New uses for uniform lifted alignments

The phylogenetic alignment problem a k a the tree alignment problem arises in e orts to deduce histories of molecular evolution and in certain methods to multiply align more than two sequences The problem is known to be NP hard but several bounded error approximation methods and polynomial time approximation schemes have been developed for the problem The rst of these approximationmethods is based on what are called lifted alignments and the second method is based on simpler uniform lifted alignments The simplicity of uniform lifted alignments compared to lifted alignments allows a deeper study of their properties and yet also gives a way to derive or compute results about lifted and optimal phylogenetic alignments In this paper we rst prove the factor of two error bound on the optimal uniform lifted alignment di erently than was previously done in Next we use uniform lifted alignments to establish error bounds on random lifted alignments Finally we use results about uniform lifted alignments to create an e cient algorithm to compute a non trivial lower bound on the cost of the optimal solution to the phylogenetic alignment problem given any problem instance We use that lower bound to gauge the accuracy of a phylogenetic alignment computed by Sanko et al AMS Subject Classi cation Primary Q Secondary C R C C B D Phylogenetic tree Alignment Evolutionary history is frequently represented by an evolutionary tree where known extant organ isms are represented at the leaves of the tree and their unknown but perhaps deduced ancestors are represented at internal nodes of the tree It is common now to deduce such evolutionary trees from molecular sequence data obtained from the organisms under study However the opposite direction of study is also possible When the evolutionary tree is already known from previous data and deductions it can be used to deduce possible ancestral molecular sequences that gave rise to the extant sequences through a series of mutational events This general problem has been called the phylogenetic alignment problem or the tree alignment problem and has been formalized as the problem of deducing sequences at the internal nodes to minimize the cost given by an objective function de ned below Partially supported by Dept of Energy grant DE FG ER In the above description the tree with its deduced internal node labels is the desired output of the problem However once the labeled tree is in hand one can also use it to nd a multiple alignment of the extant sequences which is in uenced by the hypothesized evolutionary history see or The details are a bit involved and we only mention this application as additional motivation for the phylogenetic alignment problem We will not discuss it further in this paper