In this section, we define a cooperation aware IDNC (C-IDNC) graph G(V,E) to represent both coding and transmission conflicts in one unified framework and select a set of trans- mitting devices and their XOR packet combinations in each D2D time slot. A transmission conflict occurs due to the simultaneous transmissions from multiple devices to a device in their coverage zones. Moreover, a coding conflict occurs due to the instant decodability constraint.
4.3.1
Vertex Set
To define the vertex set V of the C-IDNC graph G, given FSMF at time slott, we form an Yi×Hi local status matrix (LSM) Fi = [fk,l], ∀Uk ∈ Yi, Pl ∈ Hi, for a device Ui ∈ M such
that: fk,l =
0 if packet Pl is received by device Uk,
1 if packet Pl is missing at device Uk.
(4.7)
Note that the rows in LSM Fi represent the devices which are in the coverage zone of device
Ui and the columns in LSM Fi represent the packets in the Has set of device Ui which are
used for forming a transmitted packet from device Ui. Fig. 4.2 shows four LSMs for four
devices corresponding to SCM in (4.3) and FSM in (4.5).
We generate a vertex for a missing packet in each LSM at C-IDNC graphG. In fact, for each LSM Fi,∀Ui ∈ M, a vertex vi,kl is generated for a packet Pl ∈ {Hi ∩ Wk},∀Uk ∈ Yi.†
In other words, a vertex is generated for a missing packet of another device in Yi, which †Note that vertex v
i,kl represents a transmission from device Ui ∈ M to a neighboring device Uk ∈ Yi
also belongs to the Has set Hi of potential transmitting device Ui. Note that a missing
packet at a device can generate more than one vertex in graph G since that packet can be present in multiple LSMs. Once the vertices are generated in C-IDNC graph G, two vertices vi,kl and vr,mn are adjacent (i.e., connected) by an edge due to either a coding conflict or a
transmission conflict.
4.3.2
Coding Conflicts
Two vertices vi,kl and vr,mn are adjacent by an edge due to a coding conflict if one of the
following two conditions holds:
• C1: Pl 6= Pn and Uk = Um. In other words, two vertices are induced by different
missing packets Pl and Pn at the same device Uk.
• C2: Uk 6= Um and Pl 6= Pn but Pl ∈ H/ m or Pn ∈ H/ k. In other words, two different
devices Uk and Um require two different packets Pl and Pn, but at least one of these
two devices does not possess the other missing packet. As a result, that device cannot decode a new packet from the XOR combination of Pl⊕Pn.
4.3.3
Transmission Conflicts
Two vertices vi,kl and vr,mn are adjacent by an edge due to a transmission conflict if one of
the following three conditions holds:
• C3: Ui 6= Ur and Uk = Um ∈ {Yi∩ Yr}. In other words, two vertices representing
the transmissions from two different devices Ui and Ur to the same device Uk in the
coverage zones of both transmitting devices Ui and Ur. This prohibits transmissions
from two different devices to the same device in the common coverage zone and prevents interference at that device from multiple transmissions.
• C4: Ui 6= Ur and Uk 6= Um but Uk ∈ {Yi∩ Yr} or Um ∈ {Yi ∩ Yr}. In other words,
two vertices representing the transmissions from two different devicesUi andUr to two
different devices Uk andUm, but at least one of these two devices Uk and Um is in the
from device Ur to device Um in the case of transmission from device Ui to device Uk,
and vice versa.
• C5: Ui 6= Ur but Ui =Um or Ur = Uk. In other words, two vertices representing the
transmissions from two different devicesUiandUr, but at least one of these two devices
Ui and Ur is targeted by the other device. This prohibits transmission from a device in
the case of that device is already targeted by another device, and vice versa. In other words, a device cannot be a transmitting device and a targeted device simultaneously.
4.3.4
Maximal Independent Sets
With this graph representation, we can define all feasible coding and transmission conflict- free decisions by the set of all maximal independent sets in C-IDNC graph G.
Definition 17. (Independent Set) An independent set or a stable set in a graph is a set of pairwise non-adjacent vertices.
Definition 18. (Maximal Independent Set) A maximal independent set (denoted by κ) is an independent set that cannot be extended by including one more vertex without violating pairwise non-adjacent vertex constraint. In other words, a maximal independent set is an independent set that is not subset of any larger independent set [102].
Each device can have at most one vertex in a maximal independent set κ representing either a transmitting device or a targeted device. Moreover, the selection of a maximal independent set κ is equivalent to the selection of a set of transmitting devices Z(κ) =
{Ui|vi,kl ∈ κ} and a set of targeted devices X(κ) = {Uk|vi,kl ∈ κ}. Each of the selected
transmitting devices forms a coded packet by XORing the source packets identified by the vertices in κ representing transmission from that device.
Example 9. The new C-IDNC graph G corresponding to SCM in (4.3) and FSM in (4.5)
is shown in Fig. 4.3. An example of a maximal independent set of this graph is κ =
{v2,1,1, v3,4,1}. Here, the set of transmitting devices isZ(κ) ={U2, U3} and the set of targeted
v1,2,3 v3,2,2
v2,1,1 v3,4,1
κ1 ={v1,2,3}
κ2 ={v2,1,1, v3,4,1}
κ3 ={v3,4,1, v3,2,2}
Figure 4.3: Cooperation aware IDNC graph corresponding to SCM in (4.3) and FSM in (4.5).