DISSIMILARITY VECTORS OF TREES AND THEIR TROPICAL LINEAR SPACES. 1. Introduction

Transcription

1 DISSIMILARITY VECTORS OF TREES AND THEIR TROPICAL LINEAR SPACES BENJAMIN IRIARTE GIRALDO Abstract. We present an introduction to the combinatorics of weighted trees from the point of view of tropical algebraic geometry and tropical linear spaces. We then prove that the set of dissimilarity vectors of weighted trees is contained in the tropical Grassmannian, and further describe the tropical linear space of a dissimilarity vector and its associated family of matroids. This gives a family of complete flags of tropical linear spaces, where each flag is described by a weighted tree. 1. Introduction 1.1. Motivation. This writing is, more than anything else, an excursion into the world of weighted trees. Recently, this has become a combinatorial branch of its own, know as T -theory. We find it very motivating that weighted trees are amazingly simple objects to understand, yet, a large number of natural questions about them remain unanswered. The T -theory comes motivated by its applications in biology and computer science. However, it is not hard to notice the intrinsic combinatorial wealth and beauty of the subject. In particular, it turns out that weighted trees provide a handful of examples to tropical geometry, illustrating several central concepts and providing further intuition, conjectures and problems. Among the most common applications, we find the reconstruction of phylogenetic trees in evolutionary biology, from a set of known data on the taxa. Usually, these data are the results of analysis of DNA sequences alignments of several species. Along the lines of the writing, we provide a personal development of the subject, sometimes providing proofs of the main results. The proofs themselves will sometimes be original, sometimes not. Clearly, we will only cover the smallest part of a more developed subject, narrowed to our interests. The final sections will be devoted to present our own results and contributions to the subject. In particular, we will provide a proof of an amazing relation between weighted trees and a central object of tropical algebraic geometry, the tropical Grassmannian. Then, we will describe the tropical linear spaces that arise from weighted trees. We expect that our results provide further intuition and knowledge about weighted trees, and to contribute to future endeavors in the area Basic Definitions. To begin, we recall the significant definitions associated with this work. For every finite set E, define ( E m) as the collection of all subsets of E of size m. 1

2 2 BENJAMIN IRIARTE We adopt the somewhat standard convention under which [n] = {1, 2,..., n}. Under this convention, notice that ( [n] m) is the collection of subsets of [n] with size m. Recall that a tree can be defined as a connected graph that has no cycles. Every vertex of degree 1 in a tree is called a leaf. Every vertex of a tree that is not a leaf is called an internal vertex. Now, assume that we have a tree containing an internal vertex v satisfying the following condition: for every pair of different leaves u and w of the tree, the minimal path from u to v intersects the minimal path from w to v at v and only at v, that is, there is no common edge shared by both paths. In that case, we say that our tree is a v-path-star tree. Suppose that we have a tree T with n 1 leaves, labeled by the set [n]. Suppose also that T does not have internal vertices of degree 2. Moreover, assume that for each edge e of T, there exists a unique real number ω(e) associated to e such that: ω(e) > 0 whenever e is an internal edge of T, that is, whenever e is adjacent to two internal vertices. ω(e) R whenever e is not an internal edge of T, that is, whenever e is adjacent to a leaf. No restrictions apply to this case. We call ω(e) the weight of e and we say that our tree T is a weighted tree. We will let E(T ) be the set of (weighted) edges of T, V(T ) be the set of vertices of T, and L(T ) be the set of (labeled) leaves of T. To briefly convey all these conditions, we sometimes say that T is an n-tree, or simply that T is a tree. In fact, every time we refer to a tree T in this writing, we will always refer to a weighted tree in the sense described above. In particular, note that this convention involves the use of the letter T. For every tree T, let the total weight ω(t ) of T be defined by: ω(t ) := e E(T ) ω(e). If E(T ) =, we adopt the convention ω(t ) = 0. Now, for m [n], we define a vector d R ([n] m) associated with the tree T, which we call the m-dissimilarity vector of T. Notice that we have not indexed our vector d with the integer m. Most of the time, the value of m will be clear from the context and whenever it s not, we will explicitly refer to d as the m-dissimilarity vector of our tree. For each m-set {i 1, i 2,..., i m } ( [n] m), let di1i2...i m be the total weight of the minimal subtree (w.r.t number of edges) of T that contains the leaves i 1, i 2,..., i m. Equivalently, let d i1i 2...i m be the total weight of the subtree of T spanned by these leaves. In particular, when d is the 1-dissimilarity vector, notice that d = 0 R ([n] 1 ). This case is not very interesting. In case d is the n-dissimilarity vector, we have d = ω(t ) R ([n] n ). Neither is this very interesting.

3 TREES AND THEIR TROPICAL LINEAR SPACES 3 2. Background 2.1. The Tropical Grassmannians. We first mention the idea of classical Grassmannians. Let F be a field of characteristic 0. The space of all m-dimensional subspaces of F n is called the Grassmannian G m,n,f. Given m linearly independent vectors v 1, v 2,..., v m of F n spanning an m-dimensional subspace V, form the m n matrix M whose rows are the vectors v 1, v 2..., v m. Let P F ([n] m) be the vector of all maximal minors of M where, for all {i1, i 2,..., i m } ( [n] m), Pi1i2...i m is the minor coming from columns i 1, i 2,..., i m of M. Call P the vector of Plücker coordinates of V. The following theorem and proof can be found in [19]. Theorem 2.1. The m-dimensional linear subspaces of F n are uniquely determined by their vector of Plücker coordinates, modulo multiplication by a constant non-zero scalar. Proof. Consider the m n matrix M just defined. Any other m n matrix with equal row space is obtained as M for some invertible matrix. Each maximal minor of M is then det( ) times the corresponding maximal minor of M. Now, suppose that the maximal minors of M are a constant multiple λ of the corresponding maximal minors of N. We show that M and N have equal rowspace. For some ρ = {ρ 1 < ρ 2 < < ρ m } ( [n] m), the maximal submatrix Mρ of M obtained from columns ρ satisfies det(m ρ ) 0. Thus, det(m ρ ) = λ det(n ρ ) 0, where N ρ is the corresponding maximal submatrix of N. We can find ρ because the rank of both M and N is precisely m. Define M = Mρ 1 M and N = Nρ 1 N. Clearly, the row span of N coincides with the row span of N and the same is true for M and M. The columns ρ of M and N form the identity matrix. Consider some ϱ ( [n] m). We have det(m ϱ) = det(mρ 1 M ϱ ) = det(mρ 1 )det(m ϱ ) = 1 1 det(nρ )λdet(n ϱ ) = det(nρ 1 )det(n ϱ ) λ = det(nρ 1 N ϱ ) = det(n ϱ). Therefore, the maximal minors of M and N are equal. Take a pair 1 i m and 1 j n. Consider the columns ρ 1,..., ρ i,..., ρ m and j. The minor given by these columns is ( 1) l (M ) ij in M and ( 1) l (N ) ij in N, for l {0, 1}. But these two are equal and so (M ) ij = (N ) ij, that is, M = N. Now, suppose that we have a vector x F n and we want to check if x V. To the matrix M, append the row x after the m-th row to obtain an (m + 1) n matrix M x. If x is a point in V, then all the maximal minors of M x have to be 0, clearly. On the other hand, if x V, then M x is a rank-(m + 1) matrix, so it must have m + 1 linearly independent columns as well, which then give rise to a non-zero maximal minor. Thus, x V if and only if all the maximal minors of M x are 0. Now, notice that the maximal minors of M x can be expressed using the Plücker coordinates of V, doing cofactor expansion of the determinants along the last row. To do this, for every {i 1, i 2,..., i m } ( [n] m), recall that Pi1i 2...i m is the maximal

4 4 BENJAMIN IRIARTE minor of M coming from columns i 1, i 2,..., i m : P i1i 2...i m is a Plücker coordinate of V. Then, we can see that the maximal minors of M x are: m+1 ( 1) r P i1i 2...î r...i mi m+1 x ir for all 1 i 1 < i 2 < < i m < i m+1 n. r=0 Therefore, x is a point in V if and only if all these sums are 0. We will keep this characterization in mind when defining the completely analogous notion of tropical linear space. The tropical linear spaces will arise after tropicalizing the sums presented above. We are now ready to provide some background on tropical Grassmannians. To begin, consider the field K = C{{t}} of dual Puiseux series. Recall that this is the algebraically closed field of formal expressions k=p ω = c k t k/q where p Z, c p 0, q Z + and c k C for all k p. It is the algebraic closure of the field of dual Laurent series over C. The field comes equipped with a standard valuation val : K Q { } by which val(ω) = p/q. As a convention, val(0) =. Let X = (X ij ) be an m n matrix of indeterminates and let K[X] denote the polynomial ring over K generated by these indeterminates. Fix a second polynomial ring in ( n m) indeterminates over the same field: K[Y ] = K[Yi1i2...i m : {i 1, i 2,..., i m } ( [n] m) ]. Let φ m,n : K[Y ] K[X] be the homomorphism of rings taking Y i1...i m to the maximal minor of X obtained from columns i 1,..., i m. The Plücker ideal or ideal of Plücker relations will be the homogeneous prime ideal I m,n =ker(φ m,n ), which contains all the algebraic relations or syzygies among the m m minors of any m n matrix with entries in K. In particular, letting F = K during the first lines of this section, we can see that the Plücker coordinates P i1i 2...i m with {i 1 i 2... i m } ( [n] m) of the m-dimensional vector space V considered, lie in the variety V (I m,n ) of I m,n. For m 2, the Plücker ideal has a Gröbner basis consisting of quadrics; a comprehensive study of these ideals can be found in Chapter 14 of the book by Miller and Sturmfels [19] and in Sturmfels [29]. In particular, the tropicalization of some of these quadrics, known as the three-term Plücker relations, will give rise to the notion of tropical Plücker relations that we present in the following subsection. We note that for the purposes of studies in evolutionary biology and phylogenetics, the max convention should be adopted for tropical geometry. We follow that convention here. Now, the Plücker ideal is a polynomial ideal in K[Y ]. We can therefore define the notion of its tropical variety in the usual way, as we now recall. Let R = R { }. Consider a finite set A Z ([n] m) 0 and let: f = c α Yρ αρ K[Y ]. α A ρ ( [n] m)

5 TREES AND THEIR TROPICAL LINEAR SPACES 5 The tropicalization of f is given by: trop(f) = max α A {val(c α) + ρ ( [n] m) α ρ y ρ }. The tropical hypersurface T ( trop(f)) of f is the set of points in R ([n] m) where trop(f) attains its maximum twice or, equivalently, where trop(f) is not differentiable. It is the analogous tropical notion of the classical hypersurface of 0 s in K ([n] m) of f. By the fundamental theorem of tropical algebraic geometry [26, Theorem 2.1], the classical and tropical hypersurfaces are related in the expected way: the later is the (closure of) the pointwise valuation of the former. The actual content of the theorem says that the same will still hold for polynomial ideals and their varieties. In Theorem 2.2, we present a narrower version of the fundamental theorem, suitable for our purposes. With these tools, we now define the tropical analog of the Grassmannian G m,n,k. The tropical variety T (I m,n ) of the Plücker ideal I m,n is denoted by G m,n and is called the tropical Grassmannian. It is defined as the intersection: T (I m,n ) = T ( trop(f)). f I m,n We have the following characterization of G m,n, which is a direct application of the fundamental theorem of tropical algebraic geometry: Theorem 2.2. The following subsets of R m) ([n] coincide: The tropical Grassmannian G m,n. The closure of the set {(val(c ρ )) ρ ( [n] m) : (c ρ) ρ ( [n] m) V (I m,n) K m) ([n] }. An important consequence of this theorem will be the following. Suppose that a vector P K m) ([n] arises as the set of maximal minors of a full rank m n matrix with entries from K. That is, suppose that P is the vector of Plücker coordinates of some m-dimensional vector space in K n, as V. Then, its componentwise valuation p = val(p ) = (val(p ρ )) ρ ( [n] m) will lie in the tropical Grassmanian G m,n. Naturally, p will also satisfy the tropical Plücker relations that we will describe soon, and it will do so as much as P satisfies the classical Plücker relations, and the former are the tropicalization of a subset of the latter Tropical Linear Spaces. In this subsection we introduce one of the main objects of this writing. Let p R ([n] m). We say that p satisfies the tropical Plücker relations with the max convention if the maximum of the set {p Sij + p Skl, p Sik + p Sjl, p Sil + p jk } is attained at least twice for all S ( ) ( [n] m 2 and {i, j, k, l} [n]\s ) 4. For all {i 1, i 2,..., i m } ( [n] m), define ei1i2...i m = e i1 + e i2 + + e im, where the sum takes place in R n with its canonical basis {e 1, e 2,..., e n }.

6 6 BENJAMIN IRIARTE Let H m be the m-hypersimplex of R n. Recall that H m is the polytope obtained as the convex hull of all vectors e i1i 2...i m with {i 1, i 2,..., i m } ( [n] m). Each of these vectors is a vertex of H m. Suppose that p R ([n] m). We define a regular subdivision Dp of H m from p. Let R R n+1 be the polytope obtained as the convex hull of all points (e i1i 2...i m, p i1i 2...i m ) with {i 1, i 2,..., i m } ( [n] m). Consider the face of R consisting of all points r R that maximize a linear functional ( x, 1) r with x R n. This face is the convex hull of the vertices (e i1i 2...i m, p i1i 2...i m ) that it contains. Now, let B x be the collection of sets {i 1, i 2,..., i m } ( [n] m) for which (ei1i 2...i m, p i1i 2...i m ) is a point in our face. This is the collection of sets {i 1, i 2,..., i m } ( [n] m) for which p i1i 2...i m (x i1 + x i2 + + x im ) is maximal. Finally, construct a face P x of D p obtained from x as the convex hull of all e i1i 2...i m with {i 1, i 2,..., i m } B x. Now, let P be a subpolytope of H m. To recall, this means that the vertices of P are of the form e i1i 2...i m with {i 1, i 2,..., i m } ( [n] m). Let BP be the collection of sets {i 1, i 2,..., i m } ( [n] m) for which ei1i2...i m P. We say that P is matroidal or that P is a matroid polytope if the set B P is the collection of bases of a rank-m matroid over [n], see Oxley [20, Chapter 1] for a general introduction to the language of matroids. The following is a theorem of David Speyer [24], which motivates the introduction of tropical Plücker relations. Theorem 2.3. The following assertions are equivalent: the vector p R m) ([n] satisfies the tropical Plücker relations; the one skeleta of D p and H m are the same; every face of D p is matroidal. Now, suppose that p R m) ([n] satisfies the tropical Plücker relations. Define L(p) R n as the set of all x for which the maximum of the set {p i1i 2...î r...i mi m+1 + x ir r [m + 1]} is attained at least twice, for all {i 1, i 2,..., i m, i m+1 } ( [n] m+1). The set L(p) will be called the m-dimensional tropical linear space associated to p. For any x R n, consider the subpolytope P x of H m introduced earlier. Recall that P x is a face of D p obtained as the convex hull of all points e i1i 2...i m for which p i1i 2...i m (x i1 + x i2 + + x im ) is maximal. Also, notice that B Px = B x. As p satisfies the tropical Plücker relations, we conclude that B x is the set of bases of some rank-m matroid over [n], which we call M x. We now present a remarkable theorem of Speyer [24]. Theorem 2.4. x L(p) if and only if M x is a loopless matroid. Recall that a matroid over a finite set E is said to be loopless if every element of E is contained in at least one basis. Proof. To begin, notice that L(p) can be described as the set of all x for which the maximum of the set {p i1i 2...î r...i mi m+1 ( x i1 + x i2 + + x ir + + x im + x im+1 ) r [m + 1]}

7 TREES AND THEIR TROPICAL LINEAR SPACES 7 is attained at least twice, for all {i 1, i 2,..., i m, i m+1 } ( [n] m+1). So let x L(p) and suppose, without loss of generality, that [m] B x. Take any element of the set [n]\[m]. We can assume without loss of generality that this element is m + 1. By the definition of L(p), the maximum of the set {p r...m(m+1) ( x 1 + x x r + + x m + x (m+1) ) r [m + 1]} is attained at least twice. This maximum is attained by p 12...m (x 1 + x x m ). But then, any other choice of subindices for which the maximum is attained must contain m + 1, and this set of subindices will be an element of B x. Therefore, M x is loopless. Now, suppose that M x is loopless and we want to prove x L(p). Without loss of generality, it is enough to show that the maximum of the set {p r...m(m+1) ( x 1 + x x r + + x m + x (m+1) ) r [m + 1]} is attained at least twice. Let e [m+1] = e 1 + e e m + e m+1. For all reals t 0, let M t = M x te[m+1] and let B t = B x te[m+1]. Notice that for large enough values t > 0, we have B t [m + 1]. Let t be the infimum of these values. Then M t = M t for all t t. As t ranges from 0 to, the matroid M t changes at a finite number of times t, which we call 0 = t 0 < t 1 < t 2 < < t q = t, and remains constant throughout (t l, t l+1 ) for all 0 l < q. Let t (t l, t l+1 ). Then, notice that the set of bases of M t coincides with the subset of bases of M tl that meet [m + 1] at a maximal number of points. Now, suppose that every element of [m + 1] is contained in a basis of M tl. Let i be the maximal number of elements of [m + 1] contained in a basis of M tl. Assume that [i] I for some I B tl. If j ([m + 1]\[i]), find J B tl for which j J. Then, by the maximality of i and considering the bases I and J, we can see that ((I k) j) B tl for some k [i]. Therefore, for all j [m + 1], there exists J B tl such that j J and J [m + 1] = i. That is, every element of [m + 1] lies in basis of M tl which meets [m + 1] at a maximal number of points. Therefore, every element of [m + 1] is contained in a basis of M t. On the other hand, the set of bases of M t coincides with the subset of bases of M tl+1 that meet [m + 1] at a minimal number of points. If every element of [m + 1] is contained in a basis of M t, then every element of [m + 1] is contained in a basis of M tl+1. As M x = M t0 is loopless, so in particular every element of [m+1] is contained in a basis, inductively we obtain that every element of [m + 1] is contained in a basis of M tq = M t. But notice that x L(p) if and only if B t 2, which is what we just proved. A different approach to tropical linear spaces is fully presented now. Let p R ([n] m) satisfy the tropical Plücker relations with the max convention. Consider the nonempty unbounded n-dimensional polyhedron P p := {x R n x i1 + x i2 + + x im p i1i 2...i m for all {i 1, i 2,..., i m } ( [n] m) }.

8 8 BENJAMIN IRIARTE The polyhedron P d coincides with the notion of the envelope of H m with respect to p, as presented in [12]. Let the reduced tropical linear space of p be the set P p := {x P p If y x componentwise for some y P p, then we have y = x}. Take x (P p ) and define B x to be the collection of all sets {i 1, i 2,..., i m } ( [n] m) for which x i1 + x i2 + + x im = p i1i 2...i m. The set B x is the set of bases of a matroid M x with rank m, over the set [n]. To understand why this is true, consider the polytope R of R n+1 previously defined. For each r R, notice that ( x, 1) r 0. The set of all such points r for which ( x, 1) r = 0 is a face of R. Of course, this face is then the convex hull of all vertices (e i1i 2...i m, p i1i 2...i m ) of R with {i 1, i 2,..., i m } B x, and its projection P x back in the m-hypersimplex H m is the convex hull of the vertices e i1i 2...i m with {i 1, i 2,..., i m } B x. Given that p obeys the tropical Plücker relations with the max convention, we know that P x is matroidal, so indeed B x is the set of bases of a matroid M x. Now, notice that x P p if and only if M x is a loopless matroid. Consider the set L m,n of all loopless matroids of rank m on the set [n]. Following the spirit of our previous notation, denote the set of bases of a matroid M L m,n by B(M). For each M L m,n, define the set P p (M) := {x P p x i1 + x i2 + + x im = whenever {i 1, i 2,..., i m } B(M)}. p i1i 2...i m Clearly P p (M) is a polyhedron, which can be empty for some choices of M. Moreover, notice that P p (M) is a (closed) proper face of P p or a face of the boundary complex (P p ) of P p. Notice also that we can write P p = P p (M). M L m,n Therefore, the set P p is a polyhedral subcomplex of (P p ). It has been proved in [24] that P p is pure (m 1)-dimensional, or equivalently (as we shall see), that L(p) is pure m-dimensional. We have: For x R n, x L(p) if and only if M x is loopless. For x (P p ), x P p if and only if M x is loopless. Now, notice that for any x R n we may find a real t for which x + te [n] P p. In particular, for every x R n, there exists a unique t for which x + te [n] (P p ). On the other hand, notice that L(p) and the matroid M x are invariant under translation by e [n]. Therefore, we conlude that L(p) and P d are the same object if considered in the tropical projective space TP n 1 = R n /(1,..., 1)R, and P p arises from L(p) after a natural choice of representative for each class. Under this choice, the maximum in the definition of L(p) is precisely 0. This allows us to write P p = L(p)/(1,..., 1)R.

9 TREES AND THEIR TROPICAL LINEAR SPACES 9 Let S R n. The subset S is said to be tropically convex if (max{a + x 1, b + y 1 }, max{a + x 2, b + y 2 },..., max{a + x n, b + y n }) S Now, notice that the set {p r...m(m+1) + x r r [m + 1]} for all x, y S and a, b R. is tropically convex. By the definition, the intersection of two tropically convex sets is tropically convex. We then see that the tropical linear space L(p) is tropically convex. It is proved in [8] that tropically convex sets are contractible. Therefore, we see that L(p) is a contractible subspace of R n. It deformation retracts onto P p, which is then also contractible in R n. On the other hand, for x (P p ), notice that if M x contains a loop, then x belongs to the relative interior of an unbounded face of (P p ). To see this, consider the loop in the defining equations of the face. Therefore, the complex P p can be obtained from the polyhedron P p by removing the relative interior of a number of unbounded faces, in particular, the interior of P p. Finally, we want to emphasize the fact that P p may contain unbounded faces. In any case, it is common practice to consider only the complex of bounded faces of P p. As noted, this coincides with the complex of bounded faces of the polyhedron P p, and it will be the main object of our study. Let the tight span T d of p be the complex of bounded faces of P d. It has been proved by Herrmann and Joswig in [12, Proposition 2.3] that T p coincides with the bounded faces of the dual complex of the subdivision of H m induced by p. The tight span T p is also known to be a contractible space, see for example Dress [10] or Hirai [14]. We would now like to present a lemma from which the contractibility of L(p), P p and T p can be deduced. Lemma 2.5. Let P R n be an n-dimensional unbounded polyhedron with no lines. Let P be the complex of bounded faces of P and let F be the set of closed unbounded faces of P. For A F, define ( ) P A = P F. Then, P A is contractible. Proof. Let {µ 1, µ 2,..., µ l } be the set of extremal rays of P. For all i [l], consider the projection ρ i : P (P ) along the ray µ i. That is, for x P, let a R 0 be such that x aµ i P and x (a + δ)µ i / P if δ > 0, and define ρ i (x) = x aµ i. We claim that for each i [l], ρ i fixes pointwise P, ρ i (F ) F for all F F, and ρ := ρ 1 ρ 2... ρ l satisfies that ρ(p ) = P. Hence, ρ induces a deformation retraction of P A onto P. Write P = {x R n Γx β} for some matrix Γ and vector β. Notice Γµ i 0. To prove the first claim, take y P. Let E be a closed face of the complex P such that y E. Let Γ E be the submatrix of Γ describing E and β E the respective subvector of β, so that E = {x P Γ E x = β E }. Then, Γ E µ i must contain a F A

10 10 BENJAMIN IRIARTE strictly positive entry because otherwise E is unbounded, a contradiction. But then, Γ E (y δµ i ) β E contains a negative entry if δ > 0, so ρ i (y) = y. To prove the second claim, for F F consider the submatrix Γ F of A and the subvector β F of β describing F, so F = {x P Γ F x = β F }. Then, the result is a direct consequence of Γµ i 0. To prove the final claim, for y P let Γ y be the submatrix of Γ describing the equalities attained by ρ(y) in the descrption of P, and let β y be the respective subvector of β so that Γ y ρ(y) = β y. By our second claim Γ y µ i must contain a strictly positive entry for all i [l]. Thus, Γ y µ contains a strictly positive entry for every µ in the cone of rays of P. However, in the decomposition of P as the sum of the convex hull of its vertices and its cone of rays, this shows that ρ(y) lies in the convex hull of vertices of P, which consists of points either in P or in int P. Several studies about the combinatorial structure of the tight span have been done, in particular as it relates to an important problem in evolutionary biology: to determine trees from their metric structure. There have been important results and conjectures on the face enumeration of T p [11], among which we find most relevant to us the ones from Speyer [24]. Conjecture 2.6 (The f-vector Conjecture). Let p R m) ([n] satisfy the tropical Plücker relations. Then, the f-vector of T p satisfies the following bounds: ( )( ) n 2 2i n i 2 f i for all i with 0 i m 1. d 1 i i In particular, series-parallel tropical linear spaces realize the f-vector of the f-vector conjecture. An example of these are the tree-tropical linear spaces of Speyer [24]. These spaces can be regarded as approximations to the tropical linear spaces of dissimilarity vectors, which we study in Section 4. On the other hand, tropical linear spaces which are realizable have f-vectors which are bounded by the f-vector conjecture [25]. We present the main idea of this Section in a separate Proposition. Proposition 2.7 (Speyer, Herrmann, Joswig). Let p R m) ([n] satisfy the tropical Plücker relations with the max convention. Then P p = L(p)/(1,..., 1)R and T p coincides with the subcomplex of bounded faces of the dual complex to the subdivision of H m induced by p Results About Trees. Consider a tree T with m-dissimilarity vector d. In this writing, we prove that d satisfies the tropical Plücker relations. We will later provide a proof of this fact by obtaining d as the componentwise valuation of a vector of classical Plücker coordinates of an m-dimensional vector space of K n. By the fundamental theorem of tropical algebraic geometry, this will moreover imply that d G m,n. The present writing is then in reality an investigation of the objects that we have associated to the tropical Plücker relations, as obtained from d. To begin, we point out that a connection between phylogenetic trees and tropical geometry had been noted some time ago. That these two subjects are mathematically related can be traced back to Pachter and Speyer [21], Speyer and Sturmfels [26], and Ardila and Klivans [2]. The precise nature of this connection has been the matter of some recent papers by Bocci and Cools [3] and Cools [7]. Later,

11 TREES AND THEIR TROPICAL LINEAR SPACES 11 we will describe a concise relation between m-dissimilarity vectors and tropical Grassmannians G m,n. In order to present the classical motivating example, we develop some concepts. Consider the polynomial ring K[Y ij : {i, j} ( ) [n] 2 ]. In this ring, consider the ideal of Plücker relations, or the Plücker ideal I 2,n : I 2,n = Y ij Y kl Y ik Y jl + Y il Y jk : 1 i < j < k < l n. The generators of I 2,n presented are called the quadric relations, or simply quadrics. As remarked earlier, they form a Gröbner basis of I 2,n. Even more, it is known that the quadrics form a tropical basis of I 2,n, by which it is meant that G 2,n = T ( trop(y ij Y kl Y ik Y jl + Y il Y jk )). 1 i<j<k<l n We can restate this by saying that G 2,n is the set of points y R ([n] 2 ) for which the maximum of the set {y ij + y kl, y ik + y jl, y il + y jk } is attained at least twice, for all {i, j, k, l} ( ) [n] 4. Notice that, as promised, a vector of R ([n] 2 ) lies in the tropical Grassmannian G 2,n if and only if it satisfies the tropical Plücker relations. The main motivating example for studying these tropical Grassmannians in connection with trees comes from the following theorem, also known as the four point condition theorem. It is a theorem of Buneman [5], widely applicable in evolutionary biology. Theorem 2.8 (Pachter and Sturmfels [23]). The set of 2-dissimilarity vectors of trees is equal to the tropical Grassmannian G 2,n. The theorem of Buneman is used as a tool for reconstructing shapes of trees from their metrics. The extent of its applicability relies on the fact that we can project generic points in R ([n] 2 ) on the tropical Grassmannian G2,n. See Ardila [1]. One of the main open problems in computational biology is to device a way to use the higher m-dissimilarity vectors to carry on the same purpose. It is commonly known that such approach would allow more reliable statistical solutions of the tree reconstruction problem. From here sprouts the natural problem of characterizing m-dissimilarity vectors of trees. In the case of m = 2, there are other characterizations of dissimilarity vectors. Among these, we present a result of Dress [9] which relates tropical linear spaces and 2-dissimilarity vectors. This result is also nicely obtained in Hirai [14, Appendix A], with the additional verification that the face the author finds to prove it is indeed a bounded face of the reduced tropical linear space. Theorem 2.9. A vector d R ([n] 2 ) is a 2-dissimilarity vector if and only if Td is a tree. Describing the object T d for general m-dissimilarity vectors will be the main goal of Section 4. See also Sturmfels and Yu for pictures of tight spans of generic metrics [30].

12 12 BENJAMIN IRIARTE Now, with the theorem of Buneman as motivation, the question had been posed of whether or not the set of m-dissimilarity vectors of trees is contained in the tropical Grassmannian G m,n, for all values of m. This was initially asked by Lior Pachter and David Speyer [21]. The major obstacle in trying to prove such result comes mainly from the fact that no tropical basis is known for the Plücker ideals I m,n for m 3. Notice that the knowledge of such bases would reduce the problem to a finite number of verifications for each value of m. We have not found tropical bases for these ideals. A different approach to understanding m-dissimilarity vectors for m 3 is to find a relation between them and the 2-dissimilarity vectors, which are all well understood. Along this line of thought, a key result of Cools is the existence of a map that, from the 2-dissimilarity vector of a tree T, computes its m-dissimilarity vector. We provide a combinatorial proof of this fact. Proposition Consider a tree T and let m 3. Let C m S m be the set of cyclic permutations. For any σ C m, define d σ = d 1σ(1) + d σ(1)σ2 (1) + d σ2 (1)σ 3 (1) + d σ3 (1)σ 4 (1) + + d σ m 1 (1)σ m (1). Then d 12...m = 1 2 min σ C m d σ. Proof. Without loss of generality, assume that m = n. Consider an internal vertex v of T. This vertex induces a partition on the set of leaves of T under which two leaves i and j belong to the same class if and only if the minimal path from i to v and the minimal path from j to v share a common edge. But then, each of the sums d σ above can be seen to consider at least twice every edge adjacent to v. Thus, d σ considers every edge of T at least twice. Now, let σ C n be such that d σ is minimal. Suppose that some edge e adjacent to v is considered more than twice in d σ. Notice that e must then be internal, so ω(e) > 0. Let e be the class of leaves of T containing e in their minimal path to v. Then, there exist a, a 1,..., a i e and b, b 1,..., b j ([n]\e) such that Now, define σ = (, a, b 1,, b j, a 1,, a i, b, ). σ = σ(a, a i )(a, b j ) = (, a, a 1,, a i, b 1,, b j, b, ) C n. But then d σ d σ 2ω(e). This contradicts the minimality of d σ. Therefore, every edge is considered exactly twice in d σ, which proves the result. Remark We expand on the argument presented. Abusing the notation of the proof, for every leaf i of T, let d iv be the total weight of the minimal path from i

13 TREES AND THEIR TROPICAL LINEAR SPACES 13 to v. Then: d a b 1 = d a v + d b1v d a1b j = d a1v + d bjv d aib d aib 1 = d a iv + d b v = d aiv + d b1v. From here we obtain: ( da b 1 + d a1b j + d aib ) ( da a 1 + d aib 1 + d bjb ) = ((d a v + d a1v) d a a 1 ) + (( ) ) d bjv + d b v dbjb 2ω(e) + 0 = 2ω(e). From this result we could try to study m-dissimilarity vectors and the way in which they induce a metric on T. An initial attempt would go along the lines of trying to invert the presented map. One immediate application of the result is presented now. Corollary Suppose that we have a tree T. Then, the following identity holds for all {i, j, k, l, p} ( ) [n] 5 : d ij = 1 3 (2d ijk + 2d ijl + 2d ijp d ipl d ikl d ikp d jpl d jkl d jkp + 2d plk ). Corollary 2.12 is a direct consequence of the identity d ijk = 1 2 (d ij + d ik + d jk ) for all {i, j, k} ( ) [n] 3. In order to visualize the computation involved, it could be useful to think of the following construction. Let A = {i, j, k, l, p}. Consider the subdivision of a 3-dimensional tetrahedron into four tetrahedra, induced by selecting a point i in the interior of the tetrahedron. Then, label the remaining vertices j, k, l, p. Note that ( A 3) is the set of 2-faces of the subdivision and ( A 2) is the set of edges. Let Cl be the free Q-module with basis ( A l+1), and define a Q-module homomorphism : C2 C 1 under which [ijk] = 1 2 ([ij] + [ik] + [jk]), for all [ijk] ( A 3). Then, carry on the computation (2[ijk] + 2[ijl] + 2[ijp] [ipl] [ikl] [ikp] [jpl] [jkl] [jkp] + 2[plk]) to obtain 3[ij]. To finish, we note that for m 4 the situation is rather unclear.

14 14 BENJAMIN IRIARTE 3. Dissimilarity Vectors are Contained in the Tropical Grassmannian The result presented in this section is based on two papers of Cools [7] and Bocci and Cools [3], where the cases m = 3, m = 4 and m = 5 are handled. We answer the question of Pachter and Speyer affirmatively for all m: Theorem 3.1. Let T be a tree with m-dissimilarity vector d. Then, d G m,n. This result has more recently been proved by Manon [18], using very interesting ideas from representation theory. In order to prove Theorem 3.1, we need to introduce a new family of graphtheoretical-trees with special combinatorial properties. We will refer to this new kind of tree using the letter U. In particular, we do this to stress the difference with the conventions adopted in Section 1 for the letter T. To start, let U be a tree with n leaves labeled by the set [n], with n 1. Suppose also that U has a weight function E(U) ω R 0. The function ω will extend to a function from the subtrees of U to the nonnegative reals in the natural way. In the latter case, we will speak of total weights of subtrees, following our convention. In particular, notice that this time we are allowing internal edges of weight 0, and not allowing external edges with negative weight. The tree U will be refered to as being ultrametric if it further satisfies that: U is trivalent; U is rooted; U is l-equidistant. That is, the total weight of the minimal path from every leaf to the root is a constant l; ω induces a metric on [n]. In particular, it separates points. Note that the fourth condition directly implies l > 0. On the other hand, there is also an independent notion of an ultrametric. A metric space S with distance function d : S S R 0 is called an ultrametric space if the following inequality holds for all x, y, z S: d xz max{d xy, d yz }. It is a well known fact that finite ultrametric spaces are realized by ultrametric trees, see for example [4, Lemma 11.1]. Note that the condition on ultrametric spaces S is equivalent to saying that for all x, y, z S, the maximum of the set {d xy, d xz, d yz } is attained at least twice. Now, suppose that we have a tree T with 2-dissimilarity vector d. From here, define a vector d R ([n] 2 ) by: d ij = 2t + d ij d in d jn for all different i, j [n]. Let t > 0 be sufficiently large, so that d ij > 0 for all different i, j. Observe that d in = 2t for all i [n 1]. Now, pick {i, j, k} ( ) [n 1] 3 and consider the set {d ij d in d jn, d ik d in d kn, d jk d jn d kn }.

15 TREES AND THEIR TROPICAL LINEAR SPACES 15 We want to compare the members of this set. Comparing them pairwise, we obtain three new sets to study: {d ij + d kn, d ik + d jn } {d ij + d kn, d in + d jk } {d ik + d jn, d in + d jk }. Note that these sets involve only three numbers. By the theorem of Buneman, one of these three sets contains the same number twice, and that number is greater than or equal to the third one. This translates back to concluding that the maximum of the set {d ij d in d jn, d ik d in d kn, d jk d jn d kn } is attained at least twice. Therefore, d ( [n 1] [n 1]. Hence, d ( [n 1] 2 ) 2 ) is realized by an ultrametric tree. induces an ultrametric on the set Following this ingenious idea, Cools shows somewhat technically that in order to prove Theorem 3.1, it suffices to prove the following variation: Theorem 3.2. Let U be an ultrametric tree with m-dissimilarity vector d. Then, d G m,n. This is the theorem that we will prove. First, we will need to introduce some terminology Column Reductions. Let n 3. Suppose that we are given integers 1 a, b n with a b and let c a,b be the operator acting on Puiseux matrices for which, for any n n matrix M, c a,b (M) is the matrix obtained from M by subtracting column b to column a. We know that c a,b preserves the determinant, i.e. det (c a,b (M)) = det(m). For l 1, let (c al,b l c a2,b 2 c a1,b 1 ) (M) be the matrix obtained from M by first subtracting column b 1 to column a 1, then subtracting column b 2 to column a 2, and so on up to subtracting column b l to column a l. Call this matrix a column reduction of M if the following conditions are met: 1 a 1,..., a l, b 1,..., b l n; the numbers a 1, a 2,..., a l are pairwise different; whenever 1 k l, the number b k is different from a 1,..., a k. For simplicity, we will accept M as a column reduction of itself Main Result. We are now ready to prove Theorem 3.1. We do this by proving Theorem 3.2 for m 3. Proposition 3.3. Suppose that 3 m n. Let U be an ultrametric tree with m-dissimilarity vector d, all of whose edges have rational weight. For each edge e of U, denote by h(e) the total weight of the minimal path from the top node of e to any leaf below e. Also, let a 1 (e),..., a n 2 (e) be n 2 generic complex numbers associated to e. For all i [n 2] and j [n], let t ij be the sum of the monomials a i (e)t h(e), where e runs over all edges in the minimal path from leaf j to r. Notice that t ij K.

16 16 BENJAMIN IRIARTE Then, define a matrix: t 11 t t 1n (t 11 ) 2 (t 12 ) 2... (t 1n ) 2 M = t 21 t t 2n..... t (n 2)1 t (n 2)2... t (n 2)n Finally, for all ρ ( [n] m), let mρ be the m m upper minor of M coming from the columns ρ. Then, val(m ρ ) = d ρ. Proof. First, consider the following matrix for some p 1, p 2,..., p n 2 K: t 11 + p 1 t 12 + p 1... t 1n + p 1 (t 11 + p 1 ) 2 (t 12 + p 1 ) 2... (t 1n + p 1 ) 2 N = t 21 + p 2 t 22 + p 2... t 2n + p t (n 2)1 + p n 2 t (n 2)2 + p n 2... t (n 2)n + p n 2 Then, notice that the situation is completely analogous to proving that val (det(n)) = ω(u). However, by first expanding the quadric terms of the second row, then subtracting multiples of the first row to all rows to cancel the p-terms, and then using the second row to cancel the appropriate remaining terms in the third row, we see that det(n) = det(m). Therefore, there is no loss of generality if we prove that val (det(m)) = ω(u). This is the plan. As U is trivalent with n leaves, we know that U has n 2 internal nodes of degree three, one node (the root) of degree two and 2(n 1) edges. Let U be the tree order of U with respect to r, that is, the order on the set of nodes of U by which v U w if and only if v lies in the path from r to w in U. Let v 1, v 2,..., v n 1 be the n 1 internal nodes of U, numbered in such a way that if v i U v j, then j i. We must have v n 1 = r. Consider an injective function α : v i a i from the set of internal nodes to the leaves of U, so that v i U a i for all i with 1 i n 1. Now, for each of these values of i, let b i be the unique leaf such that b i a j for all j with 1 j i, and such that v i U b i. To show the existence of α, we construct it succesively starting with α(v 1 ), then α(v 2 ) and then continuing up until we define α(v n 1 ). Suppose that we have already defined α(v 1 ),..., α(v i 1 ) for some i < n 1. Consider the maximal subtree U i of U whose root is v i, i.e. U i is the subtree below v i. If this tree has m leaves, then it has m 1 internal nodes, including v i itself. So far, we have not defined α for nodes between r and v i, but we have defined it for all internal nodes of U i different from v i. Therefore, there are exactly m 2 leaves of the tree U i which

17 TREES AND THEIR TROPICAL LINEAR SPACES 17 have been assigned to some of v 1,..., v i 1 under α, so there are 2 leaves which we can assign to v i : α(v i ) can be either one of them. Incidentally, this also gives us the existence and uniqueness of the respective b i. Now, we want to establish the equality n 1 i=1 h(v i) = ω(u) l. This equality is clearly true when U has 2 or 3 leaves, so that n = 2 or n = 3. Let now n 4 and suppose that we have proved the result for all trees with i leaves, with i < n. Recall n is being taken as the number of leaves in U, which is rooted l-equidistant with root r = v n 1. We know that the equality holds for each of the subtrees U 1,..., U n 2 below v 1,..., v n 2, respectively. Let U n 2 be l n 2 -equidistant and let U n 3 be l n 3 -equidistant. There are two cases to distinguish. If v n 2 < U v i for all i < n 2, then n 2 i=1 h(v i) = (ω(u) l (l l n 2 )) l n 2 = ω(u) 2l by induction, so n 1 i=1 h(v i) = ω(u) l. Otherwise, suppose that v n 2 and v n 3 are incomparable in < U. Then U n 2 and U n 3 are disjoint graphs and we have v i V(U n 2) h(v i ) = ω(u) by induction. Reordering we get: v i V(U n 2) v j V(U n 3) h(v i ) + h(v j ) + l n 3 (l l n 3 ) (l l n 2 ) v j V(U n 3) h(v j ) = ω(u) 2l, so if we add h(v n 1 ) = l to both sides, we get our result. l n 2 Now, consider the column reduction M = ( ) c an 1,b n 1 c a2,b 2 c a1,b 1 (M) of M. We claim that the valuation of all the nonzero monomials n i=1 M i,σ(i) with σ S n in the sum det(m ) = ( ) n sgn(σ) σ S n i=1 M i,σ(i) is precisely n 1 i=1 h(v i) + l = ω(u). To see this, notice that for all i, 1 i n 1, we have: M 1a i = 0; the valuation of M 3a i is l + h(v i ); the valuation of M ja i is h(v i ) if j 1 and j 3; the only nonzero term in the first row of M is the 1 in column b n 1. Because of our generic choice of coefficients, we can find some monomial term in the sum det(m ) with valuation ω(u) that does not get cancelled. Example 3.4. Consider the 9-equidistant 10-tree of Figure 1 with total weight 35. The second row of the matrix M associated to this tree is the following vector with

18 18 BENJAMIN IRIARTE r = v 9 5 (p) 5 (q) 9 v 7 v 8 v 5 v 6 v 1 2 (h) v 2 v 3 v 4 1 (a) 1 (b) 3 (f) 2 (g) 1 (c) 1 (d) 1 (e) (r) 1 (s) 2 (x) 1 (z) 2 (y) Figure 1. A rooted 10-tree. The injective function α := {(v 1, 1), (v 2, 4), (v 3, 6), (v 4, 8), (v 5, 3), (v 6, 7), (v 7, 2), (v 8, 9), (v 9, 5)} is depicted, as well as the equality 9 i=1 h(v i) = (u) 1 (v) 4 (w) generic complex coefficients: [at 1 + ft 4 + pt 9,bt 1 + ft 4 + pt 9,ct 2 + gt 4 + pt 9, dt 1 + ht 2 + gt 4 + pt 9,et 1 + ht 2 + gt 4 + pt 9,rt 1 + xt 3 + zt 4 + qt 9, st 1 + xt 3 + zt 4 + qt 9,ut 1 + yt 3 + zt 4 + qt 9,vt 1 + yt 3 + zt 4 + qt 9, wt 4 + qt 9 ] Using the operator (c 5,10 c 9,10 c 2,5 c 7,9 c 3,5 c 8,9 c 6,7 c 4,5 c 1,2 ) suggested by the figure we obtain the column reduction M whose second row is the vector: [(a b)t 1, (b e)t 1 ht 2 + (f g)t 4, et + (c h)t 2, (d e)t 1, et 1 + ht 2 + (g w)t 4 + (p q)t 9, (r s)t 1, (s v)t 1 + (x y)t 3, (u v)t 1, vt 1 + yt 3 + (z w)t 4, wt 4 + qt 9 ] It has valuation vector: (1, 4, 2, 1, 9, 1, 3, 1, 4, 9) = (h(v 1 ), h(v 7 ), h(v 5 ), h(v 2 ), h(v 9 ), h(v 3 ), h(v 6 ), h(v 4 ), h(v 8 )), where v 1, v 7, v 5, v 2, v 9, v 3, v 6, v 4, v 8 are the preimages of 1, 2, 3, 4, 5, 6, 7, 8, 9 under α, respectively in that order. Also notice that 9 i=1 h(v i) = 35 9.

19 TREES AND THEIR TROPICAL LINEAR SPACES 19 We have shown that the m-dissimilarity vector of a tree T gives a point in the tropical Grassmannian G m,n, and therefore satisfies the tropical Plücker relations. Thus, it gives rise to a tropical linear space. We now study this tropical linear space, in its tight span version. 4. The Tropical Linear Space of a Dissimilarity Vector The main object of this section is a tree T with m-dissimilarity vector d. To keep our discussion interesting, from now on and for the remaining part of this writing, d will never be a 1-dissimilarity vector or an n-dissimilarity vector. The purpose of the section is to describe, from both a geometrical and combinatorial point of view, the space T d. During the whole section, we will sometimes use the symbol st to denote the relation of being a subtree of. We will also use st to denote the relation of being a proper subtree of. We consciously use these symbols sporadically, combining them with the use of the words is a subtree of. The motivation of the use of this symbol will be emphasizing the fact that we are comparing trees. Consider the tree in(t ) st T that consists of the internal edges and vertices of T. If T does not contain internal edges or vertices, as a technical convenience we let in(t ) = Λ, where Λ is the abstract tree consisting of an empty set of vertices and edges, and which we consider to be a proper subtree of every other tree. We call in(t ) the internal tree of T. If S st T, let T/ v S be the tree obtained from T by collapsing the subtree S to a single vertex v Vertices of T d and Their Matroids. Our first and main result on the combinatorial structure of the tight span T d of d is a description of its vertices. We prove that the set of vertices of T d is in bijection with the set of subtrees of in(t ) that satisfy certain conditions. To state these conditions, let S st in(t ). Let S be the maximal subtree of T such that in(s) = S. Notice that S is the subtree of T consisting of all edges and vertices of T that are in S or adjacent to S. By definition S st S. We call S the extended tree of S. It will also be convenient to refer to the set L(S) of leaves of S as the set of extended leaves of S. Computing degrees in T, the number of extended leaves L(S) of S is deg(v) 2( V(S) 1). v V(S) The number of vertices of S is deg(v) 2( V(S) 1) + V(S). v V(S) We will say that S is an m-good subtree of T if S st in(t ), L(S) m 1 and L(S) m + 1. Using these definitions, we are now ready to state the main result of this subsection. Theorem 4.1. Let T be a tree with m-dissimilarity vector d. The set of vertices of the tight span T d is in bijection with the set of m-good subtrees of T.

20 20 BENJAMIN IRIARTE In the case where T is a trivalent tree, that is, when each internal edge of T is of constant degree 3, we can enunciate our result in a more compact way. In this case deg(v) = 3 for all v V(S), so the number of extended leaves of S becomes deg(v) 2( V(S) 1) = 3 V(S) 2( V(S) 1) = V(S) + 2. v V(S) Also, the number of vertices of S becomes 2 V(S) + 2 = ( V(S) + 2) + V(S). Therefore, the main result for this case can be stated as follows. Theorem 4.2. Suppose that T is a trivalent tree with m-dissimilarity vector d. The set of vertices of the tight span T d is in bijection with the set of trees S st in(t ) such that L(S) m 1 V(S). To facilitate the exposition of the proof, we introduce some additional notation. Let S st in(t ). For each leaf i [n] of T, there is a unique minimal path from i to S made up solely of edges from E(T )\E(S). Let ω(i, S) be the sum of the weights of all edges in this minimal path. Notice that ω(i, S) coincides with the intuitive notion of distance from the leaf i to the subtree S. From now on, assume that S is an m-good subtree of T as in Theorem 4.1. For all i [n], define the entries of x R n in the following way: x i = ω(i, S) + ω(s) m. Proposition 4.3. The point x R n just defined is a vertex of T d. Proof. It has already been noted that our statement is equivalent to proving that x is a vertex of the polyhedron P d. This is what we do. The proof is divided into several steps. Notice that for each m-set {i 1, i 2,..., i m } [n] we have: x i1 + x i2 + + x im = ω(s) + ω(i 1, S) + ω(i 2, S) + + ω(i m, S). Let T i1i 2...i m be the subtree of T spanned by the leaves i 1, i 2,..., i m. Step I: We have x i1 + x i2 + + x im d i1i 2...i m with equality if and only if S st T i1i 2...i m and T i1i 2...i m / v S is a v-path-star tree. To avoid the excessive use of subscripts, let us assume without loss of generality that i 1 = 1, i 2 = 2,..., i m = m. Let T m = T 12...m. First, suppose that T m does not intersect S at an edge or a vertex. Notice that ω(t m ) = d 12...m by definition. There exists a unique path P = (v S, v m ) with v S V(S) and v m V(T m ) which consists purely of edges from the set E(T )\ (E(S) E(T m )). This is the minimal path connecting S and T m. Notice that this minimal path is made up of internal edges of T, which therefore have positive weight. As a consequence, ω(p ) > 0. Also, notice that ω(1, P ) + ω(2, P ) + + ω(m, P ) = ω(1, v m ) + ω(2, v m ) + + ω(m, v m ) d 12...m. However, ω(i, S) = ω(i, P ) + ω(p ) for all i [m], so ω(1, S) + ω(2, S) + + ω(m, S) > d 12...m.