# An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

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TL;DR: An algorithm for global alignment of two protein-protein interaction (PPI) networks is described, guided by the intuition that a protein should be matched with a protein in the other network if and only if the neighbors of the two proteins can also be well matched, and the results of global alignment are interpreted to identify functional orthologs between yeast and fly.

Abstract: We describe an algorithm, IsoRank, for global alignment of two protein-protein interaction (PPI) networks. IsoRank aims to maximize the overall match between the two networks; in contrast, much of previous work has focused on the local alignment problem-- identifying many possible alignments, each corresponding to a local region of similarity. IsoRank is guided by the intuition that a protein should be matched with a protein in the other network if and only if the neighbors of the two proteins can also be well matched. We encode this intuition as an eigenvalue problem, in a manner analogous to Google's PageRank method. We use IsoRank to compute the first known global alignment between the S. cerevisiae and D. melanogaster PPI networks. The common subgraph has 1420 edges and describes conserved functional components between the two species. Comparisons of our results with those of a well-known algorithm for local network alignment indicate that the globally optimized alignment resolves ambiguity introduced by multiple local alignments. Finally, we interpret the results of global alignment to identify functional orthologs between yeast and fly; our functional ortholog prediction method is much simpler than a recently proposed approach and yet provides results that are more comprehensive.

328 citations

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TL;DR: The maximum agreement phylogenetic subnetwork problem (MASN) is introduced and it is proved that the problem is NP-hard even if restricted to three phylogenetic networks and an O(n2)-time algorithm is given for the special case of two level-1 phylogenetics networks.

Abstract: We introduce the maximum agreement phylogenetic subnetwork problem (MASN) for finding branching structure shared by a set of phylogenetic networks We prove that the problem is NP-hard even if restricted to three phylogenetic networks and give an O(n2)-time algorithm for the special case of two level-1 phylogenetic networks, where n is the number of leaves in the input networks and where N is called a level-f phylogenetic network if every biconnected component in the underlying undirected graph induces a subgraph of N containing at most f nodes with indegree 2 We also show how to extend our technique to yield a polynomial-time algorithm for any two level-f phylogenetic networks N1, N2 satisfying f = O(log n); more precisely, its running time is O(|V (N1)| ċ |V (N2)| ċ 2f1 + f2), where V (Ni) and fi denote the set of nodes in Ni and the level of Ni, respectively, for i ∈ {1, 2}

79 citations

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TL;DR: It is proved that MASP is NP-hard for any fixed $k \geq 3$ when $D$ is unrestricted, and also NP- hard forAny fixed $D \geqi 2$ when £k is unrestricted even if each input tree is required to contain at most three leaves.

Abstract: Given a set $\T$ of rooted, unordered trees, where each $T_i \in \T$ is distinctly leaf-labeled by a set $\Lambda(T_i)$ and where the sets $\Lambda(T_i)$ may overlap, the maximum agreement supertree problem~(MASP) is to construct a distinctly leaf-labeled tree $Q$ with leaf set $\Lambda(Q) \subseteq $\cup$_{T_i \in \T} \Lambda(T_i)$ such that $|\Lambda(Q)|$ is maximized and for each $T_i \in \T$, the topological restriction of $T_i$ to $\Lambda(Q)$ is isomorphic to the topological restriction of $Q$ to $\Lambda(T_i)$. Let $n = \left| $\cup$_{T_i \in \T} \Lambda(T_i)\right|$, $k = |\T|$, and $D = \max_{T_i \in \T}\{\deg(T_i)\}$. We first show that MASP with $k = 2$ can be solved in $O(\sqrt{D} n \log (2n/D))$ time, which is $O(n \log n)$ when $D = O(1)$ and $O(n^{1.5})$ when $D$ is unrestricted. We then present an algorithm for MASP with $D = 2$ whose running time is polynomial if $k = O(1)$. On the other hand, we prove that MASP is NP-hard for any fixed $k \geq 3$ when $D$ is unrestricted, and also NP-hard for any fixed $D \geq 2$ when $k$ is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomial-time $(n/\!\log n)$-approximation algorithm for MASP.

46 citations

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TL;DR: The maximum agreement supertree problem (MASP) is proved to be NP-hard for any fixed k ≥ 3 when D is unrestricted, and also NP- hard forAny fixed D ≥ 2 when k is unrestricted even if each input tree is required to contain at most three leaves.

Abstract: Given a set T of rooted, unordered trees, where each T i ∈ T is distinctly leaf-labeled by a set A(T i ) and where the sets Λ(T i ) may overlap, the maximum agreement supertree problem (MASP) is to construct a distinctly leaf-labeled tree Q with leaf set A(Q) ⊆ ∪ Ti ∈ T Λ(T i ) such that |Λ(Q)| is maximized and for each T i ∈ T, the topological restriction of T i to A(Q) is isomorphic to the topological restriction of Q to Λ(T i ). Let n = |U Ti ∈ T Λ(T i )|, k = |T|, and D = max Ti ∈ T {deg(T i )}. We first show that MASP with k = 2 can be solved in O(√D n log(2n/D)) time, which is O(n log n) when D = O(1) and O(n 1.5 ) when D is unrestricted. We then present an algorithm for MASP with D = 2 whose running time is polynomial if k = O(1). On the other hand, we prove that MASP is NP-hard for any fixed k ≥ 3 when D is unrestricted, and also NP-hard for any fixed D ≥ 2 when k is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomial-time (n/ log n)-approximation algorithm for MASP.

39 citations

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TL;DR: This paper proposes extensions of MAST and MCT to the context of supertree inference, where input trees have non-identical leaf sets, and shows that arbitrary instances of SMAST and SMCT can be turned in polynomial time into instances composed of trees with a bounded number of leaves.

Abstract: Given a set of leaf-labelled trees with identical leaf sets, the MAST problem, respectively MCT problem, consists of finding a largest subset of leaves such that all input trees restricted to these leaves are isomorphic, respectively compatible. In this paper, we propose extensions of these problems to the context of supertree inference, where input trees have non-identical leaf sets. This situation is of particular interest in phylogenetics. The resulting problems are called SMAST and SMCT. A sufficient condition is given that identifies cases where these problems can be solved by resorting to MAST and MCT as subproblems. This condition is met, for instance, when only two input trees are considered. Then we give algorithms for SMAST and SMCT that benefit from the link with the subtree problems. These algorithms run in time linear to the time needed to solve MAST, respectively MCT, on an instance of the same or smaller size. It is shown that arbitrary instances of SMAST and SMCT can be turned in polynomial time into instances composed of trees with a bounded number of leaves. SMAST is shown to be W[2]-hard when the considered parameter is the number of input leaves that have to be removed to obtain the agreement of the input trees. A similar result holds for SMCT. Moreover, the corresponding optimization problems, that is the complements of SMAST and SMCT, cannot be approximated in polynomial time within any constant factor, unless P=NP. These results also hold when the input trees have a bounded number of leaves. The presented results apply to both collections of rooted and unrooted trees.

38 citations

##### References

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TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.

Abstract: From the Publisher:
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects.
In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage.
As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds.
Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.

21,642 citations

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TL;DR: This paper presents algorithms for the assignment problem, the transportation problem, and the minimum- cost flow problem of operations research that find a minimum-cost solution, yet run in time close to the best-known bounds for the corresponding problems without costs.

Abstract: This paper presents algorithms for the assignment problem, the transportation problem, and the minimum-cost flow problem of operations research. The algorithms find a minimum-cost solution, yet run in time close to the best-known bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimum-cost matching in a bipartite graph) can be solved in $O(\sqrt {nm} \log (nN))$ time, where $n,m$, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.

431 citations

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TL;DR: This paper presents another approach to the problem of comparing many secondary structures by utilizing a very efficient tree-matching algorithm that will compare two trees in O([T1] X [T2] X L1 X L2) in the worst case and very close to O[T1?] for average trees representing secondary structures.

Abstract: In a previous paper, an algorithm was presented for analyzing multiple RNA secondary structures utilizing a multiple string alignment algorithm. In this paper we present another approach to the problem of comparing many secondary structures by utilizing a very efficient tree-matching algorithm that will compare two trees in O([T1] X [T2] X L1 X L2) in the worst case and very close to O([T1] X [T2]) for average trees representing secondary structures. The result of the pairwise comparison algorithm is then used with a cluster algorithm to produce a multiple structure clustering which can be displayed in a taxonomy tree to show related structures.

338 citations

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TL;DR: The tree obtained by regrafting branches on to a largest common pruned tree is shown to contain all the classes present in the strict consensus tree.

Abstract: Given two or more dendrograms (rooted tree diagrams) based on the same set of objects, ways are presented of defining and obtaining common pruned trees. Bounds on the size of a largest common pruned tree are introduced, as is a categorization of objects according to whether they belong to all, some, or no largest common pruned trees. Also described is a procedure for regrafting pruned branches, yielding trees for which one can assess the reliability of the depicted relationships. The tree obtained by regrafting branches on to a largest common pruned tree is shown to contain all the classes present in the strict consensus tree. The theory is illustrated by application to two classifications of a set of forty-nine stratigraphical pollen spectra.

213 citations

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