We present the first sub-quadratic time algorithm that with high probability correctly reconstructs phylogenetic trees for short sequences generated by a Markov model of evolution. Due to rapid expansion in sequence databases, such very fast algorithms are becoming necessary. Other fast heuristics have been developed for building trees from very large alignments (, and ), but they lack theoretical performance guarantees. Our new algorithm runs in @math time, where @math is an increasing function of an upper bound on the branch lengths in the phylogeny, the upper bound @math must be below @math , and @math for all @math . For phylogenies with very short branches, the running time of our algorithm is close to linear. For example, if all branch lengths correspond to a mutation probability of less than 0.02, the running time of our algorithm is roughly @math . Via a prototype and a sequence of large-scale experiments, we show that many large phylogenies can be reconstructed fast, without compromising reconstruction accuracy.
Erd o s @cite_19 gave an @math algorithm that reconstructs a phylogeny with high probability, assuming the Cavender-Farris model of evolution, for sufficiently long sequences. For most trees, their algorithm runs in @math time and requires @math sequence length. Cs u ros @cite_15 provided a @math algorithm with similar performance guarantees. Recent papers @cite_14 @cite_0 give similar algorithms to identify parts of the tree that can be reconstructed. These approaches use quartet queries chosen so that, with high probability, only correct quartets are queried. The only sub-quadratic time algorithm with guarantees on reconstruction accuracy is by King @cite_7 ; for most trees, its running time is @math provided that the sequences are @math in length.
@cite_14 @cite_7 @cite_0 @cite_19 @cite_15
2012625662 1989541071 1928730978 2066327035 2063021676
Distance-based phylogenetic methods attempt to reconstruct an accurate phylogenetic tree from an estimated matrix of pairwise distances between taxa. This paper examines two distance-based algorithms (GreedyBME and FastME) that are based on the principle of minimizing the balanced minimum evolution score of the output tree in relation to the given estimated distance matrix. This is also the principle that underlies the neighbor-joining (NJ) algorithm. We show that GreedyBME and FastME both reconstruct the entire correct tree if the input data are quartet consistent, and also that if the maximum error of any distance estimate is @math , then both algorithms output trees containing all sufficiently long edges of the true tree: those having length at least @math . That is to say, the algorithms have edge safety radius 1 3. In contrast, quartet consistency of the data is not sufficient to guarantee the NJ algorithm reconstructs the correct tree, and moreover, the NJ algorithm has edge safety radius of 1 4: Only edges of the true tree of length at least @math can be guaranteed to appear in the output. These results give further theoretical support to the experimental evidence suggesting FastME is a more suitable distance-based phylogeny reconstruction method than the NJ algorithm. Given natural limitations on the length DNA sequences, designing phylogenetic reconstruction methods which are reliable under limited information is a crucial endeavor. There have been two approaches to this problem: reconstructing partial but reliable information about the tree ( Mo07, DMR08,DHJ06,GMS08 ), and reaching "deeper" in the tree through reconstruction of ancestral sequences. In the latter category, DMR06 settled an important conjecture of M.Steel, showing that, under the CFN model of evolution, all trees on @math leaves with edge lengths bounded by the Ising model phase transition can be recovered with high probability from genomes of length @math with a polynomial time algorithm. Their methods had a running time of @math . Here we enhance our methods from DHJ06 with the learning of ancestral sequences and provide an algorithm for reconstructing a sub-forest of the tree which is reliable given available data, without requiring a-priori known bounds on the edge lengths of the tree. Our methods are based on an intuitive minimum spanning tree approach and run in @math time. For the case of full reconstruction of trees with edges under the phase transition, we maintain the same sequence length requirements as DMR06 , despite the considerably faster running time. Recently, we have identified a randomized quartet phylogeny algorithm that has O(nlogn) runtime with high probability, which is asymptotically optimal. Our algorithm has high probability of returning the correct phylogeny when quartet errors are independent and occur with known probability, and when the algorithm uses a guide tree on O(loglogn) taxa that is correct with high probability. In practice, none of these assumptions is correct: quartet errors are positively correlated and occur with unknown probability, and the guide tree is often error prone. Here, we bring our work out of the purely theoretical setting. We present a variety of extensions which, while only slowing the algorithm down by a constant factor, make its performance nearly comparable to that of Neighbour Joining , which requires Θ(n3) runtime in existing implementations. Our results suggest a new direction for quartet-based phylogenetic reconstruction that may yield striking speed improvements at minimal accuracy cost. An early prototype implementation of our software is available at http: www.cs.uwaterloo.ca jmtruszk qtree.tar.gz. Phylogenetic reconstruction is the problem of reconstructing an evolutionary tree from sequences corresponding to leaves of that tree. A central goal in phylogenetic reconstruction is to be able to reconstruct the tree as accurately as possible from as short as possible input sequences. The sequence length required for correct topological reconstruction depends on certain properties of the tree, such as its depth and minimal edge-weight. Fast converging reconstruction algorithms are considered state-of the-art in this sense, as they require asymptotically minimal sequence length in order to guarantee (with high probability) correct topological reconstruction of the entire tree. However, when the original phylogenetic tree contains very short edges, this minimal sequence-length is still too long for practical purposes. Short edges are not only very hard to reconstruct; their presence may also prevent the correct reconstruction of long edges. In this paper we present a fast converging reconstruction algorithm which returns a partially resolved topology containing all edges of the original tree whose weight exceeds some (non-trivial) lower bound, which is determined by the input sequence length, as well as some properties of the tree, such as its depth. It does not depend, however, on the minimal edge-weight. This lower bound provides a partial reconstruction guarantee which is strictly stronger than the guarantees given by other fast converging algorithms. Our algorithm also has optimal complexity (linear space and quadratic-time) which, together with its partial reconstruction guarantee, makes it appealing for practical use.