- Email: [email protected]
Minimal superior ultrametrics under order constraint
Discrete Mathematics North-Holland
201
84 (1990) 201-204
COMMUNICATION MINIMAL SUPERIOR CONSTRAINT
ULTRAMETRICS
UNDER
ORDER
B. Van CUTSEM Laboratoire France
de Mod&ation
et Calcul, Institut IMAG,
B. P. 53X, F-38041 Grenoble
Cedex,
Communicated by C. Benz&en Received 23 February 1990 It is well known that the pointwise minimum of all the ultrametrics which are greater than a given dissimilarity on a finite set S is this dissimilarity istelf. It is proved here that the minimum of all the ultrametrics which are greater than this dissimilarity and which are compatible with an order on S is the minimum Robinsonian dissimilarity on S for this order greater than the given dissimilarity.
1. Definitions and notations A dissimilarity 6 on a finite set S is an application verifies the following conditions V(i, j) E P,
6(i, j) 2 0,
6(i, j) = S(j, i),
from S
X S
to R! which
6(i, i) = 0.
An ultrametric on S is a dissimilarity u on S such that u(i, j) < max{u(i, k), u(k, j)}.
V(i, j, k) E S3,
If 6, and & are two dissimilarities on S, we shall say that c& is greater than 6i, and then we shall write 6, G &, if V(i, j) E P,
&(i, j) 6 &(i, j).
Let ‘%(6) be the set of all ultrametrics on S which are greater than a given dissimilarity 6. This set is not empty since, if M = max{b(i, j); (i, j) E S’}, the dissimilarity uw defined by M 0
u,& j) =
ifi#j, ifi=j,
is an ultrametric which is greater than 6 and then belongs to Q(6). It is well known that generally no minimum element exists in %(S) but only minimal ones. More, if u is a minimal element of ‘%(a), the terms u(i, j) are, for every (i, j) in S*, extracted from the set of the terms of 6, i.e. V(i, j) E S*, This property difficulty. 0012-365X/90/$03.50
u(i, j) E {6(/z, k); (h, k) E S*}.
implies the following 0
1990 -
Elsevier
result
Science Publishers
the proof
of which presents
B.V. (North-Holland)
no
202
B. Van CuBem
Theorem 1.1. Let 6 be a dissimilarity on a finite set S and %(a) denote the set of all ultrametrics which are greater than 6. Then, V(i, j) E S2,
6(i, j) E min{u(i, j); u e Q(6)).
2. Dissimilarity compatible with an order Let us suppose now that the set S has n elements and that a total order, denoted s, is defined on S. The compatibility between a total order < and a dissimilarity 6 on S, is related to the idea that if j is between i and k for this order, then 6(i, j) and 6(i, k) are less than 6(i, k). Definition 2.1. A dissimilarity compatible with the order s, if V(i, j, k) E S3, i
6 on the totally
ordered
finite set (S, s)
is
max{s(i, j), S(j, k)} < 6(i, k).
that the total order
< on S is compatible
with the
The class of dissimilarities on a set S for which there exists a compatible total order can be easily identified to the class of the Robinsonian dissimilarities on S, the definition of which is the following. Definition 2.2 (Robinson [14], Diday [5] and Hubert [13]). Let p be a dissimilarity on a finite set S. The dissimilarity p is said to be a Robinsonian dissimilarity on S, if there exists a total order d on S, such that, (S, c) being identified to Z = { 1, 2, . . . , n} with its natural order, we have V(i, j) E Z*, i
p(i, j) 3 m={p(i,
j - I), p(i + 1, j)).
The proof of the l-l correspondence between these two classes is easy to check. It can be proved that the ultrametrics on S are Robinsonian dissimilarities. Theorem 2.3. Let u be an ultrametric on a finite set S. Then there exists a total order on S compatible with u. Many proofs of compatible orders Gaud [12] and B. with an ultrametric
this result have been given. So many algorithms to calculate were published. See, for instance, Benzecri [l], Diday [5], van Cutsem [15]. In fact, there are many orders compatible on S, but it is not the place here to describe them.
Minimal superior ultrametrics under order constraint
203
3. Minimal ultrametrics with order constraints Let < be a total order on the finite set S, and 6 a dissimilarity on S. Let 9..&(a) denotes the set of all ultrametrics on S which are compatible with the order < and greater than 6. This set is not empty since the ultrametric u,+, introduced in the first paragraph is compatible with any order on S, and then belongs to %!&(a). In the same way, if < is a total order on S, let C%(S) be the set of all Robinsonian dissimilarities on S which are compatible with the order < and greater than 6. It is not very difficult to prove that there exists a minimum element in 9&(d). The following algorithm calculates this minimum element pG. Algorithm 3.1. Let (S, s) b e a totally ordered finite set identified to Z = {I, . . . > n} and 6 a dissimilarity on S. (1) For i := 1 to n do p=(i, i) := 0 (2) Fork:=lton-ldo Fori:=lton-kdo p=(i, i + k) := max{p&i, i + k - l), p=(i + 1, i + k), 6(i, i + k)}. We can now establish the main result of this communication. Theorem 3.2. Let S be a finite set and s be a total order on S. Let pS be the minimum Robinsonian V(i, j) E S2,
dissimilarity in W,(6).
Then,
p<(i, j) = min{u(i, Z); u E Q,(6)}.
Proof. As all the ultrametrics in Q,(6) are Robinsonian, it is straightforward that, for every u in 9&(S), we have p-_ su. So, we have just to prove that, for every (i, j) in S*, we have p&i, j) = min{u(i, Z); u E %(a)}.
Let (i, j) E S*. Setting similarity uii by uij(h, k) =
M = max{b(h,
k); (h, k) E S*},
p<(i, j)
if i G h G j and i < k G j,
0
ifh=k,
{M
ifh
It is then easy to prove that (1) Uij is an ultrametric on S, (2) Uijis compatible with s, (3) uij is greater than 6, (4) p&, j) = Uij(i, j), and then the proof is achieved.
Cl
let us define
a dis-
204
B. Van Cutsem
4. Conclusions
As, for a given dissimilarity 6, the construction of pC is quite easy by Algorithm 3.1, this may be used to study the set of minimal ultrametrics in Q,(6) and perhaps to be able to describe all the minimal elements of a(S).
References [l] J.P. Benzecri, L’Analyse des DonnCes, I: La taxinomie (Dunod, Paris, 1973). [2] G. Brossier, Reprtsentation ordonnee des classifications hierarchiques, Statist. Anal. Donntes 5 (2) (1980) 31-44. [3] E. Diday, Croisements, ordres et ultramttriques: application a la recherche de consensus en classification automatique. Rapport de Recherche INRIA no. 141 (1983). [4] E. Diday, Croisements, ordres et ultrametriques, Mathematiques et Sciences Humaines 2ltme annee no. 83, (1983) p. 31-54. [5] E. Diday, Crossing order and ultrametrics, Compstat (Proceedings in Computer Statistics) (Physica, Vienne, 1982). [6] E. Diday, Une repesentation visuelle des classes empietantes: les pyramides, Rapport de Recherche INRIA no. 291 (1984). [7] E. Diday, Orders and overlapping clusters by pyramids, in: J. De Leeuw et al., eds., Multidimensional Data Analysis, Proceedings of a workshop, Cambridge University 1985, (DSWO Press, 1986). [8] C. Durand, Sur la representation pyramidale en analyse des donntes, Memoire de DEA, Universite de Provence (1986). [9] C. Durand and B. Fichet, One-to-one correspondences in pyramidal representation: a unified approach, in: H.H. Bock, ed., Classification and Related Methods of Data Analysis (NorthHolland, Amsterdam, 1988) 85-90. [lo] B. Fichet, Data analysis: Geometric and algebraic structures, First World Congress of Bernoulli Society, Tashkent (1986). [ll] E. Gaud, Sur la representation hierarchique en Analyse des Dontrees, Memoire de DEA en Mathematiques Appliquees, Universitt de Provence (1980). [12] E. Gaud, Representation d’une prtordonnance. Etude de ses images euclidiennes. Problemes de graphes dans sa representation hitrarchique, These de Doctorat de Troisieme Cycle en Mathtmatiques Appliquees, Universitt de Provence, 29 juin 1983. [13] L. Hubert, Some applications of graph theory and related non-metric techniques to problems of approximate seriation: the case of symmetric proximity measure. British J. Math. Statist. Psych. 27 (2) (1974) 133-153. [14] W.S. Robinson, A model for chronological ordering of archeological deposits, American Antiquity 16 (1951). [15] B. Van Cutsem, Decomposition d’une ultrametrique, Rapport de recherche no. 388 du Laboratoire TIM3-IMAG, Grenoble (1983).
201
84 (1990) 201-204
COMMUNICATION MINIMAL SUPERIOR CONSTRAINT
ULTRAMETRICS
UNDER
ORDER
B. Van CUTSEM Laboratoire France
de Mod&ation
et Calcul, Institut IMAG,
B. P. 53X, F-38041 Grenoble
Cedex,
Communicated by C. Benz&en Received 23 February 1990 It is well known that the pointwise minimum of all the ultrametrics which are greater than a given dissimilarity on a finite set S is this dissimilarity istelf. It is proved here that the minimum of all the ultrametrics which are greater than this dissimilarity and which are compatible with an order on S is the minimum Robinsonian dissimilarity on S for this order greater than the given dissimilarity.
1. Definitions and notations A dissimilarity 6 on a finite set S is an application verifies the following conditions V(i, j) E P,
6(i, j) 2 0,
6(i, j) = S(j, i),
from S
X S
to R! which
6(i, i) = 0.
An ultrametric on S is a dissimilarity u on S such that u(i, j) < max{u(i, k), u(k, j)}.
V(i, j, k) E S3,
If 6, and & are two dissimilarities on S, we shall say that c& is greater than 6i, and then we shall write 6, G &, if V(i, j) E P,
&(i, j) 6 &(i, j).
Let ‘%(6) be the set of all ultrametrics on S which are greater than a given dissimilarity 6. This set is not empty since, if M = max{b(i, j); (i, j) E S’}, the dissimilarity uw defined by M 0
u,& j) =
ifi#j, ifi=j,
is an ultrametric which is greater than 6 and then belongs to Q(6). It is well known that generally no minimum element exists in %(S) but only minimal ones. More, if u is a minimal element of ‘%(a), the terms u(i, j) are, for every (i, j) in S*, extracted from the set of the terms of 6, i.e. V(i, j) E S*, This property difficulty. 0012-365X/90/$03.50
u(i, j) E {6(/z, k); (h, k) E S*}.
implies the following 0
1990 -
Elsevier
result
Science Publishers
the proof
of which presents
B.V. (North-Holland)
no
202
B. Van CuBem
Theorem 1.1. Let 6 be a dissimilarity on a finite set S and %(a) denote the set of all ultrametrics which are greater than 6. Then, V(i, j) E S2,
6(i, j) E min{u(i, j); u e Q(6)).
2. Dissimilarity compatible with an order Let us suppose now that the set S has n elements and that a total order, denoted s, is defined on S. The compatibility between a total order < and a dissimilarity 6 on S, is related to the idea that if j is between i and k for this order, then 6(i, j) and 6(i, k) are less than 6(i, k). Definition 2.1. A dissimilarity compatible with the order s, if V(i, j, k) E S3, i
6 on the totally
ordered
finite set (S, s)
is
max{s(i, j), S(j, k)} < 6(i, k).
that the total order
< on S is compatible
with the
The class of dissimilarities on a set S for which there exists a compatible total order can be easily identified to the class of the Robinsonian dissimilarities on S, the definition of which is the following. Definition 2.2 (Robinson [14], Diday [5] and Hubert [13]). Let p be a dissimilarity on a finite set S. The dissimilarity p is said to be a Robinsonian dissimilarity on S, if there exists a total order d on S, such that, (S, c) being identified to Z = { 1, 2, . . . , n} with its natural order, we have V(i, j) E Z*, i
p(i, j) 3 m={p(i,
j - I), p(i + 1, j)).
The proof of the l-l correspondence between these two classes is easy to check. It can be proved that the ultrametrics on S are Robinsonian dissimilarities. Theorem 2.3. Let u be an ultrametric on a finite set S. Then there exists a total order on S compatible with u. Many proofs of compatible orders Gaud [12] and B. with an ultrametric
this result have been given. So many algorithms to calculate were published. See, for instance, Benzecri [l], Diday [5], van Cutsem [15]. In fact, there are many orders compatible on S, but it is not the place here to describe them.
Minimal superior ultrametrics under order constraint
203
3. Minimal ultrametrics with order constraints Let < be a total order on the finite set S, and 6 a dissimilarity on S. Let 9..&(a) denotes the set of all ultrametrics on S which are compatible with the order < and greater than 6. This set is not empty since the ultrametric u,+, introduced in the first paragraph is compatible with any order on S, and then belongs to %!&(a). In the same way, if < is a total order on S, let C%(S) be the set of all Robinsonian dissimilarities on S which are compatible with the order < and greater than 6. It is not very difficult to prove that there exists a minimum element in 9&(d). The following algorithm calculates this minimum element pG. Algorithm 3.1. Let (S, s) b e a totally ordered finite set identified to Z = {I, . . . > n} and 6 a dissimilarity on S. (1) For i := 1 to n do p=(i, i) := 0 (2) Fork:=lton-ldo Fori:=lton-kdo p=(i, i + k) := max{p&i, i + k - l), p=(i + 1, i + k), 6(i, i + k)}. We can now establish the main result of this communication. Theorem 3.2. Let S be a finite set and s be a total order on S. Let pS be the minimum Robinsonian V(i, j) E S2,
dissimilarity in W,(6).
Then,
p<(i, j) = min{u(i, Z); u E Q,(6)}.
Proof. As all the ultrametrics in Q,(6) are Robinsonian, it is straightforward that, for every u in 9&(S), we have p-_ su. So, we have just to prove that, for every (i, j) in S*, we have p&i, j) = min{u(i, Z); u E %(a)}.
Let (i, j) E S*. Setting similarity uii by uij(h, k) =
M = max{b(h,
k); (h, k) E S*},
p<(i, j)
if i G h G j and i < k G j,
0
ifh=k,
{M
ifh
It is then easy to prove that (1) Uij is an ultrametric on S, (2) Uijis compatible with s, (3) uij is greater than 6, (4) p&, j) = Uij(i, j), and then the proof is achieved.
Cl
let us define
a dis-
204
B. Van Cutsem
4. Conclusions
As, for a given dissimilarity 6, the construction of pC is quite easy by Algorithm 3.1, this may be used to study the set of minimal ultrametrics in Q,(6) and perhaps to be able to describe all the minimal elements of a(S).
References [l] J.P. Benzecri, L’Analyse des DonnCes, I: La taxinomie (Dunod, Paris, 1973). [2] G. Brossier, Reprtsentation ordonnee des classifications hierarchiques, Statist. Anal. Donntes 5 (2) (1980) 31-44. [3] E. Diday, Croisements, ordres et ultramttriques: application a la recherche de consensus en classification automatique. Rapport de Recherche INRIA no. 141 (1983). [4] E. Diday, Croisements, ordres et ultrametriques, Mathematiques et Sciences Humaines 2ltme annee no. 83, (1983) p. 31-54. [5] E. Diday, Crossing order and ultrametrics, Compstat (Proceedings in Computer Statistics) (Physica, Vienne, 1982). [6] E. Diday, Une repesentation visuelle des classes empietantes: les pyramides, Rapport de Recherche INRIA no. 291 (1984). [7] E. Diday, Orders and overlapping clusters by pyramids, in: J. De Leeuw et al., eds., Multidimensional Data Analysis, Proceedings of a workshop, Cambridge University 1985, (DSWO Press, 1986). [8] C. Durand, Sur la representation pyramidale en analyse des donntes, Memoire de DEA, Universite de Provence (1986). [9] C. Durand and B. Fichet, One-to-one correspondences in pyramidal representation: a unified approach, in: H.H. Bock, ed., Classification and Related Methods of Data Analysis (NorthHolland, Amsterdam, 1988) 85-90. [lo] B. Fichet, Data analysis: Geometric and algebraic structures, First World Congress of Bernoulli Society, Tashkent (1986). [ll] E. Gaud, Sur la representation hierarchique en Analyse des Dontrees, Memoire de DEA en Mathematiques Appliquees, Universitt de Provence (1980). [12] E. Gaud, Representation d’une prtordonnance. Etude de ses images euclidiennes. Problemes de graphes dans sa representation hitrarchique, These de Doctorat de Troisieme Cycle en Mathtmatiques Appliquees, Universitt de Provence, 29 juin 1983. [13] L. Hubert, Some applications of graph theory and related non-metric techniques to problems of approximate seriation: the case of symmetric proximity measure. British J. Math. Statist. Psych. 27 (2) (1974) 133-153. [14] W.S. Robinson, A model for chronological ordering of archeological deposits, American Antiquity 16 (1951). [15] B. Van Cutsem, Decomposition d’une ultrametrique, Rapport de recherche no. 388 du Laboratoire TIM3-IMAG, Grenoble (1983).