Internet-Scale Simulations of a Peer Selection Algorithm

Transcription

1 Internet-Scale Simulations of a Peer Selection Algorithm Ali Boudani, Yiping Chen, Gilles Straub Thomson R&D France 1, av. de Belle-Fontaine CS Cesson-Sévigné, France firstname.lastname@thomson.net Gwendal Simon GET/ENST-Bretagne laboratory of Computer Science CS Brest Cedex 3 - France gwendal.simon@enst-bretagne.fr Abstract The match between a peer-to-peer overlay and the physical Internet infrastructure is a constant issue. Time-constrained peer-to-peer applications such as live streaming systems are even more challenging because participating peers have to discover their closest neighbors as quickly as possible. We propose in this paper an approach based on landmarks and a management server. Despite its centralized architecture, this method can scale to a large population of peers. We focus in this paper on challenging this basic idea through intensive simulations, using a large-scale map of the Internet Router (IR) layer in a simulator dedicated to peer-topeer systems. We compare the results obtained by the new method with a random selection and an optimal selection and we show that this proposal makes sense. 1 Introduction Designers of peer-to-peer systems are usually concerned by the match between logical overlay and physical infrastructure: peers should preferentially be connected with the peers that are the closest in Internet [4, 12]. However determining these closest neighbors in a wide population spread on a large-scale network as Internet is still a challenge. Among the most promising ideas, coordinate-based schemes aim to fix the location of any host on Internet in a Euclidean space, to allow two peers to estimate their latency by a basic distance computation [9, 15, 2]. Previous works show that these virtual coordinates can be obtained by active probing, i.e. by collecting round-trip-time (RTT) measurements between peers and a small set of landmarks [7]. Some studies have also investigated the placements of these landmarks [14, 13]. Unfortunately, network coordinate systems require a substantial amount of time before to deliver accurate information and do not take physical network topology into consideration. We definitely need a system to estimate quickly and accurately the closest peers of a newcomer from simple probes. We envision a basic mechanism based on traceroute tool. We make the following assumption: if a central server can store the exact route between each peer and itself, it would be able to estimate accurately and quickly the closest peers of any newcomer. The reasoning behind such assumption indeed refers to the statistical regularities observed in the large-scale structure of Internet [10]. The apparent heavy-tailed degree distribution in the graph of routers makes all routes toward a same point converge through a unique sub-path. It allows us to infer that the proximity of two peers can be estimated by taking note of their first router in common within the paths toward a same landmark. In this paper, we challenge this idea with simulations. We face a usual challenge of peer-to-peer simulations: as peers are expected to be spread all over the world, the simulator should conform to a model of Internet, although this mapping is known to be a real issue [3]. Moreover, we are interested here in topology discovery that can hardly be done on a Autonomous System (AS) layer, but rather on a Internet Router (IR) layer, known to be vaster and difficult to model [10]. So we rely on a IR map [8] and the wellknown PeerSim simulator [6]. 2 Proposal Overview In this section, we give details about the tool we envision. We would like the central manager of a peerto-peer system for live streaming diffusion to be able to accurately estimate the closest peers to a newcomer. In peer-to-peer live streaming applications, the source can act as a central server because peers usually connect to the peer-to-peer network through a light communication with the source.

2 2.1 Notations A peer-to-peer system is expected to involve a large population of peers. Let V be the set of n peers in the system. The set of peers known by a peer p V is noted K(p). As in the popular bit-torrent system [11], a peer p forwards the incoming data to few peers among this set of neighbors. For simplicity, we fix to a global constant κ the number of neighbors a peer is aware of. The Internet is seen here as a set of linked routers. We note R the set of routers and G R the graph formed by the routers and their links. As shown in many previous studies [3, 10], the graph G R exhibits statistical regularities related with the so-called complex networks, i.e. small-world and scale-free properties. One of the most interesting features is related with the degree distribution: very few routers are very well connected while the vast majority of routers have less than three connections. We note τ(r) the degree of a router r in G R. Each host participating to the peer-to-peer applications is attached to a router r with τ(r) = 1. It somehow simulates a end-user connected to a Digital Subscriber Line Access Multiplexer (DSLAM). We also note L the set of landmarks. A landmark is a reference node with a well-known network address. It also responds to any request from participants to a peer-to-peer system. Finally, these landmarks are expected to be homogeneously distributed in the network and ideally located on routers having a higher degree. We assume our ability to find any shortest path in G R. We do not consider any routing policy, so G R is a unweighted graph. Moreover, the traceroute is here simplified such that the path returned by this tool is assumed to be the shortest path in G R between the originator of the traceroute and the destination host. 2.2 Implementation The approach we briefly describe in this article contains two steps: (i) a newcomer infers the path between itself and its closest landmark and (ii) the management server estimates the closest peers to this newcomer from the analysis of this path. The first step begins once the newcomer p wants to join the peer-to-peer system. This step aims to collect useful information about the infrastructure, i.e. about G R in the surroundings of p. We assume that p knows the set of landmarks L. If L contains more than one landmark, the peer p is invited to choose one landmark among L. The Round-Trip Time (RTT) metric is used to determine this landmark we note l(p) L. We propose to choose the landmark whose connection exhibits the shortest RTT. Note that RTT values can vary over p 1 r a lmk r b r 7 r 6 r r 8 c p p 4 3 r 1 d tree(p 1, p 2 ) r 2 r 3 r 5 r 4 d(p 1, p 2 ) Figure 1. Two peers p 1 and p 2 having the same closest landmark lmk time, so this choice can be due to network fluctuation rather than a real proximity. However we do not consider network fluctuations in our simulator, so results are obtained with l(p) being the actual closest landmark of p in G R. Once p and l(p) are fixed, more refined measures are done in order to estimate the network topology nearby p. We use here a traceroute-like tool: the peer p discovers the set of routers in the path between itself and l(p). This path, noted δ(p), is then transmitted to the management server. The management server is responsible of the second step. It knows all peers in V and we assume moreover that it also knows δ(p) for all p V. A central algorithm is implemented to build a list of κ closest peers for K(p). This list is the peer set of peer p. 2.3 Principles We describe a typical situation in Figure 1. The routers r a, r b and r c are within the network core and have a large number of connections with other routers (not represented in the figure). On the contrary, the routers r i with i {1... 9} are small routers with τ(r i ) < 4. The shortest path between many peers passes through the network core as well as the path to the landmark. We note d tree (p 1, p 2 ) the hop-distance of the path deduced by the landmark from the two paths δ(p 1 ) and δ(p 2 ). In this example, this path is p 1 r 3 r 1 r c r 2 r 4 p 2 so δ tree (p 1, p 2 ) = 5. Unfortunately the shortest path between these two peers does actually not goes through r c but is rather p 1 r 3 r 5 r 4 p 2. We note d(p 1, p 2 ) the hop-distance of the shortest path between two peers p 1 and p 2. As explained, we expect that most cases verify d(p 1, p 2 ) = d tree (p 1, p 2 ). In Figure 1, all paths containing p 4 verify it, so the p2

3 management server should be able to determine the closest peers of p 4 without error. 3 Evaluation 3.1 Internet Graph Simulating an algorithm committed to runs into Internet requires usually to rely on a model of the network. Many existing works have focused on modeling one Internet layer: Autonomous System (AS) level with peering relationships, or Internet Routers (IR) level which provides a fine-grained topology of connections among routers [10]. Modeling an Internet layer relates with extracting the most significant network properties from a set of measurements done on the real network. These properties are then used to generate large-scale random graphs. One of the most commonly observed properties is the heavy-tailed distribution of node degrees [10]. We are interested with IR level network because traceroute is at the router level. Many topology generators have been implemented for AS level model, but we have not heard of such generator for the IR level. An alternative to network modeling consists in re-building a snapshot of the Internet topology. Most previous works in this way base on active probing. Thus Mercador [5] software probes randomly selected destinations and gives a partial view of Internet. Another recent tool, namely Nec (Network Cartographer) [1], claims to discover a larger part of the IR graph. We use the result of one massive crawl achieved in April 2003 by this tool. This Internet IPv4 map contains 47k routers and 119k links. This graph is representative enough to produce realistic results, because the degree distribution of the IR map used shares the heavy-tailed characteristics of Internet model. 3.2 Initialization and Metrics We now describe the metrics we observe in the simulator running on the graph G R. First we initialize the overlay network by selecting some peers. We pick a randomly chosen set of routers having a degree equals to one. For each peer p in V, we pick a router r p in G R such that τ(r p ) = 1. In the same manner, the routers acting as landmarks are chosen among a set of routers with the same in-degree noted τ lmk. The default value is τ lmk = 30. The other default parameters are: number of peers (n = 1000), number of neighbors (κ = 10) and number of landmarks ( L = 5). Every peer in V runs iteratively the peer selection algorithm to select its κ neighbors. Recall that we note K(p) the set of neighbors of a peer p. Simultaneously we associate with a peer p two alternative sets of neighbors. The first one, noted C(p), is the set of the actual closest peers of p. We use a brute-force algorithm: we measure the actual distance between p and all peers in V and we select the κ shortest distances. This set is the optimal one. The second one, noted R(p), is a randomly chosen set of neighbors. This simulates the behavior of a basic implementation of Bit Torrent. This set should be outperformed by the algorithm we described in this paper. Metrics Representation D q K(p) D tree q K(p) d tree(p, q) D closest q? C(p) D random q R(p) Table 1. metrics in the simulation We measure the sum of hop-distances between a peer p and all its neighbors in these distinct sets of neighbors. The notation is summed up in Table 1. The difference between D and D tree is the way to choose the path, as shown in Fig 1. The way we initialize the peer-to-peer system is largely based on random processes: peers and landmarks are randomly chosen. To ensure the accuracy of the results, we produce around 10 runs for plotting one point. 3.3 Landmarks Placement Every peer determines its closest landmark before launching traceroute to this one. The role of landmarks is to perform a coarse-grained peer selection by separating all peers into many small landmarkscentered clusters. With this preliminary clustering of peers, further peer selection is not only faster but also more efficient because the length of the path is shorter and we expect that a closer landmark has a more accurate view of the network. The results in Fig 2 reflects somehow the radius of the cluster with the intuition that a small cluster may provide more detailed data to the management server. The average path length decreases through two ways: (i) increasing the number of landmarks and (ii) selecting higher degree routers. The first one is obvious: the ratio landmarks per peer increases, so the probability that a peer discovers a landmark in its surroundings increases. The second one is due to the structure or G R. Routers with high connectivity are in the network core. Most of the traffic should travel through this backbone, so the paths naturally tends to decrease in average.

4 3.4 Direct Path vs. Landmark-based Path Figure 2. Average path length between peer and its closest landmark when varying the number and the degree of landmarks Figure 3. Ratio of direct path and tree path We notice that different curves in Fig 2 seems to have a lower bound near to 5 hops. The impact of the degree is less and less significant when the number of landmarks increases. This result is due to network properties: high degree nodes concentrate in the network core while peers are in the edge. Increasing the degree of landmarks tends to put landmarks very close to each other, so inefficient. This lower bound seems to be actually the distance between the edges and the core. We may expect smaller distances with increasing the number of moderately smaller routers, but we observe that the higher degree routers always outperform. This result confirms that there are few shortcuts from one edge to any other node, even the middle-connected ones. Finally, the optimal trade-off seems to be few landmarks ( L = 7) with high degree (τ > 60). After probing with traceroute, the topology of every cluster is considered as a landmark-rooted tree. Our distance estimation is based on the distance in the tree instead of distance in the real topology. The difference of two distances is explained in Fig.1. In this paragraph, we compare direct path (D) and landmarkbased path (D tree ). In Fig 3, we remark that the ratio D/D tree is higher than 1. That means the estimated distance based on landmarks is imprecise compared to the distance in the real topology. As shown in Fig 1 of Section 2.3, this imprecision is because of multiple possible paths from peer to landmark and other shorter paths among peers. The topology discovering method used in our proposal has some limitations: Traceroute-like tool returns a set of paths only to the landmark, and misses some routers between peers. This estimation error decreases when we have more peers in the network. Because for fixed κ, the density of peers in the network increases with n. So the relative distances between peers decreases as n increases. As a result, we note that the landmark-based distance estimation works better for small distance values than big ones. In spite of this error on distance estimation, we notice that the difference between real and estimated distances remains small. Only an error of 10% is observed in the case n = 1000, κ = 10. That is because, the method we propose performs a preliminary clustering before a fine-grained topology discovery. Thus, a simple traceroute can provide results similar to the real topology, with an acceptable tolerance. 3.5 Performance Analysis As shown in Table 1 three metrics are used in this evaluation: D, result of our method; D closest, perfect result with absolutely closest peers, which reflects the optimal peer selection; and D random, worst case with randomly chosen peers. In Fig 4, as we expect, the ratio D/D closest is always smaller than D random /D closest. This upper bound decreases as κ increases (in Fig 4-b). The reason is, for n fixed, D random is linearly proportional to κ. But D closest is a concave function of κ, as closest neighbors will have longer distance when κ increases. In Fig 4- a, we have similar explanation: Average distance of closest set C is a convex function of n. So the ratio D random /D closest increases when n increases. We also observe that D/D closest remains steady for different κ and n values. It means that our peer selection method is stable: the selection result is guaranteed

5 acceptable and affects slightly the accuracy of peer selection. Secondly, simulation results show that our neighbor-discovering method works well: The nearby peers can be actually identified to constitute a peer set. Using this set, the involvement of routers in peerto-peer communication is reduced by 30% compared to a random peer selection. This result is close to the optimal result, with an average difference of 20%. These results of intensive simulations allow us to conclude that our proposal is simple as well as efficient. References Figure 4. Ratio of D to D closest and D random to D closest for the κ and n values we choose. We remark in both sub figures that D/D closest is approximately 1.2. Our algorithm achieves good results: to reach its neighbors, only 20% more routers are involved compared to the closest case. With our method which introduces few overhead and requires small computation capacity, we are finally not so far to the optimal result. This error of 20% in our peer selection is due to the imprecision of topology discovery. Some nearby peers are considered as far because the shortest paths to them are unknown. Although we do not have exact selection result for closest neighbors, the distance between peer and its neighbors is significantly reduced compared to the BitTorrent scenario. Thus, we can conclude that our peer selection method gives good performance. 4 Conclusion In this paper, we have challenged a peer selection algorithm that discovers nearby neighbors with Internetscale simulations. We have evaluated its performance using PeerSim simulator and a large-scale IR level map. Our study focuses our evaluation on router-level metrics such as the amount of intermediate routers between peer and its neighbors. A first interesting results of this paper is that, we can efficiently discover network topology with a small set of high degree landmarks. A difference of real and inferred topology is observed, but this difference remains [1] Network Cartographer. magoni/nec/. [2] F. Dabek, R. Cox, M. F. Kaashoek, and R. Morris. Vivaldi: a decentralized network coordinate system. In SIGCOMM, [3] L. Dall Asta, J. I. Alvarez-Hamelin, A. Barrat, A. Vázquez, and A. Vespignani. Exploring networks with traceroute-like probes: Theory and simulations. Theor. Comput. Sci., 355(1):6 24, [4] L. Garcés-Erice, E. W. Biersack, P. Felber, K. W. Ross, and G. Urvoy-Keller. Hierarchical peer-to-peer systems. In 9th Int. Euro-Par Conference, [5] R. Govindan and H. Tangmunarunkit. Heuristics for internet map discovery. In INFOCOM, [6] M. Jelasity, G. P. Jesi, A. Montresor, and S. Voulgaris. PeerSim: A Peer-to-Peer Simulator. URL: [7] J. M. Kleinberg, A. Slivkins, and T. Wexler. Triangulation and embedding using small sets of beacons. In FOCS, [8] D. Magoni and M. Hoerdt. Internet core topology mapping and analysis. Computer Communications, 28(5): , [9] T. S. E. Ng and H. Zhang. Predicting internet network distance with coordinates-based approaches. In INFOCOM, [10] R. Pastor-Satorras and A. Vespignani. Evolution and Structure of the Internet. Cambridge University Press, [11] J. Pouwelse, P. Garbacki, D. Epema, and H. Sips. The bittorrent p2p file-sharing system: Measurements and analysis. In 4th Int. Workshop on Peer-to-Peer Systems (IPTPS), [12] S. Ratnasamy, M. Handley, R. M. Karp, and S. Shenker. Topologically-aware overlay construction and server selection. In INFOCOM, [13] K. P. Shanahan and M. J. Freedman. Locality prediction for oblivious clients. In IPTPS, [14] L. Tang and M. Crovella. Geometric exploration of the landmark selection problem. In Proc. of Passive and Active Measurement Workshop (PAM), [15] B. Wong, A. Slivkins, and E. G. Sirer. Meridian: a lightweight network location service without virtual coordinates. In SIGCOMM, 2005.