IN many interactive multimedia streaming applications, such

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1 840 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 A Geoetric Approach to Server Selection for Interactive Video Streaing Yaochen Hu, Student Meber, IEEE, DiNiu, Meber, IEEE, and Zongpeng Li, Senior Meber, IEEE Abstract Many distributed interactive ultiedia applications, such as live video conferencing and video sharing, require each participating client to transit its captured video strea to other clients via relay servers. We consider connecting ultiple clients through ultiple relay servers and study the server selection proble fro a dense pool of content delivery network edge locations and datacenters to reduce the end-to-end delays between clients. To achieve scalability in the presence of a large nuber of candidate servers, we forulate server selection as a geoetric proble in a delay space instead of in a graph, which turns out to be an extension of the well-known Euclidean k-edian proble. We propose practical approxiation schees when using only one or two servers with theoretical worst-case guarantees as well as fast heuristics when using k servers. We deonstrate the benefit of our optiized ultiserver selection schees through extensive evaluation based on real-world traces collected fro the PlanetLab and Seattle platfors, containing personal obile devices as well as real network experients based on a prototype ipleentation. Index Ters End-to-end delay, geoetric optiization, interactive video streaing, server selection. I. INTRODUCTION IN any interactive ultiedia streaing applications, such as live video/audio conferencing and interactive video gaing, each of the geographically distributed clients needs to receive a video/audio strea fro all other clients in real tie. A coon objective of these applications is to reduce the end-toend delays between clients to the iniu, while aintaining a sufficiently high data throughput, especially when users are spread across different regions [] [3]. In production systes like FaceTie and Google+, relay servers [], [] are adopted to collect data fro all the participating users and distribute a ixed strea to every user. As copared to the earlier peer-topeer (PP) solutions, such a server-based solution has several benefits. First, each user can upload its data strea at the full rate to the server and does not need to split its uplink bandwidth to serve all other users like in a PP architecture. Second, ajor application providers such as Apple, Google, and Microsoft usually have their own large server clouds coposed of point of presence (POP) locations in their content delivery networks (CDNs) and datacenters, connected in well-provisioned Manuscript received Septeber, 05; revised January, 06; accepted February 7, 06. Date of publication March 4, 06; date of current version April 5, 06. The guest editor coordinating the review of this anuscript and approving it for publication was Prof. Maria Martini. Y. Hu and D. Niu are with the Departent of Electrical and Coputer Engineering, University of Alberta, Edonton, AB T6G H9, Canada (e-ail: yaochen@ualberta.ca; dniu@ualberta.ca). Z. Li is with the Departent of Coputer Science, University of Calgary, Calgary, AB TN N4, Canada (e-ail: zongpeng@ucalgary.ca). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier 0.09/TMM and high-bandwidth backbone networks. Third, a large part of ixing and processing jobs can be done by servers, relieving the coputational burden of users. In this paper, we ask the questions how can the choices of server locations affect end-to-end delays between clients in interactive video/audio streaing applications? Is a single server sufficient, or can we reduce end-to-end delays by using ultiple relay servers spread at different locations? And where should these relay servers be placed? As a toy exaple, if three clients in Paris are in a eeting with three other clients in Sao Paulo, having two inter-connected servers placed in the two cities, respectively, clearly achieves a lower ean end-to-end network distance than placing a single server at soe place in between the two cities. However, how this network-distance phenoenon can affect delay perforance in general is yet to be investigated. Forally speaking, we ai to iniize the ean end-toend delay in an interactive video streaing session between N clients, by optially choosing k servers fro a dense cloud of CDN nodes (POP locations and datacenters). This proble has been studied in [4] on a graph fored by all the clients and candidate servers, assuing the latency between every pair of nodes is known. It is shown that this new proble is different fro the well-known k-edian and k-eans probles, and is NP-coplete for any k<n. A greedy algorith perfored on the graph is given in [4] with an approxiation ratio of if triangle inequalities are assued. However, the coplexity of this graph-based solution critically depends on the nuber of candidate servers, which can increase draatically as the candidate server pool grows to a size of hundreds, thousands or even ore. For exaple, Akaai has deployed a pervasive, highly distributed CDN with over servers in ore than 00 countries within over 300 networks. In this paper, we propose a novel geoetric approach to ultiserver selection in order to achieve scalability in the presence of a large and dense server cloud consisting of CDN nodes and datacenters. By leveraging network coordinate systes (NCSs) [5], each host can be apped onto a point in a delay space. Using geoetric optiization, we copute the ideal locations of k servers for N clients in the delay space. We finally ap the coputed ideal server locations back to the closest physical servers to for server selection decisions. Unlike graph-based solutions, the coplexity of the proposed geoetric optiization in the delay space is independent of the nuber of candidate servers. Moreover, we do not need to easure the latencies fro every client to all candidate servers; it suffices to probe a few reference servers to estiate the network coordinate of each client in the delay space, which can be efficiently coputed or even pre-coputed by any existing NCSs. Akaai, [Online]. Available: IEEE. Personal use is peritted, but republication/redistribution requires IEEE perission. See standards/publications/rights/index.htl for ore inforation.

2 HU et al.: GEOMETRIC APPROACH TO SERVER SELECTION FOR INTERACTIVE VIDEO STREAMING 84 It turns out that the proposed geoetric proble in the delay space is an extension of the k-edian and facility location probles, for which no polynoial-tie solution is known [6]. We provide siple approxiation schees for this new geoetric proble, and prove that the one-server optial solution incurs a ean delay of at ost /N ties the best possible value of using N servers foring a full esh. If two servers can be used, we provide a.5-approxiation schee based on centroids, and a one-or-two-server schee with an approxiation ratio of strictly less than /N. When k (k >) servers are adopted, we further propose efficient heuristic algoriths to choose k servers based on k-eans partitioning and convex optiization. To evaluate our server selection algoriths under various rando errors, including network coordinate errors and apping errors, we perfor extensive siulations driven by latency traces collected fro 490 PlanetLab nodes over a 5-day period as well as traces collected fro the Seattle platfor [7] which include latency easureents fro personal and obile devices. We observe that despite network coordinate errors, triangle inequality violations (TIV), as well as apping errors fro ideal to true server locations, the benefit of our optiized server selection in the delay space can largely offset these errors. Moreover, the proposed server selection procedure is efficient and can finish all the tasks within s for a server pool of close to 500 nodes. To further verify real packet delays including syste processing and queueing delays, we have ipleented a prototype interactive video streaing syste leveraging ultiple servers and deployed it on the PlanetLab. We observe that when server locations are optially deterined, not only can ultiple servers reduce ean end-to-end delays, but they also help to distribute CPU workloads as the video bit rate increases. II. RELATED WORK Current production interactive video streaing systes adopt different architectures according to the easureent study in []. ichat does not use servers and adopts a PP star topology. Skype uses centralized servers to relay video traffic, yet with all servers found to be placed in the sae location []. Google+ adopts ultiple servers distributed all over the globe to relay all the data in a session. But its server selection strategy is proprietary and unknown to the public. Other easureent studies have analyzed interactive video streaing with respect to latency, bandwidth and video quality. In [8], the responsiveness of Skype video calls to bandwidth variations is easured. In [9] an extensive easureent of Skype two-party video calls is presented under different network conditions. Arefin et al. [0] shows that people will easily get ipatient when they face long end-to-end delay. In addition, Baset et al. [], [] analyze the overlay architecture, PP protocol, and VoIP traffic of Skype. In this paper, we provide an in-depth study on server location optiization specifically to iniize delays in interactive video streaing, based on a ulti-server esh topology. Sever placeent and selection have been extensively studied for a variety of applications and topics. Various placeent strategies for Web server replicas have been proposed in [3] to iprove CDN perforance. Lee et al. [4] presents a distributed algorith that selects gae servers for a group of clients in order to iniize the server resource usage with real-tie delay constraints. Ding et al. [5] proposes a server selection schee which can achieve high availability while aintaining low delays and low cost. Caeron et al. [6] considers the server allocation proble with dense servers and clients, and has developed an algorith based on the high-rate vector quantization theory. In contrast, in this paper we specifically focus on selecting the optial server locations in real-tie interactive video streaing applications. The design of interactive video streaing systes, especially live conferencing, has been extensively studied in the context of PP networks [7], [8] within a utility axiization fraework, in order to optiize the streaing rates of the clients subject to network bandwidth constraints. Recent research has used cloud coputing and datacenter networks to enhance the perforance of interactive video streaing. Airlift [9] leverages inter-datacenter networks to relay traffic and process data streas for video conferences. It axiizes the total throughput in ultiple conference sessions by choosing the optial routes to deliver and relay packets in the inter-datacenter network, subject to end-to-end delay constraints. In contrast, in this paper we consider a server pool that is uch larger than a few datacenters and ay consist of a large nuber of CDN POP locations. A piece of closely related work [0] also adopts a cloud of servers, called the Virtual Mixer, to enhance delay perforance in a video conference. In particular, it tries to iniize either the average or the axiu end-to-end delay using a heuristic based on Steiner tree optiization perfored on a graph of servers and clients. However, this heuristic has no perforance guarantee. Neither is it scalable to a graph fored by a large nuber of CDN servers. A siilar server selection proble for distributed interactive applications is studied in [4] to select k servers fro a graph fored by all candidate servers and clients. It is proved [4] that the graph version of this proble is NP-coplete when k<n or under soe other conditions. A greedy approxiation schee has been proposed to solve the proble in graph with a coplexity that grows fast as the size of the server pool increases. Our work is siilar to this work in that we also use a server cloud to iprove delay perforance of interactive ultiedia applications. However, we propose a novel geoetric optiization procedure conducted in a geoetric delay space, achieving better scalability as the nuber of utilizable servers increases. III. PROBLEM FORMULATION There are several edia distribution odels for interactive video streaing, including architectures based on centralized servers, end-syste ixing and PP. A centralized-server solution [], [] collects user edia using a relay server, perfors signal processing (e.g., silent suppression) on the data streas and ixes the to be sent out to different clients. End-syste ixing places the ixer on one of the clients. However, in any sessions, there ay be no particularly well-endowed peer. A PP solution adopts a full-esh topology, in which each client sends its data to all the other clients directly. However, a PP solution is apparently not scalable, especially for obile clients, since as the nuber of clients N grows, each client has to split its

3 84 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 Fig.. Illustration of our ulti-server topology, where ultiple servers are chosen fro the cloud to serve each session. T is a client and S is a server. The arrows illustrate the data flow paths between clients. (a) A 4-server solution for seven clients. (b) -server optial. (c) -server using centroids. (d) -server optial. liited upload bandwidth capacity to serve N other clients, leading to vanishing pairwise throughput []. A. The Multi-Server Mesh Topology In this paper, we consider using a ulti-server esh to serve the interactive video streaing session, which cobines the advantages of both centralized servers and PP architectures, as illustrated in Fig. (a). Definition (Multi-Server Mesh): Every client is connected to only one server and no other host. The servers for a full esh. In this topology, each client sends its own data to its server. For each server S, if it receives data fro a client T,the data is forwarded to all the other hosts connected to this server, including servers and clients, excluding T. IfserverS receives data fro other servers, the data is forwarded only to the clients directly connected to S. According to the above topology and protocol, a client connected to S transits a packet to another client connected to S in two hops via S, and transits a packet to another client connected to another server S in three hops via S and S. There are two benefits of the above ulti-server topology, naely throughput advantage and latency iproveent. Fro a throughput perspective, since each client only needs to upload one copy of its data to one server, it does not suffer fro the upload bottleneck as in the PP case. As servers are connected in well-provisioned backbone infrastructures, transfers between servers are also free of bottleneck. Unlike the ultiserver topology in [0] which allows a client to relay traffic for other clients, we require each client to only talk to its assigned server. If a certain client i is not connected to any server and is connected to soe other client j, client j will relay traffic for client i, leading to an upload bottleneck at client j. More forally, suppose that a client i is not connected to any server, while the other clients are connected in an arbitrary way with or without servers. In this case, we can easily give an exaple where the bandwidth capacity of this network is not fully utilized as follows. Consider the case that the download capacity D i of each node is greater than the su of upload capacities of other nodes, i.e., D i > j i U j, and the servers have high bandwidth connections. Then the total throughput supportable by the network is i j i U j, since each node can at ost download at the rate in{d i, j i U j }. Let us show that the axiu supportable total throughput i j i U j cannot be achieved. To achieve the axiu throughput, client i ust be able to download at the rate j i U j. Since it is not connected to any server, it ust use up all the upload capacities of other clients. As a result, other clients have no spare capacity to upload anything to any server. Since no server plays a role in this case at all, the usable upload capacity is only i U i, which cannot support a total download rate of i j i U j. Fro the latency perspective, the servers for a full esh to iniize the transfer delays between any pair of servers. By adjusting the choice of server locations, we can effectively reduce the end-to-end delays between all clients in the fored ulti-server topology that supports relatively high throughput. In addition, the servers can be extended to process data individually or collaboratively. Since our focus is on the network aspect, we do not discuss the details of signal processing and ixing functions. B. Server Placeent as Delay-Space Geoetric Optiization Since an interactive video streaing solution ay utilize a large pool of servers fro datacenters, CDN nodes, and dedicated servers, we can assue the utilizable servers are geographically densely distributed. It would be coputationally expensive for any graph algoriths such as packing spanning trees or Steiner trees to select servers in the existing dense graph of a large server pool. Given a set of N clients as input, this paper considers a copletely different geoetric approach to copute where the servers should be placed in a delay space, where each host has a coordinate, whether it be a server or client. The distance between two hosts in the delay space can predict their latency on the Internet. We can eploy NCSs [] to copute the coordinates for a set of hosts given their pairwise ping data. For exaple, Vivaldi [5] is a representative distributed NCS, and is deployed in any well-known Internet systes, e.g., Baboo DHT [3] and Azureus BitTorrent [4]. Let vector x i =(x i,...,xd i ) R D denote the coordinate of client i in the fored delay space. Each utilizable server has a coordinate as well. In Vivaldi, a new node only needs to collect the latency inforation fro a few other existing nodes to copute its own coordinate. In other words, Vivaldi ebeds the hosts into a delay space R D based on a relatively sparse atrix of latencies between the hosts. Although it is believed that a Euclidean space of low diension (e.g., D =, 3, 5) ay ebed the hosts with errors, due to TIV [5], [4], in Section V through large-scale easureent data, we will show that despite coordinate apping errors, our proposed server

4 HU et al.: GEOMETRIC APPROACH TO SERVER SELECTION FOR INTERACTIVE VIDEO STREAMING 843 selection algoriths can still reduce latency in interactive video streaing sessions. Suppose that X = {x,...,x N } is a given set of coordinates of N clients in the delay space. Under the topology in Definition, our proble is to find a partition P = {C,C,...,C k } of the client set X, together with a set of vectors Y = {y,y,...,y k }, where y j (j =,...,k) denotes the server location for class C j, such that the su (or the ean) of end-to-end delays between all the clients is iniized. Given P and Y, there is a apping y : X Y, where y(x i ) is the server location of the class to which client x i belongs. The proble is forally stated as in P,Y ( xi y(x i ) + y(x i ) y(x j ) + x j y(x j ) ) i j subject to Y k () where k is the axiu nuber of servers to be used. When k = N, a trivial solution is to place a server just beside (arbitrarily close to) each client and connect each client to its server. Since a full esh is fored aong the N servers that are directly connected to each other, the su of end-to-end delays between all clients reaches the iniu. However, it is costly to engage so any servers. Since deploying each server is associated with a cost of launching a VM instance on a certain physical achine in the cloud, we ai to provide a solution under a server nuber constraint k<n. Note that it is proved [4] that the graph version of Proble (), in which a nuber of candidate servers and clients are connected in a graph with pairwise distances known, is NP-coplete for any k<n. Once the ideal server locations are coputed in the delay space, we can choose the nearest (in ters of delay) real servers in the physical network, and connect the to the clients according to the topology in the optial solution. Note that the entire procedure is light-weight as long as the geoetric Proble () can be efficiently solved. Since server coordinates are usually stable and can be routinely aintained prior to the session, we only need to copute the coordinates of the few participating clients using Vivaldi when they join the session. This procedure is efficient, since for each client, we only need to easure its round-trip ties (RTTs) to a few reference servers to obtain its coordinate in an NCS. Moreover, unlike the graph version, the coplexity of solving Proble () in space is independent of the nuber of available candidate servers. Even though the NCS has errors, e.g., due to the violation of triangle inequality on the Internet and inaccuracy of network ebedding, we will show the capability of our proposed optiization procedure in ters of reducing overall latencies, despite network coordinate errors, through extensive perforance evaluation. C. Relationships to Euclidean k-median and Facility Location For k<n, as the decision variables include both the partition P and server locations Y, Proble () is a non-convex cobinatorial proble in general. Note that Proble () appears to be siilar to the faous k-edian proble and facility location proble [6], yet is even ore coplex than the. In fact, if we ignore the delays between servers in (), Proble () is reduced to the faous Euclidean k-edian clustering proble, which is proved to be NP-hard as well as being hard to approxiate to within arbitrary constant factor [6]. On the other hand, if we fix the partition P, () becoes the Euclidean ulti-facility location proble [5], which is a convex proble. However, our proble is even harder because the partition P is also a decision variable. To the best of our knowledge, there is no known efficient solutions or even approxiation schees to the proposed Proble () in space. IV. SERVER PLACEMENT IN THE DELAY SPACE We propose a nuber of algoriths to copute the ideal server locations in the ebedded delay space, fro one-server algoriths to two-server algoriths, with theoretical perforance guarantees. We further propose practical heuristics that can place ore than two servers efficiently. Given the set of clients, all the proposed schees copute the ideal server locations in the delay space by approxiately solving Proble (), and then ap each ideal server location to the nearest real physical server. A. One-Server Algoriths The current coon solution to interactive video streaing in practice uses a single server location. A siple solution is to set the server location as the centroid of all clients: Algorith (One-Server Centroid): Set the server location to be y = x i X x i/n. A direct iproveent is to solve Proble () with k =. When k =, Proble () becoes in ( x i y + x j y ) () y x i,x j X,i j where y is the location of the single server, or equivalently in(n ) x i y (3) y x i X which is a convex progra to find the geoetric edian of the client set X. Since y is the edian if and only if N i= (x i y)/ x i y =0, we can perfor a fixed-point iteration on the above equation to copute y, which is soeties referred to as Weiszfeld s algorith [6]: Algorith (One-Server Median/One-Server Optial): Use the following iteration to get the server location y: ( N )/( x i N ) y :=. (4) x i= i y x i= i y Let D opt and D denote the su of end-to-end delays achieved by One-Server Median and One-Server Centroid, respectively. Let denote the su of end-to-end delays in the full-esh topology where all the clients are directly connected. can be regarded as the iniu value of () if N servers are used, i.e., k = N. is achieved if a server is placed arbitrarily close to each client and all the servers for a full esh directly connecting each other. Thus, is the iniu possible su of end-to-end delays aong all cases, since no server placeent can beat connecting all client pairs directly in ters of delay. We will copare D opt and D against the best delay.

5 844 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 Proposition : Given any N clients x,...,x N,wehave D opt D (5) N where the left inequality achieve equality if and only if the centroid coincides with the edian, while the right inequality achieves equality when all the clients are distributed on two points. Proof: For the right inequality, using triangle inequalities, we have D =(N ) x i = (N ) N (N ) N x i X x i X x i,x j X x j X x j N x j X (x i x j ) (x i x j ) = (N ) N where the equality is achieved if and only if vector x i x j and vector x i x k i, j, k, have the sae direction or either of the is zero, which is equivalent to that all the clients are distributed on two points. The left inequality is obvious. Reark : Proposition shows that both One-Server Median and One-Server Centroid are ( /N )-approxiation schees to Proble () for any k. Moreover, One-Server Median, although being optial using one server, has the sae worst-case perforance as One-Server Centroid. Finally, it is worth noting that even if we use N servers each serving one client to achieve the best possible delay, the delay reduction as copared to using one server is no ore than. B. Two-Server Algoriths One-server algoriths ay perfor poorly when the clients tend to distribute in separable clusters. If we slightly increase the budget and use two servers, it is not hard to check that Proble () becoes ( ) in (N ) x i y + x i y {C,C,y,y } x i C x i C +n y y (6) where := C and n := C are the nubers of clients in classes C and C, respectively. Proble (6) is still a hard cobinatorial proble, in which server locations and the way we partition clients will both affect the su of end-to-end delays. We provide a centroid-based approxiate solution and bound its perforance. To siplify notations, we set A = x i x j, B = x i x j x i,x j C x i,x j C C = x i x j. x i C,x j C Clearly, we have = A + B + C. To judge how good a partition is, we define separability as β(c,c ):= C /( A n + B ) n (7) which represents the ratio between (noralized) cross-cluster distances and in-cluster distances. Intuitively, the greater the β(c,c ), the further apart the clusters C and C. We describe our two-server algorith in Algorith 3. Algorith 3 (Two-Server Centroids): First, choose the partition {C,C } to axiize β(c,c ). Then choose the centroid as the server location in each class, i.e., set y = x i C x i / and y = x i C x i /n. Since the nuber of clients in a session (e.g., a live conference) is usually no ore than a few, the first step is easy to perfor, for exaple, siply by enuerating all the partitions {C,C } to find the one with the axiu β. For exaple, even with 0 clients, there are only 5 different partitions to go through. In Section V, we will show that our proposed algoriths always finish within s for up to clients. Let D denote the su of end-to-end delays achieved by Algorith 3. Again, we analyze D as copared to. Proposition : For any client set X, wehave D <.5. Please refer to the appendix for the proof. Reark : Proposition shows that Two-Server Centroids is at least a.5-approxiation schee to Proble () for any k. However, there is really no inforation about whether Two-Server Centroids is better than one-server algoriths or not. According to the proof of Proposition, Two-Server Centroids ay not outperfor one-server algoriths in soe cases. The underlying reason is that although the two-server optial solution is certainly no worse than the one-server optial solution, Proble () with Y =is an extension of the two-edian proble, while k-edian is NP-hard [6]. Without any known efficient algorith to copute the two-server optial solution, we cannot conclude if the centroids of two servers are better than one-server optial solution: if the clients are distributed in two clusters, two servers are better; if they are ore ingled, even foring ultiple (ore than ) clusters, one server is better. We will see this effect in Section V through siulations. We now present a siple algorith that guarantees to beat Algorith in theory. Algorith 4 (One-or-Two-Server): Use either Algorith or Algorith 3, whichever produces a saller su of end-toend delays. Denote D = in{d opt,d } as the su of end-to-end delays achieved by Algorith 4. Then we have: Proposition 3: For any client set X, wehave D = in { D opt, D } < N. (8) Reark 3: The worst-case perforance of Algorith 4 is now strictly less than /N, because when D opt / achieves its axiu value of /N, all clients ust reside on two points, and in this case, it is easy to check that D / is saller than /N. Although a tighter bound on the worst-case D / is hard to derive, we can characterize the iproveent of Algorith 4 over Algorith (One-Server Median). Since D = in{d opt,d }, we only need to bound D opt /D

6 HU et al.: GEOMETRIC APPROACH TO SERVER SELECTION FOR INTERACTIVE VIDEO STREAMING 845 fro above. If D opt /D α, we can say Algorith 4 can reduce delay by a factor of at ost α. Suppose {C,C } is the partition of clients found by Algorith 3. Let y represent the centroid of X, and let y and y represent the centroids of C and C, respectively. Let = C and n = C be the nubers of clients in C and C, respectively. To siplify notation, define D, E and F as D := x i y, E:= x i y, F:= y y. x i C x i C Siilar to β, we define another separability easure β F (C,C ):= (9) D/ + E/n which will be used to derive the iproveent ratio of Algorith 4 over one-server algoriths. Due to the hardness of the proble, we are able to provide results for up to N =6clients: Proposition 4: For a given client set X, run Algorith 3. If =<n 3, wehave D D n + β n + ( ) β n + + ( ) n n + (0) and the bound is the best possible; if < n 3, then D + n D ( + n ) + nβ + ( ) n β + n + ( ) n n + n β β () and the bound is the best possible. Please refer to the appendix for the proof of Proposition 4. Since D opt D, the right hand sides of (0) and () also upper-bound D opt /D, which is the perforance iproveent of Algorith 4 over One-Server Median. We can get the following corollary iediately: Corollary 5: For a given client set X, run Algorith 3. If n 3 with + n = N and β,wehave D opt D D D N. () Proof Sketch: By analyzing the derivatives, it can be shown that the right hand sides of both (0) and () reach the sae axiu value of /N when β. Reark 4: Corollary 5 iplies that Algorith 4 suffices to achieve the best iproveent factor of /N over One- Server Optial. The best iproveent is achieved when β. In this case, copared to the distance between two clusters, the clients are alost distributed on two points, and thus using two servers is equivalent to using N servers. According to Proposition, the best iproveent of using N servers is also achieved in this extree case. However, since in reality β is usually finite, the iproveent factor of Algorith 4 over One-Server Optial will not be as significant as using ore servers. C. Multi-Server Placeent When we need to place ore than two servers, we ay use the k-eans clustering heuristic to quickly partition the clients into k classes, although k-eans ais to find a partition to iniize the su of delays fro the clients to their corresponding class centroids, which is different fro our objective of iniizing the su of end-to-end delays. Once the partition is obtained, it is not hard to check that Proble () using k servers is reduced to the convex progra of finding the server locations Y = {y,...,y k } k in (N ) {y,...,y k } j= x i C j x i y j + k i C i C j y i y j. (3) i= j= Clearly, once the partition P = {C,...,C k } is fixed, the objective of (3) is a convex function of y i. Proble (3) can be solved efficiently using standard convex proble solvers, with the initial server locations set as the class centroids. Now we have obtained the following ulti-server placeent schee: Algorith 5: (k-server Optiization Heuristic): First, partition the clients into k classes via k-eans heuristic. Then solve the convex proble (3) to get the server locations. As has been analyzed in Section III-C, the k-eans proble is a siplified version of our proble with the weights between servers reoved. Thus, it can be a good heuristic to partition the clients into any k classes. In our k-server Optiization Heuristic, we utilize the siilarity between k-eans and our proble to help obtain a sub-optial partition and further reduce the errors of ignoring the weights between servers by fine-tuning the locations of the servers under the found k-eans partition. D. Ideal-to-Real Server Mapping and Algorith Coplexity Note that all the proposed algoriths ai to find (ideal) server locations in the delay space. As a coon final step of all the proposed algoriths, we ap each coputed ideal server location to the nearest real physical server in ters of delay, which is fast. We will see in Section V that our server selection procedure can indeed reduce latency even if the real server apping phase can introduce errors. Let us have a glipse on the coplexity of different algoriths. In fact, the running tie of an algorith is ainly dictated by the tie to copute ideal server locations in the delay space, which depends only on the nuber of clients N and the nuber of servers to be adopted k, but not on the total nuber of candidate servers M. In contrast, selecting k servers on a graph of all M candidate servers and N clients [4] has a coplexity that draatically increases with M, which can be quite large in today s large-scale server cloud. The coplexity of the proposed one-server algoriths in space is linear in N. For Two-Server Centroids and One-Or-Two-Server, the coplexity is O( N ). However, the speed is still satisfactory when the nuber of clients N is liited to a few closely related people (which is coon in today s ulti-party live conferencing and other interactive video streaing applications). Even if N is large, the

7 846 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 Fig.. Locations of the 490 PlanetLab nodes. siplek-eans partition can achieve a satisfactory perforance in polynoial tie with coplexity O(k c ), c being a constant. V. TRACE-DRIVEN SIMULATIONS We conduct siulation evaluation based on two network latency datasets we have collected, one containing RTTs in the PlanetLab, the other containing RTTs fro the Seattle network [7] to PlanetLab nodes. The PlanetLab dataset contains the RTTs between 490 PlanetLab nodes (including 5 nodes in Asia, nodes in Europe, 9 nodes in North Aerica, and 5 nodes in other regions), that we collected in 04 over a 5-day period, with the geographic distribution of the nodes shown in Fig.. We have also collected RTT easureents fro 99 nodes in Seattle, an open PP coputing platfor [7] that consists of laptops, servers, and phones donated by users and institutions for research purposes, to the 490 PlanetLab nodes. While Planet- Lab nodes are ostly stable and located in university networks, the Seattle-PlanetLab RTTs are used to odel the longer delays fro personal devices to servers in Section V-D. We use the edian latencies between nodes as the input to our server placeent algorith. In real-world applications, network coordinates of the severs ay be coputed and stored a priori. When a client joins the session, its network coordinate can be coputed by a local decentralized Vivaldi algorith with pings to only a few rando servers. In our experients based on PlanetLab data, we directly ebed all the nodes into a delay space using an local iterative Vivaldi algorith. In each test run, we randoly choose a certain nuber of nodes as clients, while the reaining nodes act as the potential server pool. This way, the clients in each of our experients are essentially randoly distributed over the entire world across ultiple regions. In experients involving the Seattle data, we take the 490 PlanetLab nodes as the potential server pool and randoly select nodes fro the 99 Seattle nodes as clients. A. Algoriths With No More Than Two Servers In this section and the next one, we study the delay perforance based on the 490-node PlanetLab dataset, as the nuber of clients N ranges fro 4 to. In each run, different algoriths are applied, and their corresponding server locations and client partitions are coputed in the delay space. Then we ap these solutions to the nearest real servers in the dataset and build up the network. Finally, we evaluate the delay perforance of these networks according to the real end-to-end delays of the Fig. 3. Perforance of different algoriths in the 490-node siulation. (a) Theoretical (in the delay space). (b) After real server apping. TABLE I PERFORMANCE OF DIFFERENT ALGORITHMS IN THE 490-NODE SIMULATION WITH EIGHT RANDOM CLIENTS REPEATED FOR 000 TIMES Average Worst Best Variance Theoretical D /D n D opt /D n D /D n D /D n Real D /D n D opt /D n D opt /D n D /D n clients in the RTT traces. For every paraeter setting, we repeat the siulation for 000 ties. For each server selection solution produced by a certain algorith, we record the su of end-to-end delays between all clients D (in the real traces), and noralize it by, which is the su of pairwise distances of all the clients foring a full esh. We finally use the ratio D/ to evaluate the server selection. Fig. 3 shows the perforance of our proposed algoriths on the PlanetLab data. Fig. 3(a) shows the average (noralized) su of end-to-end delay in each session for the ideal server locations coputed in the delay space, whereas Fig 3(b) shows such value after apping the coputed ideal server locations to real servers. In both figures, the perforance ratios are uch saller than the theoretical worst-case bounds. And the ratios onotonically increase when the nuber of clients increases. The Two-Server algorith perfors worse in the real case due to the apping error. However, the proposed One-Or-Two-Server algorith is obviously better than One-Server-Optial. The reason is that we are choosing a better one fro two solutions, and despite network coordinate errors and apping errors, the algorith can pick the one with a saller error or take advantage of rando changes. Specifically, Table I shows ore details for the average, best, worst-case and variance of the perforance of 000 tests on eight rando clients. The best and worst-case perforance follows a siilar trend as the average perforance. The real ratio has a larger variance due to other errors like network coordinate errors and apping errors. Note that the real best perforance can even be slightly better than since the triangle inequality in real networks does not hold strictly. Fig. 4(a) plots the distribution of the ean and axiu endto-end delays achieved by One-Or-Two-Server, copared to a

8 HU et al.: GEOMETRIC APPROACH TO SERVER SELECTION FOR INTERACTIVE VIDEO STREAMING 847 TABLE II ALGORITHM RUNNING TIMES (MS) Nuber of Clients One-Server Centroid One-Server Optial Two-Server Centroid One-Or-Two-Server k-server Heuristic (k = ) Fig. 4. (a) Distributions of the ean and axiu end-to-end delays with One-or-Two-Server or a single fixed server. (b) The gap between the coputed ean end-to-end delay and that after apping to real servers. Tests are perfored on eight rando clients for 000 ties. Fig. 5. Delay perforance (noralized by )ofthek-server Optiization Heuristic in the 490-node siulation. fixed single server location (siilar to what Skype adopts []) for eight clients randoly chosen fro the 490-node dataset, repeated 000 ties. We can see that the average end-to-end delays of ost sessions with our siple One-Or-Two-Server algorith are around 90 s, and the axiu end-to-end delay is ostly less than 00 s, which is obviously better than the single fixed server solution. Furtherore, Fig. 4(b) plots the real server apping error in the sae experient, that is the gap between the real end-to-end delay and the estiated end-to-end delay in the delay space (without apping to real servers). Most errors are less than 5 s. Copared to the average end-to-end delay of about 90 s, the real delay is close to the estiated delay in the delay space. Through the above perforance coparisons, we have the following observations. First, One-Or-Two-Server, which is a cobination of One-Server Optial and Two-Server-Centroids, outperfors each of the. Two-Server Centroids can successfully copleent One-Server Optial, reducing delay in the case when One-Server Optial perfors poorly. With the One- Or-Two-Server algorith, we can effectively reduce the delay over one-server algoriths, including a single fixed server and one-server optial algoriths, and achieve a reasonably good solution when the nuber of clients does not exceed. Second, although One-Server Optial outperfors One-Server Centroid, yet One-Server Centroid has the shortest execution tie, aking it a valuable choice in applications that just need one server for extreely quick launching. B. Perforance of k-server Heuristics Fig. 5 shows the average perforance of the k-server optiization heuristic. The perforance generally increases when ore servers are engaged. There is a relatively large iproveent fro the -server to 3-server solutions. When using ore than three servers, the iproveent becoes arginal and not stable. It sees that the 3-server solution is the best and sufficient to handle the diversity in the geographic distribution of clients, thus leading to a large perforance increase over the -server solution, while adopting ore than 3 servers appears to be unnecessary for clients. We can also see that as the nuber of clients increases, the perforance of ore servers degrades at a slower pace than or servers. Another observation is that having ore servers ight experience bad perforance when the nuber of clients is slightly larger than the nuber of servers. Therefore, the k-server Optiization Heuristic with three servers are generally the ost cost-effective solution. We have also easured the execution ties of our algoriths, and present the in Table II for all the proposed algoriths under different nubers of clients. According to the discussions at the end of Section IV, we estiate the running tie for each algorith by keeping track of the coputation tie as well as the apping tie, and suing the up. The running ties of the two-server algoriths involving exhaustive partition search increases draatically as the nuber of clients grows. However, they are still practical when there are no ore than clients all algoriths finish in only s. The running ties of the other algoriths are consistently low regardless of the nuber of clients. C. k-means Partition Versus Max-β Partition In our solutions with two servers, the perforance critically depends on the partition. As we have shown, the ore separate the two classes are, the better the two-server solution will perfor. We have proposed to exhaustively search for the best partition to axiize the separability β, which leads to excessive coputational overhead when N is greater than a few dozens (although a rare case nowadays). We propose to reduce such coplexity using the k-eans algorith. By (9), separability is related to the su of the delays fro the clients to the class centroid in each class, and we know that the k-eans algorith is an efficient heuristic for finding the solution iniizing that su. Consequently, we can conduct partition in Two-Server Centroid and One-Or-Two- Server algorith using the -eans algorith instead of an exhaustive search. For a typical N =6, we have perfored thousands of trials with on the 490-node dataset, and record the noralized delay perforance D / of Two-Server Centroid with -eans partition versus with the ax-β partition, and plot the results

9 848 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 Fig. 6. Perforance of Two-Server Centroid: -eans partition versus ax-β partition. Fig. 7. CDF of ean end-to-end delays for the cross-network siulation and the 490-node siulation for clients. (a) Seattle and PlanetLab. (b) PlanetLab. in Fig. 6. We observe that a large group of the points are near the line y = x, which iplies that the -eans algorith is an excellent alternative to exhaustive partition search. Due to network coordinate projection errors and server apping errors, soeties -eans has even better perforance. This efficient heuristic has nearly the sae average perforance as ax-β partition in general, and can easily scale to a large nuber of clients. D. Siulations Across Seattle and PlanetLab Networks To siulate the possibly different latencies between interserver connections and client-to-server connections, we ake the 490 PlanetLab nodes the server pool and in each run, randoly choose nodes fro the 99 Seattle nodes to serve as the clients and deploy the k-server heuristics to select server(s). Note that the Seattle nodes have relatively worse network conditions than the PlanetLab nodes since the Seattle nodes coe fro a ore diverse environent. Fig. 7(a) shows the average end-to-end delay between all client pairs achieved in 000 runs and Fig. 7(b) shows the corresponding result in the PlantLabonly siulations (both servers and clients are fro the 490 PlanetLab nodes). We can see that the ulti-server solution has a larger benefit over a single server in the cross-network siulation especially for the -server heuristic, since the better network conditions aong servers reduces the delay between servers, which reduces the cost of engaging one ore transission hop with respect to the one-server solutions. The better the network conditions between servers, the ore the ulti-server solutions will help. Fig. 8. Mean end-to-end delay in ipleentation at various source rates for six clients. (a) Ipleentation versus siulation. (b) Ipact of source rates. VI. PROTOTYPE IMPLEMENTATION To verify the real-world perforance of the proposed ethods, we have developed an asynchronous ulti-threaded packet counication odule, with the Apache Thrift fraework and the Boost library, in 000 lines of C++ code. And Thrift generated about 5000 lines of code. We deployed this prototype interactive streaing syste on PlanetLab nodes. In our experients, each client sends packets to its designated server at a frequency of 300 packets/s using TCP, where the source rate is controlled by tuning the packet size: sourcerate = packetsize frequency. We ai to easure the real packet-level end-to-end delays achieved at different source rates considering both the network distance effect and syste load. Since it is hard to synchronize the clocks on different coputers, the delay between clients (on the order of s) cannot be easured by siply recording the sending tie on the sender and the receiving tie on the receiver. We propose an indirect ethod to easure the end-to-end delay of each packet. Suppose client A is sending packets to another client B via soe servers. At the very oent before A sends out a packet, it starts a tier. When the packet eventually reaches B, B will send a ping packet directly to the sender A. When A receives the ping packet, it stops its tier and record the tie span T circle, which is the end-to-end delay of the packet fro A to B plus the one-way ping tie fro B to A. When B gets the reply of the ping fro A, it records the round trip tie RTT AB. Therefore, the end-to-end delay fro A to B can be evaluated by T circle RTT AB /. In our ipleentation, each client easures the end-to-end delay to all other clients once every 300 packets. The goal of the prototype-based experients is to verify if the delays calculated by suing up the ping values really confor to the packet-level delays in a real syste where CPU and other resource usage ay also affect end-to-end delays. Therefore, we have selected a group of six clients, for which the ultiserver heuristic can indeed iprove delay perforance in the siulation and we want to see whether such benefits still exist in the ipleentation. Fig. 8(a) shows the perforance coparison between siulation and ipleentation. To eliinate the influence of source rates on latency, here we deliberately set the source rate to be kb/s. Note that as the nuber of servers increases fro to 4, the real delay (s) in the ipleentation decreases at a

10 HU et al.: GEOMETRIC APPROACH TO SERVER SELECTION FOR INTERACTIVE VIDEO STREAMING 849 siilar pace to that in the siulation (which estiates delays by suing up RTTs). The real delay is only slightly worse than the siulated result due to the existence of queuing delays and processing (CPU) delays. Fig. 8(b) illustrates the change of the ean end-to-end delays as the source rates of clients increase. When the nuber of servers is, 3 and 4, the ean end-to-end delays only increases slightly as the source sending rate increases fro kb/s all the way to 500 kb/s at each source (which can support sufficiently high video quality). However, in the one-server solution, as the source rate increases, the ean end-to-end delays has a draatic increase, which surges fro 9.5 to 38.3 s as the source rate changes fro to 500 kb/s. The reason is that the server uploading burden increases with fewer servers, since now each server needs to upload data to ore terinals. VII. CONCLUSION In this paper, we study server selection and server location optiization with a k-server esh topology in distributed interactive video streaing applications, aiing at iniizing the suation (or ean) of end-to-end delays between clients. We forulate the proble in a delay space instead of on a graph, and propose a nuber of conceptually siple one-server or two-server approxiate solutions with theoretical worst-case perforance guarantees. We further propose practical optiization heuristics to enable rapid selection of three or ore servers. We have perfored extensive trace-driven siulations based on large aounts of easureent data collected fro the Planet- Lab and Seattle platfors and ipleented a prototype syste to verify our proposed server selection schees in real networks. Despite the errors in network ebedding and real server apping, our proposed algoriths can efficiently select servers for interactive video streaing, achieving low end-to-end delays. In conclusion, by using a siple One-or-Two-Server schee, both the ean and axiu end-to-end delays in interactive video streaing sessions can be greatly reduced, as copared to always using servers at a single location. With the proposed k-server selection heuristic, ean end-to-end delays can be further reduced as ore servers are used. However, for the costeffectiveness in sessions coposed of no ore than clients, a practical size of today s interactive video streaing sessions, using ore than three servers is ostly unnecessary. Moreover, with any of the proposed algoriths, it takes less than sona coodity personal coputer to optially select the locations of ultiple servers for -client sessions, given a pool of close to 500 candidate servers. APPENDIX Proof of Proposition : We first present several leas. Lea 6: Given any N clients x,...,x N,wehave D < 4(β ). (4) +4β n + n Siilar to (Section IV-A) in the proof of Proposition, using triangle inequalities, we can get x i y x i x j = A x i C x i,x j C x i y n x i x j = n B x i C x i,x j C where each equality is achieved when the clients in that class are distributed on two points. Using triangle inequalities with soe careful edge counting, we can get n y y = n x i C x i x j C x j n = (x i x j ) x i x j = C x i C,x j C x i C,x j C where the equality is achieved when all the clients are distributed on a line and there exists a point on the line which separates the clients fro the two classes. Therefore ( D (N ) A + n B) + C A + B + C = C ( n A + n B) + ( A + n B) A + B + C < C ( n A + n B). A + B + C Eliinating C by(7), we have ( D (β ) n < A + n B) A + B +β ( n A + (5) n B). Since n A + n B AB (6) where the equality is achieved when A/B = /n. (7) Once this equality holds, we get (4). Lea 7: Given, n with + n = N, there exists a partition {C,C }, such that C =, C = n, and β(c,c ) /( n ). We exhaust all partitions {C,C } with C = and C = n and inspect the ratio β su = P : C =, C =n ( C/(n) A/ + B/n ). P : C =, C =n Due to the exhausting procedure, both the nuerator and the denoinator of β su evenly cover all the pairwise delays of the clients. Therefore, we can evaluate their ratio by counting how any pairs of delays they cover. Since there are totally ( ) +n different kinds of partitions, after cancelling the sae factor, we have β su = ( +n n ( +n ( ) ( ) n ) + n (n ) = ). n

11 850 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 8, NO. 5, MAY 06 Hence, aong all the different β(c,c ) with C = and C = n, there ust exist one such that β(c,c ) /( n ). According to Lea 7, for the partition that axiizes β, we have β 0.5. Therefore, by (4), we have D < + n + n +4 n + + n β + n + n n + n Proof of Proposition 4: When proving Propositions and, we have fixed and found upper bounds on D and D.In the following, we take a different approach to fix D and find the axiu value D ax of D, that is the delay produced by One-Server Centroid. Since D opt D, D ax is also an upper bound on D opt. The next lea gives D ax for fixed D, E, F: Lea 8: Given D, E, and F, and the nubers of clients in the two classes, n, with n 3, wehave ( D ax =( + n ) D + + E + ( n ) + n F ( ) ( ) ) n + n F + + n F. Proof Sketch: When F is given, the delays between the centroids of the two classes and the centroid of all the clients are fixed, i.e., y y and y y are both fixed, since N n y y = N y y = F. (8) Therefore we only need to consider how the clients in each class are distributed to axiize the su of their delays to the ain centroid. That is equivalent to solving ax xi C k x i y x i s.t. x i =0 and x i =T (9) x i C k x i C k where T is soe constant. We consider when C k has, or 3 clients. When C k has only client, (9) is constant and trivially axiized. When C k has clients, the two clients x and x are syetric to each other and it is easy to find that (9) achieves its axiu when x x is orthogonal to y. When C k has 3 clients, assuing x 3 has the shortest nor so that x 3 T/3, we rewrite (9) as ax f(x 3 )= x 3 y + ax x 3 x,x x i y (0) i= s.t. x + x = x 3 () x + x = T x 3. () Now we calculate ax x,x i= x i y. Wehave ( x y + x y x y + x y ) (3) where the equality is achieved when x y = x y.we also have ( x y = x x ) ( ) + x x + x + y (4) which also holds for x. Hence, by () and (4), after soe derivation, (3) can be rewritten as x y + x y ( x + x 3 + x + x 3 + x 3 + y ). If we fix x 3, x 3 / is the geoetric edian of x and x, and thus x + x 3 + x + x 3 x i = T x 3 (5) i= where the equality is achieved when x 3 =0. Also, since x + x 3 + x + x 3 =0 x + x 3 / + x + x 3 / achieves the axiu when x + x 3 / = x + x 3 /. Therefore, by (3) to (5), we can get ax x,x ( x y + x y ), so (0) can be calculated as f(x 3 ) y x 3 ( ) T x3 + + x 3 y (6) the rhs of which reaches its axiu value y + T /4+ y when x 3 =0, which can be shown by analyzing the derivatives of rhs of (6) for all x 3. When x 3 =0, all the above inequalities can achieve equality. Therefore, when C k has three clients, we can calculate D ax. Finally, using the sae analysis as above, we can calculate D ax for all and n no ore than 3. They all have the sae for as (8), copleting the proof. Now we can vary D, E, F to find the axiu values of D ax /D under different cases. Since D opt D D ax, we only need to find the axiu of D ax /D. Since D = ( + n )(D + E)+nF, by Lea 8, we can express D ax /D as a function of D,E and F. By dividing both the nuerator and denoinator by F, and denoting D/F as λ, E/F as μ, we get D ax + n = D (+n )(λ+μ)+n ( ( ) ( ) ) n λ + + μ +n + + n +n +n. (7) Fro (9), we have λ + μ n = β. We axiize D ax /D as a function of λ and μ subject to λ + μ n = β, and can find that for both =<n 3 and < n 3, D ax /D given by (7) achieves its axiu value when μ =0. Substituting μ =0into (7) for different, n proves Proposition 4.

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Chandrasekaran and A. Tair, Open questions concerning Weiszfeld s algorith for the ferat-weber location proble, Math. Prograing A, vol. 44, nos. 3, pp , May 989. Yaochen Hu (S 6) received the B.Eng. degree in electronics and inforation engineering fro the Huazhong University of Science and Technology, Wuhan, China in 0, and is currently working toward the Ph.D. degree in electrical and coputer engineering at the University of Alberta, Edonton, AB, Canada. His research interests include the fields of cloud coputing, achine learning, and algorith analysis. Di Niu (S 08 M 08) received the B.Eng. degree electronics and counications engineering fro Sun Yat-Sen University, Guangzhou, China, in 005, and the M.A.Sc. and Ph.D. degrees in electrical and coputer engineering fro the University of Toronto, Toronto, ON, Canada, in 009 and 03, respectively. Since 0, he has been with the Departent of Electrical and Coputer Engineering, University of Alberta, Edonton, AB, Canada, where he is currently an Assistant Professor. His research interests include the areas of cloud coputing and storage, ultiedia delivery systes, data ining and statistical achine learning for social and econoic coputing, distributed and parallel coputing, and network coding. Prof. Niu is a Meber of ACM. Zongpeng Li (SM ) received the B.E. degree in coputer science and technology fro Tsinghua University, Beijing, China, in 999, and the M.Sc. degree in coputer science and the Ph.D. degree in electrical and coputer engineering fro the University of Toronto, Toronto, ON, Canada, in 00 and 005, respectively. Since 005, he has been with the Departent of Coputer Science, University of Calgary, Calgary, AB, Canada, where he is currently a Professor. His research interests include coputer networks, network coding, and cloud coputing.