1 An extremal problem on labelled directed trees

An extremal problem on labelled directed trees and applications to database theory

Gyula O.H. Katona

and Kriszti´an Tichler

1Alfr´ed R´enyi Institute of Mathematics, 1053 Budapest, Hungary, e-mail:[email protected]

2Technische Universit¨at Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany, e-mail:[email protected]

We consider an extremal problem on labelled directed trees and applications to database theory. Among others, we will show explicit keysystems on an underlying set of sizen, that cannot be represented by a database of less than 2n(1−c·log logn/logn)

rows.

Keywords:labelled directed tree, relational database, minimum matrix representation, extremal problems

1 An extremal problem on labelled directed trees

A treeF is called adirected tree,if there is a direction on the edges, so that a vertexv₀(root)has only out-neighbours, and an arbitrary vertexv6=v₀has a uniquely determined in-neighbourn(v).N(v)denotes the out-neighbourhood ofv. The set of the leaves of a treeF is denoted by`(F). LetU be a (finite) set.

A treeF=F(U)is calledlabelled, if a subsetA(v)ofUis associated with each vertexvofF.

For fixed integersk≥ 1, ` ≥2andU ={1,2, ..., m}consider the family of labelled directed trees F<sub>k,`</sub>^(m), for which the vertices of each treeF ∈ F<sub>k,`</sub><sup>(m)</sup>are labelled as follows. The label of the rootv0of F isA(v0) =U. For an arbitrary vertexv ofF there is a disjoint partitionN(v) =S`

i=1Ni(v)of its out-neighbourhood satisfying the following properties.

A(v)⊆A(n(v)) (v6=v0), (1)

|A(v)| ≥k+ 1, (2)

w1, w2∈Ni(v)⇒A(w1)∩A(w2) =∅ (1≤i≤`), (3) w<sub>1</sub>∈N<sub>i</sub>(v), w<sub>2</sub>∈N<sub>j</sub>(v)⇒ |A(w<sub>1</sub>)∩A(w<sub>2</sub>)| ≤k (1≤i < j ≤`). (4) Introduce the notationT<sub>k,`</sub>(m) = max{|`(F)|

F ∈ F_k,l^(m)}. Ifk= 1, we simply writeF_{`</sub><sup>(m)</sup>forF<sub>k,`}^(m) andT_{`</sub>(m)forT<sub>k,`}(m).

Throughout the rest of the abstract we write simplylogforlog₂. We have forF_`the following:

†The work was supported by the Hungarian National Foundation for Scientific Research, grant numbers T037846 and T034702.

‡The work was supported by the Young Researcher Fellowship of the Alfr´ed R´enyi Institute and the COMBSTRU program (Marie Curie Fellowship of the Europian Union) at Universit¨at Bielefeld and Technische Universit¨at Berlin.

1365–8050 c2005 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Theorem 1.1

T2(m)≤ 1

2mlogm. (5)

and equality holds if and only ifmis a power of 2.

Theorem 1.2

T`(m) = Θ`(mlog^αm) (6)

for`≥3. Whereα=α(`) = log`.

In [7], the magnitude ofT_k,`was determined.

Proof of Theorem 1.1:We will use the concept ofentropy [2] in the proof. Entropy is a measure of a random variableX:

H(X) =−X

i

p_ilogp_i, (7)

where Prob(X =i) =p_i. It is known, that

H((X, Y))≤H(X) +H(Y). (8)

The proof is by induction onm. (5) holds form= 2. Suppose that the statement holds for every integer smaller thanm.

LetF ∈ F₂^(m)be a tree with|`(F)| = T(m). Furthermore letN(v₀) = {v₁, . . . , v_s, w₁, . . . , w_t}, N₁(v₀) ={v₁, . . . , v_s},N₂(v₀) ={w₁, . . . , w_t}. Let us use the short notationsA_i=A(v_i),a_i =|A_i|, (1 ≤ i≤ s),B_i = A(w_i),b_i =|B_i|, (1 ≤i ≤ t). The subtree ofF of rootv_i(w_j) is denoted byF_i (Fs+j),1≤i≤s(1≤j≤t).

By the induction hypothesis T(ai)≤ 1

2ailogai, (1≤i≤s), and T(bi)≤1

2bilogbi, (1≤i≤t) holds. So it is enough to prove that

s

X

i=1

ailogai+

t

X

i=1

bilogbi≤mlogm, (9)

since then

T(m) =|`(F)|=

s+t

X

i=1

|`(Fi)| ≤

s

X

i=1

T(ai) +

t

X

i=1

T(bi)

≤

s

X

i=1

1

2a_iloga_i+

t

X

i=1

1

2b_ilogb_i≤ 1

2mlogm.

Lets₁=m−Ps

i=1a_iands⁰=s+s₁. Adds₁disjoint setsA_s+1, . . . , A_s⁰of cardinality 1, such that {1,2, . . . , m}=Ss⁰

i=1AiWe definet1,t⁰ and the setsBt+1, . . . , Bt⁰ analogously. The setsAi(1≤i≤ s⁰)andB_j(1≤j≤t⁰)have the following properties:

{Ai,1≤i≤s⁰} is a partition of{1,2, . . . , m}, (10) {Bj,1≤j≤t⁰} is a partition of{1,2, . . . , m}, (11)

|A_i∩B_j| ≤1,1≤i≤s⁰,1≤j≤t⁰. (12) LetΩX = {1,2, . . . , m} be the event space of the random variableX. Furthermore, let X(ω) = ω, ω ∈ ΩX and Prob(X = ω) = 1/m. Let us define another two random variables, Y(X ∈ Ai) = i, 1≤i≤s⁰andZ(X ∈Bj) =j, 1≤j ≤t⁰. Then

Prob(Y =i) = ai

m (1≤i≤s⁰) and Prob(Z=j) = bj

m (1≤j≤t⁰).

The random variablesY andZare well defined by (10) and (11). Furthermore, by (12) we get Prob((Y, Z) = (i, j)) =

Prob(X =k) = 1/m if Ai∩Bj ={k}, 0 if Ai∩Bj =∅.

So we have for the entropies ofY, Zand(Y, Z):

H(Y) =

s⁰

X

i=1

ai

mlogm ai

, H(Z) =

t⁰

X

j=1

bj

mlogm bj

,

H((Y, Z)) =−

s⁰

X

i=1 t⁰

X

j=1

Prob((Y, Z)=(i, j)) logProb((Y, Z)=(i, j)) =

m

X

i=1

1

mlogm= logm.

Therefore, by (8) we get

logm≤

s⁰

X

i=1

a_i mlogm

ai

+

t⁰

X

j=1

b_j mlogm

bj

,

which is equivalent to (9). 2

To prove Theorem 1.2 we need somewhat more counting. We could not find a straightforward way to generalize the concept of entropy for this case, altough the idea of the proof is based on the previous proof. Note, that the constant in Theorem 1.2 for the upper bound can be roughly upperestimated by

2^2·1012`6. (13)

2 An application to database theory

A subsetK of the setΩ(|Ω| = n) of columns of a matrix, is called akey, if there are no two rows of the matrix agree in each of the columns ofK. IfKis a key, then clearly all of its supersets are keys as well. On the other hand, for every nonemptyK ⊆2^Ωhaving this property there always exists a matrix, in which the system of keys is exactlyK[1, 3]. Since the system of keys determines the system of minimal keys and vica versa, it is enough to consider Sperner families. LetKbe a Sperner system, then lets(K) denote the minimum number of rows of such a matrix.

In [6], it was shown that there exist badly representable Sperner systems, namely of size s(K)> 1

n² n

_n

2

. (14)

The proof of this theorem is not constructive. L. R´onyai’s observation is that the number of Sperner families that can be represented by a matrix of at mostrrows is quite limited, and sorshould be at least as big as in (14) to get a representation even for all antichains at the middle level of the Boolean lattice. In the following, we show an explicit badly representable Sperner system (quite far from the middle level), no worse is known up to now.

IfKis a Sperner system, letK⁻¹ denote the set of maximal elements, that are not contained in any superset ofK. LetKn^(k)denote the completek-uniform hypergraph.

Theorem 2.1 [7] Letn=n1+n2+. . .+nt,ni≤s(1≤i≤t). LetKn=Kn^(k)₁ +Kn^(k)₂ +. . .+Kn^(k)_t . Then

|K⁻¹_n | ≤T<sub>`</sub>(s(Kn)) (15) holds for`= _k−1^s

.

Proof Sketch: [7] Suppose, thatK_n is represented by a matrix ofs(K_n)rows. We can construct recur- sively a labelled directed tree,F ∈ F_`^(s(Kn))having the property, that there is an injection fromK_n⁻¹to the leaves ofF. Easy to see [4], that a representation of a Sperner familyK should contain two rows, for eachA∈ K⁻¹, that are equal inA, but should not contain two rows that are equal in an element of K. The construction is based on these properties and labelling is due to the equalities of the entries of the

matrix in a certain (set of) column(s). 2

Corollary 2.2 There exists a sequence of Sperner systemsKn, such that

s(Kn)>2n(1−(26/3) log logn/logn) (16)

holds fornlarge enough.

Proof Sketch: The elements ofK⁻¹_n are exactly those, that contain preciselyk−1elements from each clique,Kn^(k)i . We apply Theorem 2.1 and Theorem 1.2 with the estimation (13), letni=s−1orni=s, 1≤i≤ dn/se. We choosek=g(n) + 1ands= 2g(n) + 1, so

2g(n) g(n)

n/(2g(n)+1)

≤

dn/se

Y

i=1

ni

k−1

=|K⁻¹_n |. (17)

Some calculation support to chooseg(n) =dlogn/13e −4. Substituting into (17) we get (16). 2 The above theorem on labelled directed trees has been proved surprising widely applicable for proving, that certain Sperner families are badly representable. Recently, another interesting application has been found by the authors in the theory of combinatorial search.

References

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