codingtheory.cswp.cs.technion.ac.il...2016/11/01 · onn(m;c;k) explicitconstruction...

Combinatorial Batch Codes: Bounds andConstructions

Srimanta Bhattacharya

Indian Statistical Institute

Coding Theory SeminarTechnion.

November, 2016.

(Based on joint works with Bimal Roy, Sushmita Ruj, and NiranjanBalachandran)

Outline

1 Overview

2 Preliminaries: Batch Codes and CBCs

3 On N(n, k,m)

4 On n(m, c, k)

Srimanta Bhattacharya (ISI) Combinatorial Batch Codes: Bounds and Constructions

Preliminaries: Batch Codes and CBCs Batch Codes Problem and De�nition

The problem: (N,n, k,m, t = 1)-Batch code

To store n data items,into m servers,so as to retrieve k of the n items by querying each server ≤ t(= 1)times,and total storage accross m servers limited to ≤ N.

Abstraction:

De�nition (Ishai et al., 2004)An (n,N, k,m, t) batch code over an alphabet Σ is de�ned by an encodingfunction C ∶ Σn → (Σ∗)m (each output of which is called a bucket) and adecoding algorithm A such that

1 The total length of all m buckets is N.

2 For any x ∈ Σn and {i1, . . . , ik} ⫅ [n],A(C(x), i1, . . . , ik) = (xi1 , . . . ,xik), andA probes at most t symbols from each bucket in C(x) (whose positionsare determined by i1, . . . , ik ).



The problem: (N,n, k,m, t = 1)-Batch codeTo store n data items,

into m servers,so as to retrieve k of the n items by querying each server ≤ t(= 1)times,and total storage accross m servers limited to ≤ N.

Abstraction:






The problem: (N,n, k,m, t = 1)-Batch codeTo store n data items,into m servers,

so as to retrieve k of the n items by querying each server ≤ t(= 1)times,and total storage accross m servers limited to ≤ N.

Abstraction:






The problem: (N,n, k,m, t = 1)-Batch codeTo store n data items,into m servers,so as to retrieve k of the n items by querying each server ≤ t(= 1)times,

and total storage accross m servers limited to ≤ N.

Abstraction:






The problem: (N,n, k,m, t = 1)-Batch codeTo store n data items,into m servers,so as to retrieve k of the n items by querying each server ≤ t(= 1)times,and total storage accross m servers limited to ≤ N.

Abstraction:






The problem: (N,n, k,m, t = 1)-Batch codeTo store n data items,into m servers,so as to retrieve k of the n items by querying each server ≤ t(= 1)times,and total storage accross m servers limited to ≤ N.

Abstraction:



2 For any x ∈ Σn and {i1, . . . , ik} ⫅ [n],A(C(x), i1, . . . , ik) = (xi1 , . . . ,xik), andA probes at most t symbols from each bucket in C(x) (whose positionsare determined by i1, . . . , ik ).Srimanta Bhattacharya (ISI) Combinatorial Batch Codes: Bounds and Constructions

Preliminaries: Batch Codes and CBCs Batch Codes Motivation

Motivation:

Practical:

1 Amortize computational overhead in private information retrievalprotocol.

2 Server load balancing.

Theoretical:

1

Batch Code

UnbalancedExpanders

LocallyDecodableCodes

InformationDispersal

2 It is interesting!



Motivation:Practical:


2 Server load balancing.Theoretical:

1

Batch Code

UnbalancedExpanders








2 Server load balancing.

Theoretical:

1

Batch Code

UnbalancedExpanders








2 Server load balancing.Theoretical:

1

Batch Code

UnbalancedExpanders





Preliminaries: Batch Codes and CBCs Batch Codes Example

Example:(14,21,2,3)-batch code over (0, 1).


Preliminaries: Batch Codes and CBCs Combinatorial Batch Codes (CBCs) Notion and Example

Combinatorial Batch Codes (CBCs):

Items are replicated among servers.

Encoding: Store items into servers.Decoding: Retrieve items from servers.

Example:

Advantage: Encoding and decoding are simple.Disadvantage: May require more space.

(14,21,2,3)-CBC notpossible!!




Items are replicated among servers.Encoding: Store items into servers.

Decoding: Retrieve items from servers.Example:






Items are replicated among servers.Encoding: Store items into servers.Decoding: Retrieve items from servers.

Example:







Example:

Advantage: Encoding and decoding are simple.

Disadvantage: May require more space.






Example:







Example:

Advantage: Encoding and decoding are simple.Disadvantage: May require more space.(14,21,2,3)-CBC notpossible!!Srimanta Bhattacharya (ISI) Combinatorial Batch Codes: Bounds and Constructions

Preliminaries: Batch Codes and CBCs Combinatorial Batch Codes (CBCs) Setting and Formulation

u1

u2

u3

u4

un

v1

v2

v3

vmSet of data items:L, ∣L∣ = n

Set of servers:R, ∣R∣ = m

X , ∣X ∣ ≤ k

Γ(X )

Element u1 is stored in server v2

⇕

(i) ∣L∣ = n(ii) ∣R∣ = m(iii) ∑

ideg(ui) = ∑

ideg(vi) = ∣E ∣ = N

(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣

CBC is c-uniform ifdeg(ui) = c, for all ui ∈ LCBC is `-regular if deg(vj) = `,for all vj ∈R

SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣



u1

u2

u3

u4

un

v1

v2

v3



X , ∣X ∣ ≤ k

Γ(X )


⇕

(i) ∣L∣ = n

(ii) ∣R∣ = m(iii) ∑

ideg(ui) = ∑


(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣


SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣



u1

u2

u3

u4

un

v1

v2

v3



X , ∣X ∣ ≤ k

Γ(X )


⇕

(i) ∣L∣ = n(ii) ∣R∣ = m

(iii) ∑ideg(ui) = ∑


(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣


SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣



u1

u2

u3

u4

un

v1

v2

v3



X , ∣X ∣ ≤ k

Γ(X )


⇕

(i) ∣L∣ = n(ii) ∣R∣ = m(iii) ∑

ideg(ui) = ∑


(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣


SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣



u1

u2

u3

u4

un

v1

v2

v3



X , ∣X ∣ ≤ k

Γ(X )


⇕

(i) ∣L∣ = n(ii) ∣R∣ = m(iii) ∑

ideg(ui) = ∑


(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣

CBC is c-uniform ifdeg(ui) = c, for all ui ∈ L

CBC is `-regular if deg(vj) = `,for all vj ∈R

SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣



u1

u2

u3

u4

un

v1

v2

v3



X , ∣X ∣ ≤ k

Γ(X )


⇕

(i) ∣L∣ = n(ii) ∣R∣ = m(iii) ∑

ideg(ui) = ∑


(iv) ∀X ⊂ L, ∣X ∣ ≤ k, ∣Γ(X )∣ ≥ ∣X ∣


SDR⇔ ∀X ⊂ L, ∣X ∣ ≤k, ∣Γ(X )∣ ≥ ∣X ∣


Preliminaries: Batch Codes and CBCs Combinatorial Batch Codes (CBCs) Two Extremal Problems

Two Extremal Problems:1 Find N(n,k,m)

Givennumber of servers (m),number of input data items (n),retrievability parameter (k),

�nd minimum value of total storage N, denoted by N(n, k,m).

Example:

N(7,4,6) = 10




Given

number of servers (m),number of input data items (n),retrievability parameter (k),


Example:

N(7,4,6) = 10




Givennumber of servers (m),

number of input data items (n),retrievability parameter (k),


Example:

N(7,4,6) = 10




Givennumber of servers (m),number of input data items (n),

retrievability parameter (k),�nd minimum value of total storage N, denoted by N(n, k,m).

Example:

N(7,4,6) = 10






Example:

N(7,4,6) = 10





�nd minimum value of total storage N, denoted by N(n, k,m).Example:

N(7,4,6) = 10



2 Find n(m, c,k)

Givennumber of servers (m),degree of uniformity (c),retrievability parameter (k),

�nd maximum number of input data items n, denoted byn(m, c, k).

Example:

n(6,2,5) = 9



2 Find n(m, c,k)Given

number of servers (m),degree of uniformity (c),retrievability parameter (k),


Example:

n(6,2,5) = 9




number of servers (m),

degree of uniformity (c),retrievability parameter (k),


Example:

n(6,2,5) = 9




number of servers (m),degree of uniformity (c),

retrievability parameter (k),�nd maximum number of input data items n, denoted byn(m, c, k).

Example:

n(6,2,5) = 9






Example:

n(6,2,5) = 9


On N(n, k,m)

Trivial case:

For n ≤ m, N(n, k,m) = n.Optimal CBC: n items are stored in any n out of m servers.

For n ≥ m + 1, N(n, k,m) =?.


On N(n, k,m)

Trivial case:For n ≤ m, N(n, k,m) = n.

Optimal CBC: n items are stored in any n out of m servers.

For n ≥ m + 1, N(n, k,m) =?.


On N(n, k,m)

Trivial case:For n ≤ m, N(n, k,m) = n.Optimal CBC: n items are stored in any n out of m servers.

For n ≥ m + 1, N(n, k,m) =?.


On N(n, k,m) Existing Results

1 Paterson et al., 2009 N(m + 1, k,m) = m + k

Construction

Single copy of each of any m items in m distinct serversk copies of the remaining item in any k servers.

2 Paterson et al., 2009 N(n, k, k) = kn − k(k − 1)3 Brualdi et al., 2010, Bujtás and Tuza, 2011 Let k and m be integers

with 2 ≤ k ≤ m. ThenN(m+2, k,m) = { m + k − 2 + ⌈2

√k + 1⌉ if m + 1 − k ≥ ⌈

√k + 1⌉,

2m − 2 + ⌈1 + k+1m+1−k⌉ if m + 1 − k < ⌈√k + 1⌉.

4 Paterson et al., 2009 For n ≥ (k − 1)( mk−1) ,N(n, k,m) = kn − (k − 1)( mk−1).



1 Paterson et al., 2009 N(m + 1, k,m) = m + kConstruction

Single copy of each of any m items in m distinct servers

k copies of the remaining item in any k servers.2 Paterson et al., 2009 N(n, k, k) = kn − k(k − 1)3 Brualdi et al., 2010, Bujtás and Tuza, 2011 Let k and m be integers


√k + 1⌉ if m + 1 − k ≥ ⌈

√k + 1⌉,

2m − 2 + ⌈1 + k+1m+1−k⌉ if m + 1 − k < ⌈√k + 1⌉.








√k + 1⌉ if m + 1 − k ≥ ⌈

√k + 1⌉,

2m − 2 + ⌈1 + k+1m+1−k⌉ if m + 1 − k < ⌈√k + 1⌉.






2 Paterson et al., 2009 N(n, k, k) = kn − k(k − 1)

3 Brualdi et al., 2010, Bujtás and Tuza, 2011 Let k and m be integerswith 2 ≤ k ≤ m. ThenN(m+2, k,m) = { m + k − 2 + ⌈2

√k + 1⌉ if m + 1 − k ≥ ⌈

√k + 1⌉,

2m − 2 + ⌈1 + k+1m+1−k⌉ if m + 1 − k < ⌈√k + 1⌉.








√k + 1⌉ if m + 1 − k ≥ ⌈

√k + 1⌉,

2m − 2 + ⌈1 + k+1m+1−k⌉ if m + 1 − k < ⌈√k + 1⌉.



On N(n, k,m) Our Contribution

Bhattacharya et al., 20121 Let 1 ≤ n ≤ (k − 1)( mk−1), and 1 ≤ c ≤ k − 1. Then for an

(n,N, k,m)-CBC we have N ≥ nc −

⎢⎢⎢⎢⎢⎢⎢⎣

(k−c)((k−1)(mc )

(k−1c )−n)

m−k+1

⎥⎥⎥⎥⎥⎥⎥⎦

. The r.h.s.

expression attains its maximum for least c such that n ≤ (k−1)(mc)

(k−1c ).

2 Let ( mk−2) ≤ n ≤ (k − 1)(mk−1). Then we have

N(n, k,m) = n(k − 1) − ⌊ (k−1)(mk−1)−n

m−k+1 ⌋.



3 Let ( mk−2) − (m − k + 1)A(m,4, k − 3) ≤ n ≤ (mk−2). Then

N(n, k,m) = n(k − 2) −⎢⎢⎢⎢⎣

2(( mk−2) − n)m − k + 1

⎥⎥⎥⎥⎦for 0 ≤ (( m

k − 2) − n) mod (m − k + 1) <m − k + 1

2, and

N(n, k,m) ≤ n(k − 2) − 2⎢⎢⎢⎢⎣

( mk−2) − nm − k + 1

⎥⎥⎥⎥⎦for m − k + 1

2≤ (( m

k − 2) − n) mod (m − k + 1) < m − k + 1.

A(m,4, k − 3)⇒ no. of codewords in a binary constant weight codeof length m, distance 4, and weight k − 3

Di�ers fromthe optimal byat most 1.



Optimal construction: (Um,k,c =(k−1)(mc)

(k−1c ))

k

Um,k,1 = m Um,k,k−3

Um,k,k−2

Um,k,k−1

n

N = n

m + 1

N = m + k Paterson et al., 2009

m + 2

N = m + k − 2 + ⌈2√

k + 1⌉ Brualdi et al., 2010, Bujtás and Tuza, 2011

N = kn − (k − 1)( mk−1)

Paterson et al., 2009

N = nc − ⌊ (k−c)dm−k+1 ⌋

Bhattacharya et al., 2012


On N(n, k,m) Setting

Set system - (S,X )

Ground Set (S) - set of servers.Collection of subsets (X )- corresponds to items

Example:Servers A,B,C;⇒ S = {A,B,C}.A contains items a, b, c. B contains items a, b. C contains item b.X = {{A,B},{A,B,C},{A}}.

De�nition (Paterson et al., 2009)An (n,N, k,m) batch code is a set system (S,X ), such that

∣S ∣ = m.∣X ∣ = n.Any subcollection of i subsets contains i elements for 1 ≤ i ≤ k.

a b c



Set system - (S,X )Ground Set (S) - set of servers.

Collection of subsets (X )- corresponds to itemsExample:

Servers A,B,C;⇒ S = {A,B,C}.A contains items a, b, c. B contains items a, b. C contains item b.X = {{A,B},{A,B,C},{A}}.



a b c



Set system - (S,X )Ground Set (S) - set of servers.Collection of subsets (X )- corresponds to items




a b c




Example:

Servers A,B,C;⇒ S = {A,B,C}.A contains items a, b, c. B contains items a, b. C contains item b.X = {{A,B},{A,B,C},{A}}.



a b c




Example:Servers A,B,C;⇒ S = {A,B,C}.

A contains items a, b, c. B contains items a, b. C contains item b.X = {{A,B},{A,B,C},{A}}.



a b c




Example:Servers A,B,C;⇒ S = {A,B,C}.A contains items a, b, c. B contains items a, b. C contains item b.

X = {{A,B},{A,B,C},{A}}.



a b c







a b c


On N(n, k,m) Lower Bound

TheoremLet (S,X ) be an (n,N, k,m)-CBC, and Ai be the number of i element sets ofX . Then for any j, with 1 ≤ j ≤ k − 1, we have ∑ji=1 (

m−ij−i )Ai ≤ j(

mj ).

Observation

There are k − 1 inequalities for 1 ≤ j ≤ k − 1.(j + 1)-th inequality implies j-th inequlity. Su�cient to consider(k − 1)-th inequality.

Pick uniformly at random a subset S ′ ⊆ S of cardinality j ≤ k − 1.

For X ∈ X , with ∣X∣ = i, Pr{X ⊆ S ′} =(m−ij−i )(mj )

.

Let XS′ = {X ∈ X ∣X ⊆ S ′}. Then

j ≥

E[∣XS′ ∣] =j∑i=1

Ai(m−ij−i )(mj )

,

Hall’s condition


On N(n, k,m) Lower Bound

TheoremLet (S,X ) be an (n,N, k,m)-CBC, and Ai be the number of i element sets ofX . Then for any j, with 1 ≤ j ≤ k − 1, we have ∑ji=1 (

m−ij−i )Ai ≤ j(

mj ).

Observation

There are k − 1 inequalities for 1 ≤ j ≤ k − 1.(j + 1)-th inequality implies j-th inequlity. Su�cient to consider(k − 1)-th inequality.

Pick uniformly at random a subset S ′ ⊆ S of cardinality j ≤ k − 1.

For X ∈ X , with ∣X∣ = i, Pr{X ⊆ S ′} =(m−ij−i )(mj )

.

Let XS′ = {X ∈ X ∣X ⊆ S ′}. Then j ≥ E[∣XS′ ∣] =j∑i=1

Ai(m−ij−i )(mj )

,

Hall’s condition


On N(n, k,m) Construction for the range ( mk−2) ≤ n ≤ (k − 1)(mk−1)

Initial con�guration:

Con�guration corresponding to n = n(m, k − 1, k) = Um,k,k−1.k − 1 copies of each member of ( Sk−1).d = (k − 1)( mk−1) − n, i.e. d = Um,k,k−1 − n (0 ≤ d ≤ (m − k + 1)(

mk−2)).



Construction:

1 Select ⌊ dm−k+1⌋ di�erent members of (Sk−2).

Possible, since 0 ≤ d ≤ (m − k + 1)( mk−2) and there are (mk−2) members

of ( Sk−2).2 For each selected (k − 2) element subset-

1 Delete one copy of each of its m − k + 2 supersets((k − 1) elementsubsets) from the initial collection.

2 Add one copy of the (k − 2) element subset to the collection.

3 For the remaining r = d − ⌊ dm−k+1⌋(m − k + 1) elements, wherer < m − k + 1, select another (k − 2) element subset and delete its rsupersets present in the initial collection.



Construction:



of ( Sk−2).

2 For each selected (k − 2) element subset-






Construction:





2 Add one copy of the (k − 2) element subset to the collection.3 For the remaining r = d − ⌊ dm−k+1⌋(m − k + 1) elements, where

r < m − k + 1, select another (k − 2) element subset and delete its rsupersets present in the initial collection.



Construction:









Construction:





2 Add one copy of the (k − 2) element subset to the collection.3 For the remaining r = d − ⌊ dm−k+1⌋(m − k + 1) elements, where

r < m − k + 1, select another (k − 2) element subset and delete its rsupersets present in the initial collection.



Example:

Let m = 6, k = 4,n = 43.Let S = {1,2,3,4,5,6}.Um,k,k−1 = 60 and d = Um,k,k−1 − n = 17 and ⌊ dm−k+1⌋ = 5.Initial collection - k − 1 = 3 copies of each member of (S3).



Example (contd..):


On N(n, k,m) Construction for the range n ≤ ( mk−2)

Construction is similar to the earlier one.

Intial con�guration is ( Sk−2).Carefully choose the set of selected subsets X ′ ⊂ ( Sk−3).

To ensure that any S ′ ⊂ S, with ∣S ′∣ ≤ k − 1, does not contain toomany members of X ′.Maintain good distance between the members of X ′.Choose from the codewords of constant weight code of length m,weight k − 3, and distance 4.



Construction is similar to the earlier one.Intial con�guration is ( Sk−2).

Carefully choose the set of selected subsets X ′ ⊂ ( Sk−3).




Construction is similar to the earlier one.Intial con�guration is ( Sk−2).Carefully choose the set of selected subsets X ′ ⊂ ( Sk−3).





To ensure that any S ′ ⊂ S, with ∣S ′∣ ≤ k − 1, does not contain toomany members of X ′.

Maintain good distance between the members of X ′.Choose from the codewords of constant weight code of length m,weight k − 3, and distance 4.




To ensure that any S ′ ⊂ S, with ∣S ′∣ ≤ k − 1, does not contain toomany members of X ′.Maintain good distance between the members of X ′.

Choose from the codewords of constant weight code of length m,weight k − 3, and distance 4.


On N(n, k,m) Subsequent work

Silberstein and Gál, 2014. Optimal (n,N, k,m)-CBCs, withn = q2 + q − 1, k = q2 − q − 1,m = q2 − q,N = q3 − q, where q ≥ 3 is a primepower.

Sets of X have cardinality ∈ {q, q − 1}.Construction shows that lower bound is tight when sets havecardinality ∼

√k.

Possible directions:Improvement of the lower bound in the lower range (n ∼ m).

N(m + 3, k,m) =?Good construction in the higher range.





√k.


N(m + 3, k,m) =?

Good construction in the higher range.





√k.


N(m + 3, k,m) =?Good construction in the higher range.


On n(m, c, k) The Problem

Given


�nd maximum number of input data items n, denoted by n(m, c, k).

Trivial case:

n(m, 1, k) = m(1 < k ≤ m).Extremal CBC: each item is stored in a separate server.

For c ≥ 2, n(m, c, k)=?



Givennumber of servers (m),

degree of uniformity (c),retrievability parameter (k),


Trivial case:


For c ≥ 2, n(m, c, k)=?



Givennumber of servers (m),degree of uniformity (c),

retrievability parameter (k),


Trivial case:


For c ≥ 2, n(m, c, k)=?





Trivial case:


For c ≥ 2, n(m, c, k)=?





Trivial case:n(m, 1, k) = m(1 < k ≤ m).

Extremal CBC: each item is stored in a separate server.

For c ≥ 2, n(m, c, k)=?





Trivial case:n(m, 1, k) = m(1 < k ≤ m).Extremal CBC: each item is stored in a separate server.

For c ≥ 2, n(m, c, k)=?


On n(m, c, k) Our contribution

Our Contribution1 An extremal hypergraph problem (Turán type problem): bounds

on n(m, c, k) and extremal CBCs for speci�c ranges of values of c.

2 “Explicit construction” of uniform and “almost regular” CBCswith high values of n for a wide range of values of c.


On n(m, c, k) Our contribution

Our Contribution1 An extremal hypergraph problem (Turán type problem): bounds

on n(m, c, k) and extremal CBCs for speci�c ranges of values of c.2 “Explicit construction” of uniform and “almost regular” CBCs

with high values of n for a wide range of values of c.


On n(m, c, k) An Extremal Hypergraph Problem Hypergraph Preliminaries

A hypergraph is F is a tuple F ∶= (V,E).V is a set of vertices.E is a family of subsets of V ,called edges.∣E ∣ is called size of the hypergraph.A hypergraph is called simple if it does not contain repeated edges,i.e., there are no multiple copies of any edge.A hypergraph is called c-uniform if each of its edges has cardinalityc. A graph is a 2-uniform hypergraph.


On n(m, c, k) An Extremal Hypergraph Problem Turán Numbers and Turán Density

LetH be a family of c-uniform hypergraphs.

Maximum size of a c-uniform hypergraph on m vertices, that doesnot contain a copy of any of the hypergraphs ofH as asub-hypergraph, is called Turán number of the familyH, and isdenoted by exc(m,H).

Turán determined ex(m,Kt) (hence the name).Notoriously hard in general.There are extremely few exact results.

For graphs, Erodős-Stone-Simonovits theorem:

ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),

Bipartite graphs - ex(m,H) = o(m2) = Θ(m?)

chromaticnumber



LetH be a family of c-uniform hypergraphs.Maximum size of a c-uniform hypergraph on m vertices, that doesnot contain a copy of any of the hypergraphs ofH as asub-hypergraph, is called Turán number of the familyH, and isdenoted by exc(m,H).

Turán determined ex(m,Kt) (hence the name).Notoriously hard in general.There are extremely few exact results.


ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),

Bipartite graphs - ex(m,H) = o(m2) =

Θ(m?)

chromaticnumber




Turán determined ex(m,Kt) (hence the name).

Notoriously hard in general.There are extremely few exact results.For graphs, Erodős-Stone-Simonovits theorem:

ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),


Θ(m?)

chromaticnumber




Turán determined ex(m,Kt) (hence the name).ex(m,K3) = ⌊m

2

4 ⌋,K⌊ m2 ⌋,⌈ m2 ⌉ (Mantel)

Notoriously hard in general.There are extremely few exact results.For graphs, Erodős-Stone-Simonovits theorem:

ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),


Θ(m?)

chromaticnumber





2

4 ⌋,K⌊ m2 ⌋,⌈ m2 ⌉ (Mantel)Notoriously hard in general.There are extremely few exact results.


ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),


Θ(m?)

chromaticnumber





2

4 ⌋,K⌊ m2 ⌋,⌈ m2 ⌉ (Mantel)Notoriously hard in general.There are extremely few exact results.


ex(m,H) = (minG∈H

(1 − 1χ(G) − 1) + o(1))(

m2),

Bipartite graphs - ex(m,H) = o(m2) = Θ(m?) chromaticnumber



ex(m,F) for c-uniform hypergraphs, with c ≥ 3?

Famous Turán’s 3 − 4 problem, ex(m,K34)= ?

Erdős o�ered $500 for a solution ($1000 for solution of ex(m,Kct).)

It is natural to focus on the asymptotics of ex(m,H).

1 Order of magnitude of ex(m,H),

i.e., the value α such thatex(m,H) = Θ(mα).

2 Leading coe�cient of ex(m,H). very di�cult even for c-uniformfamiliesH for which ex(m,H) = Θ(mc).

Turán density:

π(H) ≜ limm→∞

ex(m,H)(mc)

.




Famous Turán’s 3 − 4 problem, ex(m,K34)= ?

Erdős o�ered $500 for a solution ($1000 for solution of ex(m,Kct).)It is natural to focus on the asymptotics of ex(m,H).




Turán density:


ex(m,H)(mc)

.




Famous Turán’s 3 − 4 problem, ex(m,K34)= ?Erdős o�ered $500 for a solution ($1000 for solution of ex(m,Kct).)

It is natural to focus on the asymptotics of ex(m,H).




Turán density:


ex(m,H)(mc)

.





It is natural to focus on the asymptotics of ex(m,H).1 Order of magnitude of ex(m,H),



Turán density:


ex(m,H)(mc)

.





It is natural to focus on the asymptotics of ex(m,H).1 Order of magnitude of ex(m,H), i.e., the value α such that

ex(m,H) = Θ(mα).

2 Leading coe�cient of ex(m,H).

very di�cult even for c-uniformfamiliesH for which ex(m,H) = Θ(mc).

Turán density:


ex(m,H)(mc)

.






ex(m,H) = Θ(mα).2 Leading coe�cient of ex(m,H).

very di�cult even for c-uniformfamiliesH for which ex(m,H) = Θ(mc).

Turán density:


ex(m,H)(mc)

.






ex(m,H) = Θ(mα).2 Leading coe�cient of ex(m,H). very di�cult even for c-uniform

familiesH for which ex(m,H) = Θ(mc).

Turán density:


ex(m,H)(mc)

.






ex(m,H) = Θ(mα).2 Leading coe�cient of ex(m,H). very di�cult even for c-uniform

familiesH for which ex(m,H) = Θ(mc).Turán density:


ex(m,H)(mc)

.


On n(m, c, k) An Extremal Hypergraph Problem Setting

We represent a c-uniform (n, cn, k,m)-CBC by a c-uniform hypergraph(V,F),

the set of vertices V , with ∣V ∣ = m, represents the set of m serversthe set of edges F , with ∣F ∣ = n, represents the set of n data items.Edge Fi ∈ F contains vertex vj ∈ V if and only if i-th data item isstored in j-th server.

Theorem (Paterson et al., 2009)A c-uniform hypergraph (V,F) represents a c-uniform (n, cn, k,m)-CBC ifand only if ∣V ∣ = m, ∣F ∣ = n, and every collection of i edges from F contains atleast i vertices for 1 ≤ i ≤ k.




the set of vertices V , with ∣V ∣ = m, represents the set of m servers

the set of edges F , with ∣F ∣ = n, represents the set of n data items.Edge Fi ∈ F contains vertex vj ∈ V if and only if i-th data item isstored in j-th server.

Theorem (Paterson et al., 2009)A c-uniform hypergraph (V,F) represents a c-uniform (n, cn, k,m)-CBC ifand only if ∣V ∣ = m, ∣F ∣ = n, and every collection of i edges from F contains atleast i vertices for 1 ≤ i ≤ k.




the set of vertices V , with ∣V ∣ = m, represents the set of m serversthe set of edges F , with ∣F ∣ = n, represents the set of n data items.

Edge Fi ∈ F contains vertex vj ∈ V if and only if i-th data item isstored in j-th

codingtheory.cswp.cs.technion.ac.il...2016/11/01 · onn(m;c;k) explicitconstruction...

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