A Randomized Linear-time Majority Tree Algorithm

Nina Amenta∗ Frederick Clarke† Katherine St. John†

Abstract

We give a randomized linear-time algorithm for computing themajority tree, a technique widely used for summarizing sets ofphylogenetic trees. We are implementing the algorithm as partof an interactive visualization system for analyzing large sets oftrees.(Keywords: phylogeny, majority consensus, visualization)

With the recent explosion in the amount of genomicdata available, and exponential increases in computingpower, biologists are now able to consider larger scaleproblems in phylogeny, such as the construction of evo-lutionary trees on hundreds or thousands of taxa, and ul-timately of the entire “Tree of Life” which would includemillions of taxa. One difficulty is that most programs usedfor phylogeny reconstruction [6, 7, 11] are based uponheuristics for NP-hard optimization problems which gen-erally output hundreds or thousands of likely candidatesfor the optimal tree, instead of producing a single optimaltree. This large volume of data is usually summarizedwith a consensus tree.

A consensus tree, for a set of input trees, is a single treewhich includes features on which all or most of the in-put trees agree. The simplest is thestrict consensus tree,which includes only subtrees that appear in all of the in-put trees. Themajority tree, or Ml tree, includes nodesfor exactly those subtrees which occur in more than halfof the input trees, or more generally in more than somefraction l of the input trees (see Figure 1). Margush andMcMorris [9] showed that this set of bipartitions does in-deed constitute a tree for any1/2 < l ≤ 1. McMorris,Meronk and Neumann [10] called this family of trees theMl trees (e.g. theM1 tree is the strict consensus tree); weshall call them all generically majority trees. While ourexample is for rooted binary trees, this can also be appliedto non-binary and unrooted trees. The majority tree is in-teresting for a much broader range of inputs than the strictconsensus tree (see [2],§6.2, for an excellent overview).

We have developed a randomized algorithm to computethe majority tree of a set of input trees, where the expected

∗amenta@cs.ucdavis.edu . Computer Science Department,University of California, Davis, CA 95616. Supported by an Alfred P.Sloan Foundation Research Fellowship and NSF ITR DEB-0121651.

†fclarke72@aol.com , stjohn@lehman.cuny.edu . Math& Computer Science, Lehman College– City University of New York,Bronx, NY 10468. Supported by NSF ITR DEB-0121682.

running time is linear both in the numbert of treesand inthe numbern of taxa. Earlier algorithms were quadraticin n, which is problematic for larger phylogenies. OurO(tn) expected running time is optimal, since just read-ing a set oft trees onn taxa requiresΩ(tn) time. Theexpectation in the running time is over random choicesmade during the course of the algorithm, independent ofthe input; thus, on any input, the running time is linearwith high probability.

Having an algorithm which is efficient int is essen-tial, and most earlier algorithms focus on this. Largesets of trees arise given any kind of input data on thetaxa (e.g. gene sequence, gene order, character) and what-ever optimization criterion is used to select the “best”tree. The heuristic searches used for maximizing parsi-mony often return large sets of trees with equal parsimonyscores. Maximum likelihood estimation, also computa-tionally hard, generally produces trees with unique scores.While technically one of these is the optimal tree, thereare many others for which the likelihood is only negligi-bly sub-optimal. The output of the computation is againmore accurately represented by a consensus tree.

As the number of taxa which can be handled increasesinto the thousands, having an algorithm which is linearin n is also becoming important, especially for interactivesoftware. We were motivated to find an efficient algorithmfor the majority tree because we wanted to compute it on-the-fly in an interactive visualization application [1]. Fig-ure 2 shows a screen shot. The window on the left showsa representation of the distribution of trees, where eachpoint corresponds to a tree. The user interactively selectssubsets of trees and, in response, the consensus tree of thesubset is computed on-the-fly and displayed. This pack-age is built as a module within Mesquite [8], a frameworkfor phylogenetic computation by Wayne and David Mad-dison. Our original version of the visualization system

s0 s1 s2 s3 s4 s0 s1 s2 s3 s4 s0 s1 s2 s3s4 s0 s1 s2 s3 s4

T1 T2 T3 Majority tree

Figure 1:Three rooted input trees and their majority tree (for a50 percent majority). The input trees need not be binary.

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Figure 2:The tree visualization module in Mesquite. The win-dow on the left shows a projection of the distribution of trees.The user interactively selects subsets of trees with the mouse,and, in response, the consensus tree of the subset is computedon-the-fly and displayed in the window on the right. Two se-lected subsets and their majority trees are shown.

computed only strict consensus trees. We found in ourprototype implementation that a simpleO(tn2) algorithmfor the strict consensus tree was unacceptably slow, andwe implemented instead theO(tn) strict consensus algo-rithm of Day [5]. This inspired our search for a linear-timemajority tree algorithm.

Prior Work: For the strict consensus tree, Day’s de-terministic algorithm uses a cleverO(1) representationfor subtrees, and also achieves an optimalO(tn) runningtime, which does not generalize easily to otherMl trees.Wareham [12], developed anO(n2 + t2n) algorithm forthe Ml tree, which only usesO(n) space. It uses Day’sdata structure to test each subtree encountered against allof the other input trees. Another algorithm for major-ity trees is implemented in PHYLIP [6] by Felsensteinetal.. The overall running time as implemented seems to beO(tn2/(lg tn) + tn lg(tn) + n3/(lg tn)). Majority treesare also computed by PAUP [11], using an unknown (tous) algorithm.

Outline of Algorithm: A linear-time majority tree al-gorithm has been somewhat elusive. Our algorithm hastwo stages, similar to past algorithms. In the first stage,we read through the input trees and count the occurrencesof each subtree, storing the counts in a hash table. Then,in the second stage, we create nodes for the subtrees thatoccur in a majority of input trees - themajority subtrees-and “hook them” together into a tree. We sketch the algo-rithm briefly below; more details and proofs will appearin the full paper.

We achieve the linear running time by reducing thespace and time used for counting the common subtrees.The natural bit-string representation for a subtree has sizeO(n/w), wherew, the number of bits in a word, is usu-

ally taken to beO(lg tn). We introduce a representationof sizeO((lg tn)/w), which givesO(1). The represen-tation of a subtree is its constant-size hash code. We usea specificuniversal hash functionfor which we can com-pute the hash code for a node in constant time, given thehash codes for its children, so that we can compute hashcodes bottom-up for all of theO(tn) subtrees inO(tn)time.

We also give anO(tn) algorithm for hooking togetherthe majority nodes. Hooking together the nodes is nolonger trivial since we do not have explict representationsof the majority subtrees that occur in morelt trees. Weagain traverse the set of input trees, keeping track of theancestor-descendant relationships of the majority subtreesencountered. Ifp is the parent of a subtrees in the major-ity tree, the Pigeonhole Principal implies that there is atleast one input tree in whichp is an ancestor ofs; we canrecognize it becausep will be the ancestor ofs of mini-mum cardinality observed during the traversal.

The hash function is randomized, so there is a smallpossibility that collisions in the hash table may cause thealgorithm to fail. We detect this event, and repeat the pro-cess with new random choices. The expected number ofattempts at building the tree is less than two, so our ex-pected running time is linear in both the number of treesand the number of taxa.

Implementation Our majority tree algorithm is beingimplemented within Mesquite [8], a framework for phy-logenetic analysis written by Wayne and David Maddison.Mesquite is designed to be portable and extensible. It iswritten in Java and runs on a variety of operating systems(Linux, MacIntosh OS 9 and X, and Windows). Mesquiteis made up of cooperating modules. The first version ofthe module for our visualization system, TreeSetVisual-ization, can be downloaded from our webpage [1]. Themodule (including only the strict consensus) was intro-duced this summer at Evolution 2002 and has since beendownloaded by hundreds of researchers. In the commu-nications we get from users, majority trees are frequentlyrequested.

We are implementing the majority tree algorithm aspart of the next version. Figure 2 shows our current proto-type. The speed of the majority tree function seems com-parable to our linear-time strict consensus tree implemen-tation; our final paper will give comparisons.

Acknowledgments: We thank Jeff Klingner for the treeset visualization module and Wayne and David Maddisonfor Mesquite, and for encouraging us to consider the ma-jority tree.

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References

[1] Nina Amenta and Jeff Klingner. Case study: Vi-sualizing sets of evolutionary trees. In8th IEEESymposium on Information Visualization (InfoVis2002), pages 71–74, 2002. Software available atwww.cs.utexas.edu/users/phylo/ .

[2] David Bryant. Hunting for trees, building trees andcomparing trees: theory and method in phylogeneticanalysis. PhD thesis, Dept. of Mathematics, Univer-sity of Canterbury, 1997.

[3] J. Lawrence Carter and Mark N. Wegman. Universalclasses of hash functions.Journal of Computer andSystems Sciences, 18(2):143–154, 1979.

[4] Thomas H. Cormen, Charles E. Leiserson,Ronald L. Rivest, and Clifford Stein. Intro-duction to algorithms. MIT Press, Cambridge, MA,second edition, 2001.

[5] William H.E. Day. Optimal algorithms for com-paring trees with labeled leaves.J. Classification,2(1):7–28, 1985.

[6] J. Felsenstein. Phylip (phylogeny inference pack-age) version 3.6, 2002. Distributed by the author.Department of Genetics, University of Washington,Seattle. The consensus tree code is inconsense.cand is co-authored by Hisashi Horino, Akiko Fuseki,Sean Lamont and Andrew Keeffe.

[7] John P. Huelsenbeck and Fredrik Ronquist. Mr-bayes: Bayesian inference of phylogeny, 2001.

[8] W.P. Maddison and D.R. Maddison. Mesquite:a modular system for evolutionary analy-sis. version 0.992, 2002. Available fromhttp://mesquiteproject.org .

[9] T. Margush and F.R. McMorris. Consensus n-trees. Bulletin of Mathematical Biology, 43:239–244, 1981.

[10] F.R. McMorris, D.B. Meronk, and D.A. Neumann.A view of some consensus methods for trees. InNumerical Taxonomy: Proceedings of the NATOAdvanced Study Institute on Numerical Taxonomy.Springer-Verlag, 1983.

[11] D.L. Swofford. PAUP*. Phylogenetic AnalysisUsing Parsimony (*and Other Methods). Version4. Sinauer Associates, Sunderland, Massachusetts,2002.

[12] H. Todd Wareham. An efficient algorithm for com-putingMl consensus trees, 1985. BS honors thesis,CS, Memorial University Newfoundland.

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