Robinson spaces and their representation in low-dimensional metric spaces
Robinson spaces are structures equipped with a total order that encodes comparative dissimilarity relationships. We study the problem of representing Robinson dissimilarity spaces into low-dimensional metric spaces. These representations aim to pre…
Authors: Francisco Arrepol, Mauricio Soto-Gomez, Christopher Thraves Caro
Robinson spaces and their represen tation in lo w-dimensional metric spaces F rancisco Arrep ol 1* † , Mauricio Soto-Gomez 2 † and Christopher Thra v es Caro 3 † 1 Departamen to de Informática y Ciencias de la Computación, F acultad de Ingeniería, Univ ersidad de Concepción, Concepción, Chile. 2 AnacletoLab, Department of Computer Science, Univ ersit y of Milan, Milan, Italy . 3 Departamen to de Ingeniería Matemática, F acultad de Ciencias Físicas y Matemáticas, Universidad de Concep ción, Concep ción, Chile. *Corresp onding author(s). E-mail(s): farrep ol2016@udec.cl ; Con tributing authors: mauricio.soto@unimi.it ; cthra ves@udec.cl ; † These authors contributed equally to this work. Abstract Robinson spaces are structures equipp ed with a total order that encodes compar- ativ e dissimilarit y relationships. W e study the problem of represen ting Robinson dissimilarit y spaces in to lo w-dimensional metric spaces. These represen tations aim to preserve the relative dissimilarit y relationships b etw een elements rather than their exact v alues. While lo w dimensional Euclidean spaces suc h as R 1 and R 2 are natural candidates for such embeddings, previous work has shown that not all Robinson spaces admit a v alid embedding in the real line that resp ects their structural constraints. Motiv ated by this limitation, we explore the broader class of r e al tr e es , whic h retain lo w-dimensional in terpretability while allo wing greater exibility . T o address the embedding problem, we develop tw o key to ols: a combinatorial represen tation of Robinson spaces and a top ological characterization of c ater- pil lars , a restricted class of real trees. These to ols enable a formulation of the em b edding problem as a linear program, providing b oth computational and the- oretical insigh ts. W e prov e that some subclasses of Robinson spaces alw a ys admit em b eddings in a caterpillar, and we establish the existence of Robinson spaces that cannot be em b edded in an y real tree. These results clarify the geometric lim- itations of representing ordered dissimilarit y structures and open new directions for studying the interaction b etw een dissimilarit y , order, and metric geometry . 1 Keyw ords: Robinson spaces, dissimilarit y spaces, metric represen tation, v alid drawing, caterpillars. MSC Classication: 68R12 , 90C90 , 52-08 1 In tro duction A dissimilarity me asur e b etw een t wo objects is a numerical v alue that indicates ho w dieren t the tw o ob jects are from one another: the smaller the v alue, the more similar the ob jects. In the case when the dissimilarity is zero, the tw o ob jects are identical. A dissimilarity sp ac e consists of a set of elements along with a dissimilarity mea- sure for each pair of distinct elemen ts within that set. Dissimilarity represen tation is crucial in pattern recognition becaus e it eectively captures structural and relational information b et ween samples [ 1 ]. In this work, we fo cus on R obinson sp ac es , a special class of dissimilarity spaces c haracterized b y the existence of a linear ordering of the elemen ts < such that, for any three elemen ts x < y < z , the dissimilarit y b et ween x and z is at least as large as b oth the dissimilarity b etw een x and y , and b etw een y and z . This combinatorial prop erty allo ws to infer similarities directly from the ordering, and is key to the structural regularit y of these spaces. A dissimilarity space is not necessarily a metric space, as a dissimilarity measure need not satisfy all the prop erties of a distance (the triangle inequality , in particular, ma y not hold) ev en in Robinson spaces. How ever, representing dissimilarity spaces via the embedding of their elements into a metric space has pro ven useful in v arious elds, suc h as pattern recognition, and classication [ 2 ]. The structural regularit y of Robinson spaces leads us to conjecture that they can b e eciently represen ted in low-dimensional metric spaces. Since an exact preserv ation of dissimilarities is generally unattainable, we fo cus instead on preserving the r elative ordering induced by the dissimilarit y v alues. That is, we aim to construct em b eddings where, from the p ersp ective of an y given elemen t, the closer of tw o others in the dissimilarit y sense is also closer in the metric space. This kind of represen tation enables external observers to b oth visually and computationally recognize structural patterns and relationships presen t in the space. In this context, a low-dimensional metric sp ac e refers to a Euclidean space R k , where k is small, typically k ∈ { 1 , 2 } . These lo w v alues of k facilitate geometric or visual interpretation of the space’s structure. How ev er, Aracena et al. in [ 3 ] show ed that not all Robinson spaces admit an embedding in R 1 that preserves the desired relational structure, which motiv ates the exploration of more exible lo w-dimensional settings. A natural generalization of R 1 is the class of r e al tr e es , which are connected, geo desic metric spaces without cycles. These spaces preserve the notion of low dimen- sionalit y in terms of Hausdor dimension (see [ 4 , 5 ]) while enabling a wider v ariety of embeddings. Imp ortantly , they retain interpretabilit y and structure, making them suitable for represen ting hierarchical or branching relationships. 2 In this work, w e study the problem of nding an embedding o f the elements of a Robinson space into metric spaces while preserving the underlying relational structure, i.e., the comparative dissimilarities enco ded by a compatible linear order. W e show that this problem can b e formulated as a linear program, and prov e that sp ecic sub classes of Robinson spaces alw ays admit suc h em b eddings. How ev er, w e also show that there exist Robinson spaces that do not admit any embedding of this kind into a real tree. 2 Denitions, notation, and our results Let S b e a nite set. A dissimilarity on S is a function ρ : S × S → R ≥ 0 suc h that ρ ( x, y ) = ρ ( y , x ) for all x, y ∈ S , and ρ ( x, y ) = 0 if and only if x = y . The pair ( S, ρ ) is called a dissimilarity sp ac e . W e sa y that a dissimilarit y space ( S, ρ ) is strict if ρ ( x, y ) = ρ ( u, v ) for an y t wo distinct pairs { x, y } = { u, v } , not both in the diagonal (i.e., x = y or u = v ). Denition 1 (Robinson Space) . A dissimilarity space ( S, ρ ) is R obinson if there exists a total order < on S suc h that, for every triple x < y < z in S , the following inequalit y holds: ρ ( x, z ) ≥ max { ρ ( x, y ) , ρ ( y , z ) } . Suc h an order is called a c omp atible or der . In the following, we assume that nite Robinson spaces are of the form S = { 1 , 2 , . . . , n } and that the iden tity order 1 < 2 < · · · < n is compatible. T o geometrically represent dissimilarity spaces, we use the notion of a valid dr aw- ing , which refers to an em b edding into a metric space that preserves the relative similarit y structure among the elements. Denition 2 (V alid Drawing) . Let ( S, ρ ) be a dissimilarit y space and ( X , d ) b e a metric space. An injection I : S → X is a valid dr awing if for all x, y , z ∈ S such that ρ ( x, y ) < ρ ( x, z ) , it holds that d ( I ( x ) , I ( y )) < d ( I ( x ) , I ( z )) . (1) A widespread intuition on Robinson spaces is that their elemen ts can b e rep- resen ted by points on a line [ 6 ]. It is known that if a dissimilarity space admits a v alid drawing in the real line, it is Robinson. How ever, not every Robinson space admits a v alid drawing in R 1 [ 3 ]. T o o vercome this limitation, we seek richer metric spaces that allow more exibility in representing such structures without sacricing in terpretability . One natural c hoice is the class of r e al tr e es . Denition 3 (Real T ree) . A metric space ( T , d ) is a r e al tr e e if it is path-connected, and for every triple x, y , z ∈ T , there exists a p oint c ∈ T (called the center) suc h that the geo desics betw een the pairs ( x, y ) , ( y , z ) , and ( z , x ) in tersect at c . 3 spine leaf leg Fig. 1 : Example of a caterpillar graph Real trees generalize the linear structure of R 1 , allo wing the representation of a broader class of dissimilarity spaces. In particular, since Robinson spaces come with an inherent order, w e fo cus on structured real trees that preserve this order, kno wn as c aterpil lars . Caterpil lars. Quoting the evocative description by Harary et al. [ 7 ], “a caterpillar is a tree whic h metamorphoses in to a path when its co co on of endpoints is remov ed. ” In our setting, the con tinuous caterpillar structure can b e naturally interpreted as a discrete one, since w e will embed a nite set S into it. This leads us to work with a weigh ted caterpillar graph, where w eights enco de the distances b etw een adjacent v ertices. W e adopt the follo wing terminology for caterpillar graphs (see Figure 1 for an illus- tration). The central path of a caterpillar is called the spine . The vertices b elonging to the spine are called spine vertic es . An edge whose b oth endp oin ts lie on the spine is called a spine e dge . Edges that connect a spine v ertex to a leaf are called le gs . The leaf v ertices attac hed to the spine through legs are simply called le aves . This terminology will allow us to distinguish clearly b etw een the structure of the cen tral path and the p eripheral attac hmen ts, and it will be used consisten tly throughout the remainder of the pap er. These structured trees strik e a balance betw een expressive p o wer and in ter- pretabilit y . Their linear backbone naturally reects the total order inherent to Robinson spaces, making them comp elling candidates for v alid drawings that preserv e comparativ e dissimilarities. L eft-Right Center. Consider the set S = { 1 , 2 , 3 , . . . , n } , and let ( S, ρ ) b e a Robinson space for whic h the iden tity order 1 < 2 < · · · < n is compatible. Let i ≤ j ≤ k be three elements of S and supp ose that ρ ( i, j ) < ρ ( j, k ) , that is, j is less dissimilar from i than from k . Since S is a Robinson space, the dissimilarities along the compatible order are monotone. Hence, for ev ery j ′ ∈ S with i ≤ j ′ ≤ j , we hav e ρ ( i, j ′ ) ≤ ρ ( i, j ) < ρ ( j, k ) ≤ ρ ( j ′ , k ) , whic h implies ρ ( i, j ′ ) < ρ ( j ′ , k ) . In other words, all elemen ts b et ween i and j are less dissimilar to i than to k . 4 Analogously , if ρ ( i, j ) > ρ ( j, k ) , meaning that j is less dissimilar to k than to i , then for ev ery j ′ with j ≤ j ′ ≤ k it holds that ρ ( i, j ′ ) > ρ ( j ′ , k ) , so all elemen ts b etw een j and k are less dissimilar to k than to i . These observ ations show that the comparison b et ween ρ ( i, j ) and ρ ( j, k ) is com- pletely determined b y the b oundary where this inequality of dissimilarities changes. More precisely , it suces to know, in the compatible order of S , the last element that is less dissimilar to i than to k , and the rst element that is less dissimilar to k than to i . This transition point captures all the information needed to compare ρ ( i, j ) and ρ ( j, k ) . The previous discussion motiv ates the denition of the following central notions. F or each pair i < k in S , the left c enter L( i, k ) is the largest index j ∈ [ i, k ] such that j is more similar to i than to k : L( i, k ) = max { j ∈ [ i, k ] : ρ ( i, j ) < ρ ( j, k ) } . Similarly , the right c enter R( k , i ) is the smallest index j ∈ [ i, k ] such that j is more similar to k than to i : R( k , i ) = min { j ∈ [ i, k ] : ρ ( i, j ) > ρ ( j, k ) } . W e extend the denitions to the diagonal by setting L( i, i ) = R( i, i ) = i for all i ∈ S . These notions enable us to enco de the relational structure of dissimilarities into a single matrix, dened as follo ws. Denition 4 (Matrix of Centers) . The matrix of c enters of a Robinson space ( S, ρ ) with n elemen ts is the n × n matrix C ( S ) dened by: C ( S ) ( i,j ) = L( i, j ) if i < j, i if i = j, R( i, j ) if i > j. Figure 2 sho ws an example of a strict Robinson space and its matrix of cen ters. W e p oin t out an imp ortant structural prop erty of the matrix of centers of a Robin- son space: F or ev ery Robinson space, the entries of C ( S ) exhibit monotone b ehavior along b oth ro ws and columns. More precisely , for an y xed index i , the entries in ro w i of C ( S ) are non-decreasing as the column index increases. Similarly , for any xed index j , the en tries in column j of C ( S ) are non-decreasing as the ro w index increases. This monotonicity follo ws directly from the denitions of left and right cen ters com- bined with the Robinson prop ert y , which enforces monotone structure with respect to the compatible order. 5 S = 0 4 4 5 5 4 0 3 4 5 4 3 0 2 5 5 4 2 0 1 5 5 5 1 0 (a) C ( S ) = 1 1 1 1 3 2 2 2 2 3 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 (b) Fig. 2 : (a) Example of a Robinson space represented b y its dissimilarity matrix S . (b) Matrix of cen ters C(S) of the space S . Our c ontributions and structur e of the do cument. The structure of the pap er and our main contributions are organized as follows. W e start b y contextualizing our work within the existing literature in Section 3 . In Section 4 , w e presen t a mixe d inte ger line ar pr o gr amming formulation for nding v alid drawings of Robinson spaces into real trees. Although algorithmically useful, this approach ma y b e computationally demanding and do es not directly address the more fundamen tal question of whether all Robinson spaces admit such embeddings. In Section 5 , we introduce a reduction that maps any Robinson space to a strict Robinson space with the property that an y v alid drawing of the strict space also yields a v alid dra wing of the original space. This reduction allo ws us to restrict our atten tion to strict Robinson spaces without loss of generalit y . In Section 6 , we in tro duce the Strong-four-p oin t-condition, which c haracter- izes exactly those metric spaces that admit isometric embeddings into caterpillars (Theorem 9 ), a result of indep endent in terest. Building on this c haracterization, in Corollary 10 we sho w that any strict Robinson space admitting a v alid drawing in a real tree m ust necessarily admit one in a caterpillar. In Section 7 , w e exploit these structural insigh ts to achiev e tw o ob jectiv es: rst, we derive an ecient linear program- ming formulation to decide whether a strict Robinson space admits a v alid drawing in a tree; and second, w e establish a framework for identifying structural prop erties that guaran tee or preclude the existence of such drawings (Lemma 12 ). In Section 8 , w e apply these tools to obtain both p ositive and negative results. On the p ositiv e side, in Theorem 13 w e show that an y strict Robinson space satisfying L( i, j ) ≤ i + j 2 ≤ R( j, i ) for all i < j admits a v alid dra wing in a caterpillar, pro viding an easily veriable sucien t condi- tion. In Theorem 14 , we establish an alternativ e sucient condition based on a more compact representation of the associated linear program. Moreov er, we use our com- putational tools to v erify empirically that all Robinson spaces on at most v e elemen ts admit v alid drawings in trees. On the negative side, in Theorem 15 , we exhibit a strict Robinson space that cannot b e em b edded into any caterpillar, thereb y pro ving that some Robinson spaces 6 are fundamen tally incompatible with tree-based representations that preserve their comparativ e dissimilarity structure. Finally , in Section 9 , we discuss future directions and conclude the paper. 3 Related w ork Robinson spaces were introduced by W.S. Robinson in [ 8 ], where he studied how to order chronologically archeological dep osits. In the same study , Robinson in tro duced the seriation problem, which consists of deciding whether a dissimilarit y space is Robinson or not, and nding a compatible order if p ossible. Several works ha v e fo cused on the recognition and structural characterization of Robinson spaces. F ortin and Préa [ 9 ] presented an O ( n 2 ) optimal recognition algorithm for recognizing Robinson spaces based on PQ-T rees, and in terv al graph properties. Alternative approac hes w ere later proposed by Lauren t and Seminaroti using Lex-BFS [ 10 ], whose time complexit y is O ( L ( m + n )) , where m is the num b er of pairs in the space whose dissimilarity is non-zero, and L is the n umber of dieren t dissimilarit y v alues in the space. Later, Lauren t and Seminaroti generalized this metho d using similarity rst search [ 11 ], resulting in a recognition algorithm with time complexity O ( n 2 + nm log n ) . F urther structural insigh t w as pro vided b y Lauren t et al. [ 12 ], who c haracterized Robinson spaces through the absence of w eighted asteroidal triples. More recen tly , Carmona et al. [ 13 ] in tro duced the notion of mo dules, and used this notion to prop ose a divide- and-conquer recognition algorithm with O ( n 2 ) complexit y . The notion of v alid drawings was introduced b y Kermarrec and Thrav es in [ 14 ] as the Sitting Closer to F riends than Enemies (SCFE) problem, focused on signed graphs. In this study , the authors pro vided a p olynomial algorithm to determine whether a complete signed graph has a v alid drawing on the line, and characterized the set of complete signed graphs that hav e a v alid dra wing on the line. A subsequent w ork b y Cygan et al. [ 15 ] established a connection b et ween v alid dra wings in the real line and prop er interv al graphs, and sho w ed that deciding the existence of suc h drawings is NP-complete when considering incomplete signed graphs. The SCFE problem was also explored as an optimization problem by P ardo et al. [ 16 ], who related the problem of nding a v alid dra wing on the line to the quadratic assignment problem, and pro vided t wo optimization algorithms to construct v alid dra wings while trying to minimize the n umber of errors, that is, the violation of the inequality that denes a v alid drawing. The SCFE problem has also b een studied for dieren t metric spaces. Spaen et al. in [ 17 ] studied the problem of nding the minimal dimension k required such that an y signed graph of n vertices has a v alid drawing in R k , denoted by L ( n ) , and pro ved that l og 5 ( n − 3) ≤ L ( n ) ≤ n − 2 . Benítez et al. in [ 18 ] studied the SCFE problem in the circumference, sho wing that it is NP-Complete, and characterized the complete signed graphs that hav e a v alid drawing on the circumference using prop er circular arc graphs. Becerra and Thra ves [ 19 ] analyzed v alid dra wings of signed graphs in real trees, showing that a complete signed graph admits such a drawing if and only if its p ositiv e subgraph is strongly chordal. An extension of v alid drawings w as prop osed by Aracena and Thrav es in [ 3 ], where they studied the existence of a v alid drawing on the real line for weigh ted 7 graphs. They show ed that the SCFE problem for dissimilarity spaces on the line is related to the seriation problem, and they also pro v ed that the t w o problems are not equiv alent. They pro ved that the Robinson prop erty is necessary , but not sucient, for the existence of v alid dra wings on the line, and that if a complete Robinson space has a v alid dra wing on the line for a compatible order, then it has a v alid drawing for an y of its compatible orders. They also formulated the weigh ted SCFE problem as an optimization problem, and pro vided a p olynomial time method to determine if a complete dissimilarit y space has a v alid drawing on the line. 4 T ree v alid dra wing problem In this section, we present a Mixed Integer Linear Program (MILP) that, given a Robinson space, determines whether it admits a v alid drawing in a real tree. The construction of this MILP relies on a structural criterion to c haracterize tree metrics, whic h can b e obtained from the follo wing results. Denition 5 (F our-p oin t-condition) . Let d b e a metric on a set of ob jects X . W e sa y that d satises the F our-p oint-condition if, for all { x, y , u, v } ⊆ X , it holds: d ( x, y ) + d ( u, v ) ≤ max { d ( x, u ) + d ( y , u ) , d ( x, v ) + d ( y , u ) } . (2) This condition is k ey to ensuring the existence of a tree represen ting a nite metric space. Theorem 6 (Buneman [ 20 ]) . Given an ( n × n ) distanc e matrix D . Ther e is an unr o ote d tr e e whose p ath metric is c omp atible with D if and only if the metric dene d by D satises F our-p oint-c ondition. Moreo ver, the tree asso ciated with a metric satisfying the F our-p oint-condition is unique [ 21 ]. MILP formulation. A common technique for the construction of a tree metric meeting a set of constraints is the use of a linear programming form ulation [ 22 – 24 ]. Giv en a Robinson space ( S, ρ ) we dene, for eac h pair of distinct elemen ts i, j ∈ S , the v ariable d ij represen ting the distance b etw een the images of i and j in a tree v alid dra wing of S , with d ii = 0 for each i ∈ S . F rom this set of v ariables, a potential formulation for the tree v alid dra wing of a Robinson space is presen ted in LP 1 , where v ariables must satisfy three types of constrain ts. First, v alid drawing conditions enforce the distances to resp ect the restrictions imp osed by the compatible order (Equation 1 ). Note that these constraints require strict inequalities; therefore, in LP 1 we introduce a small constan t ε > 0 to represent them as inclusiv e inequalities. 8 Second, metric conditions must guaran tee that v ariables dene a metric space, that is, distances b et ween elemen ts m ust b e non-negativ e, symmetric, and must satisfy the triangle inequalit y . Third, the F our-p oin t-condition ensures that the induced metric is compatible with a tree metric (Equation 2 ). These restrictions are non-con vex and require the use of binary indicator v ariables. In LP 1 this condition is mo deled using the big- M metho d (where M ≫ 0 ) together with binary v ariables x 1 ij kl , x 2 ij kl and x 3 ij kl for each { i, j, k , l } ∈ Q , where Q denotes the set of (unordered) quadruplet having distinct v alues. These decision v ariables identify whic h of the three p ossible pairwise sums in Equation 2 is minimal, forcing the t wo larger sums to b e equal. minimize f ( S, ρ ) ( LP 1 ) sub ject to d ij ≤ d ik + ε ∀ i, j, k ∈ S, ρ ( i, j ) < ρ ( i, k ) d ik ≤ d ij + ε ∀ i, j, k ∈ S, ρ ( i, j ) > ρ ( i, k ) d ik ≤ d ij + d j k ∀ i, j, k ∈ S d ij = d j i , d ij ≥ 0 ∀ { i, j } ∈ S d ik + d j l − d il + d j k ≤ M 1 − x 1 ij kl ∀ { i, j, k , l } ∈ Q d ij + d kl − d ik + d j l ≤ M 1 − x 1 ij kl ∀ { i, j, k , l } ∈ Q d ij + d kl − d il + d j k ≤ M 1 − x 1 ij kl ∀ { i, j, k , l } ∈ Q d ij + d kl − d il + d j k ≤ M 1 − x 2 ij kl ∀ { i, j, k , l } ∈ Q d ik + d j l − d ij + d kl ≤ M 1 − x 2 ij kl ∀ { i, j, k , l } ∈ Q d ik + d j l − d il + d j k ≤ M 1 − x 2 ij kl ∀ { i, j, k , l } ∈ Q d ij + d kl − d ik + d j l ≤ M 1 − x 3 ij kl ∀ { i, j, k , l } ∈ Q d il + d j k − d ij + d kl ≤ M 1 − x 3 ij kl ∀ { i, j, k , l } ∈ Q d il + d j k − d ik + d j l ≤ M 1 − x 3 ij kl ∀ { i, j, k , l } ∈ Q X s ∈{ 1 , 2 , 3 } x s ij kl = 1 ∀ { i, j, k , l } ∈ Q x s ij kl ∈ { 0 , 1 } ∀ s ∈ { 1 , 2 , 3 } , ∀ { i, j, k , l } ∈ Q Note that the formulation LP 1 is a feasibility problem. Therefore, the ob jective function can b e chosen arbitrarily . Nonetheless, the formulation requires O ( n 4 ) inte- ger v ariables and constraints, which can b e computationally demanding for large instances. In the following sections, we sho w that, for the class of strict Robinson spaces, tree v alid drawings ha ve a caterpillar top ology . This structural insigh t enables us to design a signican tly more compact linear formulation. 9 5 Reduction to strict Robinson spaces In the following, w e fo cus on the class of strict Robinson spaces. Recall that a Robinson dissimilarit y space ( S, ρ ) is said to b e strict if all its dissimilarity v alues are pairwise distinct. Although strict Robinson spaces form a prop er sub class of Robinson spaces, we sho w that this restriction is made without loss of generality . Indeed, every Robinson space can b e asso ciated with a strict Robinson space dened on the same set of elemen ts and satisfying the following prop erties: (i) the compatible order of the original space is preserved, and (ii) every v alid drawing of the strict Robinson space is also a v alid drawing of the original Robinson space. Consequen tly , by working with strict Robinson spaces, we retain full generalit y while b eneting from stronger structural properties that simplify both the analysis and the form ulation of the problem. The construction that maps a Robinson space ( S, ρ ) to a strict Robinson space ( S, ˆ ρ ) reassigns dissimilarity v alues to eac h pair according to the relativ e order of the original dissimilarities. T o dene the strict Robinson space ( S , ˆ ρ ) , let us consider the set U = { i, j } ⊂ S : i < j of unordered pairs of elemen ts of S . First, w e construct a total order π on the set U according to the following sequence of criteria: 1. the v alue of ρ ( i, j ) , in increasing order; 2. the v alue of i , in decreasing order; 3. the v alue of j , in increasing order. These criteria eliminate all p otential ties among elemen ts of U , thus ensuring that π is a total order. Second, for each pair ( i, j ) ∈ U , dissimilarit y is dened as its rank according to order π : ˆ ρ ( i, j ) = π ( i, j ) . By construction, all v alues of ˆ ρ ( i, j ) are pairwise distinct and b elong to the set { 1 , 2 , . . . , n ( n − 1) / 2 } . Finally , we dene the dissimilarit y space ( S, ˆ ρ ) , which we call the strict mapping of ( S, ρ ) , by symmetrically extending ˆ ρ to all pairs of elements of S : ˆ ρ ( i, i ) = 0 for all i ∈ S, ˆ ρ ( j, i ) = ˆ ρ ( i, j ) for all j > i. Clearly , the strict mapping ( S, ˆ ρ ) is a strict Robinson space on the same ground set S . Figure 3 depicts an example of a Robinson space and its strict mapping. Lemma 7. L et ( S, ρ ) b e a R obinson sp ac e, and let ( S, ˆ ρ ) b e its strict mapping. Then, the fol lowing statements hold: • Every c omp atible or der for ( S, ρ ) is also c omp atible for ( S, ˆ ρ ) . • L et ( X , d ) b e a metric sp ac e, and supp ose that I : S → X is a valid dr awing of ( S, ˆ ρ ) . Then I is also a valid dr awing of ( S, ρ ) . 10 S = 0 4 4 5 5 4 0 3 4 5 4 3 0 2 5 5 4 2 0 1 5 5 5 1 0 (a) S ′ = 0 5 6 9 10 5 0 3 4 8 6 3 0 2 7 9 4 2 0 1 10 8 7 1 0 (b) Fig. 3 : (a) Example of a Robinson space S represented b y its dissimilarity matrix. (b) The strict mapping for the space S represen ted by its dissimilarity matrix. Pr o of W e rst show that every compatible order for ( S, ρ ) is also compatible for ( S, ˆ ρ ) . T o this end, let i < j < k be three elements of S ordered according to a compatible order for ( S, ρ ) . By denition of a Robinson space, we hav e ρ ( i, k ) ≥ max { ρ ( i, j ) , ρ ( j, k ) } , that is, ρ ( i, k ) ≥ ρ ( i, j ) and ρ ( i, k ) ≥ ρ ( j, k ) . By construction of the strict mapping, the order π ranks pairs primarily according to increasing v alues of ρ , and breaks ties using the pair indices. Since ( i, k ) has a dissimilarity v alue greater than or equal to those of ( i, j ) and ( j, k ) , and b ecause i < j < k , the rules dening π ensure that ˆ ρ ( i, k ) > π ˆ ρ ( i, j ) and ˆ ρ ( i, k ) > π ˆ ρ ( j, k ) . Consequen tly , ˆ ρ ( i, k ) > max { ˆ ρ ( i, j ) , ˆ ρ ( j, k ) } , whic h pro ves that the order i < j < k is also compatible for ( S, ˆ ρ ) . Hence, the same order is also compatible for ( S, ˆ ρ ) . No w, for the second statemen t, let I b e a v alid drawing of ( S, ˆ ρ ) . T o prov e that I is also a v alid dra wing of ( S, ρ ) , consider i, j, k ∈ S such that ρ ( i, j ) < ρ ( i, k ) . By the denition of the dissimilarity ˆ ρ w e hav e that ˆ ρ ( i, j ) < ˆ ρ ( i, k ) . Therefore, d I ( i ) , I ( j ) < d I ( i ) , I ( k ) . Thus, for ev ery triplet of element i, j, k ∈ S ρ ( i, j ) < ρ ( i, k ) ⇒ d I ( i ) , I ( j ) < d I ( i ) , I ( k ) . Hence, I is a v alid dra wing for ( S, ρ ) . □ 6 Strong-four-p oin t-condition In this section, w e c haracterize caterpillar metric spaces in the spirit of Buneman’s Theorem by pro viding a condition similar to the classical F our-p oin t-condition, which w e call the Str ong-four-p oint-c ondition . Let ( S, ρ ) b e a strict Robinson space on n > 4 elements, and consider an y quadruple of v ertices ordered according to a compatible order. F or the sake of simplicity (and w.l.o.g. ), we denote these vertices b y 1 , 2 , 3 , and 4 . Any v alid dra wing of ( S, ρ ) in a tree must satisfy the F our-p oin t-condition. Moreov er, since ( S, ρ ) is strict Robinson, 11 w e ha ve ρ (1 , 2) < ρ (1 , 3) and ρ (3 , 4) < ρ (2 , 4) . This implies that in an y corresponding tree dra wing, the distances satisfy d (1 , 2) < d (1 , 3) and d (3 , 4) < d (2 , 4) . Consequen tly , d (1 , 2) + d (3 , 4) < d (1 , 3) + d (2 , 4) . The F our-p oint-condition implies d (1 , 2) + d (3 , 4) ≤ max { d (1 , 3) + d (2 , 4) , d (1 , 4) + d (2 , 3) } , d (1 , 3) + d (2 , 4) ≤ max { d (1 , 2) + d (3 , 4) , d (1 , 4) + d (2 , 3) } , and d (1 , 4) + d (2 , 3) ≤ max { d (1 , 3) + d (2 , 4) , d (1 , 2) + d (3 , 4) } . T ogether with the strict inequalit y d (1 , 2) + d (3 , 4) < d (1 , 3) + d (2 , 4) , these relations force d (1 , 3) + d (2 , 4) = d (1 , 4) + d (2 , 3) . This observ ation motiv ates the following denition: Denition 8 (Strong-four-p oint-condition) . Let ( X , d ) b e a nite metric space. W e sa y that d satises the Str ong-four-p oint-c ondition if d satises the F our-p oint- condition and there exists an ordering π of X suc h that for all { i, j , k , l } ⊆ X with i ≤ π j ≤ π k ≤ π l , it holds: d ( i, j ) + d ( k , l ) ≤ d ( i, k ) + d ( j, l ) = d ( i, l ) + d ( j, k ) . (3) This result has important consequences for the searc h for v alid dra wings for Robin- son spaces. First, from an algorithmic p ersp ective, the condition signicantly simplies the constrain ts describing the F our-p oint-condition. Indeed, for any quadruple, it is kno wn in adv ance whic h of the three p ossible sums is the smallest one, and therefore no binary decision v ariables are required to enco de this c hoice. Second, from a geometric p oin t of view, in Theorem 9 we pro ve that tree met- rics satisfying the Strong-four-point-condition necessarily ha ve a caterpillar top ology . T ogether, these prop erties allow us to reduce b oth the size of the form ulation and, ultimately , the computational complexity of nding a v alid drawing in a tree. Theorem 9. L et ( X , d ) b e a nite metric sp ac e. Then, ( X , d ) satises the Str ong-four- p oint-c ondition if and only if ( X , d ) c an b e isometric al ly emb e dde d in a c aterpil lar. Pr o of Let ( X , d ) b e a metric space satisfying the Strong-four-p oint-condition. W e b egin by sho wing that ( X , d ) admits an isometric embedding into a caterpillar. Assume that the ele- men ts of X are lab eled according to an order π prescrib ed b y the Strong-four-point-condition. W e pro ceed by induction on n . Assume that there exists a caterpillar compatible with the metric induced b y the rst n elements of X . W e sho w that this caterpillar can b e extended to incorp orate the ( n + 1) -th elemen t in π . 12 1 2 3 4 d 213 d 123 d 1234 d 234 d 243 Fig. 4 : Caterpillar construction for the metric space ( { 1 , 2 , 3 , 4 } , d ) satisfying the Strong-four-p oin t-condition. The distances d ij k and d ij kl in the caterpillar are dened using the distance d in the metric space as follo ws: d ij k = 1 2 ( d ( i, j ) + d ( j, k ) − d ( i, k )) , and d ij kl = d ( i, l ) + d ( j, k ) − ( d ( i, j ) + d ( k , l )) . In the base case, consisting of four elemen ts (say 1 ≤ π 2 ≤ π 3 ≤ π 4 , where π is the ordering given by the Strong-four-p oint-condition), the caterpillar can b e constructed as illustrated in Figure 4 , where the length of each edge is given explicitly . In the gure, d ij k denotes the Gromo v product, dened b y d ij k = 1 2 d ( i, j ) + d ( j, k ) − d ( i, k ) , and we set d ij kl = d ( i, k ) + d ( j, l ) − ( d ( i, j ) + d ( k , l )) . All edge lengths in the construction are non-negativ e. Indeed, b y the triangle inequality w e ha ve d ij k ≥ 0 for all triples ( i, j, k ) , and b y the Strong-four-p oin t-condition w e obtain d 1234 = d (1 , 3) + d (2 , 4) − ( d (1 , 2) + d (3 , 4)) ≥ 0 . F urthermore, b y a direct algebraic v erication, all pairwise distances realized in the caterpillar are seen to coincide with the corresponding v alues of the metric d . Hence, the caterpillar pro vides an isometric em b edding of the four-p oint metric space. By the inductive hypothesis, assume that every metric space on n objects satisfying the Strong-four-p oin t-condition can b e isometrically embedded in to a caterpillar. Let ( X , d ) b e a metric space on ( n + 1) objects satisfying the Strong-four-point-condition, and consider the caterpillar in which its rst n ob jects (according to the order π prescrib ed b y the Strong- four-p oin t-condition) are em b edded. Notice that, in this caterpillar, the length of the spine edge that is incident to the vertices corresp onding to n − 1 and n is giv en by d 1 ,n,n − 1 (see Figure 5 ). T o extend the construction so as to include the element n + 1 , we sub divide this edge b y inserting a new v ertex s , and connect s to a new leaf, which represen ts the image of n + 1 in the tree. The lengths of the newly created edges are dened analogously to the base case and are illustrated in Figure 5 . As b efore, b y direct algebraic v erication, all newly created edge lengths are non-negativ e. It remains to chec k that the distances induced b y the caterpillar are compatible with the metric d . Since all distances b etw een pairs of v ertices among { 1 , . . . , n } are preserv ed b y the inductiv e h yp othesis, it suces to verify the distances in volving the new v ertex n + 1 . F or the pair ( n, n + 1) , w e hav e d ( n − 1) n ( n +1) + d ( n − 1)( n +1) n = d (( n − 1) , n ) . F or ev ery 1 ≤ i ≤ n − 1 , the distance in the caterpillar betw een i and n + 1 is: d ( i, n ) − d ( n − 1) n ( n +1) + d ( n − 1)( n +1) n = d ( i, n ) − d (( n − 1) , n ) + d (( n − 1) , ( n + 1)) , whic h coincides with d ( i, ( n + 1)) b y the Strong-four-p oint-condition. Therefore, all pairwise distances inv olving the new v ertex n + 1 are correctly realized, and the extended caterpillar pro vides an isometric e m b edding of ( X , d ) . 13 1 2 n − 1 n s n + 1 · · · d ( n − 1)( n +1) n d 1( n − 1) n ( n +1) d ( n − 1) n ( n +1) d 1 n ( n − 1) Fig. 5 : Construction of a caterpillar for a metric satisfying the Strong-four-p oint- condition. In the opp osite direction, assume that the metric space admits an isometric embedding in to a caterpillar. Since the metric induced by a caterpillar is a tree metric, it necessarily satises the F our-p oint-condition. Therefore, in order to show that a metric compatible with a caterpillar satises the Strong-four-p oint-condition, it suces to pro vide an ordering of its elemen ts for which condition ( 3 ) holds. Consider the order π induced b y the spine of the caterpillar when tra versed from one of its ends. More precisely , a vertex i ∈ X precedes a vertex j in π if the closest spine vertex to i (p ossibly i itself ) app ears b efore the closest spine vertex to j . Ties are brok en arbitrarily . Consider any quadruple of elements of X , w.l.o.g. , 1 ≤ π 2 ≤ π 3 ≤ π 4 . As observ ed b efore, the distance b et ween v ertex 3 and its closest neighbor on the spine is equal to d 234 (see Figure 4 ). This distance coincides with d 134 . Hence, since d 234 = d 134 w e ha ve: d (2 , 3) + d (3 , 4) − d (2 , 4) = d (1 , 3) + d (3 , 4) − d (1 , 4) whic h implies d (1 , 4) + d (2 , 3) = d (1 , 3) + d (2 , 4) . Moreo ver, the distance b et ween the spine v ertices of vertices 3 and 4 is given b y d 1234 ≥ 0 (see Figure 4 ). Therefore, 0 ≤ d 1234 = d (1 , 4) + d (2 , 3) − ( d (1 , 2) + d (3 , 4)) whic h implies d (1 , 2) + d (3 , 4) ≤ d (1 , 4) + d (2 , 3) . Com bining this inequality with the equalit y obtained abov e, w e conclude that d (1 , 2) + d (3 , 4) ≤ d (1 , 4) + d (2 , 3) = d (1 , 3) + d (2 , 4) , whic h is precisely the condition required b y the Strong-four-point-condition. □ F rom the preceding discussion, any v alid drawing of a strict Robinson space in a tree m ust satisfy the Strong-four-point-condition. W e thus obtain the follo wing result. Corollary 10. If a strict R obinson sp ac e ( S, ρ ) has a valid dr awing in a tr e e T , then T is a c aterpil lar. 7 F easibilit y problem form ulation Using the characterization presented in the ab ov e section, in this section w e present a feasibilit y form ulation that allo ws us to decide whether a strict Robinson space admits a v alid drawing in a caterpillar. 14 1 2 i n − 1 n · · · · · · h i h n − h i l i Fig. 6 : V ariable denitions describing a caterpillar graph. W e use Corollary 10 to formulate the problem of nding a v alid dra wing in a tree as a feasibility problem o ver a system of linear constraints. T o this end, we in tro duce the follo wing sets of v ariables, whic h represen t the distances in a caterpillar (see Figure 6 ): • h i denotes the distance along the spine b etw een the rst spine vertex and the i -th spine v ertex. • l i denotes the length of the leg hanging from the i -th spine v ertex. Let ( X , d ) b e a metric space on n elements satisfying the Strong-four-p oin t- condition, and let π b e the corresp onding order. Consider a caterpillar with n leav es, ordered according to the app earance of the p oint where they hang, when the spine is tra versed from one of its ends. If we inject the i -th element of X (with resp ect to π ) in to the i -th leaf of the caterpillar, then, for any i < j , the distance b et ween i and j in the caterpillar is giv en by d ( i, j ) = h j − h i + l j + l i . W e may assume h 1 = l 1 = 0 . Moreo ver, since ( S, ρ ) is a strict Robinson space, we hav e ρ (1 , i ) < ρ (1 , i + 1) for all 1 ≤ i < n , and therefore d (1 , i ) < d (1 , i + 1) in any v alid drawing. This implies that, in a caterpillar dra wing of ( S , ρ ) , elemen ts m ust appear at strictly increasing distance from element 1 according to the compatible order. A symmetric argument applies when considering distances from elemen t n . Consequen tly , the conditions for a v alid drawing in a caterpillar need only b e veried using the left and right centers of eac h pair. Sp ecically , for each pair i, j ∈ S with 1 ≤ i < j ≤ n , we must ha ve d ( i, L( i, j )) < d (L( i, j ) , j ) and d ( i, R( j , i )) > d (R( j, i ) , j ) . In terms of the v ariables h i and l i , these conditions are equiv alent to h i + h j − 2 h L( i,j ) − l i + l j > 0 , − h i − h j + 2 h R( j,i ) + l i − l j > 0 . (4) Note that if a v ector x = ( h , l ) ⊤ = ( h 2 , . . . , h n , l 2 , . . . , l n ) ⊤ ∈ R 2 n − 2 + satises the system ( 4 ), then the vector α x + β 1 = ( α h 2 + β , . . . , h n + β , αl 2 + β , . . . , αl n + β ) ⊤ also satises the same system, for ev ery α ∈ R + and β ∈ R . This in v ariance prop ert y allo ws us to simplify the formulation by removing the non-negativit y constraints on the v ariables. Indeed, giv en an y solution (possibly with negative entries) satisfying 15 ( 4 ), one can add a sucien tly large constan t β to all v ariables in order to obtain a non-negativ e solution. Therefore, the existence of a v alid drawing of a strict Robinson space in a caterpillar is equiv alent to showing that the follo wing set is non-empt y: C S = x = ( h , l ) ⊤ ∈ R 2 n − 2 : h i + h j − 2 h L( i,j ) − l i + l j > 0 − h i − h j + 2 h R( j,i ) + l i − l j > 0 , ∀ 1 ≤ i < j ≤ n . (5) Matrix formulation. Since the set C S is dened by linear constraints, it can b e equiv alently describ ed in matrix form as A S x > 0 for some matrix A S ∈ R n ( n − 1) × (2 n − 2) . The columns of A S are indexed b y the v ariables h k and l k , with k ∈ { 2 , . . . , n } , while the ro ws corresp ond to the constrain ts and are indexed by the ordered pairs ( i, j ) ∈ [ n ] 2 with i = j . By the denition of C S , in the constraint indexed by a pair ( i, j ) the only v ariables that may hav e a nonzero co ecien t are those asso ciated with i , j , and with the left or righ t center of the pair. A special situation occurs when the left or right center coincides with i or j , that is, when L( i, j ) = i or R( j, i ) = i . In these cases, the inequalities in ( 4 ) simplify to − h i + h j − l i + l j > 0 when i < j , and h i − h j − l i + l j > 0 when i > j . Figure 7 depicts the matrix A S for the example presen ted in Figure 3 . This matrix form ulation allo ws us to deriv e an alternativ e c haracterization of strict Robinson spaces admitting a v alid dra wing in a caterpillar. The characterization relies on the classical theorem of alternativ es due to Gordan. Theorem 11 (Gordan’s alternative [ 25 ]) . L et A b e an m × n r e al matrix. Exactly one of the fol lowing statements holds: 1. Ther e exists x ∈ R n such that Ax > 0 . 2. Ther e exists a nonzer o ve ctor y ∈ R m such that A ⊤ y = 0 and y ≥ 0 . In our setting, Theorem 11 can b e restated as follows. Let ( S, ρ ) b e a strict Robinson space, then exactly one of the follo wing statements holds: 1’ . ( S, ρ ) admits a v alid drawing in a caterpillar. 2’ . There exists a nonzero v ector y ∈ R n ( n − 1) suc h that A S ⊤ y = 0 and y ≥ 0 . In other words, a strict Robinson space ( S, ρ ) admits a v alid drawing in a caterpillar if and only if the k ernel of A S ⊤ con tains no nonzero nonnegative vector. K ernel char acterization. Consider now a partition of the rows (constrain ts) of the matrix A S in to t wo submatrices B S and N S , where B S consists of the rows indexed b y consecutiv e pairs, { 12 , . . . , ( n − 1) n, 21 , . . . , n ( n − 1) } , and N S con tains the remaining constraints (see Figure 7 ). F or consecutive pairs w e ha ve L( i, i + 1) = i and R( i + 1 , i ) = i + 1 . Consequently , the submatrix B S has the blo c k structure shown in Figure 8 . W e emphasize that B S dep ends only on n and is 16 A S = B S N S = h 2 h 3 h 4 h 5 l 2 l 3 l 4 l 5 12 1 0 0 0 1 0 0 0 23 − 1 1 0 0 − 1 1 0 0 34 0 − 1 1 0 0 − 1 1 0 45 0 0 − 1 1 0 0 − 1 1 21 1 0 0 0 − 1 0 0 0 32 − 1 1 0 0 1 − 1 0 0 43 0 − 1 1 0 0 1 − 1 0 54 0 0 − 1 1 0 0 1 − 1 13 0 1 0 0 0 1 0 0 14 0 0 1 0 0 0 1 0 15 0 − 2 0 1 0 0 0 1 24 − 1 0 1 0 − 1 0 1 0 25 1 − 2 0 1 − 1 0 0 1 35 0 − 1 0 1 0 − 1 0 1 31 2 − 1 0 0 0 − 1 0 0 41 2 0 − 1 0 0 0 − 1 0 42 − 1 2 − 1 0 1 0 − 1 0 51 0 0 2 − 1 0 0 0 − 1 52 − 1 0 2 − 1 1 0 0 − 1 53 0 − 1 2 − 1 0 1 0 − 1 Fig. 7 : Matrix denition of the problem for the strict Robinson space example of Figure 3 (b). In the gure, matrix ro ws represent the constrain ts induced by conditions 4 . indep enden t of the specic instance ( S, ρ ) . Moreo ver, B S has full rank and is therefore in vertible. Let us consider a v ector y ∈ ker( A S ⊤ ) and partition it as y ⊤ = ( y B ⊤ | y N ⊤ ) , according to the block decomp osition of A S in to B S and N S . With this partition, every vector in ker( A S ⊤ ) is completely determined by its comp onen ts in y N . Indeed, the equation A S ⊤ y = 0 can be written as B S ⊤ y B + N S ⊤ y N = 0 , or, equiv alently , y B = Y S y N , where Y S := − ( B S ⊤ ) − 1 N S ⊤ . 17 B S = h 2 h n l 2 l n 12 ( n − 2) n 21 n ( n − 2) T n − 1 T n − 1 T n − 1 − T n − 1 T n − 1 = 1 2 n − 1 1 1 2 − 1 n − 1 − 1 1 0 0 (a) (b) Fig. 8 : (a) Structure of matrix B S . (b) Structure of matrices T n − 1 . T o compute Y S , w e observe that the in verse of the matrix B S ⊤ admits the explicit expression ( B S ⊤ ) − 1 = 1 2 T n − 1 −⊤ T n − 1 −⊤ T n − 1 −⊤ − T n − 1 −⊤ , where T n − 1 −⊤ = 1 1 1 0 . Figure 9 depicts the matrix Y S for the example in Figure 3 . W e summarize the ab o ve discussion in the following lemma. Lemma 12. A strict R obinson sp ac e ( S, ρ ) on n elements admits a valid dr awing in a c aterpil lar if and only if the fol lowing system is infe asible: y N ∈ R ( n − 1)( n − 2) : Y S y N ≥ 0 , y N ≥ 0 , y N = 0 (LP-S) Note that Lemma 12 provides an equiv alen t description of the set C S dened in ( 5 ). How ever, it yields a more explicit structural characterization of strict Robinson spaces that admit a v alid drawing in a caterpillar. 8 Applications of Lemma 12 In this section, we presen t three applications of Lemma 12 that yield general results concerning the problem of nding a v alid drawing in a tree for Robinson spaces. R e dundant triples. Recall that the v ector y N is indexed by the constraints of the matrix A S asso ciated with pairs ( i, j ) such that | i − j | > 1 . F or any pair 1 ≤ i < j ≤ n , the corresp onding column of Y S dep ends on the type of constraint represented by ( i, j ) . In fact, there are exactly four possible column structures, which are displa yed in Figure 10 . W e sa y that the column ( Y S ) ij , corresp onding to the constraint asso ciated with the pair ( i, j ) , is r e dundant if imposing the condition y N ij = 0 do es not aect the feasibilit y of problem ( LP-S ). That is, the column ( Y S ) ij is redundant if whenever there exists a solution to ( LP-S ), there also exists a solution ˜ y N suc h that ˜ y N ij = 0 . 18 Y S = y 13 y 14 y 15 y 24 y 25 y 35 y 31 y 41 y 42 y 51 y 52 y 53 h 2 − 1 − 1 0 0 0 0 0 0 0 0 0 0 h 3 − 1 − 1 0 − 1 0 0 1 1 0 0 0 0 h 4 0 − 1 − 1 − 1 − 1 − 1 0 1 1 0 0 0 h 5 0 0 − 1 0 − 1 − 1 0 0 0 1 1 1 l 2 0 0 1 0 0 0 − 1 − 1 0 − 1 0 0 l 3 0 0 1 0 1 0 0 0 − 1 − 1 − 1 0 l 4 0 0 0 0 0 0 0 0 0 − 1 − 1 − 1 l 5 0 0 0 0 0 0 0 0 0 0 0 0 Fig. 9 : Matrix Y S for the strict Robinson space example of Figure 3 (b). A rst t yp e of redundan t columns corresponds to constrain ts for whic h either the left cen ter or the right cen ter of the pair coincides with one of the indices of the pair (see Figures 10 (a) and 10 (b)). Let ( Y S ) ij b e a column of this t yp e and supp ose that the v ector y N is a solution of problem ( LP-S ). Observ e that the column ( Y S ) ij con tains only non–positive en tries. Hence, ( y N ) ij cannot b e the only nonzero component of y N , since otherwise w e would hav e Y S y N = ( Y S ) ij ( y N ) ij < 0 , whic h contradicts the condition Y S y N ≥ 0 together with y N = 0 . Therefore, setting ( y N ) ij = 0 yields a vector y ′ N ≥ 0 that is still nonzero and satises Y S y ′ N = Y S y N − ( y N ) ij ( Y S ) ij ≥ Y S y N ≥ 0 . Consequen tly , ( Y S ) ij is redundan t in the sense of the denition ab o ve. The second t yp e of redundant columns corresp onds to constrain ts asso ciated with pairs ( i, j ) with i < j for whic h there exists another pair ( i ′ , j ′ ) , with i ′ < j ′ , suc h that L( i, j ) = L( i ′ , j ′ ) , i ′ ≤ i, j ′ ≤ j (see, for instance, the columns corresp onding to (2 , 5) and (1 , 5) in Figure 9 ). In this situation, the column ( Y S ) ij is comp onen t-wise dominated by the column ( Y S ) i ′ j ′ , that is, ( Y S ) ij ≤ ( Y S ) i ′ j ′ . Let y N b e a solution of ( LP-S ). Dene a new vector y ′ N b y setting ( y ′ N ) ij = 0 , ( y ′ N ) i ′ j ′ = ( y N ) ij + ( y N ) i ′ j ′ , and leaving all other comp onen ts unchanged. Clearly , y ′ N ≥ 0 and y ′ N = 0 . Moreo ver, Y S y ′ N = Y S y N + ( Y S ) i ′ j ′ − ( Y S ) ij ( y N ) ij ≥ Y S y N ≥ 0 , 19 n 100 200 300 400 500 600 700 ˆ µ 21.86% 20.86% 20.70% 20.49% 20.43% 20.20% 20.43% ˆ σ / ˆ µ 8.78% 6.59% 5.57% 4.84% 4.36% 4.08% 3.56% T able 1 : Comparison betw een the num b er of constraints in the orig- inal and reduced linear program. The rst ro w indicates the n umber of elemen ts of the Robinson spaces ( n ). The second row rep orts the a verage, in a sample of one hundred spaces, of the prop ortion of constrain ts remained after remo ving redundant constrain ts ( ˆ µ ). Last ro w shows the co ecient of v ariation, computed as the sample stan- dard deviation normalized b y the av erage ( ˆ σ / ˆ µ ). where the inequality follows from the fact that ( Y S ) i ′ j ′ − ( Y S ) ij is comp onent-wise nonnegativ e. Hence, setting ( y N ) ij = 0 do es not aect feasibility , and the column ( Y S ) ij is redundan t. An analogous argument shows that a column ( Y S ) j i with i < j is redundan t if there exists another pair ( j ′ , i ′ ) suc h that R( j, i ) = R( j ′ , i ′ ) , i ′ ≥ i, j ′ ≥ j In this case, the column ( Y S ) j i is comp onent-wise dominated b y ( Y S ) j ′ i ′ , and the same redistribution argumen t applies. Remo ving redundant triples signicantly reduces the num b er of constraints in the linear program. T o ev aluate this reduction, we conduct an exp erimen tal analysis comparing the num b er of constraints in the original matrix Y S with those remaining after eliminating the redundant columns. F or each size in an increasing sequence of space dimensions, we use a random generator to sample one hundred distinct Robinson spaces. F or each sampled space, we compute the proportion of columns retained after remo ving redundancies. T able 1 presents the exp erimen tal results. The num b er of remaining constrain ts is appro ximately 20% of the original coun t, regardless of space size. Moreo ver, the co ecient of v ariation ranges b etw een 3-9%, demonstrating that this reduction rate is highly consisten t across all generated instances. Existenc e of V alid Dr awings in T r e es. W e no w exploit the dieren t characterizations of the existence of v alid drawings in a caterpillar in order to identify sub classes of Robinson spaces for whic h the (non- )existence of suc h a drawing is guaranteed. W e b egin b y pro viding structural conditions that are sucien t to ensure the exis- tence of a v alid drawing in a caterpillar. In particular, we sho w that strict Robinson spaces whose left and right centers are suitably correlated with the midp oints of their corresp onding in terv als alwa ys admit a v alid dra wing in a caterpillar. T o pro v e this result, w e rst establish a structural prop erty of the cen ter matrix for strict Robinson spaces: the left and right centers of any pair are necessarily adjacent. 20 h 2 0 0 − 1 ← L( i, j ) + 1 . . . − 1 ← j 0 h n 0 l 2 0 l n 0 ( a ) L( i, j ) = i 0 0 0 0 − 1 ← i + 1 . . . − 1 ← R( j, i ) 0 0 ( b ) R( j, i ) = j 0 0 − 1 ← L( i, j ) + 1 . . . − 1 ← j 0 0 0 0 1 ← i + 1 1 ← L( i, j ) 0 0 ( c ) L( i, j ) = i 0 0 1 ← R( j, i ) + 1 1 ← j 0 0 0 0 − 1 ← i + 1 . . . − 1 ← R( j, i ) 0 0 ( d ) R( j, i ) = j Fig. 10 : Structure types of columns in Y S . Sp ecically , for every pair i < j in a strict Robinson space, L( i, j ) = R( i, j ) − 1 . By denition, the right center R( i, j ) = k is the smallest index in { i, . . . , j } satisfying d ( k , j ) < d ( k , i ) . Therefore, the preceding elemen t ( k − 1) m ust satises d ( k − 1 , j ) ≥ d ( k − 1 , i ) , as otherwise this would contradict the minimality of R( i, j ) . Because the dissimilarities are strictly ordered (no ties), equalit y is imp ossible; therefore, d ( k − 1 , j ) > d ( k − 1 , i ) . This inequality means that k − 1 is closer to i than to j , thus k − 1 = L( i, j ) , establishing the adjacency prop erty . This structural constraint has an imp ortan t implication: the p osition of one cen ter completely determines the other, reducing the degrees of freedom in the center matrix to a single family of v alues. This observ ation prov es crucial in the pro of of the follo wing theorem. Theorem 13. L et ( S, ρ ) b e a strict R obinson dissimilarity sp ac e. If for every p air i < j we have L( i, j ) ≤ i + j 2 ≤ R( j, i ) , then ( S , ρ ) admits a valid dr awing in a c aterpil lar. Pr o of W e pro v e that for every vector y N ≥ 0 , the sum of the en tries of Y S y N is non-positive. In particular, this implies that Y S y N cannot b e a strictly positive vector. 21 F or every pair i < j , let m ij = ( i + j ) / 2 denote its midp oin t. Consider a pair of elements i < j − 1 . Summing all the rows of the column ( Y S ) ij corresp onding to a left–cen ter constrain t (see Figure 10 (c)) yields S ij = − ( j − L( i, j )) + (L( i, j ) − i ) = 2 L( i, j ) − m ij . Summing no w all the rows of the column ( Y S ) j i corresp onding to a righ t–center constraint (see Figure 10 (d)) gives S j i = ( j − R( j, i )) − (R( j, i ) − i ) = 2 m ij − R( j, i ) . Hence, the sum of all the en tries of the v ector Y S y N is X i j must b e zero, and therefore Y S y N m ust con tain a negative entry . Let ( i, j ) b e a pair with i < j such that ( y N ) j i > 0 . Then necessarily i ≥ ¯ ı and R( j, i ) ≤ L( ¯ ı, ¯ ȷ ) . Otherwise, there exists a ro w h r suc h that ( Y S ) h r ,j i = − 1 and ( Y S ) h r , ¯ ı ¯ ȷ = 0 (see Figures 10 (c) and 10 (d)), which would imply ( Y S y N ) h r < 0 , a contradiction. Moreo ver, we m ust ha ve j < ¯ ȷ . Indeed, if j ≥ ¯ ȷ , then together with i ≥ ¯ ı we obtain L( j, i ) < R( j, i ) ≤ L( ¯ ı, ¯ ȷ ) , whic h con tradicts the monotonicity of the center matrix. Hence, ( Y S ) h ¯ ȷ , ¯ ı ¯ ȷ = − 1 and ( Y S ) h ¯ ȷ ,j i = 0 , and therefore ( Y S y N ) h ¯ ȷ = ( Y S ) h ¯ ȷ , ¯ ı ¯ ȷ ( y N ) ¯ ı ¯ ȷ + X i
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