4-DMWM Approach for Caching Based Optimal D2D Pairing and ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2016.2640270, IEEE Access

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4-DMWM Approach for Caching based Optimal D2D Pairing and Channel Allocation: Centralized and Distributed Algorithm Design Lu Miao, Bo Bai, Member, IEEE, and Wei Chen, Senior Member, IEEE

Abstract—Caching popular contents at mobile devices can potentially improve the quality of service for mobile users and relieve traffic burden of base station (BS) in cellular networks. In this paper, we jointly consider the resource allocation, the cached contents, and the distance between two devices for the optimal device pairing problem in centralized and distributed cases, where the BS is the central controller in the centralized case. The joint optimization problem of device-to-device (D2D) caching with channel allocation is formulated as a weighted 4-uniform hypergraph model. The optimal solution for the problem is 4dimensional maximum weighted matching (4-DMWM), which is NP-hard unfortunately. To approach the 4-DMWM with lowcomplexity, we adopt the greedy algorithm and the squareIMP algorithm in the centralized case. Moreover, distributed algorithms are also designed for the caching problem in both synchronous and asynchronous cases. The simulation results will illustrate that the squareIMP algorithm can be used to get a better transmission rate with the complexity of O(n5 ), while the greedy algorithm can be used in the case with stringent latency requirement for centralized 4-DMWM problem. The sum-rate of distributed asynchronous algorithm is close to the centralized greedy algorithm with the complexity of O(n2 ) for each device. However, the simulation result of the synchronous algorithm is slightly lower than the centralized algorithm, where each device performs O(n2 ) computational operations in each iteration. Therefore, the algorithms proposed in this paper can be used in different cases for solving optimal D2D pairing and channel allocation problem. Index Terms—D2D caching, weighted 4-uniform hypergraph, 4-dimensional maximum weighted matching (4-DMWM), centralized algorithm, distributed algorithm.

I. I NTRODUCTION ITH the increasing of user number and the growth of the peak traffic in base stations (BSs), device-to-device (D2D) communication is expected to improve the resource utilization efficiency and reduce peak traffic in BS by allowing mobile devices to directly communicate with each other [1], [2]. Due to the need for the throughput gain and the scarcity of wireless spectrum resource, caching holds the promise of providing substantial capacity gains by disseminating popular

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L. Miao, B. Bai, and W. Chen are with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail: [email protected], [email protected], [email protected]). Part of this work has been submitted to IEEE ICC 2017. This research was supported in part by the National Natural Science Foundation of China under Grant No. 61671269, No. 61322111, No. 61321061, and No. 61401249, the National Basic Research Program of China (973 Program) under Grant No. 2013CB336600 and No. 2012CB316000. Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No. 20130002120001.

content in the idle spectrum when the network is off-peak. Therefore, accurately caching the user demanding contents in mobile devices is one of the key issue to exploit the advantage of D2D communication schemes [3]. In wireless networks, the content caching problem has attracted much attention from both academia and industry. Caching aims to provide the spectral and energy efficiency gain by paying storage resource as a cost. However, not all the content items can be cached due to the buffer size. The performance limit of wireless caching has been studied in [4]. The fundamental limit of D2D caching was studied in [5]. There are also many works focusing on practical caching algorithms. In [6], the cache placement problem was investigated in femtocell networks, where femtocell BSs act as helpers to cache popular files. It is shown in [7] and [8] that caching may efficiently reduce the outage probability and the average transmit power in wireless small-cell networks and deviceto-device (D2D) networks respectively. In [7], the caching performance was studied in terms of outage probability and average delivery rate in wireless small-cell networks, where the cache-enabled BSs are distributed according to a Poisson point process. In [8], the throughput-outage tradeoff performance of wireless networks was investigated by exploiting clustered device caching via D2D communications. The caching and pushing problem with a finite receiver buffer was studied in [9], [10]. There are many studies using matching algorithm for resource allocation. In OFDMA systems, the problem of optimal subcarrier allocation can be solved by H-matching approach to minimize outage probability in [11]. To offload not only traffic but also computation overhead, some distributed algorithms are adopted by pushing computation and storage to network edges for the caching problem in D2D network [12], [13]. A belief propagation based distributed algorithm was used to solve the cache placement problem in [14]. In the distributed system, the appropriate channel allocation can achieve a low outage probability with low computation complexity [15]. To the best of our knowledge, however, these studies have not considered the joint optimization problem of content caching, resource allocation and device pairing to improve the throughput in D2D networks. In this paper, we consider the problem of caching based optimal D2D pairing and channel allocation. In the considered system, each mobile device caches some popular contents. 3GPP Proximity Services (ProSe) discovery can provide registration of both services offered by devices, and application requests of users in [16]. In the same way, a device can get the

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content caching information of users in D2D network. When some users request the contents, the system will determine which device provides help and allocate the channel at the same time, so as to maximize the system throughput. Inspired by [17], the considered joint optimization problem is first formulated as a weighted 4-uniform hypergraph model. In this context, the optimal solution for the caching based D2D pairing and channel allocation problem is the 4-dimensional maximum weighted matching (4-DMWM) in the weighted 4-uniform hypergraph. Unfortunately, the multi-dimensional matching problem is NP-hard in general. Therefore, we adopt some approximation algorithms in this paper with low complexity in centralized and distributed manners. Recently proposed cloud radio access networks (CloudRAN) is a typical centralized cloud computing-based architecture [18], while fog radio access networks (Fog-RAN) is a distributed architecture that leverages caching capabilities at the wireless edge nodes for cellular networks [19]. In [20] we focused our attention on the centralized 4-DMWM problem under the control of BS. In the centralized case, based on the squareIMP algorithm proposed in [21], we first propose high performance algorithm with the complexity of O(n5 ), which nearly achieves the optimal throughput of the exhaustive search. Moreover, to fulfill the stringent latency requirement, we further proposed a greedy based algorithm, which is shown to achieve the 1/2 optimal performance in [22]. Usually, BS in centralized case is required to collect the whole information of the network, which introduces much overhead. However, coordination may not be possible with no central controller exists in some self-organized networks [14]. Moreover, the underlay caching and interference management algorithms often bring in much extra computation overhead for the central controller. Therefore, we use distributed algorithms to solve the caching based optimal D2D pairing and channel allocation problem. In the distributed algorithm, we consider synchronous and asynchronous scenarios for the caching based optimal D2D pairing and channel allocation problem. A bipartite graph matching problem can be solved in [23]. Inspired by [23], we convert the weighted 4-uniform hypergraph to a weighted bipartite graph. After the transformation, the 4DMWM problem can be formulate as a weighted bipartite graph matching problem with a little performance loss in the distributed synchronous system. To solve the weighted matching problem, belief propagation (BP) is an effective algorithm that explained in [24], [25]. In the distributed synchronous system, a distributed synchronous algorithm based on BP is proposed for the weighted bipartite graph matching problem, so that we obtain the approximate solution for the 4-DMWM problem. In the distributed synchronous algorithm, each device performs only O(n2 ) computational operations in each iteration. Moreover, in the distributed asynchronous algorithm, the 4-DMWM problem based on the weighted 4-uniform hypergraph can be transformed into a maximum independent set (MIS) problem. We search for each device and its neighbors to find the maximum independent set of devices one by one in the distributed asynchronous system. After that, we get the 4-DMWM result with the complexity of O(n2 ) for each device. The distributed algorithms in this

paper are both low-complexity for the caching based optimal D2D pairing and channel allocation problem. The main contributions of this paper are summarized as follows: • We propose the D2D pairing and channel allocation algorithm scheme based on the squareIMP and a greedy algorithm to solve the caching based optimal D2D pairing and channel allocation problem in the centralized manner. • In the distributed algorithm, there are synchronous and asynchronous scenarios. – In the distributed synchronous system, a distributed synchronous algorithm based on BP is proposed for optimal D2D pairing and caching with channel allocation with low complexity. – In distributed asynchronous system, a low complexity distributed asynchronous algorithm is proposed to solve the caching based optimal D2D pairing and channel allocation problem. The rest of this paper is organized as follows. Section II presents the system model and problem formulation. The hypergraph based centralized 4-DMWM algorithm for caching based optimal D2D pairing and channel allocation is proposed in Section III. The distributed synchronous and asynchronous algorithm for the caching problem are discussed in Section IV. Section V illustrates the simulation results. Finally, Section VI concludes the paper. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model As shown in Fig. 1, the considered D2D communication scenario consists of a BS and M devices where the index set of devices is denoted by M = {1, . . . , M }. Each device caches some popular contents. When a device requests a content item, the system will allocate a channel and choose another device to provide help at the same time. A communication connection between two devices can be established when the channel can support data exchange at a target transmission rate. After the connection is established, the transmitter of the communication link can deliver content data files to its receiver. We denote the receiver and transmitter as requester and helper respectively in our network. Each requester of the network can select a helper based on whether the helper has the target content, and whether the channel capacity between these two devices is above the target transmission rate. We select appropriate combinations of the device pairs, resource blocks and contents to meet the above requirements. In our network, many pairs of devices send requests at the same time. Therefore, we protect the device pairs from interference by considering the resource allocation. Since the underlay interference management algorithms often have much extra computation overhead for the BS, a hierarchical overlay spectrum sharing and relay selection scheme is proposed to avoid serious interference between cellular devices in [13]. The BSs allocate interleaved resource blocks to the entire D2D network at first. The interleaved subcarrier allocation scheme is adopt in both IEEE 802.16 and LTE-A standards [26] [27]. The BS allocates the resource blocks in different coherence



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array with the (r, l, b)-th element pr,l,b to present the matching between the pair of devices and the resource block: ⎧ ⎪ ⎨ 1, requester r delivers data to helper l by using resource block b, (4) pr,l,b = ⎪ ⎩ 0, otherwise,

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where the function indicates that a pair of requester and helper can communicate by using a resource block in D2D network. We set pr,l,b as 1 in the condition that the channel capacity between the pair of r and l is larger than the target transmission rate by using the resource block b. Each device caches some contents in D2D network. Suppose we have F popular contents. The index set of content can be expressed as F = {1, . . . , F }. The devices exchange contents with its neighbors to offload traffic for the BS. We define Q as an M × M × F array with the (r, l, f )-th element qr,l,f to explain the relation between a pair of devices and a cache content item: 1, requester r delivers content f to helper l, qr,l,f = (5) 0, otherwise,

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Fig. 1. Scenario for caching based optimal D2D pairing and channel allocation in D2D network. The requester can get required content from the helper via D2D links depends on whether the helper has the target content, and whether the allocated resource block can support the whole content transmission from the helper to the requester.

bandwidths, so that different resource blocks undergo independent channel fading. After that the D2D network will obtain the full frequency diversity according to [28]. We assume that the index set of resource blocks is denoted by B, where B = {1, . . . , B}. Let hi be the channel gain on the resource block b between the devices. The channel gain follows a circular symmetric Gaussian distribution [29]. Suppose the resource block has a unit bandwidth. The channel between two devices in D2D communications is Rayleigh fading channel. In this case, we have: yi = hi xi + zi , i = 1, 2, ...,

(1)

where xi , yi represent transmitter and receiver of D2D communication respectively, and zi ∼ CN (0, σ 2 ) is the additive white Gaussian noise between the transmitter and receiver. We denote Pt as the transmit power of device in dBm. P (d) is the function of the power at a distance d from the transmitter. P (d) can be represent as P (d) = Pt − α log(d) + α log(d0 ),

(2)

where d0 is the standard distance between transmitter and receiver, and α is the path loss exponent. Then the channel capacity can be represented as 2 P (d) Ci = log 1 + |hi | . (3) σ2 Let r and l denote the index of requester and helper respectively, where r, l ∈ M . We define P as an M × M × B

where the function indicates that a content helper can deliver a content to the requester in D2D network. We set qr,l,f as 1 when the content item f is a required file of the requester r and a cache file of the helper l. In this paper, caching based optimal D2D pairing and channel allocation is discussed in two scenarios as follows. It is noted that, in both scenarios, the requester will try to get the required content from adjacent helpers for traffic offloading. 1) Case 1: Centralized Control Scenario With the central controller, the BS has extra overhead for the computation of the caching based optimal D2D pairing and channel allocation scheme. 2) Case 2: Distributed Control Scenario The device exchanges information with adjacent devices to determine which adjacent device can provide help in D2D network. B. Problem Formulation In this section, we formulate the caching based optimal D2D pairing problem with channel allocation to maximize the sum of achievable date rate in D2D network. We represent potential combinations of the content requester, content helper, resource block, and content as set N . A potential combination (r, l, b, f ) means that requester r can receive content f from helper l by using the resource block b. Thus, the set of all potential combinations can be described as N = {(r, l, b, f ), pr,l,b = 1, qr,l,f = 1}.

(6)

It means that the appropriate content, resource block, content helper and requester can be linked via D2D communications. These D2D links can construct a weighted 4-uniform hypergraph model among content requester, content helper, resource block and content, as shown in Fig. 2. The transmission rate of D2D links is a key issue to find the optimal assignment scheme for the optimal D2D pairing and caching with channel allocation. We define the sum-rate



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III. H YPERGRAPH BASED C ENTRALIZED 4-DMWM A PPROACH

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In this section, 4-DMWM based the squareIMP algorithm and a greedy algorithm is proposed for the caching based optimal D2D pairing and channel allocation to maximize the sum-rate of D2D network.

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as the sum of achievable date rate for all the D2D links. The sum-rate Srate is given by Srate =

wr,l,b ,

(7)

(r,l,b)∈A

where wr,l,b is the channel capacity between requester r and helper l by using the resource block b, and A is a set of these elements (r, l, b) in D2D links. Therefore, we focus on maximizing the sum-rate for the caching based optimal D2D pairing with channel allocation. In this paper, we consider caching content, required content, resource block, and the physical distance of a device pair jointly to maximize the sum-rate in D2D networks. Let ζrlbf be decision variables. If the helper l delivers content f to requester r by using resource block b, let ζrlbf =1. Otherwise, ζrlbf =0. In this case, each resource block can only be allocated to a pair of devices. A pair of devices can deliver a content item in a transmission. Therefore, the sum-rate of caching based optimal D2D pairing and channel allocation can be formulated as the following problem: max

ζrlbf · wr,l,b ;

r∈M,l∈M,b∈B,f ∈F

s.t. pr,l,b · qr,l,f = 1; ζrtbf + r∈M,b∈B,f ∈F

ζtlbf ≤ 1, ∀t ∈ M;

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ζrlbf ≤ 1, ∀b ∈ B;

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ζrlbf ≤ 1, ∀f ∈ F.

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(8)

A hypergraph can be expressed as HG = (V, E) where V is the set of vertices and E is the set of hyperedges. A weighted hypergraph means each hyperedge e ∈ E has a weight w(e). A hyperedge can be represented as a nonempty subset of the vertices. The number of vertices is the cardinality of the hyperedge. If each hyperedge has the same cardinality n, the hypergraph is n-uniform. Therefore, an ordinary graph is a 2uniform hypergraph. It means each edge has a subset of two vertices. Content requester, content helper, resource block and content item of each D2D link can be represented as 4 vertices in a hyperedge. Each hyperedge has a weight that indicate the channel capacity of the D2D link. Therefore, the caching based optimal D2D pairing with channel allocation problem can be formulated as a weighted 4-uniform hypergraph, as illustrated in Fig. 2. A matching of an ordinary graph G is a subset of the edges in G such that no edges of the matching are incident. It means that each vertex is covered at most once in these matching edges. Similarly, a hypergraph matching of a hypergraph HG is a subset of the hyperedges in HG such that no hyperedges contain the same vertex. The cardinality of a matching is the number of matching edges. If a matching has the largest cardinality of all possible matching results, it is called the maximum matching. The multi-dimensional maximum weighted matching of a weighted hypergraph HG is a subset of the matching hyperedges in HG. In this case, our target is to solve the 4-dimensional maximum weighted matching (4-DMWM) problem. Therefore, the caching based optimal D2D pairing problem with channel allocation is to find the 4-DMWM solution in the weighted 4-uniform hypergraph. B. Centralized 4-DMWM based D2D Pairing and Channel Allocation Algorithm In this paper, we formulate the 4-DMWM problem for the caching based optimal D2D pairing and channel allocation. However, the 4-DMWM problem is a NP-hard problem. Therefore, we propose a caching joint optimal D2D pairing and channel allocation algorithm to maximize the sum-rate with low-complexity. The MIS problem of graph G is to find a maximum collection of mutually non-adjacent vertices. Since the hypergraph matching is closely related to the MIS problem, the 4-DMWM problem based on the weighted 4-uniform hypergraph can be transformed into a MIS problem. First, we create a graph G = (V , E ) where each vertex is a subset of 4 vertices in a hyperedge and the edge E captures all intersections between the vertices V . In our weighted 4-uniform hypergraph, the hyperedges of HG correspond to the vertices of graph G.



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Algorithm 1 D2D Pairing and Channel Allocation Algorithm Input: M , B, F , α Output: Srate Construct HG = (V, E, w) with M , B, F Map HG to graph G if α = 1 then A = msq (G) //Algorithm2 else if α = 0 then A = mgr (G) //Algorithm3 end if end if

Srate (A) = (wi ) i∈A

Therefore, each vertex of G has a weight that indicate the channel capacity of the D2D link. We define the maximum weighted matching set as A, where A is the set of the (r, l, b, f ) combinations. The w-MIS is to find an independent set A with maximum w(A). Therefore, the hypergraph HG can be transformed into an ordinary weighted graph G = (V , E , w). Each vertex of weighted graph G is an element of potential combinations in Fig. 2. The edge of G indicates non-empty set intersections between two vertices. In the D2D network, each vertex represents a D2D link and an edge means that two D2D links have one or more same elements in the weighted 4-uniform hypergraph. In this case, the 4-DMWM problem can be solved by finding a w-MIS solution in graph G. The w-MIS problem of an ordinary weighted graph G is NP-hard. Therefore, we need some approximation algorithms to find the w-MIS result. 1) SquareIMP Algorithm: The squareIMP algorithm needs polynomial time to find an independent set A, where there is a k-claw free graph. A k-claw W is an induced subgraph that consists of a center node and talons TW in an undirected graph. The center node is connected to an independent set of k talons. A graph that is k-claw free means it possesses no k-claws induced subgraph. Set the center set of a claw W as ZW = W − TW [21]. In the 4-DMWM problem, the weighted graph G is a 5-claw free graph. To realize the squareIMP algorithm, we search for each claw in the 5-claw free graph to update A by improving w2 (A) in Algorithm 2. We define N (TW , A) = {u ∈ A : ∃v ∈ TW such that u, v ∈ E or u = v}. If the TW of a claw W can improve w2 (A), we update the set A as follows. A ←− A ∪ TW \ N (TW , A). Therefore, the set A is the expected set of D2D links for the w-MIS problem. 2) Greedy Algorithm: The greedy algorithm is a simple algorithm to find an independent set A. In the greedy algorithm, we search for all vertices to choose the heaviest independent set of vertices in the graph G, as shown in Algorithm 3. The 4-DMWM is the subset of mutually disjoint hyper-

Algorithm 2 SquareIMP Algorithm Input: graph G = (V , E , w) Output: A A=∅ while there exists claw W such that TW improves w2 (A) do N (TW , A) = {u ∈ A : ∃v ∈ TW such that {u, v} ∈ E or u = v} for k = 1 : 4 do for k − claw in G do Anew ← A − N (TW , A) ∪ TW if w2 (Anew ) > w2 (A) then A = Anew end if end for end for for v in G do Anew ← A − N (v, A) ∪ v if w2 (Anew ) > w2 (A) then A = Anew end if end for end while Algorithm 3 Greedy Algorithm Input: graph G = (V , E , w) Output: A A=∅ Vsort =sorted(V , ’descend’) for v in Vsort do N (A, V) = {u ∈ V : ∃v ∈ A such that {u, v} ∈ E or u = v} if N (A, v) = ∅ then A←A∪v end if end for

edges. The set A corresponds to a subset of hyperedges in hypergraph. The maximum weighted independent set A is a set of vertices in graph G. The index set of A can be expressed as A = {1, . . . , A}. Each vertex a of A corresponds to a D2D link for the caching based optimal D2D pairing and channel allocation. We compare the complexity of the optimal, squareIMP, greedy and random algorithm, as shown in Table I. The squareIMP algorithm is more achievable than the exhausted search with the complexity of O(n5 ). The greedy algorithm and random algorithm are much easier to implement than the other algorithms. As shown in Algorithm 1, we maximize the sum-rate for the caching based optimal D2D pairing and channel allocation in D2D network. In the D2D pairing and channel allocation algorithm, hypergraph HG can be constructed after given the parameters M , B, F according to the Section III. Then we map the hypergraph HG to a graph G. The w-MIS of weighted graph G corresponds to the weighted 4-uniform



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TABLE I COMPLEXITY OF ALGORITHM Algorithms

Complexity

Optimal algorithm

O(2n )

SquareIMP algorithm

O(n5 )

Greedy algorithm

O(n2 )

Random algorithm

O(n2 )

hypergraph matching result. Unfortunately, the w-MIS is NPhard. Therefore, we choose the squareIMP algorithm or greedy algorithm in Algorithm 1 to find the w-MIS. α is the indicating index of algorithm. We set α = 1 when we choose the squareIMP algorithm and α = 0 for the greedy algorithm in Algorithm 1. At last, we get the 4-DMWM result for the caching based optimal D2D pairing and channel allocation. IV. D ISTRIBUTED 4-DMWM A PPROACH In this section, distributed caching algorithms are proposed for the 4-DMWM problem to maximize the sum-rate in distributed synchronous and asynchronous networks.

algorithm in bipartite graph. Usually, the distributed maximum weighted matching is not fully equivalent with the 4-DMWM problem. It will loss a small part of sum-rate in the 4-DMWM result, while it is a simple and practical method to solve the distributed 4-DMWM problem. B. BP based Distributed Synchronous Algorithm In the centralized algorithm, the BS acts as a central controller that has the network information, such as the resource allocation, the connectivity, users’ preference content statistics, and distributed caching content statistics, etc. To avoid heavy overhead of global network information and maximize the sum-rate of the D2D network, BP based distributed synchronous algorithm for 4-DMWM problem is proposed by transforming the hypergraph HG into a weighted bipartite graph BG. In the BP based distributed synchronous algorithm, the weighted matching can be formulated as follows. max w e xe ; e∈E

s.t.

xe ≤ 1 for all i ∈ V,

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e∈Ei

A. Bipartite Graph Formulation A bipartite graph can be denoted as BG = (V1 , V2 , Eb ) where V1 , V2 are two sets of vertices and Eb is the set of edges. The bipartite graph is a graph whose vertices can be divided into two disjoint sets V1 and V2 such that each edge connects a vertex in V1 to one in V2 . Moreover, the bipartite graph does not contain any odd-length cycles. A weighted bipartite graph is a graph that each edge eb ∈ Eb has a weight w(eb ). A matching of a bipartite graph is similar to that of an ordinary graph G. The maximum weighted matching of a bipartite graph BG is a subset of the matching edges in BG that are mutually disjoint to maximize the weights of the edges. In this section, our goal is to solve the 4-DMWM problem in a distributed way. However, it is hard to realize. To simplify the 4-DMWM problem, we transform the weighted 4uniform hypergraph into a weighted bipartite graph. As shown in Fig. 3, the weighted hypergraph HG can be converted to a weighted bipartite graph BG. First, we create a weighted bipartite graph BG = (V1 , V2 , Eb , w) that each hyperedge of HG can be represented as 4 vertices of V1 and each vertex of HG can be represented as a vertex of V2 . The 4 vertices of V1 connect 4 vertices of V2 corresponding to the 4 vertices of the hyperedge respectively. In our weighted bipartite graph, 4 edges corresponding to the same hyperedge have the same weight. Therefore, the hypergraph HG can be transformed into a weighted bipartite graph BG. In this paper, we find the maximum weighted matching result of the weighted bipartite graph to simplify the 4DMWM problem. When the matching result of the weighted bipartite graph contains 4 edges corresponding to a hyperedge, the hyperedge is a member of the maximum weighted matching set A. After that, the 4-DMWM problem can be solved by using a distributed maximum weighted matching

xe ∈ {0, 1} for all e ∈ E, where Ei is the set of edges incident to vertex i in bipartite graph. The linear programming relaxation can be realized by replacing the constraint xe ∈ {0, 1} with the constraint 0 ≤ xe ≤ 1 for each e ∈ E in the matching problem [24]. BP is a method to estimate the marginal distributions with respect to the variables. The method associates with a bipartite graph BG called the factor graph. With this graph, BP iteratively passes belief messages along the edges of the graph that represent estimates of the marginal distributions with respect to the variables [30]. The Max-product form of belief propagation allows us to find the most likely state of a probability distribution [24]. The max-sum algorithm is equivalent to the max-product algorithm, but work in log space. Therefore, it can avoid potential underflow problem. The max-sum algorithm works by iteratively passing messages between the variables and factors. In order to apply maxsum algorithm, we formulate maximum weighted matching on bipartite graph BG as a MAP estimation problem by constructing a suitable probability distribution. In our maxsum algorithm based caching problem, set a binary variable xe ∈ {0, 1} with each edge e ∈ E. The probability distribution is considered as follows: p(x) ∝ ψi (xEi ) w e xe , (10) i∈V

e∈E

which contains

a factor ψi (xEi ) for each vertex i ∈ V. ψi (xEi ) = 1 if e∈E xe ≤ 1, and 0 otherwise. It is noted that i is referred to both the vertices of BG and factors of p. Moreover, e is referred to both the edges of BG and variables of p. The factor ψi (xEi ) means that at most one edge incident to vertex i can be assigned the value 1. If xe = 1, e. Therefore, the edges {e | xe = 1} construct a matching in BG and p(x) = 0 otherwise.



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Algorithm 4 BP based Distributed Caching Algorithm Input: hypergraph HG Output: Srate Map HG to BG t=0 Set tmax as a sufficiently large constant Set N as the number of vertices in BG Set N as the number of hyperedges in HG t+1 t t mt+1 e→i [0] = me→i [1] = mj→e [0] = mj→e [1] = 0, ∀i, j, while Not convergent and t ≤ tmax do for i = 1 : N do for j = 1 : N do t mt+1 ] e→i [xe ] = xe we + mj→e [xe

t+1 mi→e [xe ] = max {ψi (xEi ) mte →i [xe ]} xEi \e

e ∈Ei \e

end for end for end while nte [xe ] = xe we + mti→e [xe ] + mtj→e [xe ] if nte [1] > nte [0] then x ˆte = 1 else if nte [1] < nte [0] then x ˆte = 0 end if end if for j = 1 : N do if xej1 , xej2 , xej3 , xej4 = 1 then j∈A end if end for


The max-sum algorithm passes messages between variables and the factors at each iteration t for the caching based optimal D2D pairing and channel allocation. For the p in (10), each variable is a member of exactly two factors. An estimate x ˆ is the output of the MAP of p. (i, j) and e denotes the same edge in this case. Set xeh1 , xeh2 , xeh3 , xeh4 as the 4 edges in bipartite graph corresponding to a hyperedges of

Algorithm 5 Distributed Asynchronous Algorithm Input: M , B, F Output: Srate Construct HG = (V, E, w) with M , B, F Map HG to graph G A=∅ x01 = x02 = · · · = x0n = 0 t=0 //Asynchronous search of all vertices for v in V do u=v //Search for all neighbors of vertex v for i in Vnei do if xti = 0 then if wi > wu then u=i end if end if end for t=t+1 xtu = 1 A←A∪u end for


the hypergraph HG. Then we present the max-sum update equations adapted for the p in (10). At last, BP based distributed caching algorithm can find the matching result for the distributed 4-DMWM problem based on the weighted 4uniform hypergraph, as shown in Algorithm 4. The computational complexity is measured by the number of calculations in the BP based distributed caching algorithm with a graph of size n. Each vertex of bipartite graph BG sends a vector of at most size n to each of the neighbor vertices in the other partition in each iteration of the BP based distributed synchronous algorithm. Each vertex has at most n − 1 neighbors. Thus, total number of messages exchanged in each iteration are O(n2 ). Each vertex performs O(n) basic computational operations to compute each element in a message vector of at most size n. That is, each vertex performs



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Fig. 4. Sum-rate of different centralized algorithm in D2D network with B=2, F=3.

Fig. 7. F=4.

O(n2 ) computational operations to compute a message vector in each iteration. Since each vertex sends at most n message vectors, the total cost is O(n3 ) per iteration for all vertices. By sharing computing tasks, each device does fewer calculations when running the distributed algorithm.

in the distributed asynchronous algorithm. We compare the weights of each vertex and its neighbors. Each vertex has at most n − 1 neighbors. Therefore, total number of messages exchanged for all vertices are O(n2 ). Each vertex performs O(n) basic computational comparisons in the distributed asynchronous algorithm. It is observed that the computational cost of the algorithm scales as O(n2 ). Since it is an asynchronous algorithm, the total time complexity is O(n3 ) for all vertices.

C. 4-DMWM based Distributed Asynchronous Algorithm In the distributed synchronous scenario, at each clock tick, a central scheduler activates all the devices of the network simultaneously and monitors the communications that take place between these devices. The distributed asynchronous algorithm means there is no central scheduler and any device can wake up randomly at any moment independently of the other devices. This mode of operation brings clear advantages in terms of complexity and flexibility [31]. In this subsection, the distributed asynchronous algorithm is proposed for the caching based optimal D2D pairing and channel allocation to maximize the sum-rate in distributed system, as shown in Algorithm 5. The users of distributed asynchronous network connect with their neighbors by jointly considering the content caching and channel allocation. Then these devices can be formed as D2D links. In the distributed asynchronous algorithm, we search for all D2D links in different time slot. At first, we construct a hypergraph HG with the parameters M , B, F . The hypergraph HG can be transformed into a graph G. The vertices of graph G correspond to the D2D links in the distributed network. Therefore, we search for all vertices of graph G asynchronously to find the independent set of vertices. Moreover, the neighbors of each vertex can be represent as Vnei in Algorithm 5. After that we find a maximum weighted vertex among the vertex and its neighbors to add to set A. Each vertex is searched in a different time slot. After the search, we get the set A to calculate the sumrate for the caching based optimal D2D pairing and channel allocation in D2D network. The computational complexity of the distributed asynchronous algorithm is measured by the number of calculations

Sum-rate of centralized and distributed algorithm with M=6, B=3,

V. S IMULATION R ESULTS In this section, our simulation results demonstrate the performance of our proposed centralized and distributed algorithms for caching based optimal D2D pairing and channel allocation in D2D network. We set the circular cell as a radius of 400 m [32]. All devices are moving around and randomly distribute in the cell at a time period. A device pair delivers the data file by using a resource block in D2D network. The channel between two devices is independent Rayleigh fading channel. In this paper, we consider the influence of physical distance, resource block and content item during the communication. We set the number of resource blocks and content items as B and F , respectively. Suppose the transmission time of each resource block is 20ms. Our simulation has the same result when we choose other parameters. In our simulation, we analyze different schemes for the caching based optimal D2D pairing and channel allocation. The detailed simulation parameters are presented in Table II. In one case, the BS has the network information and allocates the resource block to the pair of devices. There are optimal, squareIMP, greedy and random centralized algorithm in our simulation. The optimal algorithm exhausts search for all possible states for 4-DMWM problem in exponential time. The detailed 4-DMWM based on greedy algorithm and squareIMP algorithm is presented at Section III. We search all the vertices randomly in the random algorithm. The sumrate can verify the efficiency of the caching approach. In Fig. 4, we compare the performance of sum-rate among the



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Fig. 8. Sum-rate of centralized and distributed algorithm by Content Caching Fig. 9. Sum-rate of centralized and distributed algorithm by Content Caching scheme 1 with B=3, F=4. scheme 2 with B=3, F=4.

TABLE II SIMULATION PARAMETERS Parameters

Value

BS location

The center of the square area

Cell radius Path loss Fast fading

400 m 128.1+37.6log10 (D) dB, D in km Flat Rayleigh fading channel per RB

Channel estimation

Perfect

Resource block bandwidth

180 kHz

Transmission time length

20 ms

File size

120 kbits

σ2

-169 dBm/Hz

algorithms with B = 2, F = 3. Observed from Fig. 4, the sum-rate increases with the number of devices. It is clearly shown that the 4-DMWM based squareIMP algorithm can achieve an highly approximate performance compared with the optimal approach. The optimal approach is NP-hard which is computational prohibited. Therefore, squareIMP algorithm can guarantee the performance of sum-rate with the complexity of O(n5 ) in D2D network. Moreover, the result of greedy based algorithm is slightly lower than that of squareIMP based with a much lower computational complexity O(n2 ). Besides, the matching result of greedy based algorithm is better than that of random algorithm. Therefore, squareIMP algorithm can achieve a higher rate requirements, while the greedy algorithm is a practical method with stringent latency requirement. We plot in Fig. 5 the number of contents is 3. Then we choose different number of resource blocks to observe the variance of sum-rate for the caching based optimal D2D



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pairing and channel allocation. It is observed that the sum-rate increases with the number of resource blocks when we fix the number of content items. It implies that we can increase the number of resource blocks to improve the overall throughput. Moreover, the result of greedy algorithm is still a little lower than the squareIMP algorithm with a much lower complexity. As shown in Fig. 6, the sum-rate increases with the number of cached contents in D2D network when we set B = 2. It means that the overall throughput can be increased when we increase the number of total cached contents. Fig. 6 also illustrates that the performance of sum-rate by using the 4DMWM based squareIMP algorithm is better that that of greedy algorithm. However, the computational complexity of squareIMP algorithm is higher than that of greedy algorithm. In another case, each device does some calculations when running the distributed algorithm with no the central controller. There are synchronous and asynchronous distributed algorithms in distributed system. The detailed algorithms are present at Section IV. The sum-rate of different algorithms varies with the different cell sizes. In Fig. 7, we compare the performance of sum-rate among the centralized and distributed algorithms with different cell sizes when we set M = 6, B = 2, F = 3. Observed from Fig. 7, the sum-rate decreases with the radius of the circular cell. Fig. 7 also reveals that these centralized and distributed algorithms have similar sumrate when the radius is larger than a certain size. We assume that the file size is 120 kbits. Given a larger size of file, long transmission time is required to ensure the reliable transmission. Two different content caching schemes are used in these algorithms, as shown in Fig. 8 and Fig. 9. We fix the buffer size as 2 for each devices in the simulation of scheme 1. However, in scheme 2, the same amount of contents are cached in different numbers of devices. In this case, the number of resource blocks is 3 and there are 4 content items. Each content is cached twice in the devices of the network in scheme 2. The sum-rate also increases with the number of devices in different algorithms with different caching schemes. The simulation result of both the synchronous and asynchronous distributed algorithm are lower than that of centralized 4DMWM based algorithm. It is obvious that the sum-rate of BP based distributed synchronous algorithm is lower than the centralized greedy algorithm. However, the complexity of distributed synchronous algorithm is only O(n2 ) for each device in a iteration. Moreover, the sum-rate of distributed asynchronous algorithm is close to the centralized greedy algorithm and the computational complexity is only O(n2 ) for each device. In our simulation, we consider the channel estimation is perfect. The optimality of the proposed algorithm relies on the assumption of perfect channel estimation. VI. C ONCLUSIONS In this paper, we studied the caching based optimal D2D pairing and channel allocation problem. This joint optimization problem is formulated as a weighted 4-uniform hypergraph model, where the 4-DMWM is the optimal solution. However, the 4-DMWM problem is NP-hard. Under the control of the B-

S, 4-DMWM based squareIMP and greedy algorithm was proposed to approach the optimal solution with low-complexity. The squareIMP algorithm achieves a better performance approximate to the optimal approach with the complexity of O(n5 ). The greedy algorithm, however, is much more useful in practical caching based D2D communications with stringent latency requirement. In the distributed system, we proposed algorithms in the scenarios of synchronous and asynchronous, respectively. The sum-rate of distributed asynchronous algorithm is closer to the centralized greedy algorithm with the complexity of O(n2 ) for each device. Moreover, the simulation result of BP based distributed synchronous algorithm is slightly lower than the centralized greedy algorithm in the distributed synchronous system and the complexity is O(n2 ) per iteration for each device. The distributed algorithms have very low complexity in each devices. Therefore, the algorithms of this paper can be used in different scenarios to solve the optimal D2D pairing and channel allocation problem. R EFERENCES [1] C.-H. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resource sharing optimization for device-to-device communication underlaying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2752-2763, Aug. 2011. [2] C. Xu, L.-Y. Song, Z. Han, Q. Zhao, X.-L. Wang, X. Cheng, and B.L. Jiao, “Efficiency resource allocation for device-to-device underlay communication systems: A reverse iterative combinatorial auction based approach,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 348-358, Sep. 2013. [3] X. Wang, M. Chen, T. Taleb, A. Ksentini, and V. Leung, “Cache in the air: exploiting content caching and delivery techniques for 5G systems,” IEEE Commun. Mag., vol.52, no.2, pp.131-139, Feb. 2014. [4] U. Niesen, D. Shah, and G. W. Wornell, “Caching in wireless networks,” IEEE Trans. Inf. Theory, vol. 58, no. 10, pp. 6524-6540, Oct. 2012. [5] M.-Y. Ji, G. Caire, and A. F. Molisch, “Fundamental limits of caching in wireless D2D networks,” IEEE Trans. Inf. Theory, vol. 62, no. 2, pp. 849-869, Feb. 2016. [6] K. Shanmugam, N. Golrezaei, A. Dimakis, A. Molisch, and G. Caire, “FemtoCaching: Wireless content delivery through distributed caching helpers,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 8402-8413, Dec. 2013. [7] E. Bas¸tuˇg, M. Bennis, M. Kountouris, and M. Debbah, “Cache-enabled small cell networks: Modeling and tradeoffs,” EURASIP J. Wirel. Commun. Netw., vol. 2015, no. 1, pp. 1-11, Feb. 2015. [8] M.-Y. Ji, G. Caire, and A. Molisch, “Wireless device-to-device caching networks: Basic principles and system performance,” IEEE J. Sel. Areas Commun., vol. 34, no. 1, pp. 176-189, Jul. 2015. [9] W. Chen and H. V. Poor, “Joint pushing and caching with a finite receiver buffer: Optimal policies and throughput analysis,” in IEEE Int. Conf. Commun. (ICC), Kuala Lumpur, Malaysia, 2016. [10] Y.-W. Lu, W. Chen, and H. V. Poor, “Content Pushing Based on Physical Layer Multicasting and Request Delay Information,” in IEEE Global Commun. Conf. (GLOBECOM), Washington DC, USA, 2016. [11] B. Bai, W. Chen, K. B. Letaief, and Z.-G. Cao, “Diversity-multiplexing tradeoff in OFDMA systems: an H-matching approach,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3675-3687, Nov. 2011. [12] M. Maddah-Ali and U. Niesen, “Decentralized coded caching attains order-optimal memory-rate tradeoff,” IEEE/ACM Trans. Netw., vol. 23, no. 4, pp. 1029-1040, Aug. 2015. [13] B. Bai, W. Chen, K. B. Letaief, and Z.-G. Cao, “Distributed WRBG matching approach for multi-flow two-way D2D networks,” IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 2925-2939, Apr. 2016. [14] J. Liu, B. Bai, J. Zhang, and K. B. Letaief, “Content caching at the wireless network edge: A distributed algorithm via brief propagation,” in IEEE Int. Conf. Commun. (ICC), Kuala Lumpur, Malaysia, 2016. [15] B. Bai, W. Chen, K. B. Letaief, and Z.-G. Cao, “Low complexity outage optimal distributed channel allocation for vehicle-to-vehicle communications,” IEEE J. Sel. Areas Commun., vol. 29, no. 1, pp. 161-172, Jan. 2011.



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[16] L. Militano, G. Araniti, M. Condoluci, I. Farris, and A. Iera, “Device-toDevice Communications for 5G Internet of Things,” EAI Endorsed Trans. Internet Things, vol. 15, no. 1, Oct. 2015. [17] B. Bai, L. Wang, Z. Han, W. Chen, and T. Svensson, “Caching based socially-aware D2D communications in wireless content delivery networks: a hypergraph framework,” IEEE Wireless Commun., vol. 23, no. 4, pp. 74-81, Aug. 2016. [18] J.-H. Tang, T. Q. S. Quek, and W. P. Tay, “Joint resource segmentation and transmission rate adaptation in Cloud RAN with Caching as a Service,” in SPAWC, Aug. 2016. [19] R. Tandon and O. Simeone, “Cloud-aided wireless networks with edge caching: Fundamental latency trade-offs in fog Radio Access Networks,” in Proc. Int. Symp.Inf. Theory (ISIT), Aug. 2016. [20] L. Miao, B. Bai, and W. Chen, “4-DMWM approach for caching based optimal D2D pairing and channel allocation,” submitted to ICC 2017. [21] P. Berman, “A d/2 approximation for maximum weight independent set in d-claw free graphs,” Nordic Journal of Computing, vol. 7, no. 3, pp. 178-184, 2000. [22] D. E. Drake, and S. Hougardy, “A simple approximation algorithm for the weighted matching problem,” Inform. Process. Lett., vol. 85, no. 4, pp. 211-213, Feb. 2003. [23] B. Bai, W. Chen, Z.-G. Cao, and K. B. Letaief, “Max-matching diversity in OFDMA systems,” IEEE Trans. Commun., vol. 58, no. 4, pp. 11611171, Apr. 2010. [24] S. Sanghavi, D. Malioutov and A. Willsky, “Belief Propagation and LP Relxation for Weighted Matching in General Graphs,” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2203-2212, Apr. 2011. [25] F. Kschischang, B. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498519, Feb. 2001. [26] Evolved Universal Terrestrial Radio Access: Physical Channels and Modulation (R9). 3GPP TS 36.211 V9.0.0, 2009. [27] Air Interface for Fixed and Mobile Broadband Wireless Access Systems. IEEE Std. 802.16TM -2009, 2009. [28] B. Bai, W. Chen, K. B. Letaief, and Z.-G. Cao, “Outage exponent: a unified performance analysis framework for parallel fading channels,” IEEE Trans. Inf. Theory, vol. 59, no. 3, pp. 1657-1677, Mar. 2013. [29] B. Bai, W. Chen, K. B. Letaief, and Z.-G. Cao, “A unified matching framework for multi-flow decode-and-forward cooperative networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 2, pp. 397-406, Feb. 2012. [30] S. Rangan and R. Madan, “Belief Propagation Methods for Intercell Interference Coordination,” in Proc. IEEE INFOCOM, Apr. 2011. [31] P. Bianchi, W. Hachem, and F. Iutzeler, “A Coordinate Descent PrimalDual Algorithm and Application to Distributed Asynchronous Optimization,” IEEE Trans. Autom. Control, vol. 61, no. 10, pp. 2947-2957, Oct. 2016. [32] Y. Li, X.-F. Lei, P.-Z. Fan, and D.-G. Chen, “An SCMA-based uplink inter-cell interference cancellation technique for 5G wireless systems,” in WCSP, Nanjing, China, 2015.

Lu Miao received the B.S. degree in communication engineering electronic engineering from Jilin University, Changchun, China, in 2015. She is currently pursuing the master degree in electronic engineering from Tsinghua University, Beijing, China. Her research interests include D2D caching and maximum weighted matching problem.

Bo Bai (S’09-M’11) received his B.S. degrees in Department of Communication Engineering with the highest honor from Xidian University in 2004, Xian China. He also obtained the honor of Outstanding Graduates of Shaanxi Province. He received the Ph.D. degree in Department of Electronic Engineering from Tsinghua University in 2010, Beijing China. He also obtained the honor of Young Academic Talent of Electronic Engineering in Tsinghua University. From 2009 to 2012, he was a visiting research staff (Research Assistant from April 2009 to September 2010 and Research Associate from October 2010 to April 2012) in Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology (HKUST). Now he is an Assistant Professor in Department of Electronic Engineering, Tsinghua University. He has also obtained the support from Backbone Talents Supporting Project of Tsinghua University. From March to September 2015 and April to July 2016, he visited The City University of Hong Kong (CityU) as a senior research fellow. His research interests include hot topics in wireless communications & networking, Fog/Cloud-RAN, network big data, and random graph. He has published more than 60 papers in major IEEE journals and flagship conferences. He is IEEE member and IEEE ComSoc member. He also served in Wireless Communications Technical Committee (WTC) and Signal Processing and Communications Electronics (SPCE) Technical Committee.He has served as a TPC member for several IEEE ComSoc conferences such as ICC, Globecom, WCNC, VTC, ICCC, and ICCVE. He has also served as a reviewer for a number of major IEEE journals and conferences. He received Student Travel Grant at IEEE Globecom09. He was also invited as Young Scientist Speaker at IEEE TTM 2011. He was awarded the IEEE ICC 2016 Best Paper Award.

Wei Chen (S’05-M’07-SM’13) received his BS degree in Operations Research and PhD degree in Electronic Engineering (both with the highest honors) from Tsinghua University, Beijing, China, in 2002, and 2007, respectively. From 2005 to 2007, he was also a visiting research staff in the Hong Kong University of Science & Technology. Since July 2007, he has been with Department of Electronic Engineering, Tsinghua University, where he has been prompted to be full professor since 2012, as a special case of early promotion. After the human resource reform in Tsinghua University, he is elected as tenured full professor of the new research and teaching track in 2015. He also serves as an University council member, the director of the Academic Degree Administration Office of Tsinghua University, and deputy head of EE department currently. He visited the University of Southampton, UK, from June 2010 to Sept. 2010, Telecom ParisTech, France, from June 2014 to Sept. 2014, and Princeton University, from July 2015 to Sept. 2015 and from Jan. 2016 to March 2016. He is supported by the National 973 Youth Project, the NSFC excellent young investigator project, the 10000-talent program, the new century talent program of Ministry of Education, and the Beijing nova program. His research interests are in the areas of wireless communications and information theory. He serves as Editors for IEEE Transactions on Communications, IEEE Trans. Education, IEEE Wireless Communications Letters, IET Communications, China Communications, a co-chair of communications theory symposium in IEEE Globecom 2017, and a vice director of youth committee of China Institute of Communications. He served as a tutorial cochair of IEEE ICC 2013, a TPC co-chair of IEEE VTC 2011-Spring, and symposium co-chairs for IEEE ICC, ICCC, CCNC, Chinacom, and WOCC. He also holds guest/chair professor positions of some Universities in China and is a member of Beijing Youth Federation. Prof. Chen is the recipient of the First Prize of 14th Henry Fok YingTung Young Faculty Award, the Yi-Sheng Mao Beijing Youth Science & Technology Award, the 2010 IEEE Comsoc Asia Pacific Board Best Young Researcher Award, the 2009 IEEE Marconi Prize Paper Award, the 2015 CIE information theory new star award, the Best Paper Awards at IEEE ICC 2006, IEEE IWCLD 2007, and IEEE SmartGirdComm 2012, as well as, the First Prize in the 7th Young Faculty Teaching Competition in Beijing. He receives the 2011 Tsinghua Raising Academic Star Award and the 2012 Tsinghua Teaching Excellence Award. Prof. Chen holds the honorary titles of Beijing Outstanding Teacher and Beijing Outstanding Young Talent. He is the Champion of the First National Young Faculty Teaching Competition, and a winner of National May 1st Medal.


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