198,636 research outputs found
Phylogenetic Trees and Their Analysis
Determining the best possible evolutionary history, the lowest-cost phylogenetic tree, to fit a given set of taxa and character sequences using maximum parsimony is an active area of research due to its underlying importance in understanding biological processes. As several steps in this process are NP-Hard when using popular, biologically-motivated optimality criteria, significant amounts of resources are dedicated to both both heuristics and to making exact methods more computationally tractable. We examine both phylogenetic data and the structure of the search space in order to suggest methods to reduce the number of possible trees that must be examined to find an exact solution for any given set of taxa and associated character data. Our work on four related problems combines theoretical insight with empirical study to improve searching of the tree space. First, we show that there is a Hamiltonian path through tree space for the most common tree metrics, answering Bryant\u27s Challenge for the minimal such path. We next examine the topology of the search space under various metrics, showing that some metrics have local maxima and minima even with perfect data, while some others do not. We further characterize conditions for which sequences simulated under the Jukes-Cantor model of evolution yield well-behaved search spaces. Next, we reduce the search space needed for an exact solution by splitting the set of characters into mutually-incompatible subsets of compatible characters, building trees based on the perfect phylogenies implied by these sets, and then searching in the neighborhoods of these trees. We validate this work empirically. Finally, we compare two approaches to the generalized tree alignment problem, or GTAP: Sequence alignment followed by tree search vs. Direct Optimization, on both biological and simulated data
Optimal Binary Search Trees with Near Minimal Height
Suppose we have n keys, n access probabilities for the keys, and n+1 access
probabilities for the gaps between the keys. Let h_min(n) be the minimal height
of a binary search tree for n keys. We consider the problem to construct an
optimal binary search tree with near minimal height, i.e.\ with height h <=
h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta
optimal binary search trees with near minimal height can be constructed in time
O(n^2). This is as fast as in the unrestricted case.
So far, the best known algorithms for the construction of height-restricted
optimal binary search trees have running time O(L n^2), whereby L is the
maximal permitted height. Compared to these algorithms our algorithm is at
least faster by a factor of log n, because L is lower bounded by log n
Simplifying Random Satisfiability Problem by Removing Frustrating Interactions
How can we remove some interactions in a constraint satisfaction problem
(CSP) such that it still remains satisfiable? In this paper we study a modified
survey propagation algorithm that enables us to address this question for a
prototypical CSP, i.e. random K-satisfiability problem. The average number of
removed interactions is controlled by a tuning parameter in the algorithm. If
the original problem is satisfiable then we are able to construct satisfiable
subproblems ranging from the original one to a minimal one with minimum
possible number of interactions. The minimal satisfiable subproblems will
provide directly the solutions of the original problem.Comment: 21 pages, 16 figure
Counting smaller elements in the Tamari and m-Tamari lattices
We introduce new combinatorial objects, the interval- posets, that encode
intervals of the Tamari lattice. We then find a combinatorial interpretation of
the bilinear operator that appears in the functional equation of Tamari
intervals described by Chapoton. Thus, we retrieve this functional equation and
prove that the polynomial recursively computed from the bilinear operator on
each tree T counts the number of trees smaller than T in the Tamari order. Then
we show that a similar m + 1-linear operator is also used in the functionnal
equation of m-Tamari intervals. We explain how the m-Tamari lattices can be
interpreted in terms of m+1-ary trees or a certain class of binary trees. We
then use the interval-posets to recover the functional equation of m-Tamari
intervals and to prove a generalized formula that counts the number of elements
smaller than or equal to a given tree in the m-Tamari lattice.Comment: 46 pages + 3 pages of code appendix, 27 figures. Long version of
arXiv:1212.0751. To appear in Journal of Combinatorial Theory, Series
Connected and internal graph searching
This paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.Postprint (published version
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