closure solvability for network coding and secret sharing

7858 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 12, DECEMBER 2013

Closure Solvability for Network Codingand Secret SharingMaximilien Gadouleau, Member, IEEE

Abstract—Network coding is a new technique to transmit datathrough a network by letting the intermediate nodes combine thepackets they receive. Given a network, the network coding solv-ability problem decides whether all the packets requested by thedestinations can be transmitted. In this paper, we introduce a newapproach to this problem. We define a closure operator on a di-graph closely related to the network coding instance and we showthat the constraints for network coding can all be expressed ac-cording to that closure operator. Thus, a solution for the networkcoding problem is equivalent to a so-called solution of the closureoperator. We can then define the closure solvability problem ingeneral, which surprisingly reduces to finding secret-sharing ma-troids when the closure operator is a matroid. Based on this re-formulation, we can easily prove that any multiple unicast whereeach node receives at least as many arcs as there are sources solv-able by linear functions. We also give an alternative proof thatany nontrivial multiple unicast with two source–receiver pairs isalways solvable over all sufficiently large alphabets. Based on sin-gular properties of the closure operator, we are able to generalizethe way in which networks can be split into two distinct parts; wealso provide a new way of identifying and removing useless nodesin a network. We also introduce the concept of network sharing,where one solvable network can be used to accommodate anothersolvable network coding instance. Finally, the guessing graph ap-proach to network coding solvability is generalized to any closureoperator, which yields bounds on the amount of information thatcan be transmitted through a network.

Index Terms—Closure operators, guessing games, matroids, net-work coding, secret sharing.

I. INTRODUCTION

N ETWORK coding [1] is a protocol which outperformsrouting for multicast networks by letting the intermediate

nodes manipulate the packets they receive. In particular, linearnetwork coding [2] is optimal in the case of one source; how-ever, it is not the case for multiple sources and destinations [3],[4]. Although for large dynamic networks, good heuristics suchas random linear network coding [5], [6] can be used, maxi-mizing the amount of information that can be transmitted overa static network is fundamental but very difficult in practice.Solving this problem by brute force, i.e., considering all pos-sible operations at all nodes, is computationally prohibitive. Dif-ferent alternative approaches have been proposed to tackle this

Manuscript received November 14, 2012; revised July 26, 2013; acceptedSeptember 11, 2013. Date of publication October 17, 2013; date of current ver-sion November 19, 2013. This paper was presented at the LMS-EPSRCDurhamSymposium “Graph Theory and Interactions,” Durham, U.K., July 2013.The author is with the School of Engineering and Computing Sci-

ences, Durham University, Durham, DH1 3LE, U.K. (e-mail: [email protected]).Communicated by I. F. Blake, Associate Editor for Complexity and

Cryptography.Digital Object Identifier 10.1109/TIT.2013.2282293

problem, notably using matroids, information inequalities, andgroup theory [7]–[12]. In this paper, we provide a new approachto tackle this problem based on a closure operator defined on arelated digraph. Closure operators are fundamental and ubiqui-tous mathematical objects.The guessing number of digraphs is a concept introduced in

[13], which connects graph theory, network coding, and circuitcomplexity theory. In [13], it was proved that an instance of net-work coding with sources and sinks on an acyclic network(referred to as a multiple unicast network) is solvable over agiven alphabet if and only if the guessing number of a relateddigraph is equal to . Moreover, it is proved in [14] and [15]that any network coding instance can be reduced into a mul-tiple unicast network. Therefore, the guessing number is a di-rect criterion on the solvability of network coding. One of themain advantages of the guessing number approach is to removethe hierarchy between sources, intermediate nodes, and desti-nations. In [15], the guessing number is also used to disprovea long-standing open conjecture on circuit complexity. In [16],the guessing number of digraphs was studied, and bounds onthe guessing number of some particular digraphs were derived.The guessing number is also equal to the so-called graph en-tropy [13], [17]. This allows us to use information inequali-ties [18] to derive upper bounds on the guessing number. Theguessing number of undirected graphs is studied in [19]. More-over, in [20], the guessing number is viewed as the indepen-dence number of an undirected graph related to the digraph andthe alphabet.Shamir introduced the so-called threshold secret sharing

scheme in [21]. Suppose a sender wants to communicate asecret to parties, but that an eavesdropper mayintercept of the transmitted messages. We then requirethat given any set of messages, the eavesdropper cannotobtain any information about the secret. On the other hand, anyset of messages allows us to reconstruct the original secret. The elegant technique consists of sending evaluations of apolynomial , with and all the othercoefficients chosen secretly at random, at nonzero elementsof ; this is evidently reminiscent of Reed–Solomon codes.The threshold scheme was then generalized to ideal secretsharing schemes with different access structures, i.e., differentsets of trusted parties. Brickell and Davenport have provedthat the access structure must be the family of spanning setsof a matroid; also any linearly representable matroid is a validaccess structure [22]. However, there exist matroids (such asthe Vámos matroid [23]) which are not valid access structures.For a given access structure (or equivalently, matroid), findingthe scheme is equivalent to a representation by partitions [24].

0018-9448 © 2013 IEEE

GADOULEAU: CLOSURE SOLVABILITY FOR NETWORK CODING AND SECRET SHARING 7859

In this paper, we introduce a closure operator on digraphs, anddefine the closure solvability problem for any closure operator.This yields the following contributions.1) First of all, this framework encompasses network codingand ideal secret sharing. In particular, network coding solv-ability is equivalent to the solvability of the closure oper-ator of a digraph associated with the network. This frame-work then allows us to think of network coding solvabilityon a higher, more abstract level. The problem, which usedto be about coding functions, is now a simplified problemabout partitions.

2) This approach is particularly elegant, in different aspects.First, the adjacency relations of the graph, and hence thetopology of the network, are not visible in the closure op-erator. Therefore, the closure operator filters out some un-necessary information from the graph. Second, it is strikingthat all along the paper, most proofs will be elementary,including those of far-reaching results. Third, this frame-work highlights the relationship with matroids unveiled in[9] and [25].

3) Like the guessing number approach, the closure operatorapproach also gets rid of the source-intermediate node-des-tination hierarchy. The guessing graph machinery of [20]can then be easily generalized to any closure operator. Inother words, the interesting aspects of the guessing numberapproach can all be recast and generalized in our frame-work.

4) This approach then yields interesting results. First, it wasshown in [20] that the entropy of a digraph is equal to thesum of the entropies of its strongly connected components.Thus, one can split the solvability problem of a digraphinto multiple ones, one for each strongly connected compo-nent [20]. In this paper, we extend this way of splitting theproblem by considering the closure operators induced bythe subgraphs. We can easily exhibit a strongly connecteddigraph whose closure operator is disconnected, i.e., whichcan still be split into two smaller parts. More specifically,if the graph is strongly connected but its closure operatoris disconnected, then we can exhibit a set of vertices whichare simply useless and can be disregarded for solvability.Second, we can prove that any digraph whose closure oper-ator has rank two is solvable. This means that any multipleunicast with two source–receiver pairs is solvable, unlessthere exists an easily spotted bottleneck in the network.This has already been proved in [26]; our proof is muchshorter and highlights the relation with coding theory anddesigns. Third, we can prove that any network with min-imum in-degree equal to the number of source–receiverpairs is solvable by linear functions over all sufficientlylarge alphabets of size equal to a large prime power. Fourth,we prove an equivalence between network coding solv-ability and index coding solvability. Finally, we show howthe bidirectional union of digraphs can be viewed as net-work sharing.

The rest of the paper is organized as follows. In Section II,we review some useful background. In Section III, we definethe closure solvability problem and prove that network codingsolvability is equivalent to the solvability of a closure operator.

We then use this conversion in Section IV to prove the solv-ability of different classes of networks. We investigate how tocombine closure operators in Section V. We finally define thesolvability graph in VI and study its properties.

II. PRELIMINARIES

A. Closure Operators

Throughout this paper, is a set of elements. A closureoperator on is a mapping which satisfies thefollowing properties [27, Ch. IV]. For any , ,1) (extensive),2) if , then (isotone),3) (idempotent).A closed set is a set equal to its closure. For instance, in a groupone may define the closure of a set as the subgroup generatedby the elements of the set; the family of closed sets is simplythe family of all subgroups of the group. Another example isgiven by linear spaces, where the closure of a set of vectors isthe subspace they span.A closure operator satisfies the following properties. For any, ,1) is equal to the intersection of all closed sets con-taining .

2) , i.e., the family ofclosed sets is closed under intersection.

3) .4) if and only if .We refer to

as the rank of the closure operator. For instance, in a linearspace, this is the dimension of the space. Any set ofsize and whose closure is is referred to as a basis of .An important class of closure operators are matroids [28],

which satisfy the Mac Lane–Steinitz exchange property1: if, , and , then .

A special class consists of the uniform matroids, typicallydenoted as , where

ifotherwise.

Clearly, has rank .

B. Functions and Their Kernels

While network coding typically works with functions as-signed to vertices, it is elegant to work with partitions (for areview of their properties, the reader is invited to [29]). Recallthat a partition of a set is a collection of subsets, called parts,which are pairwise disjoint and whose union is the whole of. For instance, the partition of into singletons is the

so-called equality partition, and is denoted as . We denotethe parts of a partition as for all ; thus,for all .

1In order to simplify notation, we shall identify a singleton with its ele-ment .


If any part of is contained in a unique part of , we sayrefines . The equality partition refines any other partition, whilethe universal partition (the partition with one part) is refined byany other partition. The common refinement of two partitions ,of is given by with parts

We shall usually consider a tuple of partitionsassigned to elements of a finite set

with elements. In that case, for any , we denote thecommon refinement of all , as . Forany , we then have .Any function has a kernel denoted as

, defined by the partition of into preim-ages under . Conversely, any partition of in at most canbe viewed as the kernel of some function from to . Note thattwo functions , have the same kernel if and only iffor some permutation of .

C. Digraphs

Throughout this paper, we shall only consider digraphs [30]with no repeated arcs. Unless specified otherwise, the vertex setof the digraph will be , a set of cardinality . We shall denotethe arc set as , since the letter will be reserved for the al-phabet. However, we do allow edges in both directions betweentwo vertices, referred to as bidirectional edges (we shall abusenotations and identify a bidirectional edge with a correspondingundirected edge) and loops over vertices. In other words, thedigraphs considered here are of the form , where

. For any vertex of , its in-neighborhood isand its in-degree is the size of its

in-neighborhood. By extension, we denote forany set of vertices . Also, by analogy, the out-neighborhoodof is . We say that a digraphis strongly connected if there is a path from any vertex to anyother vertex of the digraph.The girth of a digraph is theminimum length of a cycle, where

we consider a bidirectional edge as a cycle of length 2. A di-graph is acyclic if it has no directed cycles. In this case, we canorder the vertices so that only if

(this is referred to as an acyclic ordering in [30]). The car-dinality of a maximum induced acyclic subgraph of the digraphis denoted as . A set of vertices is a feedback

vertex set if and only if any directed cycle of intersects , orequivalently if induces an acyclic subgraph.Definition 1 [20]: For any digraphs and with disjoint

vertex sets and , we denote the disjoint union, unidirec-tional union, and bidirectional union of and as the graphson and respective edge sets

In other words, the disjoint union simply places the twographs next to each other; the unidirectional union adds allpossible arcs from to only; the bidirectional union addsall possible arcs between and .

D. Guessing Game and Guessing Number

A configuration on a digraph on over a finite alphabetis simply an -tuple . A protocol

of is a mapping such thatis locally defined, i.e., for all . The

fixed configurations of are all the configurations suchthat . The guessingnumber of is then defined as the logarithm of the maximumnumber of configurations fixed by a protocol of :

We now review how to convert a multiple unicast problemin network coding to a guessing game. Note that any networkcoding instance can be converted into a multiple unicast withoutany loss of generality [14], [15]. We suppose that each sink re-quests an element from an alphabet from a correspondingsource. This network coding instance is solvable over if allthe demands of the sinks can be satisfied at the same time. Weassume the network instance is given in its circuit representa-tion, where each vertex represents a distinct coding functionand hence the same message flows every edge coming out ofthe same vertex [15]; again this loses no generality. This circuitrepresentation has source nodes, sink nodes, and inter-mediate nodes. By merging each source with its correspondingsink node into one vertex, we form the digraph onvertices. In general, we have for all and theoriginal network coding instance is solvable over if and onlyif [15]. Note that the protocol on the digraph isequivalent to the coding and decoding functions on the originalnetwork.For any digraphs , on disjoint vertex sets and ,

respectively, we have

for all alphabets [20]. Notably, we can always considerstrongly connected graphs only.We illustrate the conversion of a network coding instance to

a guessing game for the famous butterfly network in Fig. 1. Itis well known that the butterfly network is solvable over all al-phabets, and conversely it was shown that the clique hasguessing number 2 over any alphabet. The combinations anddecoding operations on the network are equivalent to the pro-tocol on the digraph. For instance, if transmits the oppositeof the sum of the two incoming messages modulo on thenetwork, the corresponding protocol lets all nodes guess minusthe sum modulo of their incoming elements.

E. Parameters of Undirected Graphs

Most results of this section are not directly interesting fornetwork coding but will be instrumental in some of the proofsof the paper. As such, a first reading of the paper can safely skipthis part.An independent set in a (simple, undirected) graph is a set of

vertices where any two vertices are nonadjacent. The indepen-dence number of an undirected graph is the maximum


Fig. 1. Butterfly network as a guessing game (a) Network coding instance (b)Guessing game.

cardinality of an independent set. The chromatic numberof is the smallest number of parts of a partition of its vertexset into independent sets [31]. An automorphism for a graph

is a permutation of such that ifand only if . A graph is vertex-transitiveif for all , , there is an automorphism of such that

. For a connected vertex-transitive graph which is nei-ther an odd cycle nor a complete graph, we have [31, Corollary7.5.2], [32]

(1)

We now review three types of products of graphs; all productsof two graphs and have as vertex set.We denote two adjacent vertices and in a graph as .1) In the conormal product , we have

if and only if or . We have.

2) In the lexicographic product (also called composition), we have if and only if eitherand , or . Although this product is not

commutative, we have .3) In the cartesian product , we have

if and only if eitherand , or and . Wehave and

.

III. CLOSURE SOLVABILITY AND NETWORK CODING

A. Closure Operators Related to Digraphs

Let be a digraph on , a set of vertices.Definition 2: The -closure of a set of vertices is defined

as follows. We let andthe -closure of is obtained by applying it repeatedlytimes: .This definition can be intuitively explained as follows. Sup-

pose we assign a function to each vertex of , which only de-pends on its in-neighborhood (the function which decides whichmessage the vertex will transmit). If we know the messages sentby the vertices of , we also know the messages which will besent by any vertex in . By applying this iteratively, we candetermine all messages sent by the vertices in . There-fore, represents everything that is determined by .

We give an alternate, easier to manipulate, definition of the-closure below.Lemma 1: For any , is the largest set

of vertices inducing an acyclic subgraph such that .Proof: First, it is clear that is a set of vertices inducing an

acyclic subgraph such that . Conversely, supposeinduces an acyclic subgraph and . Denoting

and for ,we have and hence .Example 1: Some special classes of digraphs yield famous

closure operators (all claims follow from Lemma 1).1) If is an acyclic digraph, then , i.e.,

for all . This can be intuitively explained by the factthat an acyclic digraph comes from a network coding in-stance without any source or destination: no informationcan then be transmitted.

2) If is the directed cycle , then , i.e.,and for all . Therefore,

the solutions are MDS codes, such as the repe-tition code. Intuitively, comes from a network codinginstance with one source and one destination, and a chainof intermediate nodes each transmitting a messageto the next until we reach the destination.

3) If is the clique , then , i.e.,if and for all .

Therefore, the solutions of are exactlyMDS codes, such as the parity-check code. Intuitively,comes from a generalization of the butterfly network, withone intermediate node receiving from all sources and trans-mitting to all destinations.

4) If has a loop on each vertex, then , i.e.,for all . This comes from a net-

work with a link from every source to its correspondingdestination.

Since if and only if is a feedback vertex setof , we obtain that has rank .

B. Closure Solvability

We now define the closure solvability problem. The instanceconsists of a closure operator on with rank , and of a finitealphabet with .Definition 3: A coding function for is a family of

partitions of into at most parts such that forall .The problem is to determine whether there exists a coding

function for such that has parts. That is, we wantto determine whether there exists an -tupleof partitions of in at most parts such that

For any partition of , we define its entropy as


The equality partition on is the only partition with full en-tropy . Denoting , we can recast the condi-tions above as

Therefore, is solvable if and only if for somecoding function of over .The first important case is solvability of uniform matroids,

which is equivalent to the existence of MDS codes.Proposition 1: For all , , and , is solvable over if

and only if there exists an -MDS code over analphabet of cardinality .The proof follows the classical argument that a code of lengthwith cardinality and minimum distance is

separable (hence the term MDS code). We shall formally provea much more general result in Section VI, therefore we omit theproof of Proposition 1.In particular, a solution for is then equivalent to

mutually orthogonal latin squares; they exist for all sufficientlarge alphabets. This illustrates the complexity of this problem:solving (i.e., determining the possible orders for two mutu-ally orthogonal latin squares) was wrongly conjectured by Eulerand solved in 1960 [33].Combinatorial representations [25] were recently introduced

in order to capture some of the dependency relations amongfunctions. A solution for the uniform matroid corresponds to acombinatorial representation of its family of bases; however, ingeneral this is not true. Indeed, any family of bases has a combi-natorial representation, while we shall exhibit closure operatorswhich are not solvable.

C. Closure Solvability and Network Coding Solvability

We consider a multiple unicast instance: an acyclic networkwith sources , destinations , and

intermediate nodes, where each destination requests the mes-sage sent by . We assume that the messages , along witheverything carried on one link, is an element of an alphabet .Also, any vertex transmits the same message on all its outgoinglinks, i.e., we are using the circuit representation reviewed inSection II. We denote the cumulative coding functions at thenodes as , where the first indices correspondto the destinations and the other indices to the intermediatenodes, and .We now convert the network coding solvability problem

into a closure solvability problem. Recall the digraph onvertices corresponding to the guessing game, reviewed in

Section II.Intuitively, if the destination is able to recover from the

messages it receives, it is also able to recover any functionof that message. Conversely, if it can recover for somepermutation of , then it can recover aswell. We can then relax the condition and let request anysuch . Viewing as a function from to , sending

to , we remark that has the same kernel

as for any permutation . Therefore, the correct relaxation isfor to request that the partition assigned to it be the same asthat of the source .The relaxation above is one argument to consider partitions

instead of functions. The second main argument is that the de-pendency relations are completely (and elegantly) expressed interms of partitions, as illustrated in the proof of Theorem 1.Theorem 1: The network is solvable over if and only ifhas rank and is solvable over .Proof: Let be a solution for . Then, it is easy to check

that is a family of partitions ofinto at most parts such that and .As such, for all and hence .Conversely, let be a solution for over and let on

be any collection of functions with kernelsfor all and for all .

Since , we have that only depends on ;the number of parts of indicates that ; finally,

indicates that is a solution for .We remark that the closure operator approach differs from

Riis’s guessing game approach. Although it also gets rid of thesource/intermediate node/receiver hierarchy and works on thesame digraph, the distinction is in the fact that now corre-sponds to the cumulated coding functions.

IV. MAIN RESULTS

A. How to Use Closure Solvability

So far, we have considered any possible closure operator. Letus reduce the scope of our study by generalizing some conceptsarising from matroid theory.First, we say that a vertex is a loop if it belongs to the closure

of the empty set. It is clear that removing from does notaffect solvability (any vertex from is useless). We there-fore assume that . In particular, we only consider di-graphs with positive minimum in-degree or in other words, thathave a cycle.Second, we say that is separable if for all , such

that and , we have . Anymatroid is separable; likewise it is easily seen that any -clo-sure is separable too. If is separable, we can further simplifythe problem in a more general fashion than the so-called par-allel elements in a matroid. There exists such that is par-titioned into parts ; for any , thereexists such that . Again, considering theclosure operator on defined bydoes not affect solvability (since if is solvable, then there is asolution for where for all ). Therefore, wecan always restrict ourselves to -closures wherefor all . In other words, we have just removed all vertices ofin-degree one and by-passed them instead. Clearly, these ver-tices of degree one are useless for network coding, as they donot bring any more combinations. The only thing they can dois forward the symbol they receive. As such, we might as wellby-pass them.There is a natural partial order on the family of closure oper-

ators of . We denote if for all , .This partial order has maximum element (with


for all ) and minimum element (wherefor all ).Any tuple of partitions of into at most parts naturally

yields a closure operator on : we define

Proposition 2: is a coding function for if and only if. Therefore, if have the same rank and is

solvable over , then is solvable over .Proof: If is a coding function for , then

for all and hence . Conversely,if , then denote and

.Since is solvable, there exists a coding function for

with entropy , where is the rank of and . But thenand hence is also a solution for .

If is a matroid, the solvability problem is equivalent todetermining whether they form a secret-sharing matroid, i.e.,whether there exists a scheme whose access structure is thefamily of spanning sets of that matroid.Theorem 2: If is a matroid, then is solvable over some

alphabet if and only if it is a secret-sharing matroid.Proof: By definition, a secret-sharing matroid is solvable

over some alphabet. Conversely, let be a solution for . Letbe the rank function associated with , i.e.,

and[28]. Then for any , we have

. Moreover, there exists such thatand , hence

. Thus, for all and.

B. Solvable Networks

In this section, we apply the conversion of network codingsolvability in order to closure solvability to determine that someclasses of networks are solvable. Using general closure opera-tors allows us to think outside of networks. In particular, it al-lows us to use uniform matroids, which have been proved to besolvable over many alphabets (see Proposition 1), but which donot arise from networks in general (see Proposition 3 below).Proposition 3: The uniform matroid is the -closure of

a digraph if and only if .Proof: The cases , 1, , respectively have

been illustrated in Example 1. Conversely, suppose a digraphhas -closure , where . Then, any set ofvertices induces an acyclic subgraph, while any set ofvertices induces a cycle. This implies that any set of ver-tices induces a (directed) path. Without loss, letinduce a path (in that order), then inducea cycle, and so do . Therefore, in the sub-graph induced by , the vertex has out-de-gree 2 and hence that graph is not a cycle.We can then prove that all digraphs with minimum degree

equal to the rank, or with rank 2, are solvable. Note that the caseof rank 2 has already been proved in [26] using a much longerargument.

Theorem 3: Any of rank 2 is solvable over all sufficientlylarge alphabets. Moreover, if the minimum in-degree of isequal to its rank, then is solvable by linear functions overall sufficiently large prime powers.

Proof: The simplifications above mean that we can assumefor all . This is equivalent to ,

therefore by Proposition 2, any digraph with rank 2 is solvablewhenever is.Moreover, suppose the minimum in-degree is equal to the

rank . Then for any with , we haveand hence ; therefore, . Again

Proposition 2 yields the result.

V. COMBINING CLOSURE OPERATORS

In this section, we let , with respective cardinali-ties and such that and . For any

, we denote and . We alsolet , be closure operators of rank and over and, respectively.

A. Disjoint and Unidirectional Unions

We first generalize some definitions frommatroid theory [28].Definition 4: For any closure operator and any , the

deletion of and the contraction of from are the closureoperators defined on by

for any .Proposition 4: If is the closure operator associated

with the digraph , then for any , ,where is the digraph induced by the vertices in . Thus,

for any .Proof: Let , then any subset of

induces an acyclic subgraph of if and only if itinduces an acyclic subgraph of ; moreover,in if and only if . By Lemma 1, weobtain and hence

.Definition 5: The disjoint union and unidirectional union ofand are closure operators on respectively given by

ifotherwise.

For any , we have and

Recall the definitions of unions of digraphs in Section II. Ourdefinitions were tailored such that

Moreover, if there is a loop on vertex in the digraph ,then , or in other words,

. We also remark that if


and are matroids, then is commonly referred to asthe direct sum of and [28].The disjoint and unidirectional unions are related to the con-

traction as follows.Proposition 5: For any and any , the following are

equivalent:1) , i.e., for all ,

;2) ;3) there exist , defined on and , respectively, suchthat

Proof: The first property implies the second, due to thefollowing pair of inequalities: for any ,

To prove the first inequality, we have

if and

otherwise. For the second inequality, we have

and similarly , and hence.

Clearly, the second property implies the third one. Finally, ifthere exist such and , then it is easy to check that

.

B. Application to Removing Useless Vertices

The first property of Proposition 5 indicates that has noeffect on ; thus suggesting the following notation.Definition 6: If there exists such that , we

say that is disconnected and that is weak. If is weak andacyclic, then we say is useless.The -closure of a nonstrongly connected graph is discon-

nected. However, there are strongly connected graphs whose-closure is disconnected.Example 2 (Strongly Connected Graphs With Disconnected-Closures): The canonical example of a strongly connected

graph with disconnected closure operator is given in Fig. 2(a).On that graph, is useless, for

A larger example is given by the graph in Fig. 2(b), wherefor . Each example leaves a graph

which is solvable (a clique). On the other hand, in Fig. 2(c), theuseless set is . Therefore, after removing thevertices 6 to 10, we are left with the undirected cycle of length5, , which is not solvable [15].Proposition 6: Suppose is strongly connected, then is

weak for if and only if it is useless.Proof: We first claim that all arcs from to come from

. Indeed, let such that . Then, and hence .

Since , the intersectionis not empty. By Lemma 1, induces an acyclic subgraph and

, which is equivalent to .Now, suppose is not acyclic, i.e., . But

then, by the claim above there are no arcs fromto its complement, and is not strongly connected.As a corollary, if is an undirected graph, then is con-

nected if and only if is connected.We remark that if is weak, then is closed, for

. Also, it is easy to check that theunion of two weak sets is weak, hence there exists a largestweak set. Thus, if is strongly connected, there exists a largestuseless set, referred to as the useless part of .We say a cycle is chordless if there does not exist, , , such that is a cycle.In other words, a chordless cycle does not cover another shortercycle. Then, is the set of vertices which do not belong toany chordless cycle.Theorem 4: Algorithm 1 removes the useless part of a

strongly connected digraph in polynomial time.

Algorithm 1 Remove the useless part of a strongly connecteddigraph

repeat

while and do

while and do

if then

end if

end while

if then

end if

end while

until

return

The proof of Theorem 4 is given in Appendix A.


Fig. 2. Three strongly connected graphs with disconnected -closures.

C. Bidirectional Union

Definition 7: The bidirectional union of and is definedas

ififotherwise.

It is easily shown that and. Moreover, for any and any ,

we have

The first inequality means that the bidirectional union is the wayto combine and which brings the fewest dependencies;as such it is the union of and with the highest entropy.The bidirectional union of digraphs does correspond to the

bidirectional union of closure operators

and the converse is given below.Proposition 7: If , then and

. Moreover, if is a loopless graph, thenif and only if , , and

.Proof: The first claim is easy to prove. For the second

claim, if , then and .Suppose the arc is missing between and . Then

(since is acyclic), while. The converse is

trivial.

VI. GUESSING NUMBER AND SOLVABILITY GRAPH

A. Definition and Main Results

The solvability graph extends the definition of the so-calledguessing graph to all closure operators. Most of this section

naturally extends [20]. Therefore, we shall omit certain proofswhich are very similar to their counterparts in [20].First of all, we need the counterpart of the guessing number

of a graph for closure operators. Any partition of into atmost parts is henceforth denoted as, where some parts are possibly empty. By extension,

for any tuple , the partition is denoted as, where .

We denote the set of words of indexing nonempty parts ofas the image of :

Definition 8: The guessing number of over is given by

where the maximum is taken over all coding functions forover .A coding function has an image of size if and only if

it is a solution; therefore, is solvable over if and only if.

Next, we define the solvability graph of closure operators.Definition 9: The solvability graph has vertex setand two words , are adjacent if and only if there

exists no coding function for over such that , .Proposition 8 below enumerates some properties of the solv-

ability graph. In particular, Property 2 provides a concrete andelementary description of the edge set which makes adjacencybetween two configurations easily decidable.Proposition 8: The solvability graph satisfies the

following properties:1) It has vertices.2) Its edge set is , where

.3) It is vertex-transitive.Proof: Property 1 follows from the definition. We now

prove Property 2. is a coding function if and only iffor all and any , which in turn is equiva-lent to for all . Therefore,if , and is a coding function, we have

and , or in otherwords .Conversely, for distinct we construct the following

coding function. Let be a partition of into two nonemptyparts and for all , let have two parts and

if and have one part otherwise. Then,; for all , if and

only if ; and . We now check that is indeeda coding function: let and . If ,then and hence has one part. Otherwise

.For Property 3, we remark that is a Cayley graph

[31], hence it is vertex-transitive. More explicitly, if we let, then is an automorphism of the solvability

graph which takes to for any , .Corollary 1: The solvability graph for the uniform matroidhas edge set .


The main reason to study the solvability graph is given inTheorem 5 below.Theorem 5: A set of words in is an independent set of

if and only if they are the image of by a codingfunction for over .

Proof: By definition of the solvability graph, the imageof a coding function forms an independent set. Conversely, let

be an independent set of the solvability graph .Let be a partition of into nonempty parts and let

Then, we have ; for all ,

if and only if ; and . We now justify that is acoding function. Let , then for all ,

and hence

which means .Corollary 2: We have and

hence if and only if is solvable over .In [20], we remark that the index coding problem asks for

the chromatic number of the guessing graph of a digraph.We can extend the index coding problem to any closureoperator and we say that is index-solvable over if

. We have

by (1). Therefore, although determining andis distinct over a fixed alphabet , they are asymptoticallyequivalent. More strikingly, solvability and index-solvabilityare equivalent for finite alphabets too, as seen below.Theorem 6: The closure operator is solvable over if and

only if it is index-solvable over .Proof: Let be an independent set of and be a

basis of . Without loss, let ). First, we remarkthat for all , for otherwise and

means that . Second, let , then forany and any , denote . Then,it is easily shown that forms an independent setand that the family forms a partition of intoindependent sets.Conversely, if , thenby (1).

B. Neighborhood and Girth

Note that the relation “having an arc from to ” cannot beexpressed in terms of the -closure. Indeed, all acyclic graphson vertices, from the empty graph to an acyclic tournament, allhave the same closure operator . However, the -closure ofthe in-neighborhood of a vertex can be described by means ofthe digraph closure.Lemma 2: For any and any , if

and only if .

Proof: Suppose , theninduces an acyclic subgraph and ; in particular,

. Since , we easily obtain the con-verse.We remark that if there is a loop on , then there exists no

set such that . Note that is notnecessarily an inner basis of its own closure, for instance this istrivial in nonempty acyclic digraphs.Based on our results about closure operators associated with

digraphs, we can define some concepts to any closure operatorswhich generalize those of digraphs.Definition 10: For any vertex , the degree of is

if there exists such set , or by convention is equal to 0 other-wise. We denote the minimum degree as .Note that the degree (according to the closure operator )

of a vertex of the digraph is not necessarily equal to the sizeof its in-neighborhood.Definition 11: We say a subset of vertices is acyclic if

. The girth of the closure operator as the min-imum size of a nonacyclic subset of vertices.Here, the girth of a digraph is equal to the girth of its closure

operator.We denote the maximum cardinality of a code over of

length and minimum distance as .Proposition 9: For any , we have

Since and , we have ifand only if .

C. Combining Closure Operators

Recall the definitions of unions of closure operators inSection V. The following theorem is the counterpart of Propo-sitions 6, 7, and 8 in [20].Theorem 7: For any and defined on disjoint sets

and of cardinalities and , we have

Therefore,

Corollary 3: The following are equivalent:1) and are solvable over ,2) is solvable over3) is solvable over .Therefore, when studying solvability, we can only consider

connected closure operators.


Fig. 3. Bidirectional union . The vertices of form a basis; the high-lighted disjoint cliques 127, 248, 56 show that it is solvable.

Fig. 4. Example of network sharing (a) First network (b) Second network(c) Shared network.

Corollary 4: Without loss, suppose , thenwe have the following list of properties each implying the next1) and are solvable over ;2) is solvable over and ;3) is solvable over ;4) is solvable over and .In particular, if , then is solvableover if and only if and are solvable over .An example where is not solvable, yet is solv-

able, is given in Fig. 3.The results on the bidirectional union can be viewed as “net-

work sharing,” illustrated in Fig. 4. Suppose we have two solv-able networks and , where has the same number of ormore intermediate nodes than . Then, can be plugged into, which can share its links with without compromising

its solvability. In the resulting shared network, not only eachsource–destination pair of is there, but also each interme-diate node yields an additional source–destination pair. As a re-sult, the only intermediate nodes are those coming from .

D. Combining Alphabets

Let for any positive integer . We definea closure operator on as follows. For any , let

and for any , denote. Then,

This closure operator can be intuitively explained as follows.Consider the solvability problem of over the alphabet .Each element of is a vector of length over , thenassociates according vertices to each , each newvertex corresponding to the coordinate . If forsome , then the local function depends on . We canview (and hence all its coordinate functions)as depending on all coordinates of all vertices in , hence thedefinition of the closure operator.In particular, for construct as follows: its vertex set is

and its edge set is .Then, it is easy to check that .Proposition 10: We have the following properties:1) .2) and hence

.3) If is connected, then so is for all .Proof: The proof of the first two claims is similar to that

of [20, Proposition 10]. We now prove the last claim. For any, we denote , ,

and . Note that. Then, we claim that if ,

then . For any , let; then and . We then have

and in particular, then intersections with are equal, thusproving the claim.Now suppose is disconnected, then

for some (and hence and ).Then, for any , and we have

and hence for all .

APPENDIX

Appendix A) Proof of Theorem 4: First of all, we justifywhy we only search for useless vertices in .

Lemma 3: If is useless, then .Proof: Let be a useless set. First of all, if induces

a chordless cycle, then it cannot entirely lie in , for isacyclic. Suppose does not lie entirely in either. Sinceis acyclic, we have , where

. Therefore, ; gathering, we obtain


. More precisely, and hence isacyclic, which is a contradiction.The following results ensure that we can remove useless ver-

tices one by one.Lemma 4: Let be useless in and . Once is

removed from , is useless in .Proof: is clearly acyclic. For any , we

have

Lemma 5: Let be useless in and the last vertex ofaccording to an acyclic ordering (i.e., ). Then,

is a useless set.Proof: We only need to prove that is weak, i.e., for all

, . This clearly holds if, hence let us assume that .

It is easy to show by induction on thatif and only if ; in

particular, if and only if.

We have . Since is weak,. Moreover, since , we have

. Combining, we obtainand by the paragraph above,, which yields

.Next, we indicate an efficient way to check that a singleton is

useless.Lemma 6: For any vertex , is useless if and only if

for any , .Proof: Suppose there exists such that

. There is an edge from to , henceand . Since

, we obtain andis not weak.

Otherwise, suppose there exists such that; clearly . It is easy to show

by induction on thatif and only if . Let

, thenthere exists . We have

, and hence, which is the desired contradiction.

We can now prove the correctness of Algorithm 1. Clearly,the running time is polynomial.

Proof: First of all, Lemma 3 guarantees that the set of use-less vertices lies in . At every iteration of the Repeat loop,if there exists a set of useless vertices in the new graph, thenthere exists a singleton which is useless by 5. By Lemma6, the algorithm will find a useless vertex if there exists one.Lemma 4 guarantees that after all the iterations, all the uselessvertices will be removed.

ACKNOWLEDGMENT

The author would like to thank the team from Queen Mary,University of London for fruitful discussions and the anony-mous reviewers for their valuable suggestions.

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Maximilien Gadouleau (S’06–M’10) received an equivalent to the M.S. inElectrical and Computer Engineering from Esigelec, Saint-Etienne du Rouvray,France, in 2004 and the M.S. and Ph.D. in Computer Engineering from LehighUniversity, Bethlehem, PA, in 2005 and 2009, respectively.From 2009 to 2010, he was a postdoctoral researcher at CReSTIC, Université

de Reims Champagne-Ardenne, Reims, France. From 2010 to 2011, he helda postdoctoral research assistantship at the School of Electronic Engineeringand Computer Science, Queen Mary, University of London. In 2012, he joinedthe School of Engineering and Computing Sciences at Durham University, as alecturer in Computer Science. His current research interests are coding theory,network coding, and cryptography and their relations to combinatorics, algebra,and graph theory.Dr. Gadouleau is a member of the IEEE Information Theory Society.

closure solvability for network coding and secret sharing

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