Constructing evolutionary trees from rooted triples Bang Ye Wu Dept. of Computer Science and Information Engineering Shu-Te University

An evolutionary tree  A rooted tree  Each leaf represents one species.  Internal nodes are unlabelled. (inferred common ancestors) abcdef

A (rooted) triple (triplet)  An evolutionary tree of 3 species.  A constraint in an evolutionary tree construction problem.  (c(ab)): lca(b,c)=lca(c,a)  lca(a,b) lca : lowest common ancestor  : “ is an ancestor of “  a,b should be closer than a,c or b,c. abc

A tree compatible with triples  Given a set of triples, construct a tree satisfying all the triples.  If such a tree exists, the problem is polynomial time solvable. [Aho et al, 1981]

Incompatible (conflicting) triples Two conflicting triplesThree conflicting triples (pairwise compatible)

Two optimization problems  The maximum consensus tree: –the tree satisfying maximum number of triples. –NP-hard [Jansson, 2001][Wu, to appear] –A new heuristic algorithm [this paper]  The maximum compatible set: –The compatible species subset of maximum cardinality. –NP-hard [this paper]

Previous heuristic Best-One-Split-First  If a species x is split from a set V, all triples (x(v 1 v 2 )), v 1 and v 2 in V, will be satisfied.  Repeatedly split one species from the set. Choose the split species greedily.

{a,b,d} c b {a,d} c dabc c is chosen, and the two triples is satisfied. c is split b is split

Previous heuristic Min-Cut-Split-First  Construct an auxiliary graph: –Vertex: species –Each edge is labeled by a set: for each triple (x(yz)), x is in the label set of edge (y,z).

–A bipartition corresponds to a split in the tree. –The label in the cut of the bipartition corresponds to the triples conflicting the split.  Repeatedly find the bipartition with minimum cut. a min-cut, triple (c(bd)) is conflicting

Previous heuristic Best-Pair-Merge-First  Instead of top-down splitting, BPMF uses the bottom-up merging strategy.  Starting from sets of singleton, we repeatedly merge the sets step by step.  Scoring functions are used to evaluate which pair should be merged in each step.

{a}{b}{c}{d} {a,d}{b}{c} {a,d}{b,c} {a,d,b,c} ad adbc adbc

An exact algorithm for MCTT  Dynamic programming  F(V)=max{F(V 1 )+F(V 2 )+W(V 1,V 2 )}, taken among all bipartition (V 1,V 2 ) of V. –F(V): # of satisfied triples over V. –W(V 1,V 2 ): # of (x(v 1 v 2 ) for x not in V and v 1, v 2 in V 1, V 2 respectively.  Computed with cardinality from small to large.

n=4abcd 3 n=3abc 1 abd 3 bcd 2 n=2ab 0 ac 0 ad 2 bc 1 bd 1 cd 0 n=1a0a0 b0b0 c0c0 d0d0

Our new heuristic algorithm (DPWP)  Derived from the exact algorithm.  The number of subsets of each cardinality is limited by a parameter K.  When K=infinity, it is just the exact algorithm.  Time-quality trade-off.  The time complexity is O(n 2 k 2 (n 3 +k)). –Sorry, there is a mistake in the paper.

The experiment results (time)

The MCST problem  Given triples over species set S, find a subset U of S such that all given triples over U is compatible and |U| is maximum.  We show the problem is NP-hard. –Transformed from the Feedback Vertex Set problem.

The feedback vertex set problem  Feedback vertex set: a vertex subset containing at one vertex of each cycle of the given directed graph. –In other words, removing a feedback vertex set results in an acyclic digraph.

The reduction

Concluding remarks  What is the approximation ratio? –The Best-One-Split-First algorithm is a 3- approximation algorithm, –The larger K give us better solution, but we do not know the theoretic bound of the ratio.