1 Liner Network Codes nd Systems of Polynomil Equtions Rndll Dougherty, Chris Freiling, nd Kenneth Zeger (Submitted to ISIT 2008) Abstrct If β nd γ re nonnegtive integers nd F is field, then polynomil collection {p 1,..., p β } Z[α 1,..., α γ] is sid to be solvble over F if there exist ω 1,..., ω γ F such tht for ll i = 1,..., β we hve p i(ω 1,..., ω γ) = 0. We sy tht network nd polynomil collection re solvbly equivlent if for ech field F the network hs sclr-liner solution over F if nd only if the polynomil collection is solvble over F. Koetter nd Médrd s work implies tht for ny directed cyclic network, there exists solvbly equivlent polynomil collection. We provide the converse result, nmely tht for ny polynomil collection there exists solvbly equivlent directed cyclic network. (Hence, the problems of network sclr-liner solvbility nd polynomil collection solvbility hve the sme complexity.) The construction of the network is modeled on mtroid construction using finite projective plnes, due to McLne in 1936. A set Ψ of prime numbers is set of chrcteristics of network if for every q Ψ, the network hs sclrliner solution over some finite field with chrcteristic q nd does not hve sclr-liner solution over ny finite field whose chrcteristic lies outside of Ψ. We show tht collection of primes is set of chrcteristics of some network if nd only if the collection is finite or co-finite. Two networks N nd N re ls-equivlent if for ny finite field F, N is sclr-linerly solvble over F if nd only if N is sclr-linerly solvble over F. We further show tht every network is ls-equivlent to multiple-unicst mtroidl network. I. INTRODUCTION We first demonstrte certin equivlence between networks nd collections of polynomils. Specificlly, we show tht ssocited with every finite collection of polynomils with integer coefficients is corresponding network which is sclr-linerly solvble precisely over This work ws supported by the Institute for Defense Anlyses, the Ntionl Science Foundtion, nd the UCSD Center for Wireless Communictions. R. Dougherty is with the Center for Communictions Reserch, 4320 Westerr Court, Sn Diego, CA 92121-1969 (rdough@ccrwest.org). C. Freiling is with the Deprtment of Mthemtics, Cliforni Stte University, Sn Bernrdino, 5500 University Prkwy, Sn Bernrdino, CA 92407-2397 (cfreilin@csusb.edu). K. Zeger is with the Deprtment of Electricl nd Computer Engineering, University of Cliforni, Sn Diego, L Joll, CA 92093-0407 (zeger@ucsd.edu). those finite fields where the polynomils hve common root. A consequence is tht the complexity of determining whether networks re sclr-linerly solvble over prticulr finite fields is equivlent to the complexity of determining whether collections of polynomil hve common roots over the corresponding fields. Secondly, we show tht the collections of prime numbers corresponding to the field chrcteristics of sclr-linerly solvble network lphbets re precisely those which re finite or co-finite. Finlly, we show tht for every network, there exists multiple-unicst network which is mtroidl (i.e. obtined from certin mtroid-tonetwork construction), such tht the two networks re sclr-linerly solvble over the sme finite fields. There hs been interest in determining the solvbility, sclr liner solvbility, nd vector liner solvbility of n rbitrry network with respect to chosen lphbet (e.g. [4] [7], [9] [11], [13], [15], [17], [18]). For given finite lphbet, to determine if network is solvble or sclr-linerly solvble, one cn perform finite exhustive serch of ll possible codes for the network. If vector dimension is lso fixed, finite serch cn lso estblish if network is vector-linerly solvble over tht dimension. There is presently no known lgorithm for determining the generl solvbility or vector-liner solvbility of n rbitrry network. The existence of n lgorithm (which is pprently not computtionlly efficient) to determine sclr-liner solvbility of n rbitrry network follows from work in [12]. Their technique ws to construct finite collection of polynomils from n rbitrry network, such tht for ech finite field, the polynomils hve common root over the field if nd only if the network hs sclrliner solution over the field. Throughout, polynomils will hve integer coefficients nd will use the vribles α 1, α 2,.... For nonnegtive integers β nd γ, ny finite set P = {p 1,..., p β } Z[α 1,..., α γ ] will be clled polynomil collection. If F is field, then polynomil collection is sid to be solvble over F if there exist ω 1,..., ω γ F such tht for ll i = 1,..., β we hve p i (ω 1,..., ω γ ) = 0. We sy tht network nd polynomil collection re solvbly
equivlent if for ech field F the network hs sclrliner solution over F if nd only if the polynomil collection is solvble over F. We present n lgorithm in Section II for constructing network from ny polynomil system. Our min results re tht: the network is sclr-linerly solvble over the sme fields s those for which the polynomils hve common roots (Theorem I.2), the constructed network is lwys mtroidl (Theorem I.3), every network is sclr-linerly solvble on the sme set of fields s multiple-unicst mtroidl network (Corollry I.8), nd the collections of prime numbers corresponding to the field chrcteristics of sclr-linerly solvble network lphbets re chrcterized s either finite or co-finite (in Theorem I.9). Let Ψ be n rbitrry collection of integers of the form q i, where q is prime nd i 1. We sy tht Ψ is the solvbility set for network (respectively, polynomil collection) if for every finite field F, the network is sclr-linerly solvble (respectively, polynomil collection is solvble) if nd only if F Ψ. The set of primes q, such tht q i lies in the solvbility set for some i 1, is clled the set of chrcteristics 1 for network (respectively, polynomil collection). The following theorem leds to n lgorithm (vi Gröbner bses [2]) for determining whether network hs sclr-liner solution. No such lgorithm is presently known for determining whether network hs generl nonliner solution. Theorem I.1. (follows from Koetter-Médrd [12]) Every directed cyclic network hs solvbly equivlent polynomil collection. In this pper, we provide the following converse result. Theorem I.2. (Converse to Theorem I.1) Any polynomil collection hs solvbly equivlent directed cyclic network. Furthermore, the solvbly equivlent network in Theorem I.2 is given constructively nd is mtroidl, s stted in the next theorem. Theorem I.3. If polynomil collection P is solvble over some finite field, then ny network constructed, s in Section II, from P is mtroidl. The next definition is tken from [8] ( CSLS stnds for coding solvbility, liner solvbility ). Definition I.4. Two networks N nd N re CSLSequivlent if the following two conditions hold: 1 This terminology is tken from [1]. 1) For ny finite lphbet A, N is solvble over A if nd only if N is solvble over A. 2) For ny finite field F nd ny positive integer k, N is vector-linerly solvble over F in dimension k if nd only if N is vector-linerly solvble over F in dimension k. The following definition gives type of equivlence tht is weker thn CSLS (the cronym ls stnds for (sclr) liner solvbility ). Definition I.5. Two networks N nd N re lsequivlent if for ny finite field F, N is sclr-linerly solvble over F if nd only if N is sclr-linerly solvble over F. Theorem I.6. (see [8, Theorem II.1]) Any network is CSLS-equivlent to multiple-unicst network. The next theorem shows tht if Theorem I.6 is pplied to mtroidl network, then the resulting multipleunicst network cn lso be tken to be mtroidl. Theorem I.7. (see [7, Corollry VII.8]) Any mtroidl network is CSLS-equivlent to multipleunicst mtroidl network. The next corollry follows from our min result in Theorem I.2 together with severl previous results. It demonstrtes tht, when considering which finite fields rbitrry networks re sclr-linerly solvble over, it suffices to consider to the subclss of networks which re simultneously multiple-unicst nd mtroidl. Corollry I.8. Any network is ls-equivlent to multiple-unicst mtroidl network. Theorem I.9. A set of prime numbers is the set of chrcteristics of some network if nd only if the set is finite or co-finite. Theorem I.1 nd our Theorem I.2 together indicte tht determining the sclr-liner solvbility of directed cyclic network over field F is computtionlly equivlent to determining whether collection of polynomils hs common root over F. Given ny lgorithm for determining sclr-liner network solvbility, our result gives n lgorithm for determining polynomil solvbility. This is mny-to-one reduction (i.e., it converts single instnce of the polynomil solvbility problem to single instnce of the network sclr-liner solvbility problem with the sme nswer). The reduction cuses t most liner blowup in input size, in the following sense: the number of nodes nd edges in the resulting network is t most liner function of the number Pge 2 of 5
of steps (vrible retrievls nd rithmetic opertions) needed to compute the vlues of the polynomils in the collection. In terms of bit representtions, it is t most n O(n ln n) blowup. This mny-to-one reduction hs the dditionl property tht, given sclr-liner solution to the network, we cn directly reconstruct solution to the polynomil collection. It cn be shown (vi Gröbner bses) tht the sets of chrcteristics of polynomil collections re precisely the sets of primes which re finite or co-finite. In contrst, there hs been no known chrcteriztion of the sets of chrcteristics or the solvbility sets of networks. If Ψ is the solvbility set of network nd n Ψ, then n i Ψ for ll positive integers i While there re n uncountble number of sets of powers of primes closed under exponentition, there re only countbly infinite number of solvbility sets since there re only countble number of networks nd polynomil collections. A fundmentl problem is to determine which sets of integers cn be solvbility sets nd which cn be sets of chrcteristics for networks. Theorem I.1 shows tht every network solvbility set is lso polynomil collection solvbility set. Our Theorem I.2 shows tht every polynomil collection solvbility set is lso network solvbility set. Thus, the network solvbility sets re the sme s the polynomil collection solvbility sets. Our Theorem I.9 shows tht set of primes is the set of chrcteristics of network if nd only if the set of primes is finite or co-finite. II. NETWORK CONSTRUCTION FROM POLYNOMIAL SYSTEM In this section we present n lgorithm for constructing directed cyclic network N from finite polynomil collection P = {p 1,..., p β } Z[α 1,..., α γ ], for i = 1,..., β. The network will be built piece by piece from eight building block components, C 0,..., C 7, which re shown in Figures 1 nd 2 (using Tble I). The messges will be, b, nd c. Certin nodes of the network will be lbeled by x q, y q, u q, or z q, where for ech such node, q is some polynomil in Z[α 1,..., α γ ]. For exmple, the sources for, b, c will be nodes x 0, x 1, y 1, respectively. During the construction, we will lbel vrious nodes with polynomils nd will lter demonstrte connection between these polynomils nd the lphbet symbols crried by these nodes. It will be demonstrted tht this construction lgorithm produces network such tht for ny field F, the network hs sclr-liner solution over F if nd only if the polynomil collection P hs solution over F. The network construction process consists of the steps: Step (1): Strt with component C 0 which cretes nodes x 0, x 1, y 1, z 0, nd z. (See Figure 1) Step (2): If γ > 0, then dd components C 1 (1),..., C 1 (γ), creting nodes x α1,... x αγ. Ech of these components is djoined to the network t the nodes x 0, x 1, z, which hve lredy been creted t Step (1). (See Figure 2 nd Tble I) Step (3): Repetedly dd components C 2,..., C 7 to crete nodes x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ). Steps (3)-(3d) describe the cretion of x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ) s well s mny intermedite nodes. (See Figure 2 nd Tble I) Step (3): For ny positive integer n, to crete node lbeled x n : First, dd component C 4 (1) to crete node u 1. Then, for i = 1,..., n 1, dd component C 2 (i) to crete node z i nd dd component C 6 (i, 1) to crete node x i+1. This is possible since x 1, z 0, z hve lredy been creted. Step (3b): For ny positive integer n, to crete node lbeled x n : First, dd component C 2 (1) to crete node z 1, nd dd component C 5 (1) to crete node u 1. Then, for i = 0,..., n 1, dd component C 6 ( i, 1) to crete node x i 1 nd dd component C 2 ( i 1) to crete node z i 1. Step (3c): For ny positive integer n nd ny α {α 1,..., α γ } to crete node lbeled x α n: First, dd component C 3 (α) to crete node y α. Then, for j = 1,..., n 1, dd component C 2 (α j ) to crete node z α j nd dd component C 7 (α j, α) to crete node x α j+1. Step (3d): To crete nodes lbeled by n rbitrry polynomil in Z[α 1,..., α γ ]: Add vrious instnces of components C 6 nd C 7 to crete nodes lbeled by sums nd products of lbels of existing nodes creted bove. (Some instnces of components C 2, C 3, nd C 4 my lso hve to be dded in order to use C 6 nd C 7.) Step (4): Force ech of the nodes x p1(α 1,...,α γ),..., x pβ (α 1,...,α γ) to demnd messge. To construct nodes lbeled by rbitrry polynomils in Z[α 1,..., α γ ] in Step (3) of the lgorithm, one cn use Step (3) to crete ll positive integer coefficients of the polynomils, use Step (3b) to crete ll negtive integer coefficients of the polynomils, use Step (3c) to crete ll vrible powers occurring in the polynomils, nd finlly use Step (3d) to combine the existing network nodes to crete the desired polynomils. This lgorithm converts single instnce of the poly- Pge 3 of 5
nomil solvbility problem to single instnce of the network sclr-liner solvbility problem with the sme nswer. The procedure bove is not the most efficient method to crete the network N from the polynomil collection P. A smller network cn in generl be constructed whose size is liner in the size of the representtion of the polynomil collection. c b y 1 x 0 x 1 z 0 z 3 Fig. 1. Network component C 0. The leftmost three nodes re sources, generting messges c,, nd b from top to bottom, respectively. The rightmost four nodes re receivers nd demnd messges, c, b, nd, respectively. Five of the nodes re lbeled by x 0, x 1, y 1, z 0, or z. Input 1 Input 2 Input 3 New node c b 3 New receiver Fig. 2. A generic network component C i for 1 i 7. Input 1, Input 2, nd Input 3 re existing nodes in the network nd the remining nodes nd edges in component C i re new. The rightmost node, New receiver, demnds one messge. Tble I lists seven different instntitions of this generic network component tht re used in network construction. III. MATROIDALITY OF CONSTRUCTED NETWORKS First we review the concepts of mtroids, mtroidl networks, nd the finite projective plne, ech of which New New Comp. Input 1 Input 2 Input 3 node demnd C 1 (i) x 0 x 1 z x αi C 2 (q) z 0 z x q z q c C 3 (q) x q z 1 z 0 y q C 4 (q) x q z 0 z u q c C 5 (q) x 0 z q z u q c C 6 (q, r) z q u r z x q+r C 7 (q, r) z q y r z x qr TABLE I INSTANTIATIONS OF THE GENERIC NETWORK COMPONENT SHOWN IN FIGURE 2. EACH LINE IN THE TABLE GIVES THE FIVE VALUES THAT ARE USED TO FORM A SPECIFIED COMPONENT. will be used in wht follows. A mtroid M (e.g. see [16]) is n ordered pir (S, I), where S is finite set nd I is set of subsets of S stisfying the following three conditions: (I1) I. (I2) If I I nd J I, then J I. (I3) If I, J I nd J < I, then e I J such tht J {e} I. The set S is clled the ground set, the members of I re clled independent sets, nd ny subset of S not in I is clled dependent set. For ny mtroid M = (S, I) nd ny X S, let I X = {I X : I I}, nd let M X = (X, I X). Then M X is mtroid nd the rnk of X, denoted ρ(x), is the (unique) size of mximl independent set of M X. The rnk of the mtroid M is defined to be ρ(s). Let N be network with messge set µ, node set ν, nd edge set ɛ. Let M = (S, I) be mtroid with rnk function ρ. The network N is mtroidl network (see [7]) ssocited with M if there exists function f : µ ɛ S such tht the following conditions hold: (M1) f is one-to-one on µ. (M2) f(µ) I. (M3) ρ(f(in(x))) = ρ(f(in(x) Out(x))), x ν. It ws shown in [7] tht mny interesting networks re mtroidl, including ll networks tht re sclr-linerly solvble over finite field (e.g. solvble multicst networks). The mtroid used is vector spce over the finite field (with dimension the number of messges); the function f mps the messges to elementry vectors (vectors which re ll 0 except for single 1) nd mps the edges to the corresponding globl coding vectors (see, e.g., [11]) for the given sclr-liner code. In [7], method ws presented for constructing, from given mtroids, (mtroidl) networks which reflect some Pge 4 of 5
of the mtroids properties. This construction ws used to obtin networks used to prove vrious results in the literture [5], [6], [8]. For exmple, in [7], network ws constructed from the Vámos mtroid tht demonstrtes the insufficiency of using Shnnon-type informtion inequlities to compute network coding cpcity. In wht follows, we will prove tht if network is constructed from solvble polynomil collection s in Section II, then the network is mtroidl. The network construction lgorithm given in Section II ws inspired by the 1936 work of Sunders McLne in [14]. For ny positive integer n, projective plne (e.g. see [3]) comprises set of points, set of lines, nd n incidence reltion between points nd lines stisfying: (P1) Any two points re incident to exctly one line. (P2) Any two lines re incident to exctly one point. (P3) There exist 4 points, no 3 of which re incident to the sme line. Every finite projective plne induces rnk-three mtroid s follows. Let S be the set of ll points in the projective plne, let I be the collection of subsets of S of crdinlity t most 3 tht do not contin 3 colliner points, nd let M = (S, I). It is esy to see tht M stisfies (I1) nd (I2). Suppose I, J I where I > J. Then J {0, 1, 2}. If J < 2, then for ny v I J, we trivilly hve J {v} I. If J = 2 nd if for ech v I J we hve J {v} I, then the 3 points in I re colliner, contrdicting I I. Thus, M lso stisfies (I3), nd therefore M is rnk-3 mtroid. For ny field F, one cn construct projective plne Π F (of order F if F is finite) s follows. Let Π F = (F F ) F { } where two points (, b) nd (c, d) in F F re sid to hve slope s F if c nd s = (d b)(c ) 1, nd slope s = if = c. A line in Π F consists of n element s of F { } (clled point t infinity) together with mximl set of points in F F such tht every two of them hve slope 1/s (where we mke the convention tht v/0 = nd v/ = 0, for ll nonzero v F ). The set of ll points t infinity is lso considered line nd its point t infinity is. It cn be verified tht xioms (P1) (P3) hold for Π F. McLne [14] (see lso [19, pp. 18 21]) used this construction s follows. Let P be polynomil collection nd let K be finite field such tht P hs solution over K. Then McLne constructs mtroid M tht is representble over K nd such tht, for ny finite field F, if M is representble over F, then P hs solution over F. However, it is not necessrily true tht, if P hs solution over F, then M is representble over F. Such n if-nd-only-if result is not ttinble in generl for mtroids; for instnce, it is known tht, if mtroid is representble over the 2-element field nd the 3-element field, then it is representble over ll finite fields [16, Theorem 6.6.3]. The extr flexibility of networks llows us to construct network solvbly equivlent to ny given polynomil collection. REFERENCES [1] R. Bines nd P. 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