Source-linked AI summary
A Fast Algorithm for Computing Geodesic Distances in Tree Space
Megan Owen, J. Scott Provan
TL;DR
The paper addresses the lack of a polynomial-time exact method for computing geodesics between edge-weighted phylogenetic trees. It iteratively transforms a simple path through tree-space orthants, using weighted vertex cover to identify new orthants, and presents the first polynomial-time algorithm for finding geodesics and their distances.
Problem
Existing geodesic algorithms search exponentially many candidate paths, while edge lengths remain important for comparing phylogenetic trees.
Method
The algorithm starts with a simple path and repeatedly transforms it through a new geodesic-intersecting orthant identified using weighted vertex cover.
Results
The paper presents the first polynomial-time algorithm for finding geodesics and computing geodesic distances between phylogenetic trees in tree space.
Takeaways & Limitations
The method increases the usefulness of geodesic distance as a modeling tool beyond the previous practical restriction to trees with fewer than 50 leaves.
Takeaways & Limitations
The algorithm may fail to work correctly when started from an arbitrary proper path rather than the path generated in its own algorithmic context.
Abstract
from arXiv · showhide
Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a Euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.
1 Introduction
Phylogenetic trees can differ across genes and reconstruction methods, creating a need for efficient distance measures that preserve edge-length information. The paper introduces a polynomial-time approach to compute tree-space geodesics by iteratively shortening paths through relevant orthants.
- Motivation: Different genes and reconstruction methods can produce multiple phylogenetic trees for the same organisms.This motivates quantitative comparison across alternative tree reconstructions.
- Motivation: Many existing measures emphasize topology while ignoring edge lengths, despite lengths encoding information such as mutations between speciation events.Most such measures are also inefficient for large trees.
- Tree space: Tree space embeds each topology in a Euclidean orthant, with coordinates representing edge lengths, and joins orthants when topologies can be transformed by edge exchanges.This structure supplies the setting for the geodesic distance.
- Prior work: Earlier geodesic algorithms search exponentially many candidate paths, while no polynomial-time exact algorithm was known.A polynomial-time √2-approximation had been established previously.
- Contribution: The proposed method starts with a simple path and repeatedly transforms it through newly identified orthants until the geodesic is obtained.Weighted vertex cover in a specially constructed bipartite graph identifies each new orthant, keeping the number of transformations polynomial.
2 Tree Space and Geodesic Distance
Tree space represents phylogenetic tree topologies as Euclidean orthants joined along shared-edge boundaries, with edge lengths determining positions and geodesic routes. Because tree space is CAT(0), geodesics are unique, and the paper’s algorithm finds them by iteratively shortening paths through relevant orthants.
- Tree space: An n-tree has labeled leaves, interior edges with non-negative lengths, and a topology represented by compatible splits.A set of n−2 compatible splits uniquely determines an n-tree topology, and edges in different trees correspond when their splits match.
- Tree space: Each tree topology corresponds to a Euclidean orthant whose coordinates are the positive edge lengths; lower-dimensional orthants represent boundary trees with contracted edges.Maximum-dimension orthants have dimension n−2, while orthants meet when topologies share edges or when edges are contracted to length zero.
- Geodesic distance: The cone path contracts all edges of the first tree to the origin and expands the second tree, while tree-space distance is the length of a shortest path.Paths are sequences of Euclidean line segments within orthants, and their lengths are summed across segments.
- Geodesic distance: Tree space is CAT(0), so every pair of trees has a unique geodesic, whose intermediate orthants depend on both topology and edge lengths.Geodesics may be straight within adjacent orthants, follow a cone path, or traverse intermediate orthants; larger trees can cross multiple orthant boundaries.
- Algorithmic motivation: A local geodesic is globally geodesic in CAT(0) space, motivating repeated tests for local shortestness and transformations of non-geodesic paths into shorter paths.The algorithm starts from a path, identifies bends where it can be shortened through additional orthants, and continues until the geodesic is obtained.
- Algorithmic motivation: The paper formulates the Geodesic Treepath Problem and gives a polynomial-time algorithm that adds orthants intersecting the geodesic while optimally shortening the current path.The method restricts attention to orthants intersecting the geodesic; a proper path is characterized as a geodesic by conditions that rule out further shortening.
3 A Polynomial Algorithm to Solve the Geodesic Treepath Problem
The GTP Algorithm starts with a cone path and iteratively refines it through compatible orthants, using weighted bipartite vertex-cover problems to obtain a geodesic in polynomial time.
- Algorithm: The incompatibility graph is bipartite, with vertices for edges and graph edges representing incompatible splits; compatible edge sets form independent sets.This graph formulation converts compatibility testing into a graph problem.
- Algorithm: The Extension Problem reduces to finding a sufficiently heavy independent set, equivalently a minimum-weight vertex cover in the incompatibility graph.Vertex weights are normalized squared edge lengths, and the minimum-weight cover determines how support pairs are split.
- Complexity: The Extension Problem can be solved in O(n3) time using maximum flow on bipartite graphs.Vertex weights are implemented as capacities on source and sink arcs.
- Algorithm: The algorithm begins with the cone path and repeatedly replaces support pairs when an Extension Problem yields a shorter proper path.Each successful refinement creates a new support pair and reduces the path length.
- Correctness: The algorithm preserves property (P2) at every stage, and termination implies property (P3), so Theorem 2.5 establishes that the resulting path is the geodesic.At most n −3 iterations are needed, with Extension Problems solved only on newly created support pairs.
- Complexity: The GTP Algorithm correctly solves the Geodesic Treepath Problem in O(n4) time.The polynomial bound follows from at most n −3 iterations and the O(n3) Extension Problem solver.
4 The GTP algorithm with common edges and leaf edge-lengths present
The expanded algorithm handles common edges, including leaf edges, by decomposing the trees into pairs of disjoint subtrees, applying the GTP Algorithm to each pair, and recombining the resulting paths. Common edges remain on every geodesic and change uniformly in length.
- Common edges occur on every geodesic between trees that share them, with lengths changing uniformly between the terminal trees.
- The algorithm identifies common nonleaf edges and leaf edges, then bisects common edges to separate the trees into pairs of disjoint subtrees with identical leaf sets.
- It applies the GTP Algorithm independently to each pair of disjoint subtrees and uses their supports to construct a composite path.
- The composite path coordinates subtree paths using a common parameter λ and reconnects the resulting intermediate trees at the bisected common-edge vertices.
- Common-edge lengths on the composite path are determined from the corresponding edge-length vectors of the two terminal trees.
- Because partitioning the tree takes linear time, handling common edges does not increase the algorithm's complexity.An implementation is available from the paper's cited URL.
5 Conclusion
The paper introduces the first polynomial-time algorithm for finding geodesics between phylogenetic trees in tree space, extending the approach to ordered leaves and common statistical analyses. This broadens the practical scope of geodesic distance and suggests applications to centroids and object-oriented data analysis.
- The paper presents the first polynomial-time algorithm for finding geodesics between phylogenetic trees in tree space and further characterizes geodesic properties.
- The technique also solves the geodesic treepath problem when non-root leaves have a specified right-left ordering.This is equivalent to requiring the tree to be planar under a given clockwise leaf ordering.
- For trees satisfying the noncrossing property, the geodesic in the ordered setting is the same as in the unrestricted setting.
- The framework potentially applies to other tree problems and measures that use tree space's intrinsic Euclidean structure.
- The geodesic framework could provide a more direct and efficient way to compute centroids defined through minimum summed squared distances.
- The path space concept offers a model for tree-based object-oriented data analysis, while the CAT(0) property supports efficient iterative improvement methods.