1. Technion – Israel Institute of Technology
Adaptive Fast Convergence Towards Optimal Reconstruction Guarantees for Phylogenetic Trees
Ilan Gronau
Technion – Israel Institute of Technology
Haifa, Israel
Joint work with
Shlomo Moran , Sagi Snir

2. Phylogenetic Reconstruction
reconstructed tree
“true tree” F D F B G B G A C D E A C E H I J H I J k
A: TGGAC … ATT
B: TGAAC … ATT
C: GGGAT … ACT
J: TGGAG … TCT
Goal: reconstruct the true tree as accurately as possible
December 5, 2018
PLGW03 - Cambridge, UK

3. Evaluating Reconstructed Tree
“true tree”
reconstructed tree
contraction F D D F B G B G A C E H I J A C E H I J
False Negatives: edges in the true tree which we don’t reconstruct
False Positives: edges we reconstruct which aren’t in the true tree
We’d like to reduce the number of reconstruction errors
(FP and FN)
December 5, 2018
PLGW03 - Cambridge, UK

4. The Reconstruction Threshold
weight e1 e3 e4 e5 e6 e7 e2 ? e1 e6 e2 e5 e7 e3 e4 F D B A E C G J I H
seq. length k 
Input length may be insufficient to reconstruct some edges
(short and deep)
Can we guarantee reconstruction of all edges above the threshold?
December 5, 2018
PLGW03 - Cambridge, UK

5. Near-optimal “information efficiency”
Fast Convergence
Near-optimal “information efficiency”
P. Erdos, M. Steel, L. Szekely, and T. Warnow. A few logs suffice to build (almost) all trees (I). Random Structures and Algorithms, 14:153–184, 1999.
D. Huson, S. Nettles, and T. Warnow. Disk-Covering, a fast-converging method for phylogenetic tree reconstruction. J Comp Biol, 6:369–386, 1999.
T. Warnow, B. Moret, and K. St. John. Absolute convergence: true trees from short sequences. In SODA, pages 186–195, 2001.
M. Csürös. Fast recovery of evolutionary trees with thousands of nodes. Journal of Computational Biology, 9(2):277–297, 2002.
E. Mossel. Distorted metrics on trees and phylogenetic forests. ACM Transactions on computational biology and bioinformatics, 4:108–116, 2007.
C. Daskalakis, C. Hill, A. Jaffe, R. Mihaescu, E. Mossel, and S. Rao. Maximal accurate forests from distance matrices. In RECOMB, pages 281–295, 2006.
And more…
Reconstruct the entire tree (w.h.p.) from sequences of polynomial-length.
December 5, 2018
PLGW03 - Cambridge, UK

6. The Reconstruction Threshold Fast Converging Algorithms
weight e1 e3 e4 e5 e6 e7 e2   e1 e6 e2 e5 e7 e3 e4 F D B A E C G J I H
seq. length k
Existing FC methods provide guarantees only when the threshold is lower than the weight of the shortest edge
December 5, 2018
PLGW03 - Cambridge, UK

7. Forest Reconstruction Methods
[Mossel `07] [Daskalakis et al `06]
Shallow edges are easier to reconstruct
December 5, 2018
PLGW03 - Cambridge, UK

8. Forest Reconstruction Methods
[Mossel `07] [Daskalakis et al `06]
Short edges block reconstruction of long edges deeper in the tree
December 5, 2018
PLGW03 - Cambridge, UK

9. Adaptive Fast Convergence
weight e1 e3 e4 e5 e6 e7 e2    e1 e6 e2 e5 e7 e3 e4 F
! Adaptive Fast Convergence ! D B A E C G J I H ε
seq. length k
Existing FC methods provide guarantees only when the threshold is lower than the weight of the shortest edge
December 5, 2018
PLGW03 - Cambridge, UK

10. Incremental Reconstruction
fast converging
The incremental approach: [WSSB `77] [Csuros `99] [KZZ `03]
Use directional queries to insert taxa one at a time.
Directional queries implemented using a quartet oracle.
Total time complexity of O(n2). A B A G D F D
<ε E F G C
Short edges (below reconstruction threshold) lead to false positives.
False positives lead to faulty reconstruction of long edges.
December 5, 2018
PLGW03 - Cambridge, UK

11. A Reliable Quartet Oracle
The basic building-block: a reliable quartet oracle i k
> ε
*similar oracle used also in
[Daskalakis et al `06]
O(depth) j l
Never returns a false quartet-split (may return ‘fail’).
Returns correct split if:
separating path is long enough (above the reconstruction threshold)
quartet is short enough (proportional to tree-depth)
December 5, 2018
PLGW03 - Cambridge, UK

12. A Reliable Incremental Algorithm
The idea: never reconstruct faulty edges!
Use truthful directional oracle (never wrong , may return ‘fail’).
Insertion zone: leaves point “inwards” ; internal vertices give no direction.
Contract edges already reconstructed, if necessary.
contracted short edge G A B C D E F
contraction A B ? ? D E ?
must be contracted F G C
False Positives: None. (returned tree is an edge-contraction of “true tree”)
False Negatives: Only short edges. (contracted edges are below rec. thres.)
December 5, 2018
PLGW03 - Cambridge, UK

13. Main Challenges Directional oracle on vertices of high degree
- Correctness: no faulty answers + enough correct answers
- Complexity: using O(deg(v)) quartet queries
(sustaining O(n2) time complexity of algorithm)
Querying only quartets of O(depth)-diameter
- Representing each direction with a “close” taxon
- Dealing with very large contracted subtrees v
reconstructed
tree v ε
true tree
More details in: “Fast and Reliable Reconstruction of Phylogenetic Trees with Very Short Edges”, In SODA `08
December 5, 2018
PLGW03 - Cambridge, UK

14. Towards Optimal Reconstruction Guarantees for Phylogenetic Trees
Further optimizing reconstruction threshold:
Reducing bound on diameter of quartets we query
Allowing reconstruction of short shallow edges
(using ideas from forest reconstruction)
Practical issues:
Improving reliability of directional oracle
Using reliable partial reconstruction i j k l
r = O(depth) e1  e1 e2 e3 e4 e5 e6 e7 A B C D F G I E H J e3 e4 e5 e6 ε e7
seq. length k e2 
December 5, 2018
PLGW03 - Cambridge, UK

15. December 5, 2018
PLGW03 - Cambridge, UK