Cache-Aided Interactive Multiview Video Streaming in Small Cell Wireless Networks

Transcription

1 Cache-Aided Interactive Mltiview Video Streaming in Small Cell Wireless Networks Eirina Bortsolatze, Member, IEEE, and Deniz Gündüz, Senior Member, IEEE arxiv: v [cs.ni] 3 Apr 208 Abstract The emergence of novel interactive mltimedia applications with high rate and low latency reqirements has led to a drastic increase in the video data traffic over wireless celllar networks. Endowing the small base stations of a macrocell with caches that can store some of the content is a promising technology to cope with the increasing pressre on the backhal connections, and to redce the delay for demanding video applications. In this work, delivery of an interactive mltiview video to a set of wireless sers is stdied in an heterogeneos celllar network. Differently from existing works that focs on the optimization of the delivery delay and ignore the video characteristics, the caching and schedling policies are ointly optimized, taking into accont the qality of the delivered video and the video delivery time constraints. We formlate or oint caching and schedling problem as the average expected video distortion minimization, and show that this problem is NPhard. We then provide an eqivalent formlation based on sbmodlar set fnction maximization and propose a greedy soltion with 2 e ) approximation garantee. The evalation of the proposed oint caching and schedling policy shows that it significantly otperforms benchmark algorithms based on poplarity caching and independent schedling. Another important contribtion of this paper is a new constant approximation ratio for the greedy sbmodlar set fnction maximization sbect to a d-dimensional knapsack constraint. Index Terms Joint caching and schedling, mltiview video, sbmodlar fnction maximization, d-dimensional knapsack I. INTRODUCTION Recent analyses of visal networking applications indicate a steady growth in mobile video traffic, which is expected to reach more than three qarters of the total mobile traffic by 202 []. Large share of this traffic is attribted to video content generated by emerging mltimedia applications, sch as virtal reality VR), agmented reality AR) and interactive mltiview video streaming IMVS). These novel mltimedia technologies offer sers a completely new experience throgh the possibility to interact with the application in real time. However, high qality experience and interactivity comes with low latency and high bandwidth reqirements that mst be met by the mobile data operators. To deal with the ever increasing amont of mobile video data, the se of small cell base stations SBSs) eqipped with caches to store some of the high data E. Bortsolatze is with the Department of Electronic and Electrical Engineering at University College London, London WCE 7JE, United Kingdom e.bortsolatze@cl.ac.k). D. Gündüz is with the Department of Electrical and Electronic Engineering at Imperial College London, London SW7 2AZ, United Kingdom d.gndz@imperial.ac.k). This work has been fnded by the Eropean Union s Horizon 2020 research and innovation programme nder the Marie Sklodowska-Crie grant agreement No and by the Eropean Research Concil ERC) nder the Eropean Union s Horizon 2020 research and innovation programme nder grant agreement No rate content has been proposed in [2]. SBS caches can be exploited by off-loading content to the caches dring offpeak hors, and serving sers locally throgh short-range lowlatency commnication dring peak-hors. In that way, the se of costly backhal links that connect the SBSs to the core network dring the peak-hors can be alleviated and the load on the macro cell base station MBS) can be redced [3], [4]. Or work is motivated by the new challenges arising with the emergence of immersive and interactive mltimedia technologies. In particlar, we stdy an IMVS application provided to sers over a celllar network. IMVS enables sers to freely explore the 3D scene of interest from different viewpoints in real time [5]. The challenge of offering sch interactivity to sers is the need to deliver mltiple video streams corresponding to different views, as views selected by sers are not known a priori. Ths, compared to single view conventional video streaming, an IMVS application typically reqires mch higher bandwidth to enable low-latency view switching at high qality. As in state-of-the-art wireless edge caching systems proposed for video-on-demand VoD) applications [2], [3], [4], the placement of mltiview video content in the SBS caches can bring the video content closer to wireless sers, and redce the load on the MBS and the backhal links. As a reslt, a larger set of views can be delivered to sers with a better qality of experience QoE). However, the key obective in the context of caching for real-time video streaming is different from the one considered for VoD applications. In the latter, the sers reqest a single file according to some poplarity distribtion and the aim is to place the video content in the caches in a way to minimize the average download delay. This obective is not sitable for real-time video streaming applications, as it ignores the video delivery time constraints and the qality of the delivered content. In an IMVS system, sers do not reqest a single file, bt a set of views, which ideally wold inclde all views captred by the cameras. However, when the available bandwidth is limited, only a sbset of available views can be delivered to the sers. Hence, when optimizing the caching policy one mst take into consideration the qality of the delivered content, and perform the schedling of optimal sets of views ointly with cache placement. Differently from existing caching soltions, we ointly optimize the caching and schedling policies to ensre the delivery of the optimal sets of views within the time constraints imposed by the real-time video application. Thogh oint caching and roting has been previosly considered in the literatre [4], [6], [7], [8], the time constraints and the qality of the delivered video have not been taken into accont in the proposed soltions.

2 2 #2.*.-/2$,"-./"0!"#$%"&'')*$ )7.#*-.*- 545 Fig.. Illstration of an IMVS with V p cameras captring the scene of interest from mltiple viewpoints. Captred content is transmitted to the core network. The optimal selection of the delivered views that maximizes the video qality has been stdied in [5], [9]. In these works the delivered sets of views are optimized assming the bandwidth resorces of the sers are fixed and known a priori. In a celllar network deploying mltiple SBSs, each ser may fall within the coverage range of several SBSs. Ths, there is no prior knowledge of the sers bandwidth capabilities. The sers bandwidth resorces may vary depending on the density of the SBS placement and the nmber of sers served by these SBSs, and, therefore, mst be allocated ointly with the caching policy. In this paper, we propose a novel framework for ointly optimizing the caching and schedling policies for the delivery of interactive mltiview video in a wireless celllar network consisting of an MBS and mltiple SBSs eqipped with caches. The goal of or oint policy is to optimally allocate the cache and bandwidth resorces of the wireless network in order to minimize the average expected distortion of the sers who freely navigate throgh the available set of views dring the streaming session. Differently from previos works, or framework takes into accont the time constraints of the streamed video and the qality of the video content delivered to the sers. Initially, we formlate or oint caching and schedling problem for IMVS as the average expected distortion minimization and show that this problem is NPhard. We then show that this problem can be eqivalently expressed as the maximization of the redction in the average expected distortion, and prove that the eqivalent problem involves maximizing a sbmodlar set fnction sbect to a d-dimensional knapsack constraint. In order to efficiently solve or optimization problem, we adopt a greedy algorithm and prove that it admits a constant approximation ratio of 2 e ). To the best of or knowledge, this is a novel reslt in the literatre which extends the existing reslts on greedy sbmodlar fnction optimization with a single knapsack constraint to the case with a d-dimensional knapsack constraint. Finally, we show throgh nmerical evalation that or proposed algorithm for oint caching and schedling significantly otperforms benchmark algorithms based on Fig. 2. The mltiview video is cached in the small-cell base stations and delivered to wireless sers simltaneosly dring the streaming session. poplarity caching and independent rate allocation. II. SYSTEM MODEL We consider an IMVS application as illstrated in Fig.. An array of eqally spaced cameras captre a 3D scene of interest from mltiple viewpoints. Let V p {v,v 2,...,v Vp } denote the set of views acqired by the cameras, called the anchor views, where V p = V p > 2 is the total nmber of anchor views. Each camera then encodes its acqired anchor view independently from the other cameras, and transmits it to the core network. This is a common assmption for distribted video acqisition scenarios, where the mltiple cameras typically do not commnicate with each other; and, ths, cannot ointly compress the captred video streams. In the core network, the compressed video streams are stored at the content provider s servers. From the core network, the video content is frther delivered to a set of wireless sers throgh a celllar network as shown in Fig. 2. We stdy a streaming scenario where the IMVS session is initiated some time after the video content is recorded, and not immediately after the content is acqired as in live streaming. For example, this may correspond to the broadcasting of a sports event or a concert, which took place in one part of the world and cold not be live-streamed to regions in different time zones. In this case, the content provider acqires the recorded content and makes it available according to a predefined viewing schedle, also providing an interactive mltiview experience to sers. We focs on the operation of a single macro-cell that serves a set U {,2,...,U} of U wireless sers. The macro-cell is covered by a total of N + base stations BSs); an MBS indexed by n = 0, and a set N {,2,...,N} of N SBSs distribted across the macro-cell. The MBS covers the entire macro-cell region and can commnicate with all the sers within the cell. The SBSs have limited coverage and can only commnicate with the sers located within their proximity. Let U n U denote the set of sers that are covered by the n-th BS. Given a sfficiently dense placement of SBSs within the macro-cell area, every ser will typically fall within the

3 3 commnication range of mltiple SBSs. We denote the set of BSs that can commnicate with ser as N N {0}. Note that, we have U 0 = U, and 0 N, U. We assme that the MBS and the SBSs are assigned disoint sets of sbchannels, while neighboring SBSs operate in orthogonal freqency bands [2], [4]. This permits s to ignore any interference among the BSs. Frthermore, we consider that SBS n N is eqipped with a cache of size C n bytes and can store all or part of the mltiview video, while the MBS is assmed to have nlimited cache capacity. Additionally, the total transmission capacity of each BS inclding the MBS is assmed to be limited, and eqal to R n Mbps Mbits per second), n N {0}. This total transmission capacity R n mst be allocated among the sers within the coverage area of BS n, U n. To facilitate the caching and delivery of the mltiview video content, and to se the network resorces more efficiently, the video stream from each camera is partitioned into T segments of b t bytes each, t T {,2,...,T}. We denote the t-th segment of view v as B v,t. The index t also stands for the time slot in which the segment B v,t can be schedled for delivery. Since the video content is available at the server before its dissemination to the sers, the caches available at SBSs can be exploited in order to redce the load on the congested backhal links dring the streaming session. Part of the prerecorded video content can be offloaded to the caches of the SBSs dring low-traffic hors and before the IMVS session starts. The cached content can then be directly delivered from the local caches to the sers, avoiding ths the se of costly backhal links and redcing the delivery delay which is critical in real-time video streaming. In addition to the anchor views directly acqired by the cameras, additional virtal views can be synthesized from the textre and depth map information of the anchor views via the depth image based rendering DIBR) techniqe [0]. Specifically, we assme that at the decoder side, L additional virtal views can be synthesized between two adacent views v i, v i+ V p. We denote the set of virtal views as V s {v +δ,v +2δ,...,v +Lδ,...,v Vp +δ,...,v Vp +Lδ}, is the distance between two adacent views in the set V V p V s of all available viewpoints. In order to virtally synthesize a view v V, a left and a right reference view from the set of anchor views V p are reqired. Let v l < v < v r with v l, v r V p denoting the left and the right reference anchor views, respectively. The distortion at which the virtal view v is synthesized depends generally on the qality of the reference anchor views v l,v r, and the spatial correlation between the virtal view and the anchor ones. For an array of aligned and eqally spaced cameras, the spatial correlation is proportional to the relative distance between the synthesized view v and the reference anchor views v l and v r [5]. To measre the distortion of the synthesized view v, we adopt the distortion model proposed in []: where δ = L+ ) d v v l,v r ) = γe αvvr v l) e βvmin{v v l,v r v}, ) We assme that the views are symmetrically coded. Hence, the size of a segment does not depend on the view index. where d v v l,v r ) is the distortion at which view v can be reconstrcted from reference anchor views v l and v r. The parameters γ, α v, β v define the rate at which the distortion of the virtal view increases with the distance from the reference anchor views. Note that the optimization of the distortion fnction for virtal view synthesis is beyond the scope of this paper, and the model in ) has been chosen de to its simplicity and accracy. Or framework for optimizing the cache-aided IMVS in small cell wireless networks is general and can incorporate other distortion models as well. At any given time dring the IMVS session, a ser can select any of the actal camera viewpoints in set V p, or choose to synthesize a virtal view from set V s in real time. Hence, the ser can freely navigate throgh the available set of views V and explore the 3D scene of interest from different viewpoints. To enable sch interactivity at the best possible qality, the fll set of anchor views V p mst be delivered to the ser at any given time, so that the ser can reconstrct any virtal view in set V s from the best left and right reference anchor views. However, this is not always possible in a bandwidthlimited system de to the strict delay constraints imposed by the IMVS application. Depending on the available resorces, typically only a sbset of the anchor views can be delivered to the wireless sers. The sbset of anchor views delivered to the ser determines the distortion at which viewpoints selected by the ser and not inclded in the delivered set of anchor views can be reconstrcted. Note that the sers select the desired viewpoints in real time. This implies that the views reqested by a ser at any given time dring an IMVS session are not known when the contents are placed into the caches of SBSs. Ths, the sbset of views to be stored at the SBSs has to be optimized with respect to the expected video qality based on a probabilistic model of the poplarity of video segments of each view. For each segment B v,t, we define a poplarity p v,t [0,] which represents the probability that the t-th segment of view v will be selected for viewing. The segment poplarity depends on the video content corresponding to this segment and withot loss of generality w.l.g.) can be considered the same for all sers. The content poplarity can be learned by the content provider by analyzing the mltimedia content [2], or the history of viewing reqests [3]. To garantee the reconstrction of any view within set V at a minimm qality, we consider that views v and v Vp, the leftmost and rightmost anchor views, are always delivered to all the sers by the MBS. 2 The remaining transmission capacity R 0 of the MBS, as well as the cache capacity C n and transmission capacity R n of the SBSs are sed to deliver additional anchor views to frther improve the qality of the synthesized views. In this work, we aim to find the optimal oint caching and schedling policy for the mltiview video segments B v,t that minimizes the average expected distortion at the wireless sers 2 A potential alternative for the delivery of high-rate mltiview video content is hybrid networking, which combines terrestrial digital video broadcasting DVB) with broadband celllar networks [4]. Therefore, one can consider that the two extreme anchor views are broadcasted throgh the terrestrial network to all the sers.

4 4 which participate in the IMVS session. In the next section, we provide the formal problem formlation, and prove that the oint caching and schedling problem for IMVS is NP-hard. III. JOINT CACHING AND SCHEDULING PROBLEM A. Problem formlation Let s define the binary variable x v,t n {0,}, where x v,t n =, if the t-th segment of view v V p is placed in the cache of the n-th SBS dring the caching phase, and x v,t n = 0 otherwise. Similarly, let yv,t n, {0,} be a binary decision variable which indicates whether the segment B v,t is schedled for delivery from BS n to ser ; yn, v,t = if the segment B v,t is schedled, and yn, v,t = 0 otherwise. The vector x, y) of decision variables, where x x v,t n {0,}, n N, v V p, t T ), and y y v,t n, {0,}, n N {0}, U, v V p, t T ), 2) defines a oint caching and schedling policy. From the assmption that the leftmost and rightmost anchor views v and v Vp, respectively, are always delivered to the sers by the MBS, we have x v,t n = 0, n N, v {v,v Vp }, t T, y v,t n, = 0, n N, U, v {v,v Vp }, t T, y v,t 0, =, U, v {v,v Vp }, t T. The distortion D v,t x,y) at ser for reconstrcting segment B v,t for the given caching and schedling policy x,y) can be expressed as follows: D v,t { Dv,t x,y) ½ x,y) = { n N yv,t n,>0} ), v V p D v,tx,y), v V s 4) v,t where D x, y) is the minimm distortion at which segment B v,t can be reconstrcted at ser given the oint caching and schedling policy x,y). The indicator fnction ½ {c} is if the condition c is tre, and 0 otherwise. In Eq. 4), we distingish the following two cases. When view v belongs to the set of anchor views captred by the cameras, the distortion for reconstrcting the segment B v,t at ser is 0, if the segment is delivered to ser by at least one of the BSs n N that cover ser. Otherwise, the distortion is eqal D v,t to the minimm distortion x,y) at which segment Bv,t can be reconstrcted at ser given the oint caching and schedling policy x,y). When view v is a virtal view, the segment B v,t is not delivered to ser, and is synthesized sing the corresponding segments of the closest left and right anchor views according to the oint caching and schedling D v,t policy x,y). Finally, the minimm distortion x,y) at which segment B v,t can be reconstrcted at ser when it is not delivered, is given by 3) D v,t, x,y) = v l <v d v v l,v r ) v r>v ½ { n N yv l,t n, >0} v l <v l <v ½ { n N yvr,t n, >0} v<v r <vr ) ½ { n N yv l,t n, >0} ) ½ { n N yv r,t n, >0} The second term of the prodct in Eq. 5) is eqal to if for the view indexv l V p the segmentb vl,t is delivered to ser by at least one BS, and for all other segments B v l,t delivered to ser, the view v l is farther from v than v l. Similarly, the third term of the prodct in Eq. 5) is eqal to if for the view index v r V p the segment B vr,t is delivered to ser by at least one BS, and for all other segments B v r,t delivered to ser, the view v r is farther from v than v r. Note that the prodct of the second and the third terms of the prodct in Eq. 5) is non-zero only for one niqe pair of left and right views v l and v r. Frthermore, de to the assmption that all segments for views v, v Vp are always delivered to all the sers, one sch pair always exists. Finally, from Eqs. x,y) is a fnction of only the schedling vector y. However, the distortion depends implicitly on the caching policy x since the latter determines the feasible schedles at SBSs. Ths, any attempt at minimizing the distortion mst ointly consider the caching and schedling decisions. Or goal is to devise a oint caching and schedling policy that minimizes the average expected distortion of the wireless sers. The optimization problem can be formally written as 4) and 5) we can observe that the distortion D v,t x,y ) = argmin x,y) s.t. U T t T v V p\{v,v Vp } U v V p\{v,v Vp } U t T v V 5) D v,t x,y)p v,t 6) x v,t n b t C n, n N, 7) y v,t n, r R n, n N {0}, t T, 8) y v,t n, x v,t n, U, n N, v V p, t T. 9) x v,t n,y v,t n, {0,} and constraints in 3) 0) Constraint 7) is the cache capacity constraint and garantees that the total amont of data stored in an SBS s cache does not exceed its capacity. Constraint 8) is the transmission capacity constraint, which states that the total rate delivered by an SBS in time slot t mst not exceed its capacity. The constant r denotes the video rate, and since we consider symmetric coding of the views, it is the same for all captred views. Finally, the ineqality in 9) coples the caching and schedling decisions, and ensres that only the segments that are stored in SBS caches can be schedled for transmission. The optimization problem defined in Eqs. 6)-0) is an integer program which is difficlt to solve directly de to the non-convex natre of the obective fnction and the integer

5 5 constraints. In the next sbsection, we show that this problem belongs to the class of NP-hard problems. B. Complexity We now show that the optimization problem in 6)-9) is NP-hard. To prove that, it is sfficient to show that the corresponding decision version of the problem, which we call the oint caching and schedling JCS) decision problem, is NP-hard. The proof relies on the method of restriction, which consists of placing additional restrictions on the instances of a given problem P NP so that the restricted problem is eqivalent to some known NP-complete problem P [5]. In the following, we first define the decision version of or optimization problem, and provide the definition of the set K- cover SKC) decision problem [6], which will be sed in the proof. Definition. JCS decision problem: Given the set of BSs N {0}, the set of sers U, the sets of views V p and V s, the segment size vales B = {b,b 2,...,b T }, the nmber of segments T, the segment poplarity P = {p v,t }, the SBSs cache capacity vales C = {C,C 2,...,C N }, the BSs total transmission capacity vales R = {R 0,R,...,R N }, x,y) and a positive real nmber Z, determine if there exists a feasible oint caching and schedling policy x, y) that satisfies the constraints in 7)- 0) and the distortion fnction D v,t U T U t T v V D v,t x,y)p v,t Z. ) We denote the JCS decision problem instance as JCSN {0},U,V p,v s,b,t,p,c,r,d v,t x,y),z). Definition 2. SKC decision problem [6]: Given a collection of sbsets S of a set A and a positive integer K 2, does S contain K disoint covers for A, i.e., covers S,S 2,...,S K, where S k S, sch that every element of A belongs to at least one member of each of S k? We denote the above SKC decision problem by SKCA, S, K). The SKC decision problem is known to be NP-complete [6]. Proposition. The JCS decision problem is NP-hard. Proof. Consider the SKCA, S, K) decision problem and an instance of the JCSN {0},U,V p,v s,b,t, P,C,R,D v,t x,y),z) decision problem with V p 2 = K, U = A, {U,...,U N } = S, T =, B = {b}, p v,t = V, C = {b,b,...,b}, R 0 = 0, R n = U n and Z = V s V v V s Dmin v, where Dv min is the minimm distortion at which a virtal view v V s can be synthesized, and it is achieved when view v is reconstrcted from the two closest left and right reference anchor views. This instance corresponds to the scenario in which the whole stream consists of a single segment T = ) of size b. Each SBS can cache only one single view C n = b), and the sers receive data only from the SBSs R 0 = 0). The n-th SBS can deliver data to all the sers in the set U n simltaneosly R n = U n ). Frthermore, every ser reqests one view from the set V of available views niformly at random p v,t = V ). The average expected distortion for this instance of the JCS decision problem is lower bonded by U T U t T v V D v,t x,y)pv,t V s V v V s D v min, x,y), 2) which follows immediately if we observe that D v,t x,y) 0, v V p, and D v,tx,y) Dv min, v V s. The minimm vale of the average expected distortion can only be attained if every captred view v V p \{v,v Vp } can be delivered to every ser U by at least one of the SBSs in set N that cover ser. Therefore, deciding whether there exists a oint caching and schedling policy x, y), sch that the average expected distortion is eqal to Z, redces to determining whether there exists a caching policy x sch that every view in set V p \{v,v Vp } is cached in at least one SBS in set N for every ser, since, de to the assmption that R n = U n, the view can always be delivered. This, in trn, is eqivalent to finding V p 2 = K disoint sbsets of SBSs, sch that every ser is covered by at least one SBS in each sbset. It now becomes apparent that the considered instance of the JCS decision problem is eqivalent to the SKC decision problem, which is known to be NP-complete. It, therefore, follows that the JCS decision problem is NP-hard. IV. EXPECTED DISTORTION REDUCTION MAXIMIZATION In order to deal with the comptational complexity of the optimization problem in 6)-0), we reformlate it as an eqivalent problem which aims at maximizing the average expected distortion redction. We express the eqivalent optimization problem as a maximization of a set fnction defined over sbsets of an appropriately selected grond set. We then show that the obective set fnction is a monotone non-decreasing sbmodlar fnction. This permits s to devise efficient soltions based on the greedy approach. A. Eqivalent problem formlation Let s define the grond set E as E {e v,t : n N {0},A n U n,v V p \{v,v Vp }, t T } 3) The element e v,t of the grond set E denotes the placement of the segment B v,t in the cache of BS n N {0} and its schedling for delivery to a sbset A n U n U of the sers covered by BS n. Any oint caching and schedling policy x,y) can be represented by a sbset S of the grond set E. For example, placing the element e v,t E in S can be regarded as setting the decision variables x v,t n and yv,t n,, A n, to. Recall that, by or initial assmption, all segments of views v and v Vp are delivered by the MBS to all the sers. Ths, e v,t 0,U 0 and e vvp,t 0,U 0, t T, will always be inclded in a set S that represents a oint caching and schedling policy x,y). Let s frther define the sets F v,t {e v,t : n N, A n U n s.t. A n }, U, v V p, t T. Set

6 6 E essentially represents all possible ways to deliver segment B v,t to ser. Given the grond set E and sets F v,t, we can re-write the distortion fnction in 6) in the form of a set fnction D v,ts) : 2E R as follows: ) { Dv,t D v,t S) ½ S) = {S F v,t }, v V p D v,ts), v V, s 4) F v,t where D v,t S) = v l <v d v v l,v r ) v r>v ½ {S F v l,t ½ {S F vr,t ½ v } {S F l,t ) } v l <v l <v } v<v r <v r ½ {S F v r,t } ). 5) The distortion redction at ser for reconstrcting the segment B v,t is defined as D v,t S) = D max D v,t S), S E, 6) where D max is the maximm distortion when the corresponding segment cannot be reconstrcted. The distortion redction fnction in Eq. 6) represents the redction in the distortion experienced by ser after reconstrcting the segment B v,t. The constraints defined in Eqs. 7), 8), 9) can also be expressed in terms of set fnctions defined over the grond set E. Recall that the elemente v,t represents the oint placement of the segment B v,t in the cache of BS n and its delivery to a sbset of sers A n. This implies that, when the element e v,t is inclded in the soltion set S, the segment B v,t is placed in the cache of BS n consming a total space of b t bytes and a rate of A n r Mbps is allocated by the BS n to transmit it to the sers in A n. Ths, with each element e v,t E, we associate a caching cost of b t bytes and a rate cost of A n r Mbps. We define the cache cost and rate cost fnctions c n S) : 2 E R and r t ns) : 2 E R, respectively, as: c n S) = v,t c n en,a ), n N, n 7) e v,t n S,A n r t ns) = e v,t n,a n S r t ne v,t n,a n ), n N {0}, 8) where c n e v,t n,a ) = bt n if n = n, and 0 otherwise, and r n e v,t n,a ) = A n n r if n = n,t = t, and 0 otherwise. Finally, we define the cost fnction fn v,t S) : 2 E R as: fn v,t S) = v,t,t n ev n,a n ), 9) e v,t n,a n S f where fn v,t,t ev n,a ) = if n n = n,v = v,t = t, and 0 otherwise. Essentially, fnction fn v,t S) conts the nmber of times segment B v,t is placed in the cache of BS n. We can now reformlate the minimization problem in Eqs. 6)-0) as a maximization of the average expected distortion redction as follows: S OPT = argmax S U T U t T v V D v,t S) 20) s.t. c n S) C n, n N 2) r t n S) R n, n N {0}, t T 22) fn v,t S), n N {0}, v V p\{v,v Vp }, t T. 23) Constraints 2) and 22) are the cache capacity and the transmission capacity constraints. Constraint 23) garantees that each segment is placed in the cache of a BS only once. This constraint is necessary since neither the cache cost fnction nor the rate cost fnction can distingish between two elements e v,t and e v,t n,a associated with the same segment n B v,t. In other words, for two elements e v,t, e v,t n,a S, the n reqired cache space calclated by the cache cost fnction is 2b t, and the reqired rate calclated by the rate cost fnction is A n + A n )r. In practice, however, the two elements e v,t and e v,t n,a can be replaced with an eqivalent element n e v,t n,a. Hence, the actal cache space needed is n A bt, and the n actal rate needed is A n A n r. Constraint 23) ensres that only a niqe element e v,t for segment B v,t and cache n will be inclded in the soltion set. In the following sbsection, we show that the obective fnction is a monotone non-decreasing sbmodlar set fnction. We then leverage this property to propose comptationally efficient algorithms. B. Proof of Sbmodlarity Sbmodlarity is an important property of set fnctions that permits to deploy greedy soltions with a good performancecomplexity trade-off [7]. In this sbsection, we prove that the obective fnction in the maximization problem in Eq. 20) is a monotone non-decreasing sbmodlar fnction. The definition of a monotone non-decreasing set fnction is given in Appendix A. Proposition 2. The obective fnction in 20) is a monotone non-decreasing set fnction over the grond set E. Proof. Let s consider sets S S 2 E. Monotonicity follows immediately from the observation that the distortion of a reconstrcted segmentb v,t at sercan only redce with the delivery of additional segmentsb v,t ; that is, the distortion redction can only increase with the delivery of additional data segments. It ths holds that D v,t S ) D v,t S 2 ), i.e., the distortion redction at ser for reconstrcting the segment B v,t is a monotone non-decreasing fnction. Hence, the obective fnction in 20) is also monotone non-decreasing as a linear combination of monotone non-decreasing fnctions with non-negative weights. We will now prove that the obective fnction in 20) is sbmodlar. The following lemma will be sefl in the proof.

7 7 Lemma. Let V,V 2 satisfy V p V 2 V {v,v Vp }. Consider a view v V p V s. Then, for any ṽ V p, we have D v V ṽ) D v V ) D v V 2 ṽ) D v V 2 ), 24) where the distortion redction fnction D v ˆV) : 2 Vp R is defined as D v ˆV) D max d v v l,v r ), with v l v and v r v, respectively, being the closest to v left and right anchor views in ˆV. Proof. We prove the lemma for the case where ṽ v. De to symmetry, the same argments hold for the case ṽ v. For =,2, let v l V denote the left anchor view closest to v, sch that 0 v v l < v v l, v l V with l l. Similarly, let v r V be the right anchor view closest to v, sch that 0 v r v < v r v, v r V with r r. Since V V 2, we have v l v l2 and v r v r2. We can distingish three cases depending on the relative position of view ṽ with respect to views v l and v l2 : i) ṽ v l, ii) v l < ṽ v l2, and iii) v l2 < ṽ. We now prove for each of these three cases that the ineqality in 24) holds. ) ṽ v l : In this case, the addition of view ṽ to either of sets V, V 2 does not provide any frther distortion redction since view ṽ is farther from view v than views v l and v l2 ; and ths, views v l, v l2 remain as the left anchor views closest to v. In particlar, D v V ṽ) = D v V ) = D max d v v l,v r ), and D v V 2 ṽ) = D v V 2 ) = D max d v v l2,v r2 ). Therefore, D v V ṽ) D v V ) = D v V 2 ṽ) D v V 2 ) = 0. 2) v l < ṽ v l2 : As in the previos case, the addition of ṽ to set V 2 does not provide any distortion redction since view ṽ is farther from view v than view v l2, and v l2 remains the left anchor view closest tov in setv 2 ṽ. Ths, we have D v V 2 ṽ) = D v V 2 ) = D max d v v l2,v r2 ) and D v V 2 ṽ) D v V 2 ) = 0. On the contrary, the addition of view ṽ to set V redces the distortion for view v, since view ṽ is closer to view v than v l, i.e., d v v l,v r ) d v ṽ,v r ). Therefore, D v V ṽ) D v V ) = d v v l,v r ) d v ṽ,v r ) 0, and D v V ṽ) D v V ) D v V 2 ṽ) D v V 2 ). 3) v l v l2 < ṽ: In this case, view ṽ becomes the left anchor view closest to v. Ths, we have D v V ṽ) D v V ) 0 and D v V 2 ṽ) D v V 2 ) 0. However, it is no longer possible to dedce straightforwardly which of the two gains in distortion redction is larger, and an inspection of all the sb-cases concerning the relative positions of the views v l, v l2, ṽ, v r, and v r2 with respect to view v is needed. We can distingish the following ten sbcases: v ṽ < v v l2 v v l v r2 v v r v v ṽ < v v l2 v r2 v v v l v r v v ṽ v r2 v v v l2 v v l v r v v r2 v v ṽ < v v l2 v v l v r v v ṽ < v v l2 v r2 v v r v v v l v ṽ v r2 v v v l2 v r v v v l v r2 v v ṽ < v v l2 v r v v v l v ṽ v r2 v v r v v v l2 v v l 25a) 25b) 25c) 25d) 25e) 25f) 25g) 25h) v r2 v v ṽ v r v v v l2 v v l v r2 v v r v v ṽ < v v l2 v v l 25i) 25) Here we show analytically that the ineqality in 24) holds for the case in Eq. 25a), and provide the gidelines for showing its validity for the remaining cases, omitting the details de to limited space. From the distortion model in Eq. ) and the ineqalities in 25a), we have: D v V 2 ) D v V ) D v V 2 ṽ) D v V ṽ) = eavvr v l ) e βvv v l ) ) e avvr 2 v l 2 ) e βvv v l 2 ) ) e avvr ṽ) e βvv ṽ) ) e avvr 2 ṽ) e βvv ṽ) ) v v l >v v l2 v v l2 >v ṽ e avvr v l ) e avvr 2 v l 2 ) )e βvv v l 2 ) ) e avvr ṽ) e avvr 2 ṽ) )e βvv ṽ) ) e avvr v l ) e avvr 2 v l 2 ) 26) e avvr ṽ) e avvr 2 ṽ) = eavvr vr 2 +v l 2 v l ) )e avvr 2 v l ) e avvr vr 2 ) )e avvr 2 ṽ) v r2 v l v r2 ṽ e avvr vr 2 +v l 2 v l ) e avvr vr 2 ) v l2 v l. Ineqality 24) follows immediately from 26). Using the same procedre, we can prove that 24) holds for the cases in 25b), 25c) 25e), 25f), 25h). For the rest of the cases, we form the expression D v V ṽ) D v V ) D v V 2 ṽ) D v V 2 ṽ), and sing similar argments as before, we prove that this expression is greater or eqal to. Intitively, the above reslt can be explained by the fact that the qality of the left reference anchor view improves more when adding view ṽ to set V than when adding ṽ to set V 2, since ṽ v l ṽ v l2. Ths, the gain in the distortion redction is higher when adding view ṽ to set V compared to adding it to V 2. Proposition 3. The obective fnction in 20) is a sbmodlar set fnction over the grond set E. Proof. Since a non-negative linear combination of monotone sbmodlar fnctions is also sbmodlar [7], it is sfficient to show that the distortion redction D v,ts) : 2E R is monotone sbmodlar U, v V p V s, t T. Let s consider the sets S S 2 E, and an element eṽ, t E\S 2. This element represents the oint placement of segment Bṽ, t in the cache of BS n, and its delivery from BS n to the set of sers A n. If / A n or t t, then adding eṽ, t to sets S and S 2 does not affect the distortion redction at ser for segment B v,t since ser does not receive any additional segments with respect to those received according to the oint caching and schedling policies defined by sets S and S 2. Ths, D v,t S eṽ, t ) D v,t S ) = D v,t S 2 eṽ, t ) D v,t S 2) = 0. Next, we focs on the case

8 8 A n and t = t, i.e., ser belongs to the grop of sers to which segment Bṽ,t is delivered. We associate set S, =,2, with set V V p, where v V iff e v,t S. From the definition of sets V, and since S S 2, it holds that V V 2. De to the assmption that all the segments of the leftmost and rightmost views are delivered to the sers, we also have V {v,v Vp }. From Lemma it follows that D v V ṽ) D v V ) D v V 2 ṽ) D v V 2 ); and therefore, D v,t S eṽ,t ) D v,t S ) D v,t S 2 eṽ,t ) D v,ts 2), which completes the proof. C. Greedy algorithms In the previos section, we have shown that the obective fnction in the maximization problem in 20) is a monotone non-decreasing sbmodlar fnction. We can now show that the optimization problem defined by Eqs. 20)-23) is in the form of a sbmodlar set fnction maximization problem sbect to a separable d-dimensional knapsack constraint defined in Appendix A. By inspection of the constraints in 2)-23), it is straightforward to see that they can be partitioned into d = 3 disoint sets of constrains with M = N cache constraints in the first set, M 2 = N + )T rate constraints in the second set, and M 3 = N + )V p 2)T constraints in the third set that ensre the niqeness of the selected elements. We can therefore apply the niform cost UC) and the weighted cost-benefit WCB) greedy algorithms described in Appendix B to efficiently solve the maximization problem in 20)-23). For the sake of completeness, we smmarize the UC and WCB algorithms as applied to the maximization problem in 20)-23) in Algorithms and 2, respectively. According to Theorem provided in Appendix C, at least one of the two greedy algorithms achieves the approximation ratio of 2 e ). It is worth noting that the WCB greedy algorithm was sed in [8] to maximize a sbmodlar obective fnction sbect to two knapsack constraints. However, the athors did not provide any theoretical garantees on its performance. To the best of or knowledge, or work presents the first constant approximation ratio for solving the sbmodlar set fnction maximization problem sbect to a d-dimensional knapsack constraint by means of greedy algorithms. Algorithm Uniform cost greedy algorithm : Inpt: E, vale qery oracle DS), C n, R n, cost fnctions c n, rn, t fn v,t 2: Initialization: S UC, k 0 3: while E\S UC do 4: k k + 5: e k argmax DS UC e v,t ) DS UC ) e v,t n,an E\SUC 6: if c n S UC e k ) C n, rns t UC e k ) R n, fn v,t S UC e k ) then 7: S UC S UC e k, 8: else 9: E E\e k 0: end if : end while 2: Otpt: S UC Algorithm 2 Weighted cost-benefit greedy algorithm : Inpt: E, vale qery oracle DS), C n, R n, cost fnctions c n, rn, t fn v,t, weights λ, λ 2, λ 3 2: Initialization: S WCB, k 0 3: while E\S WCB do 4: k k + 5: e k argmax λ e v,t n,an E\SWCB DS WCB e v,t n,an ) DSWCB) n c n ev,t n,an ) DS k e + λ v,t n,an ) DS k ) 2 n t rt n ev,t n,an ) + λ 3 DS k e v,t n,an ) DS k ) n v t fv,t n e v,t n,an ) 6: if c n S WCB e k ) C n, rn ts WCB e k ) R n, fn v,ts WCB e k ) then 7: S WCB S WCB e k, 8: else 9: E E\e k 0: end if : end while 2: Otpt: S WCB V. PERFORMANCE EVALUATIONS For the performance evalations, we consider a circlar cell with the MBS located at its centre. The transmission range of the MBS is set to 400m. A total nmber of 20 SBSs, each with a coverage radis of 00m, are placed niformly at random over the cell. The transmission capacity of the SBSs is set to 00Mbps. We consider 200 wireless sers niformly distribted across the macro cell. The IMVS system consists of V p = 8 cameras. Each camera generates a video stream encoded at r = 2 Mbps and divided into T = 20 segments of eqal size. We assme that L = 3 virtal viewpoints can be synthesized between any two adacent anchor views. The distortion of the synthesized views is compted based on the model in ). We assme that the sers select the first segment among the captred views niformly at random. Then, dring the streaming session, each ser can switch from view v i to a neighboring anchor or virtal view v with probability pv v i ) 2πσ e v v i )2 2σ 2 for v v i W, and 0 otherwise. For or evalations, we set W = 8 and σ 2 = 5/L+). From this model, we calclate the poplarity distribtion p v,t of the video segments. We compare the proposed greedy oint caching and schedling algorithms with a maximm poplarity algorithm. The latter fills each SBS s cache with the most poplar video segments. It then performs greedy schedling independently of the cache placement phase. We evalate both UC greedy and WCB greedy schedling for the maximm poplarity algorithm. Fig. 3 shows the average expected distortion redction verss the cache capacity of the SBSs expressed as a percentage of the total size of the mltiview video. UC-J and WCB-J

9 9 Average expected distortion redction UC-J 300Mbps) WCB-J 300Mbps) UC-MP 300Mbps) WCB-MP 300Mbps) WCB-J 200Mbps) UC-J200Mbps) WCB-MP 200Mbps) UC-MP 200Mbps) Cache capacity Average expected distortion redction UC-J 0.) 450 WCB-J 0.) UC-MP 0.) 400 WCB-MP 0.) UC-J 0.2) WCB-J 0.2) 350 UC-MP 0.2) WCB-MP 0.2) Total transmission capacity Mbps) Fig. 3. Average expected distortion redction vs the cache capacity at the SBSs, expressed as the percentage of the total size of the mltiview video. denote the UC and WCB greedy algorithms, respectively, for oint caching and schedling. UC-MP and WCB-MP denote the maximm poplarity caching algorithm with UC and WCB greedy schedling, respectively. We present reslts for a total transmission capacity of 200Mbps and 300Mbps for the MBS. For the WCB algorithm we have sed λ = 0.2, λ 2 = 0.5 and λ 3 = 0.3. The reslts indicate that the oint caching and schedling algorithms otperform the maximm poplarity conterparts for all vales of the cache capacity. For low vales of the cache capacity, the improvement in the performance is significant as the maximm poplarity algorithm caches the same content in all SBSs; ths the content diversity across the network is limited. Along with the most poplar content cached only in few SBSs, the oint caching and schedling algorithm also caches the less poplar content, which, when delivered to the sers, improves the reconstrction qality of the views. It is worth noting that this range of capacity vales is of great practical interest as SBSs are typically assmed to cache only 5-0% of the total video cataloge [9], [20]. The performance of all the algorithms becomes limited by the insfficient transmission capacity of the network. Ths, even thogh all the SBSs can cache almost all of the contents, they cannot be delivered to the sers. In Fig. 4 we show the average expected distortion redction verss the total transmission capacity of the MBS for cache capacity eqal to 0% and 20% of the total size of the video. As the total transmission capacity of the MBS increases, the average expected qality of the mltiview video delivered to the sers improves. We can see that the oint caching and schedling algorithm otperforms the maximm poplarity algorithm for all vales of the MBS transmission capacity. Althogh the cache capacity of the SBSs is limited, or algorithm performs mch better compared to the maximm poplarity algorithm de to the more efficient se of the available cache and transmission capacities. As previosly, the content diversity is higher when the caching and schedling policies are optimized ointly. It is worth noting that to achieve the same average expected distortion redction, the maximm poplarity caching algorithm reqires a mch higher transmis- Fig. 4. Average expected distortion redction vs the total transmission rate of the MBS. sion rate to be allocated by the MBS compared to the case of oint cache and schedling optimization. Finally, we note that the UC and WCB greedy algorithms in Figs. 3 and 4 perform identically. This is de to the fact that all video segments have the same size. We expect that in the case of mltiple mltiview videos encoded at different rates, or video segments of neqal dration, the performance of the two algorithms wold be different. We leave this investigation for or ftre work. VI. CONCLUSIONS AND FUTURE WORK We have presented a framework for ointly optimizing the caching and schedling policy for interactive mltiview video delivery over a wireless celllar network. Unlike existing works for wireless edge caching, or scheme takes into accont the qality of the video delivered to the sers and the rate reqirements for real-time video delivery. Nmerical evalation of or scheme shows that the oint policy performs significantly better than the independent caching and schedling policies for the case of mltiview video. In or ftre work, we will investigate ways to simplify the expression for calclating the distortion of the delivered video, with the aim of obtaining a convex problem which can be solved for optimality. A possible approach is to organize the views into embedded sets that progressively improve the qality of the delivered video. This will also relax the constraint of caching the whole segment in the same SBS, and will permit to cache parts of the same video segment encoded with a rateless code in different SBSs. The latter will allow to transform the integer optimization problem into a linear one. A. Definitions and properties APPENDIX A SUBMODULAR FUNCTIONS Here we recall some basic definitions and reslts from the theory of sbmodlar fnctions. Definition 3. [7] A fnction g : 2 W R defined over a grond set W is sbmodlar, if for every Z Z 2 W and

10 0 w W\Z 2, gz w) gz ) gz 2 w) gz 2). 27) Alternatively, g is sbmodlar, if for every Z,Z 2 W, gz Z 2)+gZ Z 2) gz )+gz 2). 28) Definition 4. [7] A fnction g : 2 W R defined over a grond set W is monotone non-decreasing if for every Z Z 2 W, gz ) gz 2 ). Proposition 4. [2] If g : 2 W R is a monotone nondecreasing sbmodlar fnction defined over the grond set W, then gz ) gz 2)+ w Z \Z 2 gz 2 w) gz 2) 29) for all Z,Z 2 W B. Sbmodlar fnction maximization with a d-dimensional knapsack constraint Let g : 2 W R be a monotone non-decreasing sbmodlar fnction defined over the grond set W. The sbmodlar fnction maximization problem sbect to a d-dimensional knapsack constraint is formlated as s.t. max gz) Z W h iz) H i, i =,2,...,d, 30) where d is the nmber of knapsack dimensions, h i Z) = z Z h iz) is the cost fnction of the ith dimension, and h i z) > 0. Each dimension of the knapsack can be viewed as a resorce, with H i being the total bdget of the ith resorce. We now consider a special case of the maximization problem in 30). Let s assme that the d knapsack constraints in 30) can be partitioned into d disoint sbsets sch that, for every sbset i =,2,...,d of the constraints, the grond set W can be partitioned into M i disoint sbsets Wi,W2 i...,wmi i, where M i is the nmber of constraints in the ith sbset and d i= M i = d. Let h m i Z) = z Z hm i z) be the cost fnction for the mth constraint in the ith sbset of constraints with m =,2,...,M i. We frther assme that h m i z) > 0 if z Wm i, and 0 if z / Wm i, only a sbset of elements in W consme a non-zero amont of the mth resorce in the ith set of resorces, and there are M i sch disoint sbsets of elements associated with the ith set of resorces for i =,2,...,d. Formally, this problem can be written as max gz) Z W s.t. h m i Z) Hi m, i =,2,...,d, m =,2,...M i, 3) where h m i Z) = z Z hm i z) = z Z W h m i m i z), and Hm i is the total bdget for the mth resorce in the ith set of constraints. We refer to this problem as the sbmodlar fnction maximization with a separable d-dimensional knapsack constraint. Algorithm 3 UC greedy algorithm : Inpt: W, vale qery oracle gz), cost fnctionsh i Z), total bdget vales H i 2: Initialization: Z UC, k 0 3: while W\Z UC do 4: k k + 5: w k argmax gz UC w) gz UC ) w W\Z UC 6: if h i Z UC w) H i, i then 7: Z UC Z UC w 8: else 9: W W\w k 0: end if : end while 2: Otpt: Z UC h m i Z UC w) H m i APPENDIX B GREEDY ALGORITHMS FOR SUBMODULAR FUNCTION MAXIMIZATION A. The niform cost UC) greedy algorithm The UC greedy algorithm [22] for the sbmodlar fnction maximization problem with d-dimensional knapsack constraint stated in 30) is described in Algorithm 3. The algorithm takes as inpt the grond set W, a vale qery oracle gz) that retrns the vale of the obective fnction in 30) for some sbset Z of the grond set, the cost fnctions h i Z) and the vales of the total bdgets H i, i =,2,...,d. The algorithm starts with an empty soltion set, and at the k-th iteration picks the element from the grond set that maximizes the gain with respect to the soltion set compted at step k step 5 of Algorithm 3). If this choice satisfies the d- dimensional knapsack constraint specified in 30), the element is added to the soltion set. Otherwise, the soltion set is not pdated and the element is removed from the grond set. This procedre is repeated ntil all elements from the grond set have been either inclded in the soltion set or removed from the grond set. When Algorithm 3 is applied to the separable d-dimensional knapsack constraint problem in 3), the condition in line 6 mst be replaced with the condition given in the parentheses. B. The WCB greedy algorithm The UC greedy algorithm presented above can perform arbitrarily poorly as it does not take into accont the cost of the element selected greedily at each iteration [7]. This shortcoming is addressed in the cost-benefit greedy algorithm for the smbodlar fnction maximization problem with a single knapsack constraint [23]. In the cost-benefit greedy algorithm, the next element to be inclded in the soltion set is the element that maximizes the gain-to-cost ratio. However, in a d-dimensional knapsack constraint problem, each element is associated with a d-dimensional cost vector. To accont for the d different costs, the athors in [8] have introdced the WCB algorithm. The WCB algorithm is smmarized in Algorithm 4. It works similarly to the UC greedy algorithm, bt instead )