Solving the Tree Containment Problem for Genetically Stable Networks in Quadratic Time
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(2) Context and motivations I. Phylogenetic trees are routinely used to represent evolution, but they cannot display exchanges of genetic material between species;. I. When these happen, we rely on phylogenetic networks instead;. Example (tree). (from Wikimedia). I. Example (network). (from The Genealogical World of Phylogenetic Networks). We still need to verify that the network “contains” a prescribed set of trees to ensure consistency with previous biological knowledge;.
(3) Phylogenetic networks and related concepts A phylogenetic network is a rooted DAG with a labelled leaf set {`1 , `2 , . . . , `k }.. `1. `5. `2. `4 `3. We only consider binary networks and trees, i.e. all internal nodes have degree three..
(4) Phylogenetic networks and related concepts A phylogenetic network is a rooted DAG with a labelled leaf set {`1 , `2 , . . . , `k }.. I. `1. root: indegree 0;. `5. `2. `4 `3. We only consider binary networks and trees, i.e. all internal nodes have degree three..
(5) Phylogenetic networks and related concepts A phylogenetic network is a rooted DAG with a labelled leaf set {`1 , `2 , . . . , `k }.. `1. I. root: indegree 0;. I. tree nodes: indegree 1, outdegree 2;. `5. `2. `4 `3. We only consider binary networks and trees, i.e. all internal nodes have degree three..
(6) Phylogenetic networks and related concepts A phylogenetic network is a rooted DAG with a labelled leaf set {`1 , `2 , . . . , `k }.. `1. `5. `2. I. root: indegree 0;. I. tree nodes: indegree 1, outdegree 2;. I. reticulations: indegree 2, outdegree 1;. `4 `3. We only consider binary networks and trees, i.e. all internal nodes have degree three..
(7) Phylogenetic networks and related concepts A phylogenetic network is a rooted DAG with a labelled leaf set {`1 , `2 , . . . , `k }.. `1. `5. `2. I. root: indegree 0;. I. tree nodes: indegree 1, outdegree 2;. I. reticulations: indegree 2, outdegree 1;. I. leaves: outdegree 0;. `4 `3. We only consider binary networks and trees, i.e. all internal nodes have degree three..
(8) Tree subdivisions A subdivision of a tree T is a tree T 0 obtained by inserting any number of vertices into the edges of T .. Example (a tree and a subdivision). `1. `2. `3. T. `4. `5. `1. `2. `3. T0. `4. `5.
(9) The tree containment problem Network N displays tree T if we can obtain a subdivision of T by removing incoming edges from reticulations and “dummy leaves”.. `1 `1 `2. `5 `3. `4. `2. `3. `4. `5.
(10) The tree containment problem Network N displays tree T if we can obtain a subdivision of T by removing incoming edges from reticulations and “dummy leaves”.. `1. `1 `1 `2. `5 `3. `4. `2. `5 `3. `4. `2. `3. `4. `5.
(11) The tree containment problem Network N displays tree T if we can obtain a subdivision of T by removing incoming edges from reticulations and “dummy leaves”.. `1. `1 `1 `2. `5 `3. `4. `2. `5 `3. `4. `2. `3. `4. `5.
(12) The tree containment problem Network N displays tree T if we can obtain a subdivision of T by removing incoming edges from reticulations and “dummy leaves”.. remove edges. `1. contract paths. `1 `1 `2. `5 `3. `2. `4. `2. `3. `4. `5 `3. `4. Problem (tree containment) Input: Question:. a phylogenetic network N, a phylogenetic tree T . does N display T ?. `5.
(13) tree containment prior to this work. A → B class A contains class B solvable in polynomial time in P by class inclusion NP-complete. binary. nested spread-k spread-3 spread-2 spread-1 leaf outerplanar. level-k level-3 level-2. compressed. k-nested 3-nested. galled network. tree-sibling. reticulation-visible genetically stable. FU-stable 2-nested. nearly stable. tree-based. distinct-cluster. nearly tree-child. regular. tree-child. galled tree. normal. unicyclic. phylogenetic tree. time-consistent. (adapted from http://phylnet.univ-mlv.fr/isiphync by Philippe Gambette).
(14) Our contributions A → B class A contains class B solvable in polynomial time in P by class inclusion NP-complete. 1. genetically stable (GS) networks; 2. inclusion relations w.r.t. other classes; 3. tree containment in P for GS networks;. binary. nested spread-k spread-3 spread-2 spread-1 leaf outerplanar. level-k level-3 level-2. compressed. k-nested 3-nested. galled network. tree-sibling. reticulation-visible genetically stable. FU-stable 2-nested. nearly stable. tree-based. distinct-cluster. nearly tree-child. regular. tree-child. galled tree. normal. unicyclic. phylogenetic tree. time-consistent. (adapted from http://phylnet.univ-mlv.fr/isiphync by Philippe Gambette).
(15) Genetically stable networks A node v in a network N is stable on a leaf ` if every path from the root to ` contains v ..
(16) Genetically stable networks A node v in a network N is stable on a leaf ` if every path from the root to ` contains v . A network N is genetically stable if every reticulation has a stable parent (on any leaf)..
(17) Genetically stable networks A node v in a network N is stable on a leaf ` if every path from the root to ` contains v . A network N is genetically stable if every reticulation has a stable parent (on any leaf).. A GS network a. d. b. `3. c. `4. `2 `1. a, b, c stable on `2 d stable on `4.
(18) Genetically stable networks A node v in a network N is stable on a leaf ` if every path from the root to ` contains v . A network N is genetically stable if every reticulation has a stable parent (on any leaf).. A GS network. A non-GS network. a. d a. b. `3. c. `4. b. `1 `2. `2 `1. a, b, c stable on `2 d stable on `4. `5 `3. `4. `2 can be reached through either a or b no other leaf “needs” a or b.
(19) Overview of the algorithm The subtree induced by two sibling leaves `, `0 and their parent α in a tree is called a cherry, and is denoted by {α, `, `0 }. `1. `2. `3. `4. `5.
(20) Overview of the algorithm The subtree induced by two sibling leaves `, `0 and their parent α in a tree is called a cherry, and is denoted by {α, `, `0 }. `1. `2. `3. `4. `5. Algorithm for tree containment in GS networks 1. Select a cherry C = {α, `, `0 } in T ; 2. If there is no match for C in N, report no; 3. Otherwise, remove the match from N and C from T ; 4. If T is now a single node, report yes, otherwise go back to 1;. Matches and removals are such that N displays T if and only if N 0 displays T 0 ..
(21) Matching cherries: stability helps Stability narrows down choices for matching α, (α, `1 ) and (α, `2 ) in N: p α T :. P1. P2. `1. `2. N: `1. `2. Lemma (1) If N displays T through some subdivision T 0 , then α must be matched to a node p such that: 1. `1 and `2 are the only leaves on which p can be stable; 2. `1 is the only leaf on which vertices in P1 \ {p} can be stable; 3. `2 is the only leaf on which vertices in P2 \ {p} can be stable..
(22) Matching cherries: genetic stability helps Lemma (1) allows us to focus on specific paths, i.e. paths P from x to ` such that each vertex in P \ {x} is either stable only on ` or not stable at all. What if several choices exist? y x P2. Q1. P1. Q2. `1. `2. Lemma (2) If N is genetically stable and contains vertices x and y connected to leaves `1 and `2 through specific paths that only intersect at x (resp. y ), then either y ∈ P1 ∪ P2 or x ∈ Q1 ∪ Q2 ..
(23) Modifying N and T when N is genetically stable Lemma (2) allows us to restrict our search to the lowest common ancestor p of `1 and `2 such that paths p `1 and p `2 in N are specific. p P1. α T :. P2. N: `1. `2. `1. `2. Lemma (3) If p, P1 and P2 match α, (α, `1 ) and (α, `2 ) in a GS network N, then N displays T if and only if N \ P1 \ P2 displays T \ {`1 , `2 }..
(24) Finding a match for α, (α, `1 ) and (α, `2 ) in N 1. Move up from `1 until we find a lowest common ancestor of `1 and `2 connected to `2 by a path free of nodes stable on other leaves; α T :. N: `1. `2. w1 `1. `2. 2. Move up from `2 to w1 while remaining in a specific path to `2 ; w1 α T :. w2. N: `1. `2. `1. `2. 3. If we succeed, we obtain two specific paths to `1 and `2 in N;.
(25) Correctness and running time The previous lemmas prove the correctness of the algorithm.. Algorithm for tree containment in GS networks 1. Select a cherry C = {α, `, `0 } in T ; 2. If there is no match for C in N, report no; 3. Otherwise, remove the match from N and C from T ; 4. If T is now a single node, report yes, otherwise go back to 1;. The running time is dominated by checking stability, which implies a running time of O(|V | · (|E | + |V |)) = O(|L|2 ) where |L| is the number of leaves of N..
(26) Relevance of GS networks A fair amount of real-world networks could be genetically stable:.
(27) Future work binary nested spread-k spread-3 spread-2 spread-1 leaf outerplanar. level-k level-3 level-2. compressed. k-nested 3-nested. nearly stable. tree-based. tree-sibling. reticulation-visible genetically stable. FU-stable 2-nested galled network. distinct-cluster nearly tree-child regular. tree-child. galled tree. normal. unicyclic. phylogenetic tree. time-consistent. I. Major open problem: complexity for reticulation-visible networks;. I. Refine hardness results;. I. Improve the complexity for tractable cases;.
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