Parallel and Successive Resource Allocation for V2V Communications in Overlapping Clusters

1 P arallel and Successi v e Resource Allocation for V2V Communications in Ov erlapping Clusters Luis F . Abanto-Leon, Arie K oppelaar , Sonia Heemstra de Groot Abstract The 3rd Generation Partnership Project (3GPP) has introduced in Rel. 14 a novel technology referred to as vehicle–to–v ehicle (V2V) mode-3 . Under this scheme, the eNodeB assists in the resource allocation process allotting sidelink subchannels to vehicles. Thereupon, vehicles transmit their signals in a broadcast manner without the interv ention of the former one. eNodeBs will thereby play a determinativ e role in the assignment of subchannels as they can effecti vely manage V2V traffic and prev ent allocation conflicts. The latter is a crucial aspect to be enforced in order for the signals to be recei ved reliably by other vehicles. T o this purpose, we propose two resource allocation schemes namely bipartite graph matching-based successiv e allocation (BGM-SA) and bipartite graph matching-based parallel allocation (BGM-P A) which are suboptimal approaches with lesser complexity than exhausti ve search. Both schemes incorporate constraints to pre vent allocation conflicts from emer ging. In this research, we consider ov erlapping clusters only , which could be formed at intersections or merging highw ays. W e show through simulations that BGM-SA can attain near-optimal performance whereas BGM-P A is subpar but less complex. Additionally , since BGM-P A is based on inter-cluster vehicle pre-grouping, we explore different metrics that could ef fecti vely portray the o verall channel conditions of pre-grouped vehicles. This is of course not optimal in terms of maximizing the system capacity—since the allocation process would be based on simplified surrogate information—but it reduces the computational complexity . Index T erms weighted bipartite graph matching, radio resource allocation, broadcast vehicular communications, sidelink I . I N T RO D U C T I O N In the last months we hav e been witness to an enormous effort from academia and industry in dev eloping nov el techniques across the many fronts of vehicle–to–v ehicle (V2V) communica- tions, which is to become a piv otal role player in the fifth generation of wireless systems. W ithin 2 the many use cases of V2V communications, safety-related services are unquestionably among the most important and challenging. Further enhancements capable of guaranteeing low latency and high reliability would become inestimable assets for deployment of fully-connected vehicle systems with the potential to reduce the amount of road traf fic accidents [1]. Nev ertheless, due to extreme mobility and highly v arying channel conditions, the stringent requirements for this type of scenario are not so straightforward to fulfill [5]. Hence, V2V communications calls for further research and comprehensiv e field tests before it can become a trustworthy technology . In this work, we consider that vehicles periodically broadcast short-term signals called coop- erati ve a wareness messages (CAMs) [2]. A CAM message—which is transported over a sidelink subchannel—contains meaningful information of a vehicle, e.g. speed, position, direction, that dri vers and /or autonomous vehicles can harness for making improv ed and more rational deci- sions. In V2V mode-3 , a crucial tar get that eNodeBs must guarantee is a time-domain conflict-free assignment of subchannels [4]. Con versely to traditional cellular systems where communications are controlled by the eNodeB and are virtually point–to–point links between mobile users, in V2V mode-3 data traffic is not subject to management. For instance, if we consider a cellular system with 4 users and therefore two point–to–point links, the eNodeB can allocate the two uplink transmit users in the same time subframe but in different frequency subchannels. Afterwards, via do wnlink the other two users may ev en receiv e in the same subframe the corresponding data from the senders. On the other hand, V2V mode-3 operates in a broadcast manner where transmission and reception are implemented without intervention of eNodeBs. Therefore, due to the absence of a controller that dictates the uplink and downlink instants, only one vehicle in the cluster can transmit at a time while the others receiv e. If two or more vehicles transmit concurrently , the data sent by one will not reach the other , thus originating a conflict. Nev ertheless, a subchannel that serves a vehicle in a certain cluster can be repurposed by other, if the latter vehicle belongs to a different cluster . Thus, eNodeBs will play a determinati ve role in ef fectiv ely allocating subchannels to in-coverage vehicles. W e formulate the resource allocation problem as a weighted bipartite graph matching where the aim is to find a perfect one–to–one vertex assignment with maximal sum-rate capacity . W e propose two suboptimal resource allocation approaches, namely ( i ) bipartite graph matching- based successi ve allocation (BGM-SA) and ( ii ) bipartite graph matching-based parallel allocation (BGM-P A). The former one is a cluster-wise sequential scheme that performs allocation with priority , from the most to the least constrained cluster . The latter algorithm is based on a primary 3 stage of random vehicle pre-grouping follo wed by a secondary resource allocation stage. In BGM- P A, we hav e experimented with dif ferent metrics in order to discov er one that could effecti vely depict the channel conditions of a set of pre-grouped vehicles, while still providing an acceptable sum-rate capacity value. W e have employed the Kuhn-Munkres algorithm [6] as a basis for both algorithms. Moreover , modifications hav e been considered to enforce intra-cluster constraints and thus prev ent conflicts. Our paper is structured as follows. In Section II, we explain the motiv ation of our work and succinctly describe our contributions. In Section III, we describe the sidelink channel structure for V2V broadcast communications. In Section IV , we formulate the resource allocation problem. In Section V and Section VI, the proposed approaches BGM-SA and BGM-P A are presented, respecti vely . In Section VII, we discuss simulation results in detail for sev eral scenarios. Finally , Section VIII is dev oted to summarizing our conclusions. I I . M OT I V A T I O N A N D C O N T R I B U T I O N S The moti v ation of this paper can be clearly explained through Fig. 1. W e observe two communications clusters; one consisting of 7 vehicles, namely { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 } , whereas the remaining cluster consists of 6, i.e. { v 5 , v 6 , v 7 , v 8 , v 9 , v 10 } . While there are no conflicts in the 7-vehicle cluster—as vehicles hav e been assigned orthogonal time-domain subchannels—in the remaining cluster we can identify a conflict. Observe that in subframe t = 4 , vehicles v 8 and v 10 hav e been assigned subchannels located in the same subframe. Thus, these subchannels are non-orthogonal in time domain and therefore, v 8 and v 10 will not be able to receiv e each other’ s information (assuming that vehicles are equipped with half-duplex PHY). In order to prev ent this kind of issues from occurring, we propose two resource allocation schemes. Our contributions are summarized in the follo wing points. • In Section IV , we introduce a compact matrix formulation for the resource allocation problem when multiple clusters are considered. • The mentioned formulation includes additional constraints to prev ent intra-cluster time- domain conflicts. It also contemplates a notation for representing vehicles with multiple cluster memberships, which facilitates modeling of vehicles at intersections. • In Section V , we propose a scheme called BGM-SA which allocates subchannels to vehicles in a sequential and hierarchical manner . BGM-SA is capable of attaining near-optimal performance at lower complexity than exhausti ve search. 4 V ehicle v 1 V ehicle v 2 V ehicle v 3 V ehicle v 4 V ehicle v 5 V ehicle v 6 V ehicle v 7 V ehicle v 8 V ehicle v 9 V ehicle v 10 v 10 v 8 v 2 v 4 v 5 v 7 v 6 v 1 v 9 v 3 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 Fig. 1: Sidelink V2V broadcast communications scenario • In Section VI, we introduce a second approach called BGM-P A which is based on ( i ) inter-cluster vehicle pre-grouping and ( ii ) subchannel assignment. Due to pre-grouping, the performance of BGM-P A is modest compared to BGM-SA but with lower complexity . • W e also devise six simple metrics to optimize the allocation of subchannels in BGM-P A and the performance of each is ev aluated. I I I . S I D E L I N K R E S O U R C E S C H A N N E L I Z A T I O N W e consider that uplink/downlink and sidelink spectrum resources are decoupled from each other . W e assume that the resources utilized for V2V sidelink communications are located in the intelligent transportation systems (ITS) band [3] whereas uplink/downlink spectrum resources are located in bands that usually serve cellular users. As mentioned before, in V2V mode-3 vehicles periodically broadcast CAM messages to their counterparts via sidelink [7]. Howe ver , uplink is used by vehicles to report their own channel conditions to the eNodeB. Downlink is employed for ( i ) signaling and for ( ii ) notifying vehicles on the subchannels they hav e been assigned. The channelization of sidelink spectrum resources can be regarded as a time-frequency arrangement 5 Control Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T L (ms) Frequency (MHz) T T T r 1 r 2 r K r K +1 r 2 K r K ( L − 1)+1 r K ( L − 1)+2 r K L B = 1 . 26 B = 1 . 26 B = 1 . 26 Fig. 2: Channelization for V2V communications of non-overlapping subchannels as sho wn in Fig. 2. The dimensions of each subchannel are T = 1 ms in time and B = 1 . 26 MHz in frequency , which to the best of our understanding is suf ficient for con v eying a CAM message. Moreover , there are L subframes and each contains K subchannels. Therefore, the total number of subchannels in this formation is K L . Furthermore, each subchannel r k (for k = 1 , 2 , . . . , K L ) consists of 7 resource blocks (RBs), where 5 RBs are used for data and 2 RBs for control. I V . P R O B L E M F O R M U L A T I O N Let J denote the total number of partially ov erlapping clusters. Thus, each cluster can be denoted as a set of vehicles V ( j ) , each consisting of N j vehicles (for j = 1 , 2 , . . . , J ). T o illustrate this description, consider Fig. 3, where the scenario is constituted by J = 4 partially overlapping clusters such that V (1) = { v 1 , v 2 , v 3 , v 4 , v 5 } , V (2) = { v 1 , v 2 , v 6 , v 7 } , V (3) = { v 1 , v 2 , v 8 , v 9 } , V (4) = { v 1 , v 2 , v 10 } and cardinalities N 1 = |V (1) | = 5 , N 2 = |V (2) | = 4 , N 3 = |V (3) | = 4 , N 4 = |V (4) | = 3 with vehicles { v 1 , v 2 } lying at the intersection. Notice that each vehicle has an absolute labeling and a corresponding relativ e one which is with respect to the clusters a vehicle is members of 1 . In addition, there exists a set of allotable subchannels which are managed by the eNodeB. In sum, there exists a whole set of vehicles V distributed into J clusters which are seeking to be assigned 1 In this section only the absolute labeling is employed. The relative notation will be used in Section V , where Fig. 3 is repurposed to illustrate an example. 6 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 Cluster V (1) Cluster V (2) Cluster V (3) Cluster V (4) v (1) 1 = v (2) 1 = v (3) 1 = v (4) 1 v (1) 2 = v (2) 2 = v (3) 2 = v (4) 2 v (1) 3 v (1) 4 v (1) 5 v (2) 3 v (2) 4 v (3) 3 v (3) 4 v (4) 3 Fig. 3: Overlapping vehicular clusters a resource from a set of allotable subchannels R . Considering the absolute labeling, this problem can be represented as a weighted bipartite graph matching between two disjoint sets: vehicles and subchannels. Such a graph is denoted by G ( V , R , E ) , where V = ∪ j V ( j ) = { v 1 , v 2 , . . . , v N } , R = { r 1 , r 2 , . . . , r K L } and E = V × R = { e 11 , e 12 , . . . , e N ( K L ) } is the set of edges. The total number of vehicles is denoted by N = P j |V ( j ) | − P j 0 |V ( j ) ∩ V ( j 0 ) | for j 6 = j 0 , whereas ˆ N = | T j V ( j ) | represents the number of vehicles at the intersection. W e can thereby represent vehicles and subchannels as vertices. Thus, the line connecting two vertices—a vehicle v i ∈ V with a subchannel r k ∈ R —is called an edge e ik . Each edge e ik has a corresponding weight c ik that in our case represents the achiev able capacity that vehicle v i can attain in subchannel r k . Therefore, c ik = B log 2 (1 + SINR ik ) , where B is the subchannel bandwidth and SINR ik is the signal–to–interference–plus–noise ratio (SINR) that vehicle v i senses in subchannel r k . The objecti ve function is the maximization of the system sum-rate capacity subject to satisfying the allocation constraints. The two types of constraints that must be enforced are ( a ) the intra-cluster allocation restrictions, which prev ent time-domain conflicts and ( b ) the one–to–one vertex matching conditions, which impose that each vehicle is assigned exactly one subchannel. This is equiv alent to finding a vector x that maximizes (1a) while satisfying the 7 constraints (1b). Thus, max c T x (1a) sub ject to   I N × N ⊗ 1 1 × L Q J × N ⊗ I L × L   ⊗ 1 1 × K ! | {z } constraint matrix x = 1 (1b) where ⊗ represents the tensor product operator , c ∈ R M , x ∈ B M with M = N LK . I N × N and I L × L are identity matrices whereas 1 1 × L and 1 1 × K are vectors whose elements are all 1. Q ∈ B J × N is the membership matrix which portrays the association of vehicles to sev eral clusters. Thus, if a vehicle v i belongs to cluster V ( j ) , the element q j i is set to 1; otherwise it is zero. Also, x = [ x 1 , 1 , . . . , x 1 ,K L , . . . , x N , 1 , . . . , x N ,K L ] T , c = [ c 1 , 1 , . . . , c 1 ,K L , . . . , c N , 1 , . . . , c N ,K L ] T are the solution vector and weight vector , respectiv ely . The relation between the graph edges e ik and the solution v ector x is the follo wing. First, we hav e assumed that the graph vertices are fully connected, i.e. there are no prohibited assignments at the beginning of the resource allocation process, and therefore e ik = 1 ∀ i, k . The solution to the problem x is a subset of edges e ik called matching whose weights c ik provide a maximal sum while respecting the constraints. Therefore, if the edge e ik is part of such optimal matching , then x ik = 1 otherwise x ik = 0 . V . P R O P O S E D A L G O R I T H M B G M - S A W ithout recurring to exhausti ve search to solve (1), we propose to perform the allocation process in an ordered and sequential manner , which will lead to a suboptimal solution. It should be noted that, the degree of constrainedness in allocating subchannels is related to the cardinality of the cluster . Hence, the assignment of subchannels becomes more complicated when the number of vehicles in the cluster is large. Considering the foregoing facts, the allocation process in BGM- SA starts by assigning subchannels to the cluster with largest cardinality and terminates when the cluster with smallest cardinality has been processed. T o illustrate this idea with an example, we consider Fig. 3. Based on the cardinality criterion, the ordered clusters are |V (1) |≥ |V (3) |≥ |V (2) |≥ |V (4) | . Thus, once each of the 5 vehicles in V (1) has been alloted a subchannel, the process will continue with cluster V (3) , where only v 8 and v 9 should be allocated since v 1 and v 2 obtained their own subchannels when V (1) was processed. Afterwards, v 6 and v 7 will receiv e their respecti ve subchannels. And the last vehicle to be serviced is v 10 . At ev ery allocation phase, vehicles must be accommodated such that they do not generate conflicts to vehicles already alloted. 8 v ( j ) 1 v ( j ) 2 . . . v ( j ) N j r 1 r 2 . . . r K r K +1 r K +2 . . . r 2 K . . . r K ( L − 1)+1 r K ( L − 1)+2 . . . r K L macro- macro- macro- verte x R 1 verte x R 2 verte x R L V ehicles: V ( j ) Resources: R Fig. 4: Constrained weighted bipartite graph T o prepare the ground for the formulation of BGM-SA, we start by isolating a single cluster V ( j ) as shown in Fig. 4, where vehicles and subchannels are represented by black and white vertices, respectively . The set R is constituted by K L v ertices which are grouped into L disjoint verte x subsets {R l } L l =1 that we call macro-vertices, i.e. R = ∪ L l =1 R l , R l ∩ R l 0 = ∅ , ∀ l 6 = l 0 . Each macro-verte x R l is a congregation of K vertices, i.e. a collection of K subchannels in the same time subframe. Considering the relati ve labeling, the bipartite graph shown in Fig. 4 is denoted by G ( V ( j ) , R , E ( j ) ) . Thus, the edge e ( j ) ik connects vehicle v ( j ) i ∈ V ( j ) with a subchannel r k ∈ R . Also, the edge weights are defined as c ( j ) ik = B log 2 (1 + SINR ( j ) ik ) . Instead of solving the allocation for the whole system in (1), we solve a graph matching subproblem for each cluster V ( j ) , for j = 1 , 2 , . . . , J . Therefore, we optimize an objectiv e function that maximizes the sum-rate capacity of each cluster V ( j ) , which is expressed by max c T j x j (2a) sub ject to   I N j × N j ⊗ 1 1 × L 1 1 × N j ⊗ I L × L   ⊗ 1 1 × K ! | {z } constraint matrix x j = 1 (2b) where c j ∈ R M j , x j ∈ B M j with M j = N j K L and L ≥ N j . For completeness, we add a number of virtual v ehicles with zero-v alued edge weights, such that N j = L and M j = M = K L 2 ∀ j . 9 Therefore, the solution and weight vectors are gi ven by x j = [ x ( j ) 1 , 1 , . . . , x ( j ) 1 ,K L , . . . , x ( j ) L, 1 , . . . , x ( j ) L,K L ] T and c j = [ c ( j ) 1 , 1 , . . . , c ( j ) 1 ,K L , . . . , c ( j ) L, 1 , . . . , c ( j ) L,K L ] T , respectiv ely . It is important to notice that the two types of allocation constraints mentioned in Section IV are also enforced in (2b). This means that each vehicle will be alloted exactly one subchannel and the resource allocation will guarantee that no two vehicles—in the same cluster—transmit in subchannels of same subframe. Although the constraint matrices (1b) and (2b) are similar , it is possible to exploit the structure of (2b) and further simplify the allocation problem. Recall that the time-domain orthogonality requirement on alloted subchannels is compulsory for vehicles in the same communication cluster only . It can be shown that enforcing this requirement is equiv alent to aggregating vertices into macro- vertices, which in addition simplifies the complexity of (2), since the dimensionality is reduced. Such said vertex aggregation can be modeled as a matrix transformation, which is depicted in Fig. 5. Thus, the problem in (2) can be recast as (3) max d T j y j sub ject to   I L × L ⊗ 1 1 × L 1 1 × L ⊗ I L × L   y j = 1 (3) where y j = [( y j ) 1 , 1 , . . . , ( y j ) 1 ,L , . . . , ( y j ) L, 1 , . . . , ( y j ) L,L ] T ∈ B L 2 and d j = lim β →∞ 1 β ◦ log  ( I M × M ⊗ 1 1 × K )e ◦ β c j  ∈ R L 2 . The function ◦ log {·} represents the element-wise natural logarithm whereas e ◦{·} is the Hadamard exponential [8]. Note that (3) is equi valent to finding a maximal matching in a graph e G ( j ) = ( V ( j ) , e R , e E ( j ) ) where e R = { ˜ r 1 , ˜ r 2 , . . . , ˜ r L } . Also, the edge weights between vertices in this resultant problem is d j , whose elements d ( j ) il depict the weight between vertices v ( j ) i and ˜ r l , for l = 1 , 2 , . . . , L . Approaching (3) by means of finding a maximal matching in e G ( j ) is less complex than solving (2) through G ( j ) because | e R| is K times smaller than |R| . Thus, instead of solving either (1) via exhausti ve search in an optimal manner or (2) sub-optimally through any av ailable method, we can attain the same performance as (2) by solving (3) at lesser computational complexity . I M × M ⊗ 1 1 × K I M × M ⊗ 1 1 × K × diag ( · ) x j c j y j d j Fig. 5: T ransformation process 10 Algorithm 1: Bipartite Graph Matching-based Successi ve Allocation (BGM-SA) Input: A bipartite graph e G ( j ) = ( V ( j ) , e R , e E ( j ) ) for each cluster , such that   V ( j )   =   e R   for completeness. Output: A set of perfect matchings M ( j ) , j = 1 , . . . , J . begin f or j = 1 : J do Step 1a: Generate an initial feasible label- ing l j . Step 1b: Compute the equality subgraph G ( j ) l = { e v r | l j ( v ) + l j ( r ) = d v r } for ∃ v ∈ V ( j ) , ∃ r ∈ e R , e v r ∈ e E ( j ) . Step 1c: Find an arbitrary matching M ( j ) in G ( j ) l . Step 2: T erminate the algorithm if the matching M ( j ) is perfect. Step 3: Find a vertex v 0 ∈ V ( j ) that has not been matched in M ( j ) and set S ( j ) = { v 0 } , T ( j ) = {∅} . Step 4: Go to Step 6 if N ( S ( j ) ) 6 = T ( j ) . Step 5a: Compute the labeling l 0 j , ∀ verte x z l 0 j ( z ) =          l j ( z ) − ε, if z ∈ S ( j ) l j ( z ) + ε, if z ∈ T ( j ) l j ( z ) , otherwise where ε = min v ∈S ( j ) r ∈ e R\T ( j )  l j ( v ) + l j ( r ) − d v r  Step 5b: Compute the equality subgraph G 0 ( j ) l . Step 5c: Update the equality subgraph and labeling: G ( j ) l ← G 0 ( j ) l , l j ← l 0 j . Step 6a: Find a vertex r ∈ N ( S ( j ) ) \ T ( j ) . Step 6b: Perform S ( j ) ← S ( j ) ∪{ u } , T ( j ) ← T ( j ) ∪ { r } and go to Step 4 if ∃ e ur ∈ M ( j ) such that u ∈ V ( j ) . Step 7a: Find an alternating path h e ˆ v 0 ˆ r 0 7→ e ˆ v 1 ˆ r 1 7→ . . . 7→ e ˆ v m ˆ r m i such that ˆ v n ∈ V ( j ) , ˆ r n ∈ e R , ˆ r m = r , e ˆ v n ˆ r n ∈ { G ( j ) l \M ( j ) } for n = 0 , 1 , . . . , m , e ˆ v n ˆ r n − 1 ∈ M ( j ) for n = 1 , 2 , . . . , m . Step 7b: Augment the previous matching M ( j ) ←  M ( j ) ∪ { e ˆ v n ˆ r n } n = m n =0  \{ e ˆ v n ˆ r n − 1 } n = m n =1 . Step 7c: Go to Step 2 . Step 8: Update the edges in e R such that e v 0 r 0 ← 0 , ∀ r 0 ∈ R , ∀ v 0 ∈  {V ( j k 1 ) ∩ V ( j k 2 ) ∩· · ·∩ V ( j k q ) }\V ( j )  if e v 0 r 0 ∈ M ( j ) . 11 In order to solve (3), we propose BGM-SA which is based on [6] and sho wn in Algorithm 1. Recall that since the allocation is performed in a hierarchical and sequential manner , we first sort the clusters according to their cardinality . Thus, we assume that the clusters hav e been labeled such that |V ( j ) |≥ |V ( j +1) | . W e believ e that the algorithm is self-explanatory and therefore we will not discuss the steps in detail. Instead, we introduce the follo wing definitions in case they were necessary for its understanding. Labeling : A feasible vertex labeling in the bipartite graph e G ( j ) is a real-v alued function l j : V ( j ) ∪ e R → R such that l j ( v ) + l j ( r ) ≥ d v r , ∀ v ∈ V ( j ) , ∀ r ∈ e R . An initial feasible labeling l j can be obtained by assigning l j ( v ) = max r ∈ e R d v r and l j ( r ) = 0 . Because Algorithm 1 operates in a sequential manner processing one cluster e V ( j ) at a time, the j indexing has been dropped to simplify the notation and thus d v r is equiv alent to d ( j ) v r . Equality subgraph : An equality subgraph G ( j ) l obtained from a labeling l j contains edges e v r ∈ e E ( j ) such that l j ( v ) + l j ( r ) = d v r holds, as described in Step 1b . P erfect matching : A matching M ( j ) is said to be perfect when e very vertex of a graph is linked to only one edge of the matching. Neighborhood of a set : In a bipartite graph, the neighborhood of a vertex v ∈ V ( j ) is defined by N ( v ) = { r | e v r ∈ G ( j ) l } . Therefore, N ( S ) = ∪ t N ( s t ) , ∀ s t ∈ S (See Step 6 ). For each cluster V ( j ) , the input is a bipartite graph e G ( j ) = ( V ( j ) , e R , e E ( j ) ) and the output is a matching M ( j ) that will contain the association of vehicles  in V ( j )  and subchannels  in e R  . Such matching M ( j ) is a collection of edges e ( j ) il that can be mapped to y j . Thus, if e ( j ) il ∈ M ( j ) , then ( y j ) il = 1 or ( y j ) il = 0 otherwise. V I . P RO P O S E D A L G O R I T H M B G M - P A The target of inter-cluster vehicle pre-grouping in BGM-P A is to decrease the computational complexity of BGM-SA. In this regard, the allocation problem can be completed in one run by forming a virtual single cluster of vehicles, in contrast to BGM-SA that requires to allot subchannels for each cluster on a consecutiv e basis. Thus, each vehicle group is denoted by W u , such that W = ∪ u W u and W u ∩ W u 0 = ∅ , ∀ u 6 = u 0 , u = 1 , 2 , . . . , |W | . Such scenario is depicted in Fig. 6, where the outcome of pre-grouping is shown. e R is the set of subchannels whereas 12 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 V (1) V (2) V (3) V (4) W 1 W 2 W 3 W 4 W 5 W 1 W 2 W 3 W 4 W 5 ˜ r 1 ˜ r 2 . . . ˜ r L W e R Pre- grouping W Fig. 6: Inter-cluster vehicle pre-grouping for BGM-P A ˜ r l ( l = 1 , 2 , . . . , L ) are the same resources we referred to in (3). Hence, there are |W | = 5 groups of vehicles: W 1 = { v 1 } , W 2 = { v 2 } , W 3 = { v 5 , v 6 , v 8 , v 10 } , W 4 = { v 4 , v 7 , v 9 } , W 5 = { v 3 } . Note that grouping is applied only to those vehicles that do not lie at the intersection. The selection of vehicles per cluster is done randomly but aiming at assembling as many vehicles as possible. For instance, W 3 contains 4 vehicles because that was the maximum number allowable, i.e. one vehicle per each cluster at most. On the other hand, W 4 consists of 3 v ehicles. Finally , v 3 was the last vehicle remaining and therefore, it by itself constitutes W 5 . Although pre-grouping is beneficial for decreasing the allocation complexity , it also originates difficulties on how to represent the overall channel conditions of each collection of vehicles W u . The formulation of this problem is similar to (2), except that J = 1 , because after pre-grouping there will exist one cluster only . Therefore, the problem can be further reduced and thus adopt a form identical to (3). Nev ertheless, instead of employing Algorithm 1, we use Algorithm 2, which in essence is similar . A central issue to take into consideration is that the resultant edge weights ˜ d ul between W u and ˜ r l , must be a joint metric that can fairly represent the ov erall channel conditions of a group of vehicles. Therefore, if such a group is defined by W u = { w u 1 , w u 2 , . . . , w u ( m u ) } with m u representing the number of vehicles in the group, then the resultant edge weight is ˜ d ul = metr ic ( d ( u 1) l , d ( u 2) l , . . . , d ( um u ) l ) . T o this purpose, we hav e devised six dif ferent metrics that are defined as follo ws. Minimum (MIN) : For each ˜ r l , select the smallest edge weight ˜ d ul among all the vehicles v 0 ∈ W u . This is a plausible metric because if fair allocation can be guaranteed for the least 13 fa vored vehicle, then the other vehicles in the group will at least experience equal or better channel conditions. Maximum (MAX) : This metric is similar to the previous one, except that the maximum value is chosen instead of the minimum. A vera ge (A VE) : This metric considers the av erage channel conditions of all the vehicles in the group. In verse of variance (IV AR) : This metric measures the de viation of the channel conditions in a group of vehicles. If IV AR is large, then the channel conditions span a large range of qualities. When IV AR is small, we can only infer that the channel conditions are similar for the vehicles but it is difficult to know whether these are good or not. Minimum plus maximum (MPM) : This is a merged metric that considers the ov erall effect of MIN and MAX metrics. Combined metrics (COMB) : This metric combines some of the metrics described above. Specif- ically to o vercome the shortcoming of IV AR and exploit the reasoning behind MIN , we define COMB = A VE + MIN - √ V AR , where V AR denotes variance. The computational complexity of exhausti ve search is O ( |R| ! / ( |R| − |V | )!) . On the other hand, when BGM-SA is solved through Algorithm 1 after dimensionality reduction via (3), the complexity is O (max { J |V | , J | e R|} 3 ) = O (max { J |V | , J K |R|} 3 ) whereas the complexity of BGM-P A is O (max {|V | , 1 K |R|} 3 ) . V I I . S I M U L AT I O N S In this section, we experiment with sev eral configurations considering different number of clusters and v ehicles. W e also v ary the number of v ehicles at the intersection in order to understand the impact on the allocation performance. W e ev aluate exhausti ve search, BGM- SA and BGM-P A using its six v ariants. In our system, we consider a message rate of 10 Hz and therefore, a new allocation is performed ev ery 100 ms for all the vehicles. In all the experiments shown onwards, we hav e averaged the results over 1000 simulations. In Fig. 7, we hav e considered J = 3 clusters with N 1 = 100 , N 2 = 90 and N 3 = 80 vehicles. The number of 14 Algorithm 2: Bipartite Graph Matching-based Parallel Allocation (BGM-P A) Input: A bipartite graph G = ( W , e R , E ) . Output: A perfect matching M . begin Drop the index j from Algorithm 1. Perform random pre-grouping of vehicles. Select an edge metric for the grouped sets of vehicles. Perform from Step 1 to Step 7 . Highest-Rate V ehicle System A verage Rate W orst-Rate V ehicle Second W orst- Rate V ehicle System Rate Standard Deviation 0 5 10 15 8 . 97 8 . 19 6 . 89 7 . 03 0 . 51 8 . 97 8 . 16 6 . 88 7 . 01 0 . 51 8 . 97 6 . 64 5 . 32 5 . 38 0 . 86 8 . 97 6 . 19 1 . 65 1 . 95 2 . 17 8 . 97 6 . 86 3 . 81 4 . 12 1 . 13 8 . 97 5 . 02 1 . 54 1 . 82 1 . 71 8 . 97 6 . 79 4 . 11 4 . 31 1 . 23 8 . 97 6 . 65 5 . 37 5 . 44 0 . 81 Rate [Mbits / s / subchannel] Exhaustiv e Search Proposed BGM-SA Proposed BGM-P A-MIN Proposed BGM-P A-MAX Proposed BGM-P A-A VE Proposed BGM-P A-IV AR Proposed BGM-P A-MPM Proposed BGM-P A-COMB Fig. 7: Data rate for N = 210 , L = 100 and K = 7 with J = 3 , N 1 = 100 , N 2 = 90 , N 3 = 80 , ˆ N = 30 Highest-Rate V ehicle System A verage Rate W orst-Rate V ehicle Second W orst- Rate V ehicle System Rate Standard Deviation 0 5 10 15 8 . 97 8 . 12 6 . 74 6 . 91 0 . 53 8 . 97 8 . 06 6 . 54 6 . 73 0 . 57 8 . 97 7 . 21 5 . 29 5 . 41 1 . 09 8 . 97 6 . 92 1 . 94 2 . 31 2 . 02 8 . 97 7 . 34 4 . 05 4 . 41 1 . 18 8 . 97 6 . 21 1 . 81 2 . 13 2 . 09 8 . 97 7 . 31 4 . 32 4 . 55 1 . 24 8 . 97 7 . 23 5 . 49 5 . 61 1 . 02 Rate [Mbits / s / subchannel] Exhaustiv e Search Proposed BGM-SA Proposed BGM-P A-MIN Proposed BGM-P A-MAX Proposed BGM-P A-A VE Proposed BGM-P A-IV AR Proposed BGM-P A-MPM Proposed BGM-P A-COMB Fig. 8: Data rate for N = 130 , L = 100 and K = 7 with J = 3 , N 1 = 100 , N 2 = 90 , N 3 = 80 , ˆ N = 70 vehicles at the intersection is ˆ N = 30 whereas the amount of vehicles in the system is N = 210 . W e hav e also chosen K = 7 and L = 100 . 15 In Fig. 7, we show 5 different criteria to e valuate the performance of the approaches. W e can observe that BGM-SA attains near-optimality as its performance is within 0 . 5% of error . As we had presumed, BGM-P A-MIN exhibits an acceptable performance compared to all other variants, being surpassed only by BGM-P A-COMB in most cases. Because BGM-P A-COMB is based on BGM-P A-MIN and in addition employs statistical information of the group of vehicles, it can in general achie ve superior performance under all the fiv e criteria. Howe ver , under the criterion system averag e rate , the best performance within the BGM-P A v ariants is attained by BGM- P A-A VE. This behavior results logical because BGM-P A-A VE considers—by definition—the av erage channel conditions. Therefore, if BGM-P A-COMB had not been introduced, we could hav e expected BGM-P A-MIN to perform best under the worst-r ate vehicle criterion, for the same reasons explained above. The v ariant BGM-P A-MPM, which is based on BGM-P A-MIN, can also attain acceptable performance under most of the criteria. On the other hand, BGM-P A-MAX and BGM-P A-IV AR are not capable of attaining good performance under worst-rate vehicle and system rate standar d deviation . These two criteria would usually exhibit a fa vorable behavior when the method can provide fairness. Nev ertheless, since BGM-P A-MAX is based on a greedy principle and BGM-P A-IV AR is by itself insufficient, both variants perform poorly . Fig. 8 illustrates a setup similar to Fig. 7 but with a change in the number of vehicles at the intersection, namely ˆ N = 70 . Thus, the number of vehicles in the system is N = 130 . W e can observe that because of the increment of ˆ N , the performance of all the approaches hav e changed. In some cases the performance improv es whereas in others degradation can be identified. Notice that BGM-SA still attains near -optimality but with a comparati vely increased error of 3% in contrast to the pre vious case. Howe ver , some BGM-P A variants hav e undergone a considerable upturn. The reason why the performance of BGM-SA has suffered degradation, is essentially due to the increase of number of vehicles at the intersection. More specifically , this means that when the first cluster V (1) is processed, the best subchannels will be selected for its N 1 = 100 vehicles. When the turn of V (2) comes, there will be ˆ N = 70 time subframes already in use, leaving only N 1 − ˆ N = 30 av ailable. Thus, the N 2 − ˆ N = 20 unalloted v ehicles of V (2) must be accommodated in those 30 remaining subframes. Notice that the remaining free subframes may not necessarily hav e subchannels with high SINR for the vehicles in V (2) , as this was nev er enforced during the allocation of V (1) . If there were fewer v ehicles at the intersection, e.g. ˆ N = 30 as in Fig. 7, BGM-SA would be able to achie ve higher performance as more unused subframes would be av ailable. On the other hand, we observe the opposite effect in BGM-P A. When the number 16 0 . 2 0 . 4 0 . 6 0 . 8 1 1 2 3 4 5 6 7 8 ˆ N /N j Rate [Mbits / s / subchannel] Exhaustiv e Search Proposed BGM-SA Proposed BGM-P A-MIN Proposed BGM-P A-MAX Proposed BGM-P A-A VE Proposed BGM-P A-IV AR Proposed BGM-P A-MPM Proposed BGM-P A-COMB Fig. 9: W orst-rate vehicle for L = 100 , K = 7 with J = 4 , N 1 = 100 , N 2 = 100 , N 3 = 100 , N 4 = 100 and varying ˆ N . of vehicles at the intersection ˆ N increases, its performance is boosted. The explanation to this outcome is that vehicles at the intersection are not grouped (this is done in order to pre vent conflicts). Thus, there are ˆ N = 70 vehicles at the intersection and at most 30 lying outside the that area (prior to pre-grouping). And as we may infer, the main performance degradation source for BGM-P A is grouping due to the difficulty of representing channel conditions of a group with a single metric. Thus, since there are fewer groups of vehicles compared to the pre vious case, the performance is improved. If we had considered a larger number of vehicles at the intersection such as ˆ N = 95 with N 1 = N 2 = N 3 = 100 , the performance of both BGM-P A-MIN and BGM-P A-COMB would hav e been within 6% of optimality . Fig. 9 shows the data rate experienced by the worst-r ate vehicle . In the abscissa, we v ary the ratio ˆ N /N j which represents the proportion of vehicles at the intersection to vehicles in each cluster . In this setup, we hav e considered that N 1 = N 2 = N 3 = N 4 = 100 and J = 4 clusters. For the reasons explained abov e, we expect that as the ratio ˆ N /N j approaches unity the performance of BGM-SA will decrease whereas the performance of BGM-P A will increase. In our opinion, lev eraging the worst-rate vehicle is a most important criterion as it guarantees a minimum achie v able rate for the least fav ored vehicle. Thus, judging from the results, we can say that the proposed BGM-SA, BGM-P A-MIN and BGM-P A-COMB are robust allocation schemes that are not prone to influence stemming from the div ersity of possible scenarios. Fig. 10 shows the cumulative distrib ution function (CDF) of the achie v able rates. In this 17 5 5 . 5 6 6 . 5 7 7 . 5 8 8 . 5 9 0 . 1 0 . 4 0 . 7 1 Rate x [bits / s / Hz] Pr  Rate < Rate x  Exhaustiv e Search Proposed BGM-SA Proposed BGM-P A-MIN Proposed BGM-P A-MAX Proposed BGM-P A-A VE Proposed BGM-P A-IV AR Proposed BGM-P A-MPM Proposed BGM-P A-COMB Fig. 10: Cumulativ e distribution function (CDF) of rate values for L = 100 , K = 7 with J = 3 , N 1 = 100 , N 2 = 90 , N 3 = 80 and ˆ N = 50 . scenario, we hav e considered J = 3 clusters with N 1 = 100 , N 2 = 90 , N 3 = 80 . Also, we have chosen ˆ N = 50 as it is an intermediate value between the most and least fa v orable scenarios for BGM-SA. W e observe that BGM-SA is similar in performance to exhausti ve search, and is undoubtedly superior to all other approaches. W e know , howe ver , that such additional gain is achie ved at the expense of higher complexity . W e also observe that the second and third best schemes are BGM-P A-COMB and BGM-P A-MIN, respectiv ely . Specifically , these two variants perform well in the low regime whereas they do not excel in the large regime. On th other hand, BGM-P A-MAX only performs well in the lar ge regime. For this reason, BGM-P A-MPM—which uses both the MAX and MIN metrics—also performs acceptably right in the whole range. V I I I . C O N C L U S I O N W e ha ve presented tw o resource allocation schemes for V2V broadcast communications. BGM- SA is based on successi ve matchings of weighted bipartite graphs whereas BGM-P A is capable of accomplishing the allocation—for all the clusters in the system—in a parallel fashion. W e showed through simulations that BGM-SA can attain near-optimality with a complexity that increases proportionally to the number of clusters. On the other hand, BGM-P A has a lower complexity b ut achie ves inferior performance. W e also presented six different metrics to improv e the matching performance of BGM-P A. Thus, the v ariants BGM-P A-COMB and BGM-P A-MIN are the most robust since they are not influenced by the system setup. In the allocation process, we always considered the enforcement of constraints in order to av oid intra-cluster allocation conflicts. A 18 nai ve assumption of this work is that clusters can always be perfectly defined although in practice this might be complicated to guarantee. R E F E R E N C E S [1] ”Crash Data Analyses for V ehicle-to-Infrastructure Communications for Safety Applications, ” FHW A-HR T -11-040, Nov . 2012. [2] ”ETSI TR 102 861; Intelligent Transport Systems (ITS); STDMA recommended parameters and settings for cooperative ITS; Access Layer Part, ” January 2012. [3] ”3GPP TS 36.213; T echnical Specification Group Radio Access Network; Evolved Universal T errestrial Radio Access (E-UTRA); Physical layer procedures; (Release 14) v14.2.0, ” March 2017. [4] L. Gallo and J. Harri, ”Self Organizing TDMA over L TE Sidelink, ” Research Report RR-17-329, Eurecom, January 2017. [5] W . Sun, E. G. Str ¨ om, F . Br ¨ annstr ¨ om, Y . Sui and K. C. Sou, ”D2D-based V2V communications with latency and reliability constraints, ” IEEE Globecom Wksps, pp. 1414-1419, Dec. 2014. [6] J. Munkres, ”Algorithms for the Assignment and Transportation Problems, ” Journal of the Society for Industrial and Applied Mathematics, V ol. 5, No. 1, pp. 32-38, 1957. [7] ”3GPP TR 36.885; T echnical Specification Group Radio Access Network; Study on L TE-based V2X Services; (Release 14) v14.0.0, ” June 2016. [8] F . Hiai, ”Monotonicity for entrywise functions of matrices, ” Journal of Linear Algebra and its Applications, V ol. 431, No. 8, pp. 1125-1146, September 2009.