A Faster Reconstruction of Binary Near-Perfect Phylogenetic Trees

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A Faster Reconstruction of Binary Near-Perfect
Phylogenetic Trees

• Srinath Sridhar
• Joint work with Kedar Dhamdhere, Guy E.
  Blelloch, Eran Halperin, R. Ravi and Russell
  Schwartz

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Steiner Tree Problem

• Input Graph G(V, E) with edge weights w E ? R
  and a terminal set S ? V
• Output Subtree T of G connecting all vertices in
  S
• Objective Minimize w(T)
• Informally MST with intermediate vertices
• NP-complete, even if G is m-dimensional hypercube
  with unit edge weights

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Near-Perfect Phylogenetic Trees

• Input set S of n points on an m-dimensional
  hypercube (n bit-strings of length m)
• Output Steiner (unrooted) tree T connecting S
  using intermediate nodes (Steiner nodes) of
  hypercube
• Objective Minimize T
• Assumption Topt ? m q, constant q

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Why is this important? (Foster et al., 98)
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Why is this important? (Wirth et al., 04)
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Typical Input Data

• Rows different species, languages etc
• Columns yes/no, 0/1 properties of rows
• Phenotypes Each column can represent binary
  questions thumbs? color-blind?
• DNA Each position has 2 possibilities (almost
  always)

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Example
0001 Boggart

• H W RS B/NB
• Basilisk 1 1 0 0
• Boggart 0 0 0 1
• Centaur 1 0 1 1
• Goblin 1 0 0 1
• H Head
• W Wings
• RS Can read stars
• B/NB Bad/not-so-bad

1001Goblin
1000 Steiner
1011 Centaur
Basilisk 1100
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Perfectness
0001 Boggart
1
1001Goblin

• Annotate tree T with the column flip
• Tree T perfect annotations occur only once
• Evolution is assumed to be (nearly) perfect

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1000 Steiner
1011 Centaur
2
Basilisk 1100
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Perfectness
0001 Boggart
1
1001Goblin

• Annotate tree T with the column flip
• Tree T perfect annotations occur only once
• Evolution is assumed to be (nearly) perfect
• q-near-perfect Topt ? m q, constant q

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3
1000 Dementor
1011 Centaur
2
Basilisk 1100
1-near perfect
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Hippogriff 1101
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General Phylogeny Problem

• Input S set of n strings in 1, , km
• Output Steiner tree T connecting all of S
  (Hamming distance)
• Objective Minimize T
• Variants
• k is bounded by a constant, k is 2
• Tree T is perfect
• Tree T is near-perfect

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Some Prior Work
States k Perfect-ness Time Work
2 perfect O(n m) Gusfield, 92
unbounded perfect NP-complete Bodlaender et al.92, Steel 92
constant perfect O(23k(n m3m4))O(22kn m2) Agarwala, Fernandez-Baca, 93, Kannan, Warnow, 97
constant q-near Fernandez-Baca, Lagergren 03
2 q-near qO(q)nmO(nm2) Our work
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Overview
Discover O(q) edges, induced topology
Optimal Tree
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Overview
Discover assignment of rows to super nodes
Optimal Tree
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Overview
Grow perfect phylogeny within Each super node
Optimal Tree
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Overview
Link the super nodes
Optimal Tree
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Current/Future Work

• Simpler algorithm
• States k gt 2, near-perfect
• Experimental evaluation, useable code
• Related harder problem Input is mixture of 2
  strings over 0, 1mInput 2 0 1 1
  2Output 1 0 1 1 0 0 0 1 1 1