Constructing Optimal Trees from Quartets BY BRYANT AND STEEL, - - PowerPoint PPT Presentation

Constructing Optimal Trees from Quartets

BY BRYANT AND STEEL, JOURNAL OF ALGORITHMS 2001, 38, 237-259

General Problem

• How to best construct a large phylogenetic tree from a set of smaller subtrees
• Having a collection of subtrees is common
• Computational Complexity
• Biological and Statistical Advantages
• Finding a phylogenetic tree that agrees with the largest number of quartets is NP-Hard

?

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Split Constrained Quartet Optimization Problem

• Input:
• Weighting w for quartets on leaf set L
• Set S of allowed splits (i.e. bipartitions) of L
• Parameter:
• Degree bound d
• Output:
• Return a tree T with vertex degree bounded by d

and splits(T) ⊆ S such that w(T) is maximized

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Main Result

• An algorithm for split constrained quartet optimization
• Can be solved in polynomial time for bounded degree d
• Time Complexity of O(n4k + n2dkd-1) where n = |L| and k = |S|

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Algorithm: Overview

1. Construct an ordered set of clusters C from input S, the set of allowed splits 2. Precompute certain weight values associated with each cluster for later use 3. Compute a complete decomposition table D for C with degree bound d 4. Compute another decomposition table Doptcontaining the optimal trees in D

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Algorithm: Example Inputs

• Input:
• Leaf set {a, b, c, d, e}
• Weight equal to 1 for the quartet AB|CD, zero otherwise
• Parameter
• Degree bound d equal to 3 (binary tree)

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{i} Set of Allowed Splits 1 A|BCDE 2 B|ACDE 3 C|ABDE 4 D|ABCE 5 AB|CDE 6 AC|BDE 7 AD|BCE 8 ABC|DE 9 E|ABCD

Algorithm: Constructing Clusters

• Construct an ordered set of clusters C from input S
• Let x be an arbitrary leaf in L (e.g. choose E)
• C = {X : X|Y ∈ S, x ∈ Y}

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{i} Set of Allowed Splits Ci 1 A|BCDE A 2 B|ACDE B 3 C|ABDE C 4 D|ABCE D 5 AB|CDE AB 6 AC|BDE AC 7 AD|BCE AD 8 ABC|DE ABC 9 E|ABCD ABCD

Algorithm: Decomposition Table for C

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{i} Set of Allowed Splits Ci Di 1 A|BCDE A

• 2

B|ACDE B

• 3

C|ABDE C

• 4

D|ABCE D

• 5

AB|CDE AB [1,2] 6 AC|BDE AC [1,3] 7 AD|BCE AD [1,4] 8 ABC|DE ABC [3,5] [2,6] 9 E|ABCD ABCD [4,8]

Algorithm: Maximum Quartet Weight

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{i} Set of Allowed Splits Ci Di mi Dopt 1 A|BCDE A

• 2

B|ACDE B

• 3

C|ABDE C

• 4

D|ABCE D

• 5

AB|CDE AB [1,2] [1,2] 6 AC|BDE AC [1,3] [1,3] 7 AD|BCE AD [1,4] [1,4] 8 ABC|DE ABC [3,5] [2,6] 1 [3,5] 9 E|ABCD ABCD [4,8] 1 [4,8]

Algorithm: Tree Achieving Maximum Quartet Weight with Allowed Splits

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A B C D A B C D E Unroot

Additional Results

• Algorithm can be improved to O(n5) time when
• S is weakly compatible
• d equals 3 or 4
• Polynomial time solvable without degree bound when
• all quartet weights are non-negative
• S is maximum weakly compatible

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Additional Results (cont.)

• Split constrained quartet optimization is NP-Complete when
• Degree d is not bounded
• Even if S is weakly compatible and all quartet weights are non-negative
• Degree d is not bounded and some quartet weights are negative
• Even if Splits(S) = T for some tree T

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Limitations and Extensions

• Understanding and generating the search space of splits
• Relationship between quartet optimization criteria and other measures of accuracy
• Importance of the weighting function
• Implementation of algorithm to study performance in practice
• Especially compared to other leading methods
• Progress since 2001

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