DOI 10.1007/s10878-012-9536-1
Are there more almost separable partitions
than separable partitions?
Fei-Huang Chang· Hong-Bin Chen · Frank K. Hwang
Published online: 16 August 2012
© Springer Science+Business Media, LLC 2012
Abstract A partition of a set of n points in d-dimensional space into p parts is called an (almost) separable partition if the convex hulls formed by the parts are (almost) pairwise disjoint. These two partition classes are the most encountered ones in clustering and other partition problems for high-dimensional points and their use-fulness depends critically on the issue whether their numbers are small. The problem of bounding separable partitions has been well studied in the literature (Alon and Onn in Discrete Appl. Math. 91:39–51,1999; Barnes et al. in Math. Program. 54:69–86,
1992; Harding in Proc. Edinb. Math. Soc. 15:285–289,1967; Hwang et al. in SIAM J. Optim. 10:70–81,1999; Hwang and Rothblum in J. Comb. Optim. 21:423–433,
2011a). In this paper, we prove that for d≤ 2 or p ≤ 2, the maximum number of almost separable partitions is equal to the maximum number of separable partitions. Keywords Partition· Separable partition · Optimal partition · Almost separable partition
1 Introduction
Let d≥ 1 and let A = {A1, A2, . . . , An} be a multi-set of n points (not necessarily
distinct) inRd (denoted by A∈ Rd×n). For p≥ 1, a p-partition of A is an ordered
F.-H. Chang was supported in part by the National Science Council under grants NSC100-2115-M-003-006.
F.-H. Chang
Department of Mathematics and Science, National Taiwan Normal University, New Taipei, Taiwan H.-B. Chen (
)Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan e-mail:hbchen@math.sinica.edu.tw
F.K. Hwang
p-tuple π= (π1, . . . , πp), where π1, . . . , πp are (possibly empty) pairwise disjoint subsets of A whose union is A. Note that when points are not necessarily distinct, then two partitions are considered the same if they differ only in equivalent elements. For example, if A= {A1, A2, A3, A4} with A1= A2= A3= A4, then π1= (π11, π21, π31)
and π2= (π12, π22, π32)are the same partition for π11= {A1}, π21= {A2, A4}, π31= {A3} and π2 1= {A 2}, π2 2= {A 3, A4}, π2 3 = {A 1}. We refer to π1, π2, . . . , π p as the parts of π , to p as the size of π and to (|π1|, . . . , |πp|), where |πi| is the cardinality of πi, as the shape of π . A typical partition problem is to find an optimal partition over a given family for a given objective function F (π ). The most general family is the constrained-shape family which requires the shape of each member partition to be in a given set Ω of shapes. This family includes four special cases which almost cover all partition families studied in the literature:
(i) A single-shape family. Ω consists of a single shape.
(ii) A bounded-shape family. For a fixed positive integer p, Ω consists of all shapes (|π1|, . . . , |πp|) satisfying
p
j=1|πj| = n and j ≤ |πj| ≤ uj for all
j= 1, . . . , p where {j} and {uj} are given.
(iii) A size family. Given a fixed p, it is indeed a bounded-shape family with j = 0 and uj= n for all 1 ≤ j ≤ p.
(iv) An open family. Ω contains all shapes, i.e., without any constraint on size or shape.
If a partition problem is to maximize an objective over an X-family, then we call the problem an X problem.
In this paper we treat d and p as constants but n can be large, as is usually the case in applications. It is impractical to solve a partition problem by brute force since even for d= 1 and p = 2, the number of single-shape partitions is exponential in n as long as min{|π1|, |π2|} > αn with α > 0 a constant, not to mention for larger d and p and the other three types of problem (see the book by Hwang and Rothblum
2011b). Therefore, it is useful to identify a class of partitions known to contain an optimal partition so that we need only to search this class. A crucial condition is of course this class must be small, i.e., its cardinality is polynomial in n. Such a class is usually characterized by a partition property which every member must observe.
Barnes et al. (1992) first studied the property of “separability” with the above pur-pose in mind (the special cases p= 2 and d = 1 were studied by Harding (1967) and Hwang et al. (1985), respectively, earlier), which has become the most studied prop-erty in the literature of partition problems. A partition π= (π1, . . . , πp)is separable if for any two parts πj and πh of π , their convex hulls conv(πj)and conv(πh)are disjoint, i.e., conv(πj)∩ conv(πh)= ∅. Let TSA(n, p, d)be the number of separable
p-partitions of A∈ Rd. Call A generic if no k+ 1 points of A lie in a common j-flat for any j < k≤ d. Define
H (n, d)= d j=0 n− 1 j . Harding (1967) proved TA
S (n,2, d)= 2H(n, d), thus establishing an upper bound of O(nd)for separable 2-partitions of generic A. Hwang et al. (1999) used this upper
bound to obtain an upper bound O(nd(p2)) for separable p-partitions, still
assum-ing A is generic. Recently, Hwang and Rothblum (2012) extended this upper bound
O(nd(p2)) to arbitrary A in Rdwhere A does not have to be generic and its members
do not have to be distinct.
Note that the Harding’s result not only yields a bound, but it is an equality. Is the equality preserved in the extension to arbitrary A? Unfortunately, it is not. To see this, call a point multi-point if it appears more than once in the multi-set A. Note that when
Ais not distinct, two partitions are considered the same (hence counted only once) if one can be obtained from the other by interchanging the same number (including 0) of copies of each multi-point. Now let A be generic and let Aconsist of n copies of the same point. Then
TSA(n,2, d)= n + 1 = 2H(n, d) = TSA(n,2, d).
To study the equality issue, we have to modify the Harding’s result somewhat. Define
TS(n, p, d)= max
A T
A
S (n, p, d). Hwang and Rothblum (2011a) proved TS(n,2, d)= 2H (n, d).
In real world problems, there is no reason to expect the points satisfy the general position assumption, in particular, two points can be the same due to the discreteness of most measurements. When A contains multi-points, then a separable partition must have all copies of a multi-point go to the same part to preserve “disjointness”. But this is too strong a requirement for a practical partition to satisfy. In particular, this requirement makes it difficult, sometimes impossible, to meet the shape constraint. Therefore, it is desirable to weaken the “strictly disjoint” condition in separable par-titions to “almost disjoint” in the following sense: A partition π= (π1, . . . , πp)is called almost separable if for any two parts πjand πhof π , conv(πj)∩conv(πh)= ∅ or{v} where v in A is a vertex1of both conv(πj)and conv(πh).
We use the same notation of separable partitions to denote numbers of almost separable partitions by substituting the subscript S by AS. Hwang and Rothblum (2012) proved the number of almost separable partitions has the same bound as the number of separable partitions for any of shape, size or open partitions. In particular, they proved TAS(n,2, d)= TS(n,2, d) by an algebraic approach and raised the open problem whether the equality holds for other values of p and d. In this paper we will give a geometric approach to study the equality problem. Our approach works for
d≤ 2 or p ≤ 2, but not for p > 2 and d > 2.
2 Main results
While our apparent goal is to prove the equality for the size family, as confirming with the literature, we will actually prove a stronger result by proving the equality for the single-shape family. Note that the equality for the single-shape family not only implies the same for the size family, but also for the constrained-shape family since
the cardinality of each of these families is simply the sum of cardinalities of all its component shapes.
Let TSA(n, (n1, . . . , np), d)denote the number of separable partitions of A with a given shape (n1, . . . , np), and define
TS n, (n1, . . . , np), d = max A T A S n, (n1, . . . , np), d .
Similarly, we define these terms for almost separable partitions by simply changing the subscript S to AS.
By definition, every separable partition is also an almost separable partition. Thus, we obtain the following results immediately.
Lemma 1 TS(n, (n1, . . . , np), d)≤ TAS(n, (n1, . . . , np), d).
We now prove TS(n, (n1, . . . , np), d)≥ TAS(n, (n1, . . . , np), d) when d≤ 2 or
p≤ 2, and then, together with Lemma1, obtain the equality between the two terms. Let A= {A1, A2, . . . , An} be a multi-set. Suppose v is a point which appears in A mv>1 times. Then mvis the multiplicity of v. For any > 0 and any fixed point
v∈ Rd, denote by B(v, ) a d-dimension ball centered at the point v with radius and denote by v an arbitrary point in B(v, ). Without loss of generality, assume
A1= A2= · · · = Amv = v. Notice again that there is no difference among those
partitions induced by interchanging A1, A2, . . . , Amv.
Lemma 2 Suppose π= (π1, . . . , πp)is an almost separable partition of A∈ Rd×n and v is a multi-point with a unique πj containing all copies of v. Then there exists a sufficiently small real number π>0 such that π= (π1, . . . , πj, . . . , πp), where
πj is obtained from πj by replacing one copy A∗of v in πj with vπ, is an almost
separable partition of A= A ∪ {vπ} \ {A∗}.
Proof To prove the lemma, it suffices to show that there exists a sufficiently small
π>0 such that conv(πj)∩ conv(πh)⊆ conv(πj)∩ conv(πh)for all h= j.
Since π= (π1, . . . , πp)is an almost separable partition of A, for each πh with
h= j we can find a separating hyperplane Hj,h of πh and πj and Hj,h does not contain the point v (this can be done by perturbing Hj,hslightly if Hj,hcontains v). Let π>0 be the minimum distance from v to all these Hj,h’s. Then it is easy to see that, for any fixed point vπ ∈ B(v, π), conv(πj)∩ conv(πh)⊆ conv(πj)∩ conv(πh)
for all h= j.
Lemma 3 Let π= (π1, . . . , πp)be an almost separable partition of A∈ Rd×nand let v belong to t parts, say π1, π2, . . . , πt, with 2≤ t ≤ mv. If d ≤ 2 or t = 2, then there exists a sufficiently small real number π >0 such that A, obtained from A by replacing a copy A∗ of v with vπ, has an almost separable partition π= (π1, . . . , πj, . . . , πp)where πj is obtained from πj, for some 1≤ j ≤ t, by re-placing A∗with vπ.
Proof Since v belongs to at least two parts, we cannot simply apply Lemma2here. However, if we consider only one part πj, j∈ {1, 2, . . . , t}, and its separability with
parts not containing v, then the proof of Lemma2applies to conclude the existence of a small enough π>0 such that conv(πj)∩ conv(πh)⊆ conv(πj)∩ conv(πh)for all h with πhnot containing v.
Next, we consider the separability between πj and other parts containing v. Our argument does not apply to any such πj, but to a specific πj chosen in the following way. It suffices to show that for an arbitrary point vπ ∈ B(v, π)there exists a part πj∈ {π1, π2, . . . , πt} such that conv(πj)∩ conv(πh)⊆ conv(πj)∩ conv(πh)for all 1≤ h ≤ t and h = j. This, together with conv(πj)∩conv(πh)⊆ conv(πj)∩conv(πh) for all h∈ {1, 2, . . . , t}, implies π = (π1, . . . , πj, . . . , πp)is an almost separable par-tition of A.
To this aim, we claim the following and prove later: For d≤ 2 or t = 2, Rd can be split at v into t convex spaces γ1, . . . , γt with overlapping boundaries but disjoint interiors such that every conv(πh)is contained in the corresponding convex space
γh for h= 1, 2, . . . , t. Consequently, the point vπ inR
d must lie in at least one of these t convex spaces, say γj (no matter whether it lies in the boundary or in the interior). Then conv(πj ∪ {vπ}) ∩ conv(πh)\ {v} = ∅, which implies conv(πj)∩
conv(πh)\ {v} = ∅, for all 1 ≤ h ≤ t and h = j. Therefore the resultant partition π= (π1, . . . , πj, . . . , πp)is an almost separable partition of A.
We now prove the above claim to complete the proof. In the case t= 2, one hyper-plane is sufficient to separate conv(π1)and conv(π2); henceRd can be split into two half spaces γ1 and γ2such that conv(πh)is contained in the corresponding convex
space γh for h= 1, 2. In the case d ≤ 2, the space considered here is either a line or a plane. For d= 1, then necessarily t = 2 and we are done. For d = 2, then there exists a clockwise ordering of these t convex hulls (meeting at v) such that we can completely splitRdinto t convex spaces as desired by using separating hyperplanes between each neighbor pair of convex hulls in this ordering. This completes the proof
of this lemma.
The restrictions on d and t are necessary in Lemma3. Figure1demonstrates a counterexample for the case t= d = 3. With Lemma3, we are now ready to prove the main result that the equality holds in Lemma1when d≤ 2 or p ≤ 2.
Denote n∗(A)as the number of distinct points of A= {A1, A2, . . . , An}.
Theorem 1 TS(n, (n1, . . . , np), d)= TAS(n, (n1, . . . , np), d), when d≤ 2 or p ≤ 2. Proof By Lemma1, it suffices to prove that TS(n, (n1, . . . , np), d)≥ TAS(n, (n1, . . . , np), d) when d≤ 2 or p ≤ 2. Let A = {A1, A2, . . . , An} satisfy TASA(n, (n1, . . . , np), d)= TAS(n, (n1, . . . , np), d)and n∗(A)is maximum among all such A. If
n∗(A)= n, then all points of A are distinct, which implies that any almost
separa-ble partition of A is also a separasepara-ble partition of A. Hence, TS(n, (n1, . . . , np), d)≥
TSA(n, (n1, . . . , np), d)≥ TASA(n, (n1, . . . , np), d)= TAS(n, (n1, . . . , np), d), as de-sired.
Suppose that n∗(A) < n, which implies the existence of a multi-point v with mul-tiplicity mv>1. For each almost separable partition π = (π1, . . . , πp)of A with shape (n1, . . . , np), there are only two cases: all copies of v belong to either ex-actly one part πj for some j , or to t parts with t ≥ 2. By Lemmas2and3, for each
Fig. 1 Consider the multi-set A= {(0, 0, 1), (0, 0, 1), (0, 0, 1), (−1, 4, 0), (−2, −3, 0), (−3, −4, 0), (3,−2, 0), (−1, 3, 0), (4, −3, 0)}. Let π1 = {(0, 0, 1), (−1, 4, 0), (−2, −3, 0)}, π2 = {(0, 0, 1),
(−3, −4, 0), (3, −2, 0)}, π3= {(0, 0, 1), (−1, 3, 0), (4, −3, 0)}. This figure shows only the plane Z = 0
for convenience. Obviously, π= (π1, π2, π3)is an almost separable partition. But one cannot find a
partition ofR3into 3 convex subspaces each containing a part exclusively, as described in the proof of Lemma3, since no γj, j= 1, 2, 3, can move into the center area of the cone-like shape (encircled by
π1, π2and π3)
almost separable partition π of A we can find an almost separable partition π of
A= A ∪ {vπ} \ A∗, where A∗ is a copy of v, for sufficiently small π. Moreover,
the shape is preserved. Since there are finitely many almost separable partitions of A, we can choose the smallest π over all almost separable partitions to be . Accord-ingly, for every almost separable partition π of A= {A1, A2, . . . , An} there exists a
corresponding π which is an almost separable partition of A= A ∪ {v} \ A∗and the shape is preserved. Finally, the corresponding partitions of A must be all distinct, thus implies a one-to-one correspondence. To see this, suppose that two distinct almost separable partitions π1= (π11, . . . , πp1)and π2= (π12, . . . , πp2)of A map to two al-most separable partitions π1= (π1
1, . . . , πr1, . . . , πp1)and π2= (π12, . . . , πs2, . . . , πp2) of A and π1= π2. Then we have r = s since π1
r and πs2 are the only two parts containing vwhich is unique. Further, πh1= πh2for all h= r. Obviously, if we iden-tify v with A∗, then πr1= πs2because of πr1= πs2 (notice that v and A∗ are the only difference between π and π ). It follows π1= π2, a contradiction to the
as-sumption that they are distinct. As a result, there is a one-to-one correspondence from almost separable partitions of A to almost separable partitions of A. Then
TASA(n, (n1, . . . , np), d)≥ TASA(n, (n1, . . . , np), d)with n∗(A)= n∗(A)+ 1 > n∗(A), a contradiction to the assumption that n∗(A)is maximum. Thus n∗(A)= n and
The-orem1is proved.
Fig. 2 This figure shows a partition π= (π1, π2) with a separating line H1,2, where π1= {a, b, v}
and π2= {c, d, v} shares a multi-point v with two copies. The boundary of the two corresponding
convex spaces is H1,2. Suppose v1= v lies on the separating line H1,2 and is assigned to the π1
side, as determined in Lemma3. Then π is mapped to π= ({a, b, v1}, {c, d, v}). In this case the
par-tition π= ({a, b, v}, {c, d, v1}), which is almost separable, is left unmapped. Instead, suppose v2= v
lies on the π2 side. Then π is mapped to π= ({a, b, v}, {c, d, v2}) while the unmapped partition
π= ({a, b, v2}, {c, d, v}) violates the almost-separability property
Remarks In the proof of Theorem1, for any multi-point v, there exists a one to one mapping from the set of almost separable partitions on A to the set of almost sep-arable partitions on A which differs from A by replacing a copy A∗ of v with v. An interesting question is whether this mapping is bijective, i.e., whether a mem-ber of A is left unmapped. The example given in Fig.2gives a positive answer and also shows that the solution depends greatly on the position where vis placed. Ac-tually, for a specific partition we can put the point v∈ B(v, ) in the boundary of two (out of t ) convex spaces as described in Lemma3. If v is assigned to the left space, we obtain an almost separable partition; if to the right space, we obtain an-other. Since we can choose only one such space to place v, one of the two almost separable partitions is left unmapped (see Fig.2as an example). As a result, we can conclude that a set A satisfying TASA(n, (n1, . . . , np), d)= TAS(n, (n1, . . . , np), d) contain no multi-points when d= 2 and when p = 2 and d = 1. The condition d = 1 is necessary for otherwise the two 1-dimensional convex spaces must be two intervals (allowing the degenerating interval{v}) meeting exactly at v, such that v must lie in only one such interval. In other words, maximum of TAS(n, (n1, . . . , np),2) is al-ways achieved by a set A consisting of distinct Ai’s. Hence all elements contributing to TAS(n, (n1, . . . , np),2) are separable partitions.
References
Alon N, Onn S (1999) Separable partitions. Discrete Appl Math 91:39–51
Barnes ER, Hoffman AJ, Rothblum UG (1992) Optimal partitions having disjoint convex and conic hulls. Math Program 54:69–86
Harding EF (1967) The number of partitions of a set of n points in k dimensions induced by hyperplanes. Proc Edinb Math Soc 15:285–289
Hwang FK, Rothblum UG (2011a) On the number of separable partitions. J Comb Optim 21:423–433 Hwang FK, Rothblum UG (2011b) Partitions: optimality and clustering. World Scientific, Singapore Hwang FK, Rothblum UG (2012) Bounding the number of almost separable partitions (to appear) Hwang FK, Sun J, Yao EY (1985) Optimal set partitioning. SIAM J Algebr Discrete Methods 6:163–170 Hwang FK, Onn S, Rothblum UG (1999) A polynomial time algorithm for shaped partition problems.
