Order distances and split systems

Given a distance $D$ on a finite set $X$ with $n$ elements, it is interesting to understand how the ranking $R_x = z_1,z_2,\dots,z_n$ obtained by ordering the elements in $X$ according to increasing distance $D(x,z_i)$ from $x$, varies with different choices of $x \in X$. The order distance $O_{p,q}(D)$ is a distance on $X$ associated to $D$ which quantifies these variations, where $q \geq \frac{p}{2}>0$ are parameters that control how ties in the rankings are handled. The order distance $O_{p,q}(D)$ of a distance $D$ has been intensively studied in case $D$ is a treelike distance (that is, $D$ arises as the shortest path distances in an edge-weighted tree with leaves labeled by $X$), but relatively little is known about properties of $O_{p,q}(D)$ for general $D$. In this paper we study the order distance for various types of distances that naturally generalize treelike distances in that they can be generated by split systems, i.e. they are examples of so-called $l_1$-distances. In particular we show how and to what extent properties of the split systems associated to the distances $D$ that we study can be used to infer properties of $O_{p,q}(D)$.


Introduction
A distance D on a finite, non-empty set X is a symmetric map D : X × X → R with D(x, x) = 0 and D(x, y) ≥ 0 for all x, y ∈ X.Following [5] we associate a new distance O p,q (D) on X to the distance D for p, q ∈ R with q ≥ p 2 > 0 as follows.For each u, v ∈ X with u = v we define the sets can also be represented by the tree T by adjusting the weights of its edges.Note that each edge of T corresponds to a split of X.
Note that the sets X u,v , X v,u and E {u,v} are pairwise disjoint and that their union is X.Now, for any bipartition or split {A, B} of X, let D {A,B} be the distance on X given by taking D {A,B} (x, y) = 1 if |A ∩ {x, y}| = |B ∩ {x, y}| = 1 and D {A,B} (x, y) = 0 otherwise for all x, y ∈ X.The order distance O p,q (D) associated to D is then defined as where X 2 denotes the set of all 2-element subsets of X.The order distance O p,q (D)(x, y) can be regarded as the amount by which the two rankings z 1 , z 2 , . . ., z n and z 1 , z 2 , . . ., z n of the elements in X generated by ordering these elements according to increasing distances D(x, z i ) from x and D(y, z i ) from y, respectively, differ [5].Note that, as pointed out in [5], to ensure that O p,q (D) satisfies the triangle inequality we must require q ≥ p 2 , and that, for any c ∈ R with c > 0, O c•p,c•q (D) = c • O p,q (D).
Most previous work concerning order distances has focused on their properties for treelike distances, that is, distances that arise by taking lengths of shortest paths between pairs of leaves of a tree (see e.g.[5,14,15,16]).In Figure 1 we present an example of the order distance O = O p,q (D) with p = 2 and q = 1 associated to a treelike distance D. Note that in this example the order distance O = O p,q (D) is also treelike since, as we can see in the figure, it can be represented by adjusting the weights of the edges in T .In fact, this is no coincidence: The main result of Bonnot et al. in [5] establishes that if D is a treelike distance which can be represented by giving non-negative weights to the edges in some tree, then for q = p 2 , after possibly adjusting the edge weights, the order distance O p,q (D) can also be represented by the same tree.
In this paper, we aim to better understand to what extent this result can be extended to more general distances.In particular we shall focus on order distances for so-called 1 -distances (cf.[11,Ch. 4]), that is, distances D on X for which there exists a set S of splits of X or split system, together with a non-negative weighting ω : S → R such that also referred to as an l 1 -decomposition of D. It is natural to consider 1 -distances in the context of order distances since it follows directly from Equation (1) that the order distance O = O p,q (D) for any distance D is an 1 -distance.
Interestingly, Bonnot et al.'s result can be re-expressed in 1 -terminology as follows.To any tree T with leaves labeled by X we can associate the split system S = S T consisting of the splits of X that correspond to the edges of the tree (e.g. the edge in T in Figure 1 with weight 6 corresponds to the split {{a, b}, {c, d, e}}).Bonnot et al.'s result then states that in case q = p 2 , for every non-negative weighting ω of the splits in S, there exists a non-negative weighting ω of the splits in S such that Since the split system S arises from a tree, it has a special combinatorial property known as compatibility (see Section 2).In this paper, we will explore under what conditions Equation (2) might hold for other split systems S that are not necessarily compatible.The rest of the paper is structured as follows.In Section 2 we present some preliminaries concerning the relationship between treelike distances and compatible split systems.Then, in Section 3, we prove a variant of Bonnot et al.'s result for arbitrary q ≥ p 2 > 0 in the special case where the split system underlying a tree is maximal (Theorem 1).In Section 4 we focus on the split system associated to a distance D on X that forms the index set for the first sum in Equation (1).In particular, we give a tight upper bound on its size (Theorem 2), and also a characterization for when it is compatible (Theorem 3).
In Section 5 we introduce the concept of an orderly split system.These are essentially split systems for which Equation (2) holds in case p = q 2 .Compatible split systems are special examples of orderly split systems, and we show that the more general circular split systems [3] also enjoy this property in case they have maximum size (Theorem 5).In Sections 6 and 7 we then explore to what extent this latter result can be extended to the even more general class of so-called flat split systems [6,26].In particular, we show that within the class of maximum sized flat split systems, the orderly split systems are precisely those that are circular (Theorem 12).In Section 8 we briefly look into consequences of our results on efficiently computing order distances.We conclude in Section 9 with some possible directions for future work.

Preliminaries
For the rest of this paper X will denote a finite non-empty set with |X| = n, and S(X) the set consisting of all possible splits of X.We also use A|B = B|A to denote a split {A, B} of X into two non-empty subsets A and B. We call any non-empty subset S ⊆ S(X) a split system on X.A pair (S, ω) consisting of a split system S on X and a weighting ω : S → R ≥0 is called a weighted split system on X and we denote by D (S,ω) = S∈S ω(S) • D S the distance generated by the weighted split system (S, ω).We emphasize that throughout this paper the weights of the splits in a weighted split systems will always be non-negative.
A split system S on X is compatible if, for any two splits The splits in a compatible split system S on X are in one-to-one correspondence with the edges of a (necessarily) unique X-tree T = (V, E, ϕ), that is, a graph theoretic tree T = (V, E) with vertex set V and edge set E ⊆ V 2 together with a map ϕ : X → V such that the full image ϕ(X) contains all vertices of degree at most two.We denote the compatible split system represented by the edges of an X-tree T by S T .The edges of the X-tree T in Figure 1, for example, yield the following collection of splits of X = {a, b, c, d, e}: {a}|{b, c, d, e}, {b}|{a, c, d, e}, {c}|{a, b, d, e}, {d}|{a, b, c, e}, {e}|{a, b, c, d}, {a, b}|{c, d, e} and {a, b, c}|{d, e} (which can be visualized by removal of each edge from T ).Assigning to each of these splits as its weight the weight of the corresponding edge in T yields a weighted compatible split system that generates D.
Note that a compatible split system S on X is maximal, that is, adding any further split to S yields a split system that is no longer compatible, precisely in case it contains 2n − 3 splits, in which case it corresponds to a binary X-tree where the elements of X are in one-to-one correspondence with the leaves of the tree.Hence, maximal compatible split systems are precisely the maximum-sized or maximum, for short, compatible split systems on X.
In proofs we will make use of these facts concerning compatible split systems and will sometimes switch between a weighted compatible split system and its equivalent unique representation as an edge-weighted X-tree.Full details concerning this correspondence can be found in [23].

Treelike distances revisited
In this section, we consider properties of the order distance O p,q (D) of a treelike distance D for arbitrary values q ≥ p 2 > 0. Note that most previous results for treelike distances, such as those in mentioned in the introduction, focus mainly on the case q = p 2 .We first consider the case where D is generated by a maximum compatible split system.
Theorem 1 For any maximum compatible split system (S, ω) on X with strictly positive weighting ω, the order distance O = O p,q (D) associated to D = D (S,ω) can be expressed as O = D (S,ω ) for some non-negative weighting ω .
Proof: Let (S, ω) be a maximum compatible split system on X with strictly positive weighting ω.First note that in [19] it is shown that if S is compatible then the splits X u,v |X − X u,v and X v,u |X − X v,u are contained in S for all (u, v) ∈ X ×X with ∅ X u,v X.Moreover, in view of the fact that maximum compatible split systems with strictly positive weighting are precisely those that can be represented by binary X-trees with strictly positive edge weights whose leaves are in one-to-one correspondence with the elements in X, we must have D(u, v) > 0 for all u, v ∈ X with u = v and, for any {u, v} ∈ X 2 with ∅ E {u,v} X, the splits X u,v |X − X u,v , X v,u |X − X v,u and E {u,v} |X − E {u,v} correspond to three edges that share a single vertex in the binary X-tree that represents S. In particular, the split E {u,v} |X − E {u,v} must be contained in S. Hence, from Equation (1) and in view of the assumption q ≥ p 2 we have for some suitable non-negative weighting ω of the splits in S, as required.
Note that the assumption in Theorem 1 that the compatible split system is maximum is necessary.Indeed, in [5, p. 258], an example is presented which provides a weighted, non-maximum compatible split system S on X, |X| = 4, such that for any q > p 2 , the order distance O = O p,q (D) associated to D = D (S,ω) cannot be expressed as O = D (S,ω ) for any non-negative weighting ω of the splits in S.There exists, however, a compatible superset S ⊇ S of splits such that O = D (S ,ω ) for some non-negative weighting ω in this example.
In general, even expressing the order distance by a suitable superset of splits that belong to the same class of split systems (here compatible) as the split system that generates D requires, in general, that we restrict to q = p 2 .To illustrate this in the following example, we make use of the fact (see e.g.[23]) that treelike distances D on a set X are characterized by the following 4 holds.But this implies q ≤ p 2 and, thus q = p 2 .Note that the distance D in the previous example is not only treelike but even an ultrametric, that is, D(x, z) ≤ max(D(x, y), D(y, z)) holds for all x, y, z ∈ X.In [14] it is shown that for any ultrametric D generated by a weighted compatible split system (S, ω) with strictly positive weighting ω, the associated order distance O = O p, p 2 (D) can be expressed as O = D (S,ω ) for some strictly positive weighting ω of the splits in S, that is, all splits in S are used to generate O.As pointed out in [16], however, this property does not characterize ultrametrics: there are examples of distances D that have this property but are not ultrametrics.Moreover, it is shown in [14] that the order distance of an ultrametric is, in general, not an ultrametric.

The midpath split system of a distance
Given a distance D on X we define the midpath split system S D associated to D to be the set of splits of X of the form S X.Note that the splits in S D are precisely those appearing in the index set of the first sum in Equation (1).We chose the name for S D since it is closely related to the midpath phylogeny introduced in [20].In this section, we consider general properties of the split system S D , including a characterization in terms of D for when this split system is compatible.
First note that, as a direct consequence of the definition of S D , it follows that S D contains at most n(n − 1) splits.As |S(X)| = 2 n−1 − 1, we immediately see that for small n this upper bound is not tight for |S D |. Nevertheless we have the following result: and, for all sufficiently large n ∈ N, this bound is tight.
Proof: It remains to show that the upper bound n(n − 1) is tight for all sufficiently large n.To this end, consider a distance D on X such that, for all {u, v} ∈ X 2 , the value D(u, v) = D(v, u) is selected randomly from the set {1, 2}, with both values having the same probability of being selected.We now argue that, for sufficiently large n, the probability that |S D | = n(n − 1) is strictly greater than 0. Note that D satisfies the triangle inequality and that D(x, y) > 0 for all {x, y} ∈ X 2 , implying that the splits S x,y and S y,x exist.Moreover, in order to have |S D | = n(n − 1), for any two distinct {u, v}, {a, b} ∈ X 2 , the splits S u,v , S v,u , S a,b and S b,a must be pairwise distinct.Now, it follows immediately from the definition of D that the probability of S u,v = S v,u is 4  2 n .More generally, the probability that at least two of the splits S u,v , S v,u , S a,b and S b,a coincide is bounded by d • c n for some constants 0 < d and 0 < c < 1.This implies that the probability of which is strictly less than 1 for sufficiently large n, as required.
The remainder of this section is devoted to giving a characterization of those distances D for which S D is compatible.In [19], a 6-point condition is given that characterizes for a distance D when (i) the split system S D is compatible and splits S a,e = {a}|{b, c, d, e} and S e,a = {a, b, c}|{d, e} in our example, however, do not share a vertex.This implies that the 6-point condition given in [19] does not provide the characterization for arbitrary distances that we are looking for.Another aspect illustrated in Figure 2 is that distances D 1 and D 2 on the same set X with S D1 = S D2 do not necessarily yield the same ranking of X 2 , not even if D 1 and D 2 yield the same set X u,v for all u, v ∈ X.
Our characterization for when S D is compatible will also be a 6-point condition.Actually, it is more convenient to state and prove a characterization for when S D is not compatible.The structure of the proof of Theorem 3 is similar to the proof of the 6-point condition in [19].
Theorem 3 Let D be a distance on a set X.The split system S D is not compatible if and only if there exist a, b, s, t, x, y ∈ X with a = b, s = t, x = y such that one of the following holds: Proof: First assume that D satisfies (1) or (2).Then the split S x,y as well as one of the splits S s,t or S t,s are contained in S D .Now, if D satisfies (1) and S s,t ∈ S D , then S x,y and S s,t are not compatible in view of a ∈ X and, if D satisfies (1) and S t,s ∈ S D , then S x,y and S t,s are not compatible in view of a ∈ X Similarly, if D satisfies (2) and S s,t ∈ S D , then S x,y and S s,t are not compatible in view of ) and, if D satisfies (2) and S t,s ∈ S D , then S x,y and S t,s are not compatible in view of Hence, S D is not compatible, as required.Now assume that S D is not compatible.Let Order the pairs in P arbitrarily and let (u 1 , v 1 ), (u 2 , v 2 ), . . ., (x m , v m ) denote the resulting sequence.Put S i = S ui,vi and S i = {S 1 , S 2 , . . ., S i } for all 1 ≤ i ≤ m.Note that the split system S 1 is compatible.Let 2 ≤ k ≤ m be the smallest index such that S k is not compatible.Such an index must exist in view of S m = S D and our assumption that S D is not compatible.The split system S k−1 , however, is compatible and there exists an X-tree T such that is not compatible implies that there must exist a vertex w on the path in T from the vertex labeled by x to the vertex labeled by y such that the split S , corresponding to some edge e in T which has one endpoint at w, is not compatible with S k = S x,y .Let {s, t} ∈ X 2 be such that S = S s,t or S = S t,s .The two possible configurations, depending on whether or not edge e lies on the path from x to y in T , are depicted in Figure 3, where a ∈ X x,y and b ∈ X − X x,y .It can be checked that the configuration depicted in Figure 3(a) implies that (1) holds while the configuration depicted in Figure 3(b) implies that (2) holds, as required.
To illustrate that a characterization as in Theorem 3 with a k-point condition for k < 6 is not possible, one can employ the following distance D that was presented in [19]: It is shown in [19] that the restriction of D to any 5-element subset of X = {a, b, s, t, x, y} yields a distance D such that S D is compatible.The split system S D , however, contains the splits S x,y = {b, t, y}|{a, s, x} and S s,t = {b, s, x}|{a, t, y} which are not compatible.
Before continuing we briefly mention some previous work that is related to the midpath phylogeny mentioned above and that is concerned with situations where the distance D on X is not known and only the rankings of the elements in X generated by D are available.The aim then is to find a weighted compatible split system that represents these rankings.Methods that follow this approach and which heavily rely on the midpath phylogeny are presented in [15,18,20,21].Moreover, in [19] it is shown (see also [21]) that any compatible split system that represents the rankings of the elements in X generated by D must contain the splits of the midpath phylogeny.However, as shown in [24], in general, if the rankings can be represented at all, further splits must be added.The decision problem of whether the rankings can be represented or not can be solved in polynomial time if restricted to representations by compatible split systems in which every split is assigned the same positive weight [18,21].However, if the splits in the compatible split system can have arbitrary positive weights, then the problem is NP-hard [25].

Orderly split systems
Motivated by the main result of Bonnot et al. in [5], we call a split system S orderly if for all q = p 2 > 0 and all non-negative weightings ω of the splits in S there exists a non-negative weighting ω of the splits in S such that O p,q (D (S,ω) ) = D (S,ω ) .The result of Bonnot et al. can then be restated as follows: Every compatible split system is orderly.In this section we show that another important class of split systems also enjoys this property.
We begin by recalling that a split system S on X is circular if there exists an ordering θ = x 1 , x 2 , . . ., x n of the elements in X such that, for any split S ∈ S, there exist 1 ≤ i ≤ j < n with S = {x i , x i+1 , . . ., x j }|X − {x i , x i+1 , . . ., x j }, in which case we will say then that S fits on θ.Circular split systems naturally appear in the context of the so-called split decomposition of a distance (see [3,Sec. 3], where they are introduced), and have applications in phylogenetics (see e.g.[7]).Note that every compatible split system is circular but not vice versa [3].
From the definition it follows immediately that a maximal circular split system on a set with n elements contains precisely n 2 splits.Hence, for fixed n, just like for compatible split systems, the maximal circular split systems are precisely the maximum circular split systems.Moreover, while in general the ordering θ of the elements in X onto which a circular split system fits might not be unique, it follows again immediately from the definition that any maximum circular split system uniquely determines this ordering up to reversing and shifting it.Now, a distance D on X is circular if there exists a circular split system S with a non-negative weighting ω such that D = D (S,ω) .It is shown in [9] (see also [10]) that a distance D on X is circular if and only if there exists an ordering θ = x 1 , x 2 , . . ., x n of the elements in X such that holds for all 1 ≤ i < j < k < l ≤ n so that, in particular, circular distances are equivalent to so-called Kalmanson distances [17].Note that an ordering θ of the elements in X satisfies condition (4) for a circular distance D if and only if the necessarily unique circular split system with strictly positive weighting that generates D fits on θ.We now make a useful observation concerning circular split systems.
Lemma 4 Let (S, ω) be a circular split system on X with non-negative weighting ω such that S fits on the ordering θ = x 1 , x 2 , . . ., x n of the elements in X.Then, for D = D (S,ω) , the split system S D is circular and fits on θ.
Proof: If S D = ∅ then S D is circular and it fits on θ.So assume that S D = ∅ and consider an arbitrary split S ∈ S D .By the definition of S D there exist u, v ∈ X with u = v and S = X u,v |X − X u,v .Note that we must have u ∈ X u,v and v ∈ X −X u,v .Assume for a contradiction that, even after possibly shifting θ, the elements in X u,v do not form an interval of consecutive elements in θ.This implies that there exist u ∈ X u,v with u = u and v ∈ X − X u,v with v = v such that, after possibly shifting and/or reversing θ, the restriction of θ to {u, v, u , v } is u, v , u , v.Then, in view of the definition of X u,v , we must have But this implies contradicting condition (4).As a consequence of Lemma 4 we immediately obtain the main result of this section: Theorem 5 Every maximum circular split system is orderly.
Proof: Let S be a maximum circular split system together with a nonnegative weighting ω and put D = D (S,ω) .In view of q = p 2 , Equation (1) can be written as where, for each S ∈ S D , the weight ω (S) is a certain non-negative integer multiple of p 2 .By Lemma 4, S D fits onto the unique ordering θ of the elements in X onto which the maximum circular split system S fits.Hence, we have S D ⊆ S, as required.
Corollary 6 For q = p 2 , the order distance O p,q (D) of a circular distance D is always circular.
Note that in Theorem 5 we assume that the circular split system is maximum.To illustrate that, in contrast to compatible split systems, we cannot remove this assumption, consider, for example, the non-maximum circular split system S = {{b}|{a, c, d}, {a, b}|{c, d}, {a, d}|{b, c}} on X = {a, b, c, d} and the weighting ω that assigns weight 1 to every split in S. So, in general, if D is generated by a non-maximum circular split system S on X the order distance associated to D may be generated only by a proper superset of S.

Linearly independent split systems
Circular split systems S on X are examples of linearly independent split systems as introduced in [6], that is, the set of split distances {D S : S ∈ S} arising from S is linearly independent when viewed as elements of the vector space of all symmetric bivariate maps D : X × X → R with D(x, x) = 0 for all x ∈ X.Note that the dimension of this vector space clearly is n 2 and, in view of the fact that there exist linearly independent split systems of size n 2 (e.g.maximum circular split systems), this implies that a linearly independent split system S on X is maximal with respect to set inclusion if and only if S has maximum size n 2 .Linearly independent split systems therefore provide a natural generalization of circular split systems, and in this section we explore to what extent Theorem 5 can be generalized to maximum linearly independent split systems that are not circular.
In the following we always assume q = p 2 and, to avoid fractions in computations, put p = 2.The technical lemma we state next provides a useful link between the combinatorial structure of a linearly independent split system and the order distances that it generates.To describe this link, we call a split system S on a set X with n ≥ 4 elements closed if for any two incompatible splits A 1 |B 1 and A 2 |B 2 in S at least one of the following holds: (a) S also contains the splits Lemma 7 Let X be a set with n ≥ 4 elements and S a linearly independent split system on X.If S is orderly then S is closed.
Proof: If S is compatible then it is trivially closed.So assume that S is orderly and contains two incompatible splits Let ω be the weighting of S with ω(S 1 ) = ω(S 2 ) = 2 and ω(S) = 0 for all other S ∈ S. We consider the order distance O = O(D) with D = D (S,ω) .Then, putting and, for all other u, v ∈ X, we have O(u, v) = 0.This implies that, for any non-negative weighting ω of the splits in S(X) with only the splits S in can have a weight ω (S) > 0. Note that S * is not linearly independent but every 6-element subset of S * is.Now, can be viewed as a system of linear equations for the weights ω (S).Assuming without loss of generality that n 2 n 3 ≤ n 1 n 4 , it can be checked that there are only the following two solutions (i) and (ii) of this system with ω(S) ≥ 0 for all S ∈ S * and |{S ∈ S * : ω (S) > 0}| ≤ 6: But this implies, in view of the assumption that O = D (S,ω ) for some nonnegative weighting ω of S, that S must be closed, as required.Now we use Lemma 7 to obtain the following property of non-circular maximum linearly independent split systems on sets with 5 elements.Lemma 8 Let X = {a, b, c, d, e} be a set with 5 elements.Then every maximum linearly independent split system S on X that is not circular is not orderly.
Proof: Let S be a maximum linearly independent split system on X that is not circular and assume for a contradiction that S is orderly and, therefore, by Lemma 7, closed.Since S is a maximum linearly independent split system, we have |S| = 10.Thus, as any compatible split system on X contains at most 7 splits, S must contain two incompatible splits S 1 and S 2 .Relabeling the elements in X, if necessary, we assume without loss of generality that S 1 = {a, b}|{c, d, e} and S 2 = {b, c}|{a, d, e}.Then, since S is closed, it must also contain the splits {b}|{a, c, d, e} and {d, e}|{a, b, c}.Moreover, in view of |S| = 10, S must contain an additional split S 3 = {x, y}|X − {x, y} for a 2-element subset {x, y} ⊆ X with {x, y} ∈ {{a, b}, {b, c}, {d, e}}.We consider three cases.
Case 1: S 3 = {a, c}|{b, d, e}.Then, in view of the fact that S is closed, S 1 , S 3 ∈ S implies that {a}|{b, c, d, e} ∈ S and, similarly, S 2 , S 3 ∈ S implies that {c}|{a, b, d, e} ∈ S.But then it can be checked that S is not a linearly independent split system, a contradiction.
Case 2: S 3 = {b, e}|{a, c, d}.(Note that the case S 3 = {b, d}|{a, c, e} is symmetric.)Then, in view of the fact that S is closed, {d, e}|{a, b, c}, S 3 ∈ S implies {a, c}|{b, d, e} ∈ S. From this we obtain a contradiction as in Case 1.
Case 3: S 3 = {c, d}|{a, b, e}.(Note that the cases S 3 = {c, e}|{a, b, d}, S 3 = {a, d}|{b, c, e} and S 3 = {a, e}|{b, c, d} are symmetric.)Then, using again that S is closed, S 2 , S 3 ∈ S implies {c}|{a, b, d, e}, {a, e}|{b, c, d} ∈ S. Three further applications of the definition of closedness each yield that the split {z}|X − {z} is also contained in S for z ∈ {a, d, e}.But this implies that S is a maximum circular split system on X, a contradiction.
Lemma 8 together with Theorem 5 yields the following characterization of orderly split systems amongst all maximum linearly independent split systems on sets with 5 elements: Proposition 9 A maximum linearly independent split system S on a set X with 5 elements is orderly if and only if it is circular.
It can be checked that maximum circular split systems on sets with 4 elements cannot be characterized as in Proposition 9.

Flat split systems
In the last section we showed that a maximum linearly independent split system S on a set X with 5 elements is circular if and only if it is orderly.We would like to know if this can be extended to all values of n ≥ 5, but have not been able to find a proof (or counter-example).However, in this section we show that the result can be extended to a certain subclass of linearly independent split systems called flat split systems (see e.g.[26]) that includes all circular split systems, and that was first introduced in [6] under the name of pseudoaffine split systems.Just like circular split systems, flat split systems have found applications in phylogenetics (see e.g.[2]).
Then, putting m = n 2 , a pair (π, κ) consisting of an ordering π of X and a sequence κ = (k 1 , . . ., k m ) ∈ {1, 2, . . ., n − 1} m is allowable if the sequence π 0 , π 1 , . . ., π m of orderings of X defined by putting π 0 = π and π i = π i−1 (k i ), 1 ≤ i ≤ m, has the property that any two elements in X swap their positions precisely once, that is, sw(π i−1 , k i ) = sw(π j−1 , k j ) holds for all 1 ≤ i < j ≤ m.Note that such a sequence of orderings is commonly called a simple allowable sequence [13].Now, a split system S ⊆ S(X) is called flat if there exists an allowable pair (π, κ) with Here we are only concerned with maximal flat split systems which, similarly to circular split systems, are precisely the flat split systems with maximum size n 2 .
Note that in [1,Theorem 14] it is shown that a maximum linearly independent split system S on a set X with n ≥ 2 elements is a maximum flat split system if and only if, for every 4-element subset Y ⊆ X, the restriction S |Y contains precisely 6 splits, where the restriction of a split system S on a set X to a subset Y ⊆ X is the split system Our goal is to provide a characterization of orderly split systems amongst all maximum flat split systems similar to Proposition 9.The following technical lemma will be used to obtain this characterization.A split system S on X has the pairwise separation property if, for any two distinct elements x, y ∈ X, there exist subsets A and B of X − {x, y} such that A ∪ {x, y}|B, A ∪ {x}|B ∪ {y}, A ∪ {y}|B ∪ {x} and A|B ∪ {x, y}, if they form splits of X, are contained in S. Note that every circular split system satisfies the pairwise separation property [6].
Lemma 10 Let S be a maximum linearly independent split system on a set X with n ≥ 3 elements that satisfies the pairwise separation property.Then, for any y ∈ X, the restriction of S to X − {y} is a maximum linearly independent split system that satisfies the pairwise separation property.
Proof: Fix an arbitrary y ∈ X.For every c ∈ X − {y} let A c and B c denote the two subsets of X − {y, c} that must exist for the pair {y, c} according to the pairwise separation property.Consider the set of splits on the set X − {y}.Note that in the context of this proof it will be convenient to consider ∅|X − {y} as a split of X that may be contained in S ↔ .
We claim that |S ↔ | ≥ n − 1.To see this, consider the graph G ↔ with vertex set S ↔ in which there is an edge between two distinct splits S and S if there exists a c ∈ X −{y} such that S = A c |X −(A c ∪{y}) and S = B c |X −(B c ∪{y}).Note that G ↔ has the following properties: Property (i) follows directly from the definition of G ↔ .Property (ii) follows from the fact that, in order to arrive along a cycle in G ↔ at the same split S = A|B again, all elements in A ∪ B must move from one side of the split to the other.Since A ∪ B = X − {y}, this requires n − 1 moves.Now, note that properties (i) and (ii) imply that G ↔ must have at least n − 1 vertices.Hence, we have |S ↔ | ≥ n − 1, as claimed.
Next note that |S ↔ | ≥ n − 1 implies that, when restricting S to X − {y}, we obtain at most n 2 − (n − 1) splits of X − {y} in view of the fact that, for every S ∈ S ↔ , there are, according to the pairwise separation property, at The base case of the induction for n = 5 is established by directly checking the following two split systems S 1 and S 2 on the set X = {a, b, c, d, e} that are, up to isomorphism, the only maximum flat split systems on X that are not circular: For n ≥ 6 select an element y ∈ X such that the restriction S = S |X−{y} is not circular.Since S is not circular, such an element must exist.By Lemma 10 and Corollary 11, we have that S is a maximum flat split system on X − {y} that is not circular.Thus, by induction, S satisfies (e).Then S must contain two incompatible splits S 1 and S 2 such their restriction to X − {y} satisfies (e).But then S must satisfy (e) for the two incompatible splits S 1 and S 2 as well, finishing the inductive proof that S satisfies (e).
To finish the proof of the theorem, it suffices to note that (e) implies that S is not closed, which yields, in view of Lemma 7, that S is not orderly.

Computing the order distance
In this section we briefly look into algorithms for computing the order distance O = O(D) = O p,q (D) from an input distance D on a set X with n elements.In [16] it is noted that a run time in O(n 4 ) can be achieved without any restrictions on the input distance and the values of p and q.Moreover, clearly no algorithm for computing O(D) can run faster than the size of the output, that is, we have a lower bound of Ω(n 2 ).
As mentioned in Section 1, the order distance O associated to a distance D on X can be viewed as a way to quantify, for any two elements u, v ∈ X, the differences between the rankings R u and R v of the elements in X generated by D when sorting according to non-decreasing distances from u and v, respectively.For non-generic distances D the rankings R u and R v are partial in the sense that ties between elements can occur when they have the same distance from u or v.In fact, it can be checked that the value O(u, v) coincides with the so-called Kendall distance K π (R u , R v ) with penalty parameter π = q p between the partial rankings R u and R v that was introduced in [12].In [12] the range of the penalty parameter π is restricted to 0 ≤ π ≤ 1 and it is also established that the Kendall distance K π on partial rankings of a fixed set X satisfies the triangle inequality if and only if 1  2 ≤ π ≤ 1.In [4] it is noted that, for any π ≥ 0, the Kendall distance K π (R 1 , R 2 ) between two partial rankings R 1 and R 2 of a set with n elements can be computed in O(n log n) time in a purely comparison based model of computation and it is established that in more powerful models of computation a run time in O(n log n/ log(log n)) can be achieved (see also [8] for further related results).Applying the algorithm from [4] to each pair of elements in X we therefore obtain: Proposition 13 The order distance O p,q (D) of a distance D on a set with n elements can be computed in O(n 3 log n) time.
An immediate question is whether the gap to the lower bound Ω(n 2 ) on the run time can be further narrowed.In the following we will show that this is the case for circular distances when we set q = p 2 .For the special case of weighted compatible split systems a run time in O(n 2 log n) follows immediately from the fact that, as established in [19], the midpath phylogeny associated to an input distance D on a set with n elements can be computed in O(n 2 log n) time in a purely comparison based model of computation and in O(n 2 ) in models of computation that allow for sorting in linear time.We now show that a similar run-time can also be achieved for circular distances: Proof: The first step in the computation of O(D) from D is to obtain an ordering θ = x 1 , x 2 , . . ., x n of the elements in X such that, D = D (S,ω) for a suitable weighting ω of the circular split system S consisting of the splits S i,j for which there exist 1 ≤ i ≤ j < n with S i,j = {x i , x i+1 , . . ., x j }|X − {x i , x i+1 , . . ., x j }.
In view of the assumption that D is circular such an ordering must exist and it can be computed in O(n 2 ) time with the algorithm presented in [10].
Next note that, in view of Lemma 4, for all u, v ∈ X with D(u, v) > 0, there exist 1 ≤ i(u) ≤ j(u) < n such that S i(u),j(u) = S u,v as well as 1 ≤ i(v) ≤ j(v) < n such that S i(v),j(v) = S v,u .Thus, using binary search, we can compute i(u), j(u), i(v) and j(v) each in O(log n) time for fixed u, v ∈ X. Doing this for all u, v ∈ X with D(u, v) > 0, we obtain in O(n 2 log n) time a weight ω(S) for each split S ∈ S such that O = O(D) = D (S,ω) .
Once we have the weighted split system (S, ω), encoded as the ordering θ and a non-negative number representing ω(S i,j ) for all 1 ≤ i ≤ j < n, it remains to compute O(x, y) = D (S,ω) (x, y) for all x, y ∈ X.It is known that this can done in O(n 2 ) time (see e.g.[22] where this and related computational problems on split systems are discussed).

Concluding remarks
In this paper, we have shed some light on the relationship between split systems and the order distances generated by them.A specific question that remains open is whether or not the generalization of Lemma 8 to n ≥ 6 is true, which would for example yield an interesting new characterization of maximum circular split systems among all maximum linearly independent split systems.Computational experiments that we have performed on a large number of randomly generated maximum linearly independent split systems seem to indicate that, at least for n = 6, if counterexamples exist they are very rare.
To explain another aspect of order distances and maximum linearly independent split systems, note that, usually, an 1 -distance D on X can be generated by many different maximum linearly independent split systems on X.Therefore, for any 1 -distance D, one might hope to at least always find some such split system that would generate both D and O(D).However, through an exhaustive search through the 34 isomorphism classes of maximum linearly independent split systems on a set X with n = 5 elements with a computer program we found that, for every maximum linearly independent split system S on X that is not maximum flat, there exists a non-negative weighting ω such that for the distance D = D (S,ω) and its associated order distance O(D) there is no maximum linearly independent split system that generates both D and O(D).In future work, it would be interesting to explore this further and see if it yields a characterization of maximum flat split systems.
Another interesting direction could be to further explore properties of the split system S D .Note that from Theorem 2 it follows that, in general, S D will not be linearly independent because it contains more than n 2 splits.In contrast, if D is circular then, in view of Lemma 4, S D is circular too and, therefore, linearly independent.So, a specific question one can ask is for which distances D the split system S D is linearly independent?
Finally, in future work it could also be interesting to study variants of the order distance that are obtained by employing, instead of the Kendall distance with penalty parameter q p , any of the other distance measures on partial rankings introduced in [12].

Figure 1 :
Figure 1: A tree T with non-negative edge weights whose leaves are labeled by the elements in the set X = {a, b, c, d, e} and the distance D of shortest path distances between the leaves of T .The associated order distance O = O 2,1 (D)can also be represented by the tree T by adjusting the weights of its edges.Note that each edge of T corresponds to a split of X.

Figure 2 :
Figure 2: A distance D 1 on X = {a, b, c, d, e} such that S D1 is compatible but D 1 has no midpath phylogeny.The edges of the X-tree T represent the splits in S D1 .For the distance D 2 we obtain S D2 = S D1 but D 1 and D 2 do not yield the same ranking of X 2 .

Figure 3 :
Figure 3: The two possible configurations in the X-tree T referred to in the proof of Theorem 3. The gray triangles indicate the two subtrees of T connected to the endpoints of the edge e, one of them being the vertex w that lies on the path from x to y.The configuration in (a) yields condition (1) and the configuration in (b) yields condition (2) stated in Theorem 3.

Theorem 14
The order distance O(D) = O p, p 2 (D) of a circular distance D on a set X with n elements can be computed in O(n 2 log n) time.