, that are mappings among the nodes of two trees. The
typical application is the visualization of the co-evolution of hosts and
parasites in biology. Our strategy combines a space-filling and a node-link
approach. Differently from traditional methods, it guarantees an unambiguous
and `downward' representation whenever the reconciliation is time-consistent
(i.e., meaningful). We address the problem of the minimization of the number of
crossings in the representation, by giving a characterization of planar
instances and by establishing the complexity of the problem. Finally, we
propose heuristics for computing representations with few crossings.Comment: This paper appears in the Proceedings of the 25th International
Symposium on Graph Drawing and Network Visualization (GD 2017
Exact Bipartite Crossing Minimization under Tree Constraints
A tanglegram consists of a pair of (not necessarily binary) trees T_1, T_2 with leaf sets L_1, L_2. Additional edges, called tangles, may connect nodes in L_1 with those in L_2. The task is to draw the tanglegram with a minimum number of tangle edge crossings while making sure that no crossing occurs between edges within each tree. This problem has relevant applications in computational biology, e.g., for the comparison of phylogenetic trees. In this work, we show that the problem can be formulated as a quadratic linear ordering problem (QLO) with additional side constraints. It was already shown that, appropriately reformulated, the QLO polytope is a face of some cut polytope. It turns out that the additional side constraints arising in our application do not destroy this property. Therefore, any polyhedral approach to max-cut can be used in our context. We present experimental results for drawing random and realistic tanglegrams using both linear and semidefinite programming techniques, showing that our approach is very efficient in practice
On trees, tanglegrams, and tangled chains
International audienceTanglegrams are a class of graphs arising in computer science and in biological research on cospeciation and coevolution. They are formed by identifying the leaves of two rooted binary trees. The embedding of the trees in the plane is irrelevant for this application. We give an explicit formula to count the number of distinct binary rooted tanglegrams with n matched leaves, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This work gives a new formula for the number of binary trees with n leaves. Several open problems and conjectures are included along with pointers to several followup articles that have already appeared
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