Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology

Conference · Tue Dec 31 04:00:00 EST 1996

OSTI ID:416816

Henzinger, M R ^[1]; King, V ^[2]; Warnow, T ^[3]

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1. Cornell Univ., Ithaca, NY (United States)
2. Univ. of Victoria, British Columbia (Canada)
3. Univ. of Pennyslvania, Philadelphia, PA (United States)

We are given a set T = (T{sub 1}, T{sub 2},...,T{sub k}) of rooted binary trees, each T{sub i} leaf-labeled by a subset (T{sub i}) {contained_in}(1, 2,..., n). If T is a tree on (1,2,..., n), we let TJC denote the subtree of T induced by the nodes of C and all their ancestors. The consensus tree problem asks whether there exists a tree T* such that for every i, T*{vert_bar}(T{sub i}) is homeomorphic to Ti. We present algorithms which test if a given set of trees has a consensus tree and if so, con- struct one. The deterministic algorithm takes time min (O(mn{sup 1/2}), O(m + n{sup 2} log n)), where m = {Sigma}{sub i} {vert_bar}T{sub i}{vert_bar} and uses linear space. The randomized algorithm takes time O(m log{sup 3} n) and uses linear space. The previous best for this problem was an 1981 O(mn) algorithm by Aho et al. Our faster deterministic algorithm uses a new efficient algorithm for the following interesting dynamic graph problem: Given a graph G with n nodes and m edges and a sequence of b batches of one or more edge deletions, then after each batch, either find a new component that has just been created or determine that there is no such component. For this problem, we have a simple algorithm with running time O(n{sup 2} log n + b{sub 0} min(n{sup 2}, m log n)), where b{sub 0} is the number of batches which do not result in a new component. For our particular application, b{sub 0} {le} 1. If all edges are deleted, then the best previously known deterministic algorithm requires time 0(m{radical}n) to solve this problem.

• OSTI ID: 416816
• Report Number(s): CONF-960121--
• Country of Publication: United States
• Language: English

Related Subjects

55 BIOLOGY AND MEDICINE
BASIC STUDIES
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
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DIAGRAMS
MAXIMUM-LIKELIHOOD FIT