Corollary 19. For any rational number kn and for any arbitrarily large number x, there exists a network which has a rate kn fractional linear network coding solution only if the message dimension is larger than x.
Proof. Let abe any given number. Letx be any positive integer greater thana. Then, settingd=x in Theorem 18, guarantees existence of such a network.
3.3 MDim-m network
s21 s11
s21 s11 s32
s21 s11 s33
s22 s11 s31
s22 s11 s32
s22 s11 s33
s21 s11 s31
s21 s11 s32
s21 s11 s33
s21 s12 s31 s31
s21 s13 s33
s23 s11 s31
s23 s11 s32
s23 s11 s33 s21
s12 s32
s21 s12 s33
s21 s13
s21 s13 s32
T1* T2*
T1 T2 T*
s31 v4
v1 v
2 v
3 s11 s
12 s
13 s
21 s 22 s
23 s
31 s
s 33 32
u1 u
2 u
3
Figure 3.6: The MDim-m network form= 3. The elements of the vector under each terminal are the sources from which messages are demanded. Note that T1 and T2 have 3 terminals demanding the same sources. A terminal t∈T is connected to node the vi for 1≤i≤m by m−1 edges. To protect clarity we represent these m−1 edges with a thicker but single edge.
terminalt inTi demands a unique tuple ofm source messages (but two different terminals belonging respectively toTi andTj fori6=j may demand the same set of sources). This unique tuple is decided following these rules: (1) all t ∈ Ti demands the message generated by the source si1 (2) for each j∈ {1,2, . . . , m−1},j6=i, terminalt∈Timust demand one source from the set{sj1, sj2, . . . , sj(m−1)} (note it doesn’t demand sjm) (3) t ∈ Ti demands any one source from set Sm. It can be seen that each Ti hasm(m−1)(m−2) terminals (to choose one element from (m−1) elements of each setSj for 1≤j≤m−1,j6=i, and one element frommelements ofSm). The setT∗ is further partitioned into m−1 subsets: T1∗, T2∗, . . . , Tm−1∗ . Each of Ti∗ for 1≤i≤m−1 containsm terminals. For any fixed ordering thejth terminal for 1≤j≤m inTi∗ demands messages from the sources: sim,smj, and all sources in the set{sk1|1≤k≤m−1,k6=i}. (Note that MDim-m is the M-network for m= 2.) Theorem 20. For any positive integer m ≥ 3, the MDim-m network has a w-dimensional vector linear solution if and only if w is greater than or equal to m−1.
Proof. First we show the ‘only if’ part. Let f be the function that maps the network MDim-m to a discrete polymatroid D such that the conditions of Definition 6 are satisfied. Let ρ be the rank function of D, and letg=ρ◦f. Say that the MDim-mhas a d-dimensional vector linear solution.
TH-2118_136102023
Claim 3. g(E1) =g(E2) =· · ·=g(Em) = (m−1)d Proof. As per D3 of Definition 6, g(sij) =d. So,
m2d=
m
X
i,j=1
g(sij) =g(s11, s12, . . . , smm) [using Lemma 4]
≤g(s11, . . . , smm, E1, . . . , Em, e1(m+1), . . . , em(m+1))
=g(E1, E2, . . . , Em, e1(m+1), . . . , em(m+1)) (3.16)
≤g(E1)+· · ·+g(Em) +g(e1(m+1))+· · ·+g(em(m+1)). (3.17) Equation (3.16) holds because of D4 and the fact that every source message is demanded by some terminal, which must be retrieved from Ei ∪ {ei(m+1)|1 ≤ i ≤ m}. Since Ei contains m−1 edges, using D3 of Definition 6 we get g(Ei) ≤ (m−1)d, and also g(ei(m+1)) ≤ d for 1 ≤ i ≤ m. So for equation (3.17) to hold we must have: g(Ei) = (m−1)dfor 1≤i≤m.
Claim 4. for 1≤i≤m:
g(Ei, si1) +g(Ei, si2) +· · ·+g(Ei, sim)≥(m2−m+ 1)d.
Proof. This is proved by using the Definition 4.
g(Ei, si1) +g(Ei, si2) +· · ·+g(Ei, sim)≥g(Ei, si1, si2, . . . , sim) + (m−1)g(Ei) (3.18)
=g(si1, si2, . . . , sim) + (m−1)g(Ei)
=md+ (m−1)(m−1)d= (m2−m+ 1)d.
equation (3.18) is obtained by using P3 of Definition. 4 (m−1) times.
As per Definition 6 and Definition 5, for anyX ⊆ {S∪E},f(X) mapsXto a subset ofG, and for each j =f(xi), wherexi ∈ X, there exists a vector space Vj such that dim(P
∀xi∈XVf(xi)) =g(X).
Then, for any 1≤i≤m:
md=g(si1, . . . , sim) =g(ei, si1, . . . , sim) [from D3, D4]
or, md=dim(X
l=1,...,m
Vf(sil)) =dim(X
l=1,...,m
Vf(sil)+Vf(ei)). (3.19) Since for any two vector spaceU and V, dim(U) +dim(V) =dim(U∪V) +dim(U ∩V). Then, for
3.3 MDim-m network
any 1≤i, j≤m,i6=j, dim( X
l=1,...,m
Vf(sil)+Vf(ei)) +dim( X
l=1,...,m
Vf(sjl)+Vf(ej))
= dim(X
l=1,...,m
Vf(sil)+Vf(ei)+X
l=1,...,m
Vf(sjl)+Vf(ej)) +dim((X
l=1,...,m
Vf(sil)+Vf(ei))∩(X
l=1,...,m
Vf(sjl)+Vf(ej))).
Then from D3 of Definition 6 and using equation (3.19):
dim(( X
l=1,...,m
Vf(sil)+Vf(ei))∩( X
l=1,...,m
Vf(sjl)+Vf(ej))) = 0.
This implies, for any 1≤i, j, l, k≤m:
dim((Vf(sil)+Vf(ei))∩(Vf(sjk)+Vf(ej))) = 0.
So,dim(Vf(sil)+Vf(ei)) +dim(Vf(sjk)+Vf(ej)) =dim(Vf(sil)+Vf(ei)+Vf(sjk)+Vf(ej))
or, g(sil, ei) +g(sjk, ej) =g(sil, ei, sjk, ej). (3.20) It can be seen that equation (3.20) can also be obtained by using Lemma 5.
Claim 5. Let 1≤j1, j2, . . . , jm−1 ≤m−1 and 1≤jm ≤m. Then for1≤i≤m−1 the eqns. (3.21) and (3.22) shown below hold
g(E1, s1j1) +· · ·+g(Ei−1, s(i−1)ji−1) +g(Ei, si1) +g(Ei+1, s(i+1)ji+1) +· · ·
· · ·+ g(Em, smjm)≤(m2−m+ 1)d (3.21) g(E1, s11) +· · ·+g(Ei−1, s(i−1)1) +g(Ei, sim) +g(Ei+1, s(i+1)1) +· · ·
· · ·+ g(Em−1, s(m−1)1) +g(Em, smjm)≤(m2−m+ 1)d. (3.22) Proof. According to the network description there exists a terminal t ∈ Ti that demands the mes- sages from sources in {s1j1, s2j2, . . ., s(i−1)j(i−1), si1, s(i+1)j(i+1), . . ., smjm}, and the (jm)th terminal in Ti∗ demands the messages from sources{s11, .., s(i−1)1, sim, s(i+1)1, .., s(m−1)1, smjm}. Proof of (3.21):
g(E1, s1j1) +· · ·+g(Ei−1, s(i−1)ji−1) +g(Ei, si1) +g(Ei+1, s(i+1)ji+1) +· · ·+g(Em, smjm)
=g(E1, s1j1, . . . , Ei, si1, . . . , Em, smjm) [using equation (3.20)].
≤g(E1, s1j1, . . . , Ei, si1, . . . , Em, smjm,(vm+1, t))
=g(E1, . . . , Ei, . . . , Em,(vm+1, t)) [Due to t∈Ti]
≤m(m−1)d+d= (m2−m+ 1)d. [D3 of Definition 6].
TH-2118_136102023
Proof of equation (3.22) is similar (due to demands of t∈T∗).
Claim 6. For each i ∈ {1,2, . . . , m−1} there exist at least two distinct values of j in the range 1≤j≤m such that g(Ei, sij)≥(m−1)d+ 1. And there exists at least one value of j∈ {1,2, . . . , m}
such that g(Em, smj)≥(m−1)d+ 1.
Proof. Say for some value of ithere exists no such j. Then g(Ei, sij) = (m−1)d for 1≤j ≤m (as g(Ei) = (m−1)dandg(Ei, sij)≥g(Ei)). But if these values are substituted in the equation given by Claim 4, we get: m(m−1)d≥(m2−m+ 1)d, which is a contradiction.
We show that there exist at least 2 suchj for 1≤i≤m−1. Say, for some 1≤i≤m−1 there is only one value ofjfor whichg(Ei, sij)≥(m−1)d+1. Hence,g(Ei, sij0) = (m−1)dfor anyj0 6=j. Then to satisfy equation of Claim (4) and equation of Claim (3) we must have: g(Ei, sij) = (m−1)d+d.
We show this is not possible. Let jm ∈ {1,2, . . . , m} be such that g(Em, smjm) ≥ (m−1)d+ 1.
Case I: j∈ {1,2, . . . , m−1}.
equation (3.21) tells us that there exist an l∈ {1,2, . . ., m−1}, l6= i, and 1 ≤ j1, . . . , jm−1 ≤ m−1 such that
g(El, sl1) +
m
X
k=1,k6=l,i
g(Ek, skjk) +g(Ei, sij)≤(m2−m+1)d.
Substituting g(El, sl1) ≥(m−1)d, for 1 ≤k ≤ m−1, k 6=i, l, g(Ek, skjk) ≥(m−1)d, g(Ei, sij) = (m−1)d+d, andg(Em, smjm)≥(m−1)d+ 1, we have:
(m−1)(m−1)d+ (m−1)d+d+ 1≤(m2−m+ 1)d
or, (m2−m+ 1)d+ 1≤(m2−m+ 1)d. (3.23)
equation (3.23) is a contradiction.
Case II: j=m.
From equation (3.22), we have:
m−1
X
k=1,k6=i,m
g(Ek, sk1) +g(Ei, sim) +g(Em, smjm)≤(m2−m+1)d.
Substituting g(Ek, sk1)≥(m−1)d for 1 ≤ k ≤ m −1, k 6= i, l, g(Ei, sim) = (m−1)d+d, and g(Em, smjm)≥(m−1)d+ 1, same contradiction as equation (3.23) can be obtained.
Claim 6 guarantees that there exist 1≤j2, ..., jm−1 ≤m−1 and 1≤jm≤msuch thatg(Ei, sji)≥
3.3 MDim-m network
(m−1)d+ 1 for 2≤i≤m. Now, from equation (3.21) of Claim 5, we have:
g(E1, s11) +g(E2, s2j2) +g(E3, s3j3) +· · ·+g(Em, smjm)≤(m2−m+ 1)d. (3.24) In Equation (3.24), substituting g(E1, s11) ≥(m−1)d, and for i6= 1: g(Ei, siji) ≥(m−1)d+ 1, we get:
(m−1)d+ (m−1)((m−1)d+ 1)≤(m2−m+ 1)d or, (m−1)(md+ 1)≤(m−1)md+d
or, d≥m−1.
For any positive integerm≥3, we now show that the MDim-mnetwork has aw-dimensional vector routing solution for any w ≥m−1 (‘if part’). We first give an (m−1)-dimensional vector routing solution. We denote the message vector generated at the source sij by Xij, and thekth component of XijbyXijk. Let the edges inEifor 1≤i≤mcarry all components of the vectorXi1, and the lastm−2 components of each vectorXij for 2≤j≤m(Note thatm−1 + (m−2)(m−1) = (m−1)(m−1)).
The messages carried byei(m+1) for 1≤i≤mare shown in Table 3.1. It can be seen that if Table 3.1 is followed then a terminalt∈Ti needs no more thanm−1 symbols through (vm+1, t), and ift∈T∗, then (vm+1, t) carries no more than 2 symbols.
Table 3.1: Messages carried by {(ui, vm+1)|1≤i≤m} whend=m−1.
e1(m+1) e2(m+1) · · · e(m−1)(m+1) em(m+1) X121 X221 · · · X(m−1)21 Xm21
X131 X231 · · · X(m−1)31 Xm31
... ... · · · ... ... X1m1 X2m1 · · · X(m−1)m1 Xmm1
Now, if the terminals that demand the exact same set of sources are considered as one (since they receive information from the same set of edges, this does not affect solvability), then MDim-m network is a sub-network of the Dim-m network of [44], and hence it has an m-dimensional vector routing solution. Furthermore, an (m−1)-dimensional vector routing solution and anm-dimensional vector routing solution guarantees an (2m−1)-dimensional and (2m−2)-dimensional vector routing solution. So the case for m = 3 is already solved. We now show that MDim-m has an (m+k)- dimensional vector routing solution for 1≤k≤m−3 andm≥4. For 1≤i≤m−1, let the edges in
TH-2118_136102023
Table 3.2: Messages carried by {(ui, vm+1)|1≤i≤m}when d=m+k.
e1(m+1) e2(m+1) · · · e(m−1)(m+1) em(m+1) X121 X221 · · · X(m−1)21 Xm11 X131 X231 · · · X(m−1)31 Xm21
... ... · · · ... ... X1(m−1)1 X2(m−1)1 · · · X(m−1)(m−1)1 Xm(m−2)1 X1m1 X2m1 · · · X(m−1)m1 Xm(m−1)1 X1m2 X2m2 · · · X(m−1)m1 Xmm1 X1m3 X2m3 · · · X(m−1)m3 Xm12
... ... · · · ... ... X1m(k+2) X2m(k+2) · · · X(m−1)m(k+2) Xmk2
Ei carry all components of the vector Xi1, lastm+k−1 components of Xij for 2≤j≤m−1, and the lastm−2 components ofXim(Note m+k+ (m−2)(m+k−1) +m−2 = (m−1)(m+k)). Edges inEm carries the lastm+k−2 components ofXmj for 1≤j ≤k, and the lastm+k−1 components of Xmj fork+ 1≤j ≤m (Notek(m+k−2) + (m−k)(m+k−1) = (m−1)(m−k)). The symbols not sent through{Ei|1≤i≤m} are carried by{ei(m+1)|1≤i≤m} as shown in Table 3.2.
It can be seen that for a terminal t∈Ti, the edge (vm+1, t) carries no more than m+ 1 symbols.
Notice that no terminal demands any two vectors from the set {Xim|1 ≤ i≤ m−1}. So if t ∈ T∗ demandsXim for some 1≤i≤m−1, then vm+1 carries no more than k+ 4 symbols (k+ 2 symbols of Xij, and maximum 2 symbols of Xml for some 1≤ l ≤ k). So we must have k+ 4 ≤m+k, or, m≥4.
We now show that there exists a network which has a (wk, wn) fractional linear solution if and only if w is greater than or equal tod.
Theorem 21. Consider the MDim-m network for m=dn+ 1. The n-factored network of MDim-m has a (w, wn) fractional linear network coding solution if and only ifw is greater than or equal to d.
Proof. Let then-factored network of MDim-m be denoted by MDimn-m. From Theorem 14 it can be seen that MDimn-mhas a (w, wn) fractional linear network coding solution if and only if MDim-mhas a (wn, wn) fractional linear network coding solution. However, from Theorem 20 we know that for the latter to holdwnmust be greater than or equal tom−1. Now,wn≥m−1⇒wn≥dn⇒w≥d.