Navigation in the Space of Hierarchies using NNI Moves

Transcription

1 ESE TECHNICAL REPORT - MAY 3, 203 Navigation in the Space of Hierarchies using NNI Moves Omur Arslan, Dan P. Guralnik and Daniel E. Koditschek Abstract A fundamental classification problem common to both computational biology and engineering pattern recognition is the efficient and informative comparison of different hierarchies. In this paper, we introduce a discrete dynamical system in the space of nondegenerate hierarchies rooted binary trees over n leaves connected through Nearest Neighbor Interchange NNI moves computed at the cost of On each step to navigate any initial hierarchy towards a desired goal hierarchy in O n 2 steps. Our construction and stability analysis combines in a novel manner aspects of two traditional approaches to comparing combinatorial trees: edge comparison and edit distance. The length of the resulting NNI navigation path between any pair of trees is given by a simple closed form, symmetric function, but it does not define a metric. On the other hand, we introduce a simple new metric over tree space by imposing an ultrametric on hierarchies and we show that the distance to a desired tree in this sense is non-increasing during the evolution of our dynamical system. Index Terms Evolutionary trees, Nearest Neighbor Interchange, Comparison of Classifications, Tree Space, Tree Metric, Rotation Distance, Diagonal-Flip Distance. INTRODUCTION Comparison of trees is a common, essential problem of interest in both bioinformatics and engineering classification. In this paper, we introduce a novel discrete dynamical system on the space of non-degenerate hierarchies rooted binary trees over a fixed finite leaf set J, denoted by BT J, to navigate from any hierarchy, τ BT J, towards some selected final desired hierarchy, τ BT J, using a widely applicable type of tree rearrangement, the so-called Nearest Neighbor Interchange NNI moves [8], [9]. Each such move is computed at the cost of On each step to navigate any initial hierarchy towards a desired goal hierarchy in O n 2 steps, for n = J. Our construction and stability analysis combines in a novel manner aspects of two traditional approaches to comparing combinatorial trees: edge comparison and edit distance.. Motivation.. Coordination of Particle Swarms Notwithstanding the familiar problems of hierarchical classification and comparison, our primary motivation for this work arises from the problem of particle swarm navigation. A fundamental open problem of multi-agent coordination is the generalization of navigation functions [] for collision-free steering of a single disk-shaped robot to the multi-agent case. The reader may wish to consult [5] for a up to date review of the many papers that have recently addressed this problem, among them [2] [4]. In a slightly different setting, a significant number of researches focuses on the control of relative Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 904 organizations of particles within a swarm based on neighbor rules [6] or leader-follower schemes [7], where each agent tries to locate itself at a desired position relative to either its local neighbors for example, within certain range of sensing or its global or local leader. Recently, we have combined both aspects of particle swarm coordination by introducing a new approach to navigation via hierarchical clustering [8], [9]. In [8], we introduce a cover H of the configuration space of distinct Euclidean particles Conf R d, J using closed sets where each element, a hierarchical stratum Sτ, of H is associated with a non-degenerate hierarchy τ BT J and contains particle configurations that might result in the same hierarchical model τ from a certain type of clustering method. Additionally, we propose Sτ- invariant navigation rules with provable global stability properties. As an extension of [8], we study how to perform hierarchical transitions between these local hierarchy-invariant controllers for the complete coverage of the configuration space Conf R d, J [9]. In [9] we define and show how to compute a crucial connection between NNI neighborhood of a hierarchy in BT J and neighborhood, non-empty intersections, of a hierarchical stratum Sτ of H. In summary, our motivation for navigating over tree space arises from the problem of coordinating swarms via hierarchical transitions. We are hopeful that the new methods we develop may also be of interest to researchers in the fields of bioinformatics and classification...2 The Significance of NNI moves Various notions of local hierarchy restructuring navigation in BT J have appeared in the bioinformatics and classification literature. Arguably, the most commonly

2 ESE TECHNICAL REPORT - MAY 3, encountered tree rearrangements are: Nearest Neighbor Interchange NNI, Subtree Pruning and RegraftingSPR and Tree Bisection and Reconnection TBR operations or moves [2]. Among these, NNI moves are particularly important for us in coordination of particle swarms because a pair of trees related by a single NNI move turns out to yield an intersecting pair of elements of the hierarchical cover H over the configuration space Conf R d, J [9]. Another critical property of NNI moves is that they define the neighbourhood relation between tree orthants of the metric tree space, T n, of Billera, Holmes and Vogtmann [3]. From a broader perspective, they are also the simplest natural way of transforming trees with the property that NNI SBR TBR, that is to say every NNI walk is also a SBR and TBR operation. Finally, NNI moves are related in a particular way to rotations of binary search trees BST -a binary rooted tree with a fixed taxa order, e.g. ascending or descending and diagonal-flip moves of triangulation of convex polygons see the classical reference [4] for further motivation and discussion..2 Contribution.2. Formal Problem Statement In this paper we study the problem of navigating any non-degenerate hierarchy τ in the NNI-graph see Section G J = BT J, E, a graphical representation of the space of nondegenerate hierarhies BT J over a fixed finite leaf set J connected through Nearest Neighbor Interchange operations or moves of trees, towards any selected desired hierarchy τ BT J..2.2 Formal Statement of Results Theorem The NNI control law u τ in Section 4.2 defines an abstract closed loop discrete dynamical system 50 in the NNI-graph G J = BT J, E on a fixed finite leaf set J. A Lyapunov function V τ 5, which is positive definite with respect to the desired hierarchy, τ BT J, strictly decreases at each iteration of this dynamical system, hence all initial conditions eventually converge to the desired goal τ. Proof: See Lemma 6 and its proof. Corollary The NNI control policy u τ in Section 4.2 at every hierarchy τ BT J towards any desired hierarchy τ BT J can be computed in linear time, On, in the number of leaves, J = n. Proof: See Section 5.. Corollary 2 The complete sequence of trees and associated NNI moves along the NNI navigation path joining τ BT J to τ BT J resulting from the NNI control law u τ can be efficiently computed in polynomial time, O n 2, and the NNI navigation path length is tightly bounded by 2 n n 2. Proof: See Section 5.3 and Lemma 27. Corollary 3 For any τ, τ BT J, even though a NNI navigation path joining τ to τ result of the NNI control law u τ is not necessarily unique, the lengths of all NNI navigation paths joining them are the same and symmetric i.e. starting from τ or τ. The length can be efficiently computed using 95a in time O n 2 in the number of leaves, J = n. Proof: See Lemma 26, Lemma 28 and Section Remarks Concerning the Results In addition to main results listed above, we now continue with a list of other relevant contributions and important properties of our proposed method: We introduce a new ultrametric representation of hierarchies Lemma 6 and associated norm-induced tree metric, the cluster-cardinality metric, Definition 7. We show that the cluster-cardinality distance to a desired hierarchy τ BT J is non-increasing along the NNI navigation path resulting from the NNI control law u τ Lemma 9. It is evident from both the Lyapunov function 5 and the cluster cardinality distance 69 Lemma 7 that there is an internal hierarchical weighting of clusters edges of trees such that clusters closer to the root have more priority than the ones closer to the leaves. The ability to eliminate higher level incompatibilities between hierarchical structures before lower level reconciliation might represent a valuable side benefit of our approach, for example, akin to matching split distance of [0]. The NNI control law in Section 4.2 satisfies a modularity [5], affording decomposition of the original problem into disjoint trees. Hence, the Robinson- Foulds distance d RF to any desired hierarchy τ BT J is non-increasing along all NNI navigation paths ending at τ Lemma 20. The proposed NNI control law in Section 4.2 can be directly applied to the rotation graph of binary search tree BSTs Lemma 8, whereby distinct triangulations of convex polygons are traversed though diagonal-flip operations for further reading see [4]. 2 BACKGROUND & NOTATION 2. Hierarchies A hierarchy τ over a fixed finite index set J, say J = [n] : = {, 2,..., n}, uniquely determines and henceforth will be interchangeably identified with a rooted semilabelled tree : a directed acyclic graph G τ = V τ, E τ all of whose edges E τ are directed away from the root, a unique vertex of degree two, and with leaves, vertices of degree one, bijectively labeled from J [3]. The cluster C v of a vertex v V τ is defined to be the set of leaves reachable from v by a directed path in τ. Correspondingly, the cluster set C τ of τ is defined to be the set of all its vertex clusters, C τ : = {C v v V τ }.

3 ESE TECHNICAL REPORT - MAY 3, AncI,τ root PrI,τ ChI,τ I DesI,τ interior node leaf node local complementary cluster I LC of cluster I C τ as I LC : =P r I, τ\i, not to be confused with the standard global complement, I C = J I, which is distinct unless P r I, τ = J. Last but not the least, the depth or level, l τ : C τ N, of a cluster, I C τ, of hierarchy τ is simply equal to the number of its ancestors, Fig.. Hierarchical Relations: ancestors - Anc I, τ, parent - P r I, τ, children - ChI, τ, and descendants - Des I, τ of cluster I of a rooted binary phylogenetic tree, τ BT. Filled and unfilled circles represent interior and leaf nodes, respectively. An interior node is referred by its cluster, the list of leaves below it; for example, I = {4, 5, 6, 7}. C τ is also known as the unique laminar family, a collection of nested compatible subsets of J, associated with τ [6]. 2.. Cluster Compatibility Definition [2], [6] Let A, B be finite sets, then A and B are said to be compatible or nested, denoted by A B, if they are disjoint or one is a subset of other, A B = A B B A. 2 If sets A and B are incompatible, denoted by A B, then they are said to cross each other. In general, we shall abuse notation for the compatibility of subsets A and B of the power set PJ of a fixed finite set J as A B = A B. 3 A A B B 2..2 Hierarchical Relations Note that C τ of a hierarchy τ determines G τ =V τ, E τ completely as it stands in bijective correspondence with V τ, and v, v E τ for all v, v V τ if C v C v and ṽ V τ such that C v C ṽ C v. In particular, we adopt the following notation Anc I, τ = { V C τ I V }, P r I, τ Anc I, τ \ A AncI,τ ChI, τ = {V C τ P r V, τ = I}, Des I, τ = { V C τ V I }, Anc A, τ, for the standard notions of, respectively, the set of ancestors, parents, children and descendants of every cluster I C τ. Because the children comprise a partition of each parent, we find it useful to define the. For completeness parents of the coarsest and empty cluster is declared empty, P r J, τ = and P r, τ =. l τ I = Anc I, τ Nondegeneracy & Certain Types of Trees A rooted tree with all interior vertices of out-degree two is said to be binary or, equivalently, non-degenerate, and all other trees are said to be degenerate. In this paper T J and BT J 2 denote the sets of rooted trees and rooted nondegenerate trees, respectively, over a fixed finite leaf set J. Remark [6], [7] The maximum cardinality of a laminar family over a fixed finite set J is 2 J the empty cluster, J singleton clusters and J nonsingleton clusters. Further, every nested subset of the power set PJ of J of cardinality 2 J corresponds to a unique nondegenerate rooted tree τ BT J. Definition 2 For an ordered fixed finite set J 3, a nondegenerate hierarchy τ BT J is a binary search tree BST if children {I L, I R } = ChI, τ, with mini L < mini R, of every cluster I C τ satisfy maxi L < mini R. We shall denote the set of binary search trees by BST J, which is, by definition, a subset of BT J. 4 An important property of BSTs enabling efficient computation of cluster inclusions Appendix E. is: Remark 2 Clusters of a binary search tree τ BST J over a fixed finite ordered set J are intervals of J. Definition 3 Hierarchies τ, τ T J, over a fixed finite index set J are said to be disjoint if they have no { mutual nonsingleton clusters, except the root, i.e. I C τ C τ I >, I J } =. 2.2 Tree Operations We now recall standard definitions of a certain local restructuring of trees and a relevant tree operation, tree projections onto a subset of leaves. 2. For completeness, all the one-leaf trees are assumed to be nondegenerate. 3. Note that the sequence of leaves during any depth-first tree traversal of a BST is the same as the order of leaf set J. 4. The subgraph of the NNI-graph, defined later, containing only BSTs is known as the rotation graph, and the associated operations between BSTs are called rotations [4].

4 ESE TECHNICAL REPORT - MAY 3, τ B C A τ,c τ,a τ,b τ,a τ τ A B C τ,c τ,b Fig. 2. An illustration of NNI moves between binary trees, each arrow is labeled by a source tree and associated grandchild defining the move NNI Moves One can define different notions of neighborhood of a nondegenerate hierarchy τ in BT J using a number of tree restructuring operations or moves [2], and we are particularly interested in NNI moves for reasons discussed in Section..2. A convenient restatement of the standard definition of NNI walks of unrooted binary trees [8], [9] for rooted binary trees, illustrated in Figure 2 and Figure 3, is: Definition 4 Let τ BT J. A grandchild in τ is a cluster G C τ having a grandparent Pr 2 I, τ : = P r P r I, τ, τ in τ. We say that τ BT J is the result of performing a Nearest Neighbor Interchange NNI move on τ at G if C τ satisfies the following C τ = C τ \ {P r G, τ} Pr 2 G, τ \ G. 5 It will be convenient to have Gτ denote the set of all grandchildren in τ, Gτ : = { G C τ Pr 2 G, τ }. 6 Note that after an NNI move at cluster G of τ, grandchild G of grandparent P = Pr 2 G, τ with respect to τ becomes child G of parent P = P r G, τ with respect to the adjacent tree τ The NNI-Graph We define the NNI-graph G J = BT J, E to have vertex set BT J, with two trees connected by an edge if and only if one can be obtained from the other by a single NNI move. The NNI-graph on n leaves is a regular graph of degree 2n 2 [8]. 5 The consensus tree τ of a set of trees T is defined to be the tree whose cluster set contains precisely the clusters 5. It is clear that G τ = 2 J 2 for any τ BT J directly illustrating this result. A C B Fig. 3. The NNI Graph: a graphical representation of the space of rooted binary trees, BT J, with NNI connectivity, for J = [n] = {, 2, 3, 4}. shared by all trees of T, i.e. C τ = τ T C τ [2], and the relation between adjacent trees in the NNI graph and their consensus tree is a direct consequence of 5 that can be stated as follows. Remark 3 Clusters of each tree belonging to an edge, τ, τ E, in the NNI-graph G J = BT J, E and their consensus tree differ only by one cluster, i.e. C τ \ τ τ,τ C τ = C τ \ C τ =. Proof: By Definition 4, the NNI move associated with an edge, τ, τ, of the NNI-graph replaces a cluster of the source hierarchy, τ, with a new compatible cluster to generate the target hierarchy, τ, and so the result follows. Lemma [9] Let T be an edge, τ, τ, in the NNI- graph G J = BT J, E. Then, there exists one and only one common triple of clusters, {A, B, C} C τ C τ, such that each cluster has a different parent in each tree of T, i.e. P r Q, τ P r Q, τ for all Q {A, B, C} Geodesics in the NNI-graph The number of nondegenerate trees in the NNI-graph grows super exponentially with the number of leaves, n, [3], BT [n] = 2n 3!! = 2n 32n n 2! = 2 n, for n 2. 7 n! Clearly, an exploration of the entire NNI-Graph for example, searching for the shortest path between hierarchies becomes rapidly more impractical and costly 6. Note that the consensus tree of trees belonging to T has a degeneracy at grandparent cluster P = A B C of the triplet {A, B, C}.

5 ESE TECHNICAL REPORT - MAY 3, with increasing number of leaves. In fact, computation of the shortest path geodesic whose length is known as the NNI distance [8], [9] in the NNI graph is NPcomplete [20], and a number of works in the literature address the problem of finding upper bounds on the diameter, the length of the longest geodesic [5], [8], [2] [23]. We also note that an approximation of the shortest path length in the NNI-graph on n leaves can be obtained within On log n time using the algorithm from [24], based on divide-and-conquer approach with a tree balancing strategy. 2.3 Tree Metrics A restatement of a simple and widely used tree metric for rooted trees is: Definition 6 The Robinson-Foulds, a.k.a symmetric difference, distance [25] d RF : T J T J R + between a pair of hierarchies τ, τ T J is defined to be the average number of their unshared clusters, d RF τ, τ = C τ \ C τ + C τ \ C τ Tree Projection Definition 5 Tree Projection Let J be a fixed finite set and K J. Tree projection π K : T J T K by restricting the leaf set J to subset K of a hierarchy τ T J yields another hierarchy τ T K with cluster set C τ = { } I I = I K, I C τ. 8 Remark 4 Let τ BT J and {J L, J R } = ChJ, τ. Then, C τ is the disjoint union of the root cluster J and the cluster sets of subtrees τ L = π JL τ and τ R = π JR τ, C τ = C τ L {J} C τ R. 9 Lemma 2 For any finite set J and nonempty subset K J, π K : BT J BT K is surjective. Proof: We first show that for any τ BT J every non-singleton cluster I C τ of τ = π K τ admits a bipartition I = I L I R of clusters I L, I R C τ, i.e., that π K BTJ BTK, and then we shall show that for every τ BT K there exists τ BT J such that π K τ = τ. If K =, the result simply follows since all one-leaf trees are assumed to be nondegenerate. 2 Otherwise, for any τ BT J and every interior cluster I C τ of τ = π K τ, there exists a unique cluster I C τ with the property that I K = I, I L K and I R K, where {I L, I R } = ChI, τ. To see the uniqueness, notice the following facts: i A K = I for all A Anc I, τ, but P K = for a child P ChA, τ, ii D K I for all D Des I, τ, and iii the rest of clusters, Q C τ\anc I, τ Des I {I} are disjoint with I, and so Q K I. Therefore, clusters I L = I L K C τ and I R = I L K C τ define the bipartition of I = I L I R in τ, and so π K BTJ BTK. Finally, to show that it is surjective, simply consider any nondegenerate trees τ BT K and τ BT J\K, and let τ BT J be a nondegenerate tree with cluster set C τ = C τ {J} C τ. 0 3 PRELIMINARY RESULTS 3. The NNI Complex We now define the abstract NNI complex of nondegenerate hierarchies of BT J to have faces whose each pair of elements joined by a single NNI move. Note that 0- skeleton of the NNI complex is the set of nondegenerate hierarchies in BT J and its -skeleton is the NNI-graph G J = BT J, E ; in other words, its zero-dimensional facesvertices are nondegenerate hierarchies in BT J and one-dimensional facesedges are the NNI moves. A important remark related to the topological structure of the abstract NNI-complex is as follows: Lemma 3 The abstract NNI complex is pure simplicial 2- complex. That is to say, each edge of the NNI-complex G J = BTJ, E is adjacent to a unique face, thus it has no higher order skeleta beyond the faces. Proof: The result directly holds for J = 3. For J > 3, let τ, τ, τ be a face of the NNI-graph, anda, B, C be the associate NNI triplet. Without loss of generality, let A B C τ, B C C τ and A C C τ ; in other words, NNI moves on τ at A and B result with τ and τ, respectively. Hence, C τ\c τ = C τ \ C τ = {A B}. Now, let D G τ \ {A, B}, and τ be result of following the NNI move on τ at D. Then, observe that C τ \ C τ = C τ \ C τ = C τ \ C τ = { Pr 2 G, τ \ G }, and also remember that A B C τ. Therefore, C τ \ C τ = C τ \ C τ = { A B, Pr 2 D, τ \ D } and, by Remark 3, τ is adjacent to neither τ nor τ, which completes the proof. We also find it useful to emphasize a number of important properties of a facet of the NNI-complex. Lemma 4 Clusters of each hierarchy belonging to a face, τ, τ, τ, of the abstract NNI-complex and 7. It is evident from Remark that the Robinson-Foulds metric d RF : BT J BT J R + between a pair of nondegenerate hierarchies τ, τ BT J is simply equal to the number of cluster difference of one binary hierarchy from the other, Now, observe that τ proof. = π K τ, which completes the d RF τ, τ = C τ \ C τ = C τ \ C τ. 2

6 ESE TECHNICAL REPORT - MAY 3, their consensus tree 8 differ only by one cluster, i.e. C τ \ τ τ,τ,τ C τ =. Proof: Let {A, B, C} be the associated triplet of edge τ, τ defined in Lemma. Without loss of generality, let A B C τ and B C C τ, that is to say τ and τ result from NNI moves on τ and τ at B and A, respectively. Hence, the only possible NNI moves joining τ and τ to τ occur at B and A, respectively, and so A C C τ. Therefore, one can clearly observe that C τ\ C τ = {A B}, C τ \ C τ = {B C} and C τ \ C τ = {A C} where C τ = τ τ,τ,τ C τ. Moreover, similar to Lemma, we have: Lemma 5 Trees belonging to a face,τ, τ, τ, of the abstract NNI-complex share only one common set of triple clusters, {A, B, C} τ τ,τ,τ C τ, such that the parent of each cluster differs in each tree, P r G, τ a P r G, τ b for all G {A, B, C} and τ a τ b τ, τ, τ. Proof: The result is evident from a similar argument to used in the proofs of Lemma and Lemma 4. Alternatively: Let {A, B, C } be the common triplet associated with edgeτ, τ described in Lemma. Without loss of generality, let A B C τ and NNI moves on τ at A joins τ to τ, and so B C C τ. Here, the only possible transition from τ to τ while guaranteeing τ, τ, τ is a face of the NNI-graph is NNI move on τ at C, and so A C C τ. Now, one can easily verify that the triplet A, B, C associated with edge τ, τ is also the only common set of triple clusters of trees of face τ, τ, τ, and the result follows. Note that sets of triple clusters associated with a face of the NNI complex and its edges, described in Lemma and Lemma 5, are the same, and it is convenient to call this common triplet NNI triplet associated with a face and its edges. 3.2 An Ultrametric Representation of Hierarchies We now introduce a simple monotone 9 ultrametric representation of hierarchies using their cluster set as follows: Lemma 6 Let τ T J be a hierarchy over a fixed finite index set J, then an ultrametric between elements of J associated with τ is d τ i, j : = min A Cτ i,j A ha 0 i, j J, 3 8. Note that consensus trees associated with a face and each of its edges are the same. 9. All the intra-cluster distances are smaller than all the inter-cluster distances where the height, h : PJ N, of a cluster I J is defined to be hi : = I. 4 and P. denotes the power set of its operand. Proof: By definition, d τ clearly satisfies d τ i, j 0, non-negativity 5 d τ i, j = 0 i = j, identity of indiscernible 6 d τ i, j = d τ j, i, symmetry 7 for all i, j J. Now, let i j k J, and A C τ be the common ancestor of i, j with the minimal cardinality, i.e i, j A and d τ i, j = ha. Further, let {A i, A j } = ChA, τ with the property that i A i and j A j. Hence, we have k A i d τ j, k = d τ i, j 8 k A j d τ i, k = d τ i, j 9 k A C d τ i, k = d τ j, k > d τ i, j. 20 Therefore, d τ satisfies the ultrametric inequality, d τ i, j max d τ i, k, d τ j, k, 2 and this completes the proof. Accordingly, we define a norm-induced tree metric as Definition 7 The cluster-cardinality metric between any pair of hierarchies, τ and τ, in T J is defined to be d cc τ, τ : = Uτ Uτ, 22 where Uτ denotes an ultrametric representation of τ defined to be Uτ ij : = d τ i, j. 23 Lemma 7 Let τ, τ, τ be an face of the abstract NNIcomplex and {A, B, C} be the associated NNI triplet. Then, the cluster-cardinality distance between trees belonging to an edge τ a, τ b of the face is d cc τ a, τ b = 2 A B C. 24 Proof: Let P = A B C and, without loss of generality, A B C τ a and A C C τ b. Here, note that P C τ for all τ τ, τ, τ. Since the NNI moves between τ a and τ b only change the relative relations of clusters A, B and C, the distance 0. Note that the ultrametric between leaves i, j J is the height of their common ancestor with the minimal cardinality... denotes element-wise -norm of a matrix, i.e. U : = n i= n j= U ij for all U R n n.

7 ESE TECHNICAL REPORT - MAY 3, between τ a and τ b can be simply rewritten as d cc τ a, τ b = Uτ a Uτ b, 25 = + Uτ a ij Uτ b ij Uτ a ij Uτ b ij i A i A j B j C = i A j B + i B j C Uτ a ij Uτ b ij, 26 ha B hp = C + i A j C hp ha C = B + hp hp, i B j C =0 27 = 2 A B C, 28 and the result follows. 3.3 Special Crossings of Clusters We now define a particular set of cluster crossings used in the design of the NNI control law in Section 4.2. Lemma 8 For any τ, τ BT J, let Kτ, τ 2 denote the set of common clusters with crossing splits i.e. different children, { } Kτ, τ = K C τ C τ ChK, τ ChK, τ. Then Kτ, τ = if and only if τ = τ. 29 Proof: First, observe that Kτ, τ = for all τ BT J, then it is clear to see that τ = τ C τ = C τ 30 and the lemma follows. Kτ, τ = Kτ, τ = Kτ, τ =, 3 Lemma 9 Let τ τ BT J and K Kτ, τ. Then, any cluster I C τ incompatible with split ChK, τ is always a descendant of K in τ, I ChK, τ I Des K, τ, I C τ. 32 Proof: Since any cluster I C τ \ Des K, τ is either a super set of K or disjoint with K, it is clearly compatible with each element of ChK, τ. Thus, an incompatible clusters I C τ with split ChK, τ is contained in Des K, τ, which completes the proof. Lemma 0 Let {K L, K R } be a bipartition of K, i.e. K L K R = K and K L K R =, and I K. Then, the following equivalences hold I K L, I K R i I K L, I K R 2. Note that K 3 for all K Kτ, τ. ii I {K L, K R }. 33 Proof: Let us start with i which is simply evident from I K L, I K R a K L, b K R : a, b I I K = KL KR, since K L K R = Further, by Definition, we have I K L I K L, I K R. 34 I K L, K L I, I K L, I K L I K K L I, }{{ R } I K R. since K L K R = 35 Therefore, the incompatibility of cluster I with split {K L, K R } of K can be rewritten as I {K L, K R } =I K L I K R, 36 =I K L I K R K L I K R I, 37 } {{ } true since I K=K L K R =I K L I K R, 38 and the lemma follows. Lemma Let K be a fixed finite set with any two bipartitions {K L, K R } and {KL, K R }. Then, the sum of incompatible elements of one bipartition with the other bipartition is symmetric, I {KL, KR} = I {K L, K R}, 3 39 I {K L,K R } I {K L,K R} and is only zero when the bipartitions are the same. Proof: If the bipartitions are the same, both sides of 39 simply sum to zero. Otherwise, since {K L, K R } and {KL, K R } are distinct binary partitions of K, at least an element of {K L, K R } is not a proper subset of an element of {KL, K R } and vice versa. If not i.e. each element of {K L, K R } is a proper subset of an element of {KL, K R }, K L KL K R and K R KL K R and we have K L K R KL K R = K, which is a contradiction. Now, if, without loss of generality, K L KL, that is to say K R KL and K R KR, then using Lemma 0 one can obtain that both sides of 39 sum to one. Otherwise, the summations on both side of 39 are equal to two since every pair of elements of the bipartitions are not subset of each other. This completes the proof. Lemma 2 For any τ τ BT J and K Kτ, τ, if I Iτ, τ ; K, then A Iτ, τ ; K for all A Anc I, τ Des K, τ. Proof: By Lemma 0, the incompatibility of cluster I with split {K L, K R } = ChK, τ is equivalent to I {K L, K R} I K L, I K R is the standard indicator function which returns unity if its argument is true; otherwise returns zero.

8 ESE TECHNICAL REPORT - MAY 3, Since, A Anc I, τ Des K, τ is a super set of I, i.e. I A, we have A K L I K L, A K R I K R, A {K L, K R}, 4 which completes the proof. Definition 8 For τ τ BT J and K Kτ, τ, let Iτ, τ ; K denote the nonempty 4 set of clusters of τ incompatible with split ChK, τ, { } Iτ, τ ; K : = I Des K, τ I ChK, τ, 5 42 and denote the nonempty 6 subset of deep incompatible clusters as { Dτ, τ ; K: = I Iτ, τ ; K D Iτ, τ ; K, D Des I, τ Des I LC, τ }. 44 Definition 9 For any τ τ BT J and K Kτ, τ, a deepest incompatible cluster, I Dτ, τ ; K, is said to be single if its sibling is compatible, I LC ChK, τ, and otherwise it is said to be jointly incompatible 7 see Figure 4. G I K a τ G I τ τ I LC τ,g b τ τ τ τ τ,g I LC c Fig. 4. An illustration of deep incompatible clusters Dτ, τ; K: single b and joint c incompatibilities with split ChK, τ a of a common cluster K Kτ, τ, and the associated NNI navigation moves until solving the incompatibilities with split ChK, τ. Clusters are colored based on their inclusion relation, and the thickened line show a portion of the incompatible edges. Looking ahead toward the stability analysis of the NNI control law in Section 4.2., we find it useful to remark an important property of joint incompatibilities: 4. By Lemma, at least an element of ChK, τ is incompatible with split ChK, τ, and vice versa. 5. Note that, by Lemma 9, for all I C τ if I ChK, τ then I Des K, τ. 6. This directly follows from Lemma 2. Alternatively, one might consider an incompatible cluster at maximal depth, I = arg max I, 43 I Iτ,τ ;K l τ and it is evident that I Dτ, τ ; K. 7. Due to symmetry of Definition 9, if I Des K, τ is jointly incompatible with split ChK, τ, then its sibling I LC is, too. Lemma 3 For τ τ BT J and K Kτ, τ, sibling descendants I, I LC Des K, τ of τ are incompatible with split {KL, K R } = ChK, τ if and only if they are both incompatible with both children of K in τ, I, I LC Iτ, τ ; K I K L, I K R, I LC K L, ILC K R. 45 Proof: For any descendant siblings I, I LC Des K, τ incompatible with split ChK, τ, we have the following equivalence I ChK, τ, I LC ChK, τ and the lemma follows. 4 PRINCIPAL RESULTS I K L K R ILC, I K R K L ILC, I K L, I K R, 46 I LC KL K R I, I LC KR K L I, I LC KL, ILC KR }{{ } by Lemma 0, and I K = K L K R, I I LC =, K L K R =. I K L, I K R, I LC K L, ILC K R, 47 We now introduce an abstract discrete dynamical system in the NNI graph G J = BT J, E of binary hierarchies over a fixed finite leaf set J. First, we shall propose a new NNI control policy to navigate toward any desired goal hierarchy, τ BT J, from any arbitrary hierarchy, τ BT J, with provable convergence guarantees. Next, we will continue with a list of significant properties of the resultant NNI navigation path between a pair of trees, τ and τ. 4. A Discrete-Time Dynamical System Perspective Let NNI : Ê BT J denote the NNI move on a nondegenerate hierarchy τ BT J at a grandchild cluster G Gτ, τ : = NNIτ, G, 48 where Ê denotes the set of directed edges of the NNI graph G J = BT J, E and every directed edge is referenced by a source tree and an associated NNI move, Ê : = Ê τ, Ê τ : = {τ} Gτ τ BT J Here, note that the NNI move at the empty cluster is corresponds to the identity map in BT J, i.e. τ = NNIτ, for all τ BT J. Therefore, the notion of identity operation in BT J generates self cycles in the NNI graph which 8. Every directed edge τ, τ of the NNI graph can be uniquely labeled, using the associated NNI move on τ at G Gτ yielding τ, by the pair τ, G as illustrated in Figure 2.

9 ESE TECHNICAL REPORT - MAY 3, is a necessity for a discrete-time dynamical system in BT J to have fixed points. Accordingly, one can consider an abstract discrete-time dynamical system in BT J using NNI moves described as τ k+ = NNI τ k, G k, 50a τ k, G k = uτ k, 50b where NNI : Ê BT J 48 denotes the NNI move on τ k at grandchild G k G τ k, and u τ : BT J Ê is a control policy of τ k BT J and returns an NNI move from the directed edge set Ê 49 of the NNI graph. 4.2 NNI Control Law To navigate from an arbitrary hierarchy τ BT J towards any selected desired hierarchy τ BT J in the NNIgraph, we propose an NNI control policy u τ : BT J Ê that returns an NNI move on τ at a grandchild G Gτ associated with a deep incompatible cluster, I Dτ, τ; K 44, as follows: If τ = τ, then just return the identity move at the empty cluster G =. 2 Otherwise, a Select a common cluster K Kτ, τ 29. b Find a deep incompatible cluster I Dτ, τ ; K 44. c Return a proper NNI navigation move on τ at grandchild G {I L, I R } = ChI, τ selected as below: i If I LC ChK, τ single incompatibility, Figure 4a, then return G ChI, τ with the property that G LC, I LC M for some M ChK, τ. ii Otherwise joint incompatibility, Figure 4b, return arbitrary NNI walk at a child of I in τ, default G = I L. In brief, our NNI control scheme resolves incompatibilities between clusters of τ and τ level by level, depending on on the selected common cluster K and one of its deep incompatible clusters I in Step 2, while preserving K and all other common clusters. More precisely, for a fixed K Kτ, τ, the incompatible clusters of τ with split ChK, τ are replaced by compatible ones starting from bottom to top using deep incompatibilities. If desired, one can choose the highest common cluster, K = arg min K Kτ,τ l τ K + l τ K a top-down strategy, to obtain common splits at higher levels first, yielding higher priority resolution of incompatibilities for clusters closer to the root. Since the NNI control law preserves common clusters of hierarchies, the navigation problem of trees can be divided into subproblems of disjoint trees and solved simultaneously, which is known as the decomposability property [5]. Remark 5 For τ τ BT J and K Kτ, τ, the NNI control law u τ replaces any singly incompatible cluster I Dτ, τ ; K of τ by a compatible one after a single NNI move see Figure 4a. On the other hand, a replacement of jointly incompatible clusters I, I LC Dτ, τ ; K with compatible clusters requires three NNI moves see Figure 4b. In the following section, we shall continue with the characterization of the convergence properties of our NNI control law using Lyapunov theory [26] Stability Properties For a desired binary hierarchy τ BT J over a fixed finite label set J, a candidate Lyapunov function V τ : BT J R + can be defined as V τ τ : = ρ I lτi+l τ I I, 5 I Cτ I Cτ where ρ is a hierarchical attenuation constant, l τ 4 returns the level or depth of a cluster of τ. Note that since each nondegenerate hierarchy corresponds to a unique set of maximum cardinality compatible clusters Remark, it is clear that V τ τ = 0 and V τ τ > 0 for all τ BT J \ {τ }. An important observation associated with the behaviour of hierarchical attenuation constant ρ below a certain level of a nondegenerate hierarchy is: Lemma 4 For any cluster K C τ of a binary hierarchy τ BT J, the hierarchical attenuation constant ρ satisfies I DesK,τ ρ < 2 l τi ρ 2 Proof: Proof by induction. ρ l τk, ρ For K = base case : It is trivially true since Des K, τ =. For K > induction: Using the clustering identity between descendants of cluster K and its children, {K L, K R } = ChK, τ in binary hierarchy τ, we can factor the left hand side and find an upper bound as I DesK,τ ρ l τi = ρ l τi I ChK,τ 2 = ρ lτk+ + < 2 ρ lτk = 2 ρ 2 + ρ l τi I DesK R,τ ρ l τi I DesK L,τ < 2 ρ 2 ρ l τk L < 2 ρ 2 ρ l τk R ρ + 4 ρρ 2, 53, 54 ρ l τk, 55

10 ESE TECHNICAL REPORT - MAY 3, where the depth of children and parent clusters are related by l τ K L = l τ K R = l τ K +. Thus, the result follows. Moreover, using the result of Lemma 4, one can obtain an upper bound on the change of Lyapunov function after an NNI move as follows: Lemma 5 For any desired hierarchy τ BT J and hierarchical attenuation constant ρ 2, the change V τ τ, τ in the value of Lyapunv function V τ 5 after the NNI move on τ BT J at G Gτ towards τ = NNIτ, G BT J is bounded above as V τ τ, τ : = V τ τ V τ τ 56 6ρ < ρ lτp +l τ K+2 ρ Ψ τ,τ P, K Ψ τ,τ P, K, 57 where P = Pr 2 G, τ = P r G, τ and K C τ satisfying P K 9, and Ψ τ,τ P, K denotes the total number of crossing between the children of P in τ and K in τ, Ψ τ,τ P, K : = I I. 58 Proof: See Appendix A.. I ChP,τ I ChK,τ Lemma 6 For any desired hierarhy τ BT J and hierarchical attenuation constant ρ , the discrete dynamical system in 50 following the NNI control law u τ in Section 4.2 strictly decreases the value of Lyapunov function V τ 5 at any hierarchy τ BT J away from τ, V τ NNI u τ τ V τ τ < Proof: For any τ τ BT J, let K Kτ, τ and I Dτ, τ ; K be the clusters selected by the NNI control policy u τ of Section 4.2, respectively common compatible and deep incompatible with ChK, τ, while determining the NNI move on τ at G ChI, τ towards τ = NNI u τ τ. Here, note that P = P r I, τ = P r G, τ K. The upper bound in the change of Lyapunov function V τ τ, τ in 57 is given as a function of Ψ τ,τ P, K and Ψ τ,τ P, K 58. Depending on the incompatibility of I with ChK, τ, the values of Ψ τ,τ P, K and Ψ τ,τ P, K can be bounded above as follows: Case : I LC ChK, τ If I Dτ, τ ; K is single, then, by Definition 9, we have I ChK, τ and I LC ChK, τ for the children { I, I LC} = ChP, τ of the parent cluster 9. Such a cluster K C τ always exists since P J and J C τ P, and so Ψ τ,τ P, K = = E ChK,τ E ChK,τ E ChP,τ E E =I E, since I LC ChK,τ } {{ }, since I ChK,τ, 60 I E. 6 Moreover, the NNI control rule u τ replaces singly incompatible cluster I by a compatible cluster I = Pr 2 G, τ \ G whose local complement I LC in τ is G and also compatible with split ChK, τ since G ChI, τ and, by Definition 9, ChI, τ ChK, τ {. Therefore, } we have E ChK, τ for all E I, I LC = ChP, τ, and so Ψ τ,τ P, K = E E E ChP,τ E ChK,τ =0,since E ChK,τ = As a result, for singly incompatible I, we always have Ψ τ,τ P, K Ψ τ,τ P, K. 63 Case 2 - I LC ChK, τ : In this case, by Definition 9, both of siblings I, I LC Dτ, τ ; K in τ are jointly incompatible with split ChK, τ. Further, as stated in Lemma 3, they both cross both children of K and vice versa, that is to say E E for all E ChP, τ and E ChK, τ, and so Ψ τ,τ P, K = E E E ChP,τ E ChK,τ =,by Lemma 3 = On the other hand, any arbitrary NNI move G ChI, τ replaces cluster I with another incompatible cluster I = Pr 2 G, τ \ G whose local complement I LC in τ is G and compatible with split ChK, τ. Hence, we have I ChK, { τ } and I LC ChK, τ for children clusters I, I LC = ChP, τ, which yields Ψ τ,τ P, K = E E, 65 = E ChK,τ E ChP,τ E ChK,τ =I E, since I LC ChK,τ I E Therefore, for a joint incompatibility, we always have Ψ τ,τ P, K Ψ τ,τ P, K To sum up, the NNI control policy u τ always guarantees that Ψ τ,τ P, K Ψ τ,τ P, K after each evolution of the dynamical system 50 at every τ away

11 ESE TECHNICAL REPORT - MAY 3, 203 from τ. Using 26 an upper bound on the corresponding change in the value of Lyapunov function is found to be V τ τ, τ 6ρ < ρ l τp +l τ K+2 ρ 2 2 0, 68 for ρ , which completes the proof Additional Properties of the Discrete Dynamics We now continue with relations between the NNI control law of Section 4.2 and binary search trees. Let us start with an observation stating possible types of navigation moves in BST J : Lemma 7 No Joint Incompatibility for BSTs Let τ BST J be a BST over a fixed finite ordered index set J, and {JL, J R } be a bipartition of J whose elements are intervals of J. If any cluster I C τ is incompatible with split {JL, J R }, then its sibling I LC is always compatible with {JL, J R }. That is to say, there exist no cluster of τ jointly incompatible with {JL, J R }. Proof: Proof by contradiction. Recall that I and I LC are disjoint intervals of J Remark 2 and, without loss of generality, let maxjl < minjr. Now, suppose that both of clusters I, I LC C τ are incompatible with {JL, J R }. Hence, by Lemma 0, I JL, I J R and ILC JL, ILC JR. Thus, maxi > min I LC and mini < max I LC which is a contradiction and completes the proof. Lemma 8 The subspace BST J of BT J is positive invariant for the discrete dynamical system 50 with the NNI control law of Section 4.2. Proof: For distinct initial and desired hierarchies τ τ BST J, let K Kτ, τ be a common cluster of τ and τ with crossing splits and I Dτ, τ ; K be a singly incompatible cluster of τ with {K L, K R } = ChK, τ since there is no joint incompatibility for BSTs Lemma 7. Now, the NNI move result from the control law u τ on τ at G ChI, τ replaces parent I = P r G, τ by Pr 2 G, τ\g in the next hierarchy τ = NNIτ, G BT J. Here, note that G K A and G LC, I LC K B for some A B {L, R}. Hence, Pr 2 G, τ \ G = G LC I LC = Pr 2 G, τ K B which is another nonempty interval of J since clusters of τ and τ are intervals of the index set J Remark 2. Thus, the result follows. We now continue with relation between the NNI control law in Section 4.2 and several tree metrics. First, a critical observation associated with the clustercardinality metric of trees of Section 3.2 is: Lemma 9 The cluster-cardinality distance d cc τ, τ to any desired hierarchy τ BT J during the evolution of the discrete dynamical system 50 at every hierarchy τ BT J with the NNI control rule u τ in Section 4.2 is non-increasing, d cc NNI u τ τ, τ d cc τ, τ Proof: Let τ = NNI u τ τ BT J. If τ = τ, then we simply have τ = τ = τ and so the result holds. Otherwise, let K Kτ, τ and I Dτ, τ ; K be the selected common clusters of τ and τ and deep incompatible cluster of τ with split {KL, K R } = ChK, τ, respectively, by the NNI control law while determining the NNI move on τ at G ChI, τ yielding τ. Note that the restructuring of τ only changes relative relations of G, G LC and I LC below P = Pr 2 G, τ K. Moreover, by Definition 9, G KA and GLC KB for some A B {L, R}. Hence, the change in the cluster distance to τ by the transition from τ to τ can be simply written as d cc τ, τ d cc τ, τ = U τ Uτ Uτ Uτ, 70 = U τ Uτ Uτij ij ij Uτ ij 7 i G j G LC + U τ Uτ Uτij ij ij Uτ ij 72 i G j I LC + U τ Uτ Uτij ij ij Uτ, ij 73 i G LC j I LC = i G j G LC hp hk hi hk = hp +hi= I LC 74 hp Uτ ij hp Uτ ij 75 + i G j I LC + hg LC I LC Uτ hp ij Uτ ij, 76 i G LC j I LC [hg LC I LC hp,hp hg LC I LC ]=[ G, G ] G G LC I LC + G G LC I LC = Thus, the result follows. As many other similarity measures [23], [24], [27], [28] based on tree rearrangements and having the decomposability property [5], we have the following property: Lemma 20 The Robinson-Fould distance d RF to any desired hierarchy τ along all NNI navigation paths ending at τ resulting from the NNI control law u τ in Section 4.2 is non-increasing, d RF NNI u τ τ, τ d RF τ, τ 0, τ BT J. 78 Proof: The result simply follows from Remark??. 20. Here, one can easily verify that d cc τ, τ d cc τ, τ < 0 if I is jointly incompatible with split ChK, τ.

12 ESE TECHNICAL REPORT - MAY 3, Properties of NNI Navigation Paths We define an NNI navigation path, denoted by Γ τ τ, joining τ BT J to τ BT J to be a path a sequence of trees connected by NNI-edges resulting from the dynamical system 50 obeying the NNI control law u τ of Section 4.2 starting from τ and ending at τ. Accordingly, the length, Γ τ τ, of an NNI navigation path is the number of NNI moves along it, which is simply equivalent to one less than the number of trees visited along the path since its consecutive trees are associated with a single NNI move. Before stating the salient properties of NNI navigation paths in the NNI graph, we need to introduce some notation and lemmas. Let {JL, J R } be any bipartition of the leaf set J, and G {J L,JR} denote the subset of BT J containing nondegenerate hierarchies with the root split {JL, J R }, } ChJ, G {J L,J {τ R} = BT J τ = {J L, JR}. 79 Now, instead of our original problem stated in Section.2.2, navigating towards a single desired hierarchy in the NNI graph, we consider a related and simpler problem of navigating hierarchies towards the goal set G {J L,J R} associated with any binary partition {J L, J R } of J. One can simply observe that the NNI control law in Section 4.2 can be used to solve this new problem by selecting any desired hierarchy τ G {J L,J R} and fixing the common cluster K Kτ, τ, in Step 2a of the NNI control policy, as K = J. We denote this version of the NNI control policy by u {J L,J, and it is convenient R} to have Γ {J L,J τ denote a resultant NNI navigation R} path from the NNI control law u {J L,J starting from R} any nondegenerate hierarchy τ BT J and ending in G {J L,JR}. A critical observation related to the resultant hierarchy from the NNI control law u {J L,J at the end of solving R} cluster crossings of any initial nondegenerate hierarchy τ BT J with any bipartition {JL, J R } of leaf set J is: Lemma 2 Let τ BT J be any nondegenerate hierarchy over a fixed finite leaf set J and {JL, J R } be any bipartition of J. Then, the dynamical system 50 following the NNI control rule u {J L,JR} converges to the nondegenerate hierarchy τ G {J L,J with cluster set R} C τ = C π J L τ {J} C π J R τ, 80 where π. 8 denotes the tree projection onto a certain subset of J. Proof: If ChJ, τ = {JL, J R }, then the results above directly holds with τ = τ. Otherwise, let τ G {J L,JR}, and I Dτ, τ ; J 44 be a deep incompatible cluster selected by the NNI control law u {J L,J. Note that after a certain number of R} proper NNI navigation moves, one transition for single incompatibility and three transitions for joint incompatibility see Figure 4 and Remark 5, incompatible cluster I and its sibling I LC are replaced with compatible clusters P r I, τ JL and P r I, τ J R in an intermediate tree τ BT J, C τ=c τ\ { I, I LC} { P r I, τ J L, P r I, τ J R}, 8 along an NNI navigation path Γ {J L,J τ while the R} rest of incompatibilities of τ with split {JL, J R } stay the same. Thus, in general, to solve all cluster incompatibilities of τ with {JL, J R }, any cluster I C τ \ {J} and its sibling I LC are replaced as C τ = { I I = I J L, I C τ } {J} { I I = I JR, I C τ }, 82 = C π J L τ {J} C π J R τ. 83 Here, note that the compatible clusters with split {JL, J R } = ChJ, τ are always preserved since it is a proper subset of either JL or J R Lemma 24. Moreover, the finite time convergence directly follows from Lemma 6, and this completes the proof. Lemma 22 For any bipartition {JL, J R } of a fixed finite leaf set J, the length 2 Γ{J L,JR} τ of all NNI navigation paths, resulting from the NNI control law u {J L,JR}, starting from any nondegenerate hierarchy τ BT J and ending in G {J L,J is given by R} = Γ {J L,JR} τ γ R} A {JL, J, 84 where I Cτ A ChI,τ γ : {0,, 2} {0,, 3} 85a x 2 x 2 + x 85b encodes the required number of NNI moves to replace singly and jointly incompatible clusters, which are one and three Remark 5, respectively. Proof: Let τ G {J L,JR} 79 and I Dτ, τ ; J 44 be a cluster of τ deep incompatible with {JL, J R }. After a number of NNI navigation moves resulting from the NNI control policy u {J L,J, one NNI move for R} single incompatibilities and three NNI moves for joint incompatibilities Remark 5, incompatible cluster I and its sibling I LC is replaced with compatible clusters while keeping the rest of clusters of τ the same. Consequently, based on Lemma 2, any incompatible cluster I in Iτ, τ ; J with I LC ChJ, τ is singly incompatible or becomes singly incompatible after a number of NNI navigation transitions solving incompatibilities of its descendants. Similarly, cluster I Iτ, τ ; J with I LC 2. Here, the length of a path in the NNI graph is defined to be the number of NNI moves along the path, which is one less the number of visited hierarchies.

13 ESE TECHNICAL REPORT - MAY 3, ChJ, τ is jointly incompatible or eventually becomes jointly incompatible. Therefore, the length of all NNI navigation paths until solving cluster incompatibilities of τ with split {JL, J R } can be directly written as in 84, which completes the proof. Lemma 23 For any bipartition {JL, J R } of a fixed finite index set J and any nondegenerate hierarchy τ BT J, the NNI navigation path length Γ {J L,JR} τ in 84 is tightly bounded above as Γ {J L,JR} τ J +min J L, JR 3, J Proof: One can easily observe from 84 that the NNI navigation path length Γ {J L,JR} τ is maximized if all interior clusters, except the root, of τ is incompatible with {JL, J R } and the number of cluster that eventually become jointly incompatible is maximized. Recall that the number of interior edges, except the root, of a nondegenerate hierarchy τ BT J is J 2 [8], [8]. Now, consider the set of cluster I C τ whose children are or eventually become jointly incompatible with {JL, J R { }, } J = I C τ D {JL, JR}, D ChI, τ. 87 Recall from Lemma 3 that D D for all D ChI, τ, D {JL, J R } and I J. Hence, such a cluster I J with the minimal cardinality requires two leaves from each element of {JL, J R }, any other cluster A J has at least one extra leaf labels from each element of {JL, J R }. Thus, one can easily observe that J min JL, J R. As a result, an upper bound on 84 can be obtained as Γ {JL,J R} τ J 2 + min J L, JR, 88 maximum number of maximum number of incompatible interior jointly incompatible clusters cluster candidate pairs for all J 2. One can simply verify that this upper bound is actually the tightest upper bound using the hierarchy τ =, 2, 3,..., n BT [n] given in Newick format [2] and the bipartition {JL, J R } = {{}, {2, 3,..., n}}, where Γ {J L,JR} τ = n 2. Let us introduce a list of useful notation before continue with NNI navigation paths of the NNI graph. Lemma 24 Let {K L, K R } be a bipartition of a fixed finite set K, and I and A be sets with the property that I A, I K, then I {K L, K R } A = I {K L, K R }, 89 where denotes the element-wise intersection of sets, {K L, K R } A : = {K L A, K R A}. 90 Proof: The result is evident from Lemma 0 as follows, I {K L, K R } I K L I K L A I K R I K R A, 9 I {K L A, K R A}. 92 Here, note that X Y X Y Z for any sets X Z and Y. Lemma 25 For τ, τ BT J and clusters I C τ and I C τ, the following symmetry holds κτ, τ ; I, I : = A ChI, τ I, 93a A ChI,τ I = A ChI, τ I, 93b A ChI,τ I = κτ, τ; I, I. 93c It is convenient to abuse the notation as κτ, τ : = κτ, τ; J, J for the total numbers of crossings between the root splits of τ and τ. Proof: Let us start with a special case where at least one of ChI, τ I = {I I, } and ChI, τ I = {I I, } holds. Then, it is clear that κτ, τ ; I, I = κτ, τ; I, I = 0 since the empty cluster and I I are always compatible with any cluster A I I. Otherwise, observe that ChI, τ I = ChI I, π I I τ and ChI, τ I = ChI I, π I I τ, and so 93 takes a special form below, κτ, τ ; I, I = κ π I I τ, π I I τ, 94 and the result directly follows from Lemma. Lemma 26 Length of NNI Navigation Path Lengths of all NNI navigation paths Γ τ τ result from the NNI control law u τ in Section 4.2 starting from any nondegenerate hierarchy τ BT J ending at any desired nondegenerate hierarchy τ BT J are the same and symmetric, and it is given by Γ τ τ = ΓChJ,τ τ ΓπJ + τ π J L L τ + Γ πj τ π J R R τ, 22 95a = γ A ChI, τ, 95b I Cτ I Cπ I τ = I Cτ I Cτ A ChI,π I τ γ κτ, τ; I, I = Γ τ τ, 95c where {J L, J R } = ChJ, τ. Here, Γ ChJ,τ τ denotes the navigation path until solving incompatibility of clusters of τ 22. Note that Γ τ τ is always zero for J 2, which is the base case for the recursion.