## Graph-Theoretic Analysis of Structured Peer-to-Peer Systems: Routing Distances and Fault Resilience (2003)

### Cached

### Download Links

Citations: | 111 - 7 self |

### BibTeX

@TECHREPORT{Loguinov03graph-theoreticanalysis,

author = {Dmitri Loguinov and Anuj Kumar and Vivek Rai and Sai Ganesh},

title = {Graph-Theoretic Analysis of Structured Peer-to-Peer Systems: Routing Distances and Fault Resilience },

institution = {},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper examines graph-theoretic properties of existing peer-to-peer architectures and proposes a new infrastructure based on optimal-diameter de Bruijn graphs. Since generalized de Bruijn graphs possess very short average routing distances and high resilience to node failure, they are well suited for structured peer-to-peer networks. Using the example of Chord, CAN, and de Bruijn, we first study routing performance, graph expansion, and clustering properties of each graph. We then examine bisection width, path overlap, and several other properties that affect routing and resilience of peer-to-peer networks. Having confirmed that de Bruijn graphs offer the best diameter and highest connectivity among the existing peer-to-peer structures, we offer a very simple incremental building process that preserves optimal properties of de Bruijn graphs under uniform user joins/departures. We call the combined peer-to-peer architecture

### Citations

3613 | Chord: A scalable peer-to-peer lookup service for Internet applications
- Stoica, Karger, et al.
- 2001
(Show Context)
Citation Context ...outing efficiency and resilience to node failure. Graphs in peer-to-peer networks range from star-like trees (centralized approaches such as Napster) to complex k-node-connected graphs (such as Chord =-=[40], -=-CAN [31] and Pastry [35]). The performance of each peer-to-peer architecture is determined by the properties of these graphs, which typically possess Θ(logN) diameter and Θ(logN) degree at each node... |

2784 | A Scalable Content-Addressable Network
- Ratnasawmy, Francis, et al.
- 2001
(Show Context)
Citation Context ...ficiency and resilience to node failure. Graphs in peer-to-peer networks range from star-like trees (centralized approaches such as Napster) to complex -node-connected graphs (such as Chord [42], CAN =-=[33]-=-, and Pastry [37]). The performance of each peer-to-peer architecture is determined by the properties of these graphs, which typically possess diameter and degree at each node (where is the number of ... |

2324 | The capacity of wireless networks
- Gupta, Kumar
- 2000
(Show Context)
Citation Context ... requests). Assuming fixed transmission bandwidth and discounting interference effects, the average capacity c(G) of wireless ad-hoc networks is O( 1/ √ N) due to spatial restrictions on connectivit=-=y [16],-=- while both Chord and logarithmic CAN maintain an average capacity of 1 = µ d 2 . Compared to log2 N wireless networks, this is a much better bound; however, it is still several times lower than that... |

1490 | Pastry: Scalable, decentralized object location, and routing for large-scale peer-to-peer systems
- Rowstron, Druschel
(Show Context)
Citation Context ...lience to node failure. Graphs in peer-to-peer networks range from star-like trees (centralized approaches such as Napster) to complex k-node-connected graphs (such as Chord [40], CAN [31] and Pastry =-=[35]).-=- The performance of each peer-to-peer architecture is determined by the properties of these graphs, which typically possess Θ(logN) diameter and Θ(logN) degree at each node (where N is the number of... |

1332 |
Introduction to parallel algorithms and architectures: arrays, trees, hypercubes
- Leighton
- 1992
(Show Context)
Citation Context ...me for simplicity of notation that the total number of nodes N is a power of node degree and omit ceiling functions whenever appropriate. 3.2 Structure of de Bruijn Graphs De Bruijn graphs [5], [19], =-=[22]-=-, [38] are nearly optimal, fixed-degree digraphs of diameter logkN, where k is the fixed degree of each node and N is the total number of nodes. Note that de Bruijn graphs are directed graphs with k o... |

1068 | Tapestry: An Infrastructure for Fault-tolerant Wide-area Location and Routing,UCB
- Zhao, Kubiatowicz, et al.
(Show Context)
Citation Context ...er). Finally, the trie maintains its average degree over all nodes equal to only 2 (since approximately k−1 k 1 Strictly speaking, Tapestry’s routing table size is b logb N instead of (b − 1) lo=-=g b N [43]. k de Trie Chord CAN Pastry-=- Classic Bruijn butterfly 2 20 – – huge – 31 3 13 40 – – – 20 4 10 26 – 1,000 – 16 10 6 13 – 40 – 10 20 5 10 20 20 20 8 50 4 8 – – 7 7 100 3 6 – – 5 5 Table 2: Graph diamet... |

488 | Accessing nearby copies of replicated objects in a distributed environment
- Plaxton, Rajaraman, et al.
- 1997
(Show Context)
Citation Context ...e make DHTs highly suitable for peer-to-peer networks. 2.1 Peer-to-Peer DHTs Many current DHTs [17], [35], [37], [43], [44] rely on the concept of prefix-based routing introduced by Plaxton et al. in =-=[29]. -=-Plaxton’s framework is extended in Tapestry [43] and Bayeux [17], [44] to accommodate dynamic join/departure of peers and provide necessary failure-recovery mechanisms. A similar method based on gen... |

402 | Bayeux: An architecture for scalable and fault-tolerant wide-area data dissemination
- Zhuang, Zhao, et al.
- 2001
(Show Context)
Citation Context ... structure, excellent scalability, short routing distances, and failure resilience make DHTs highly suitable for peer-to-peer networks. 2.1 Peer-to-Peer DHTs Many current DHTs [17], [35], [37], [43], =-=[44] r-=-ely on the concept of prefix-based routing introduced by Plaxton et al. in [29]. Plaxton’s framework is extended in Tapestry [43] and Bayeux [17], [44] to accommodate dynamic join/departure of peers... |

306 | Viceroy: A scalable and dynamic emulation of the butterfly
- Malkhi, Naor, et al.
- 2002
(Show Context)
Citation Context ... al. [46] study diameter-degree tradeoffs of current DHTs and propose a graph based on a modified static butterfly. Another peer-to-peer architecture based on butterfly networks (Viceroy) is shown in =-=[26]-=-. Independently of this work, several recent papers have also proposed de Bruijn graphs for peer-to-peer networks [13], [20], [30]. These developments are complementary to our investigation and provid... |

257 | Balanced allocations
- Azar, Broder, et al.
- 1999
(Show Context)
Citation Context ... N-node application-layer de Bruijn graph with diameter ⌈log k N⌉. 7.2 Random Zones Achieving constant-size zones using distributed join and leave processes is a non-trivial, but well-studied prob=-=lem [2]-=-, [27], [30]. Equal zone sizes are desirable as they maintain a fixed out-degree at the application layer and provide better balancing of user objects between the peers. Assuming uniform random hashin... |

257 | The impact of dht routing geometry on resilience and proximity
- Gummadi, Gummadi, et al.
- 2003
(Show Context)
Citation Context ...assive adversarial node failures and random object deletions. Saia et al. [38] create another highly fault-resilient structure with state at each node and per-message routing overhead. Gummadi et al. =-=[16]-=- find that ring-based graphs (such as Chord) offer more flexibility with route selection and provide better performance under random node failure compared to several other traditional DHTs. C. Random ... |

237 | Scale-free characteristics of random networks: the topology of the World-wide web, Physica A
- Barabási, Albert, et al.
- 2000
(Show Context)
Citation Context ... Gaussian as illustrated by an almost-perfect fit of a Gaussian distribution in the figure. It has been noted before that certain real-world graphs (such as those describing webpage linkage structure =-=[3]) -=-exhibit Gaussian distributions of d(x,y), but no explanation of why this happens has been offered. Below, we analyze Chord’s distribution of shortest distances, understand why it appears to be Gauss... |

224 | ZIGZAG: An Efficient Peer-to-Peer Scheme for Media Streaming - Tran, Hua, et al. - 2003 |

211 | On distinguishin between internet power law topology generators
- Bu, Towsley
- 2002
(Show Context)
Citation Context ...of the previously discovered nodes. This leads to a situation where the new neighbors “know” many of the old neighbors, which is often called the smallworld property (or clustering) of the graph [=-=3], [4]-=-. In graph theory, the growth in the number of new neighbors discovered at a certain distance is related to node expansion of the graph. Quickly expanding graphs maintain an exponentially increasing n... |

182 | Symphony: Distributed Hashing in a Small World
- Manku, Bawa, et al.
- 2008
(Show Context)
Citation Context ...22] build random expander graphs based on Hamiltonian cycles with diameter and degree. Manku et al. [29] analyze several randomized systems (i.e., Randomized Chord, Randomized Hypercube, and Symphony =-=[28]-=-) and conclude that the usage of neighbors-of-neighbors (NoNs) in routing decisions reduces the average distance in the corresponding graph from to with high probability. D. Optimal-Diameter Graphs Th... |

172 | Distributed object location in a dynamic network
- Hildrum, Kubiatowicz, et al.
- 2002
(Show Context)
Citation Context ...ashes. Their distributed structure, excellent scalability, short routing distances, and failure resilience make DHTs highly suitable for peer-to-peer networks. 2.1 Peer-to-Peer DHTs Many current DHTs =-=[17], -=-[35], [37], [43], [44] rely on the concept of prefix-based routing introduced by Plaxton et al. in [29]. Plaxton’s framework is extended in Tapestry [43] and Bayeux [17], [44] to accommodate dynamic... |

147 | Novel architectures for p2p applications: the continuous-discrete approach
- Naor, Wieder
- 2006
(Show Context)
Citation Context ...to-peer architecture is determined by the properties of these graphs, which typically possess diameter and degree at each node (where is the number of peers in the system). Until recently [13], [20], =-=[30]-=-, understanding whether these bounds were optimal and whether there existed fixed-degree graphs with diameter was an important topic of distributed hash table (DHT) research [34], [46]. In the first p... |

134 | Routing algorithms for dhts: Some open questions. Peer-to-Peer Systems - Ratnasamy, Stoica, et al. - 2002 |

110 | Building low-diameter p2p networks - Pandurangan, Raghavan, et al. - 2001 |

95 |
Probability Theory. An Analytic View
- Stroock
- 1993
(Show Context)
Citation Context ... The Gaussian shape in Fig. 1(b) follows from the Central Limit Theorem as the sum of i.i.d. random variables in (7) tends to a Gaussian distribution. The formal proof utilizes Berry-Esseen’s theorem =-=[43]-=- and shows that a -fold convolution of (9) tends to a Gaussian distribution as . We skip the details for brevity. We now have sufficient evidence that demonstrates that both Chord and CAN exhibit Gaus... |

92 |
Koorde: a simple degree-optimal hash table
- Kaashoek, Karger
- 2003
(Show Context)
Citation Context ...oined node requires only message complexity as it can be copied from the previous owner of the zone. Notification of existing neighbors has another message overhead. Further notice that unlike Koorde =-=[20]-=-, each existing zone is split in half, which leads to significantly better bounds on the size of the smallest zone [45]. Peer-to-peer linking rules are also straightforward. Consider node that owns zo... |

91 | Know thy Neighbor’s Neighbor: The Power of Look Ahead
- Manku, Naor, et al.
- 2004
(Show Context)
Citation Context ...nku [27] considers random graphs of degree and asymptotically optimal expected distance .Lawet al. [22] build random expander graphs based on Hamiltonian cycles with diameter and degree. Manku et al. =-=[29]-=- analyze several randomized systems (i.e., Randomized Chord, Randomized Hypercube, and Symphony [28]) and conclude that the usage of neighbors-of-neighbors (NoNs) in routing decisions reduces the aver... |

80 | Distributed construction of random expander networks
- Law, Siu
- 2003
(Show Context)
Citation Context ...xpected routing distance in such graphs. Their results show that both bounds are proportional to . Manku [27] considers random graphs of degree and asymptotically optimal expected distance .Lawet al. =-=[22]-=- build random expander graphs based on Hamiltonian cycles with diameter and degree. Manku et al. [29] analyze several randomized systems (i.e., Randomized Chord, Randomized Hypercube, and Symphony [28... |

70 |
Shufflenet: An Application of Generalized Perfect Shuffles to Multihop Lightwave Networks
- Hluchyj, Karol
- 1999
(Show Context)
Citation Context ...ameter and average distance are close to optimal, they are always higher than those in (nontrivial) de Bruijn graphs. Recall that the average distance in the butterfly graph is given by the following =-=[18]: µd = 3m -=-− 1 2 − 1 k − 1 + m N(k − 1) ≈ 3 logk N , (8) 2 which, for large N and k, is asymptotically 50% larger than the same metric in de Bruijn graphs. 4.5 Discussion The results of this section in... |

67 | On the fundamental tradeoffs between routing table size and network diameter in peer-to-peer networks
- Xu
- 2003
(Show Context)
Citation Context ... [20], [27], understanding whether these bounds were optimal and whether there existed fixed-degree graphs with logN diameter was believed to be the fundamental question of peer-topeer research [32], =-=[42]-=-. Besides the diameter and degree, a third very important property of a peer-to-peer structure is its resilience to simultaneous node failure. Without fault resilience, the answer to logarithmic routi... |

65 | Lightwave Networks Based on De Brujin Graphs
- Sivarajan, Ramaswami
- 1994
(Show Context)
Citation Context ...the related work does not). The lower bound on µd in any k-regular graph is given by the average distance in the corresponding Moore graph and is also not achievable for non-trivial values of N and k=-= [38]: µd ≥ DM − k-=-(kD M − 1) DM + N(k − 1) 2 N(k − 1) ≈ DM − 1 . (3) k − 1 With respect to µd, de Bruijn graphs are again asymptotically optimal and converge to the bound in (3) for sufficiently large N an... |

62 | Fault tolerant routing in peer-to-peer systems
- Aspnes, Diamadi, et al.
- 2002
(Show Context)
Citation Context ...ly) logarithmic diameter; however, the paper does not provide an efficient routing algorithm for the proposed structure that can deterministically explore the low diameter of the graph. Aspnes et al. =-=[1]-=- examine random graphs of fixed degree and derive upper and lower bounds on the expected routing distance in such graphs. Their results show that both bounds are proportional to . Manku [27] considers... |

54 | Dynamically fault-tolerant content addressable networks
- Saia, Fiat, et al.
- 2002
(Show Context)
Citation Context ...to maintain a connected graph with high probability. Fiat et al. [12] build a censorship-resistant network that can tolerate massive adversarial node failures and random object deletions. Saia et al. =-=[38]-=- create another highly fault-resilient structure with state at each node and per-message routing overhead. Gummadi et al. [16] find that ring-based graphs (such as Chord) offer more flexibility with r... |

50 | The Content-Addressable Network D2B
- Fraigniaud, Gauron
- 2003
(Show Context)
Citation Context ...f each peer-to-peer architecture is determined by the properties of these graphs, which typically possess diameter and degree at each node (where is the number of peers in the system). Until recently =-=[13]-=-, [20], [30], understanding whether these bounds were optimal and whether there existed fixed-degree graphs with diameter was an important topic of distributed hash table (DHT) research [34], [46]. In... |

48 | Yappers: A peer-to-peer lookup service over arbitray topology
- Ganesan, Sun, et al.
(Show Context)
Citation Context ... modeled in ways that can provide a clear quantitative assessment of each graph’s resilience to node failure. In addition to these classical approaches, there are proposals based on heuristics (e.g.=-=, [14]-=-, [26], [41]) with no provablyoptimal underlying foundation for choosing one or another graph structure. Our work supplements such proposals with a more fundamental insight into the problem and offers... |

46 | Routing networks for distributed hash tables
- MANKU
(Show Context)
Citation Context ...nes et al. [1] examine random graphs of fixed degree and derive upper and lower bounds on the expected routing distance in such graphs. Their results show that both bounds are proportional to . Manku =-=[27]-=- considers random graphs of degree and asymptotically optimal expected distance .Lawet al. [22] build random expander graphs based on Hamiltonian cycles with diameter and degree. Manku et al. [29] ana... |

40 | Censorship resistant peer-to-peer content addressable networks
- Fiat, Saia
- 2002
(Show Context)
Citation Context ...t al. [23] examine error resilience dynamics of Chord when nodes join/leave the system and derive lower bounds on the degree necessary to maintain a connected graph with high probability. Fiat et al. =-=[11]-=- build a Censorship Resistant Network that can tolerate massive adversarial node failures and random object deletions. Saia et al. [36] create another highly fault-resilient structure with O(log 3 N) ... |

35 |
resistant peer-to-peer content addressable networks
- Fiat, Saia
- 2002
(Show Context)
Citation Context ...t al. [23] examine error resilience dynamics of Chord when nodes join/leave the system and derive lower bounds on the degree necessary to maintain a connected graph with high probability. Fiat et al. =-=[11]-=- build a Censorship Resistant Network that can tolerate massive adversarial node failures and random object deletions. Saia et al. [36] create another highly fault-resilient structure with O(log 3 N) ... |

34 |
On the impossibility of directed Moore graphs
- Bridges, Toueg
- 1980
(Show Context)
Citation Context ... degree k and diameter D (the maximum distance between any two nodes in the graph). What is the maximum number of nodes N that can be packed into any such graph? A wellknown result is the Moore bound =-=[6], [7]:-=- N ≤ 1 + k + k 2 + ... + k D = kD+1 − 1 k − 1 = NM . (1) Interestingly, Moore bound NM is only achievable for trivial values of k and D. In fact, the Moore bound is provably not achievable for a... |

28 |
Design to minimize diameter on building-block network
- Imase, Itoh
- 1981
(Show Context)
Citation Context ...rchitectures. The paper is organized as follows. We first examine the problem of obtaining logarithmic routing diameter in fixeddegree (sparse) graphs. Our work relies on generalized de Bruijn graphs =-=[19]-=- of fixed degree k and asymptotically optimal diameter logkN. However, since the diameter itself does not tell the whole story, we also study the average distances between all pairs of nodes since thi... |

25 | Routing in multihop packet switching networks: Gb/s challenge
- Baransel, Dobosiewicz, et al.
- 1995
(Show Context)
Citation Context ...in Section VII). We further assume, for simplicity of notation, that the total number of nodes is a power of node degree and omit ceiling functions whenever appropriate. B. Structure De Bruijn graphs =-=[6]-=-, [19], [23], [40] are nearly optimal, fixed-degree digraphs of diameter , where is the fixed degree of each node and is the total number of nodes. Note that de Bruijn graphs are directed graphs with ... |

18 | Efficient peer-to-peer lookup based on a distributed trie
- Freedman, Vingralek
- 2002
(Show Context)
Citation Context ...rected graph) of fixed degree; however, the proposed structure needs to estimate the number of active nodes to properly build the application-layer graph. Among tree-based structures, Freedman et al. =-=[13]-=- propose a DHT based on distributed tries and Tran et al. [41] organize peers into a multicast tree of degree O(k 2 ) and diameter O(logkN). Xu et al. [42] study diameter-degree tradeoffs of current D... |

18 | Ulysses: A robust, low-diameter, low-latency peer-to-peer network,” European transaction on telecommunications - Kumar, Merugu, et al. - 2004 |

14 | Analysis of the Evolution of Peer-to-Peer Networks - Liben-Nowell, Balakrishnan, et al. - 2002 |

13 | On zone-balancing of peer-to-peer networks: analysis of random node join
- Wang, Zhang, et al.
- 2004
(Show Context)
Citation Context ...isting neighbors has another message overhead. Further notice that unlike Koorde [20], each existing zone is split in half, which leads to significantly better bounds on the size of the smallest zone =-=[45]-=-. Peer-to-peer linking rules are also straightforward. Consider node that owns zone . Each of the integer values in corresponds to the virtual de Bruijn graph of size . Hence, to preserve de Bruijn li... |

12 |
Fault-Tolerant Network Based on De Bruijn Graph
- Sridhar, Raghavendra
- 1991
(Show Context)
Citation Context ...oth CAN and Chord are knode-connected 3 , while de Bruijn graphs are not due to several “weak” nodes with self-loops. This classical form of de Bruijn graphs has been shown to be (k−1)-node conn=-=ected [39]; howev-=-er, we seek to achieve maximum fault tolerance, which leads us to removing the loops and linking these k “weak” nodes to each other. Consider node (h,h,. . . ,h), h ∈ Σ, with a self-loop. A cha... |

7 |
Fault-Tolerant Routing
- Aspnes, Diamadi, et al.
- 2002
(Show Context)
Citation Context ...ly) logarithmic diameter; however, the paper does not provide an efficient routing algorithm for the proposed structure that can deterministically explore the low diameter of the graph. Aspnes et al. =-=[1]-=- examine random graphs of fixed degree l + 1 and derive upper and lower bounds on the expected routing distance in such graphs. Their results show that both bounds are proportional to ln2 N . Law et a... |

6 |
Symphony: Distributed Hashing
- Manku, Bawa, et al.
- 2003
(Show Context)
Citation Context ...ed in ways that can provide a clear quantitative assessment of each graph’s resilience to node failure. In addition to these classical approaches, there are proposals based on heuristics (e.g., [14]=-=, [26]-=-, [41]) with no provablyoptimal underlying foundation for choosing one or another graph structure. Our work supplements such proposals with a more fundamental insight into the problem and offers analy... |

6 |
Diameters of communication networks
- Chung
- 1986
(Show Context)
Citation Context ... degree and diameter (the maximum distance between any two nodes in the graph). What is the maximum number of nodes that can be packed into any such graph? A well-known result is the Moore bound [7], =-=[8]-=- Interestingly, the Moore bound is only achievable for trivial values of and . In fact, the Moore bound is provably not achievable for any nontrivial graph [7]. Directed de Bruijn graphs come close to... |

6 |
On digraphs with minimum diameter and maximum connectivity
- Reddy, Kuhl, et al.
- 1982
(Show Context)
Citation Context ... of and . In fact, the Moore bound is provably not achievable for any nontrivial graph [7]. Directed de Bruijn graphs come close to the Moore bound and can be built with nodes [19] or even with nodes =-=[35]-=-; however, in general, it is not known how close we can approach the upper bound for nontrivial graphs [8]. In the context of peer-to-peer DHTs, we are concerned with a different formulation of the pr... |

5 | Koorde: A Simple Degree-optimal Hash - Kaashoek, Karger |

5 | S.: A better upper bound on the bisection width of de Bruijn networks
- Feldmann, Monien, et al.
- 1997
(Show Context)
Citation Context ...no worse than that in the butterfly. It is further conjectured that the actual bisection width of de Bruijn graphs is at least 40% higher than the pessimistic lower bound used in the above comparison =-=[11]-=-. In summary, larger values of bisection width in de Bruijn graphs point toward higher resilience against graph partitioning and lower congestion in the bisection cut in addition to their optimal rout... |

5 | Bisecting de bruijn and kautz graphs
- Rolim, Tvrdík, et al.
- 1998
(Show Context)
Citation Context ... this bisection becomes , although for small it is slightly better. Finally, the exact value of of de Bruijn graphs is unknown and the best available upper and lower bounds differ by a factor of four =-=[36]-=- (35) Using the lower bound in (35), the bisection width of de Bruijn graphs for is larger than that in Chord or CAN by a factor of (which is 2.2 for ) and is generally no worse than that in the butte... |

4 | Scalable peer-to-peer indexing with constant state
- Considine, Florio
- 2002
(Show Context)
Citation Context ... modulo field (ring) with the diameter and degree both equal to log2N. Recent proposals start to address the issue of routing in logarithmic time in fixed-degree graphs. For example, Considine et al. =-=[8] e-=-xpand on Chord’s ring structure by constructing a digraph (directed graph) of fixed degree; however, the proposed structure needs to estimate the number of active nodes to properly build the applica... |

4 |
Viceroy: A Scalable and Dynamic
- Malkhi, Naor, et al.
- 2002
(Show Context)
Citation Context ...he details (some of which we skip) are almost identical to those in Chord. We also feel that the algorithmic structure of ODRI is much simpler than that of other recently proposed fixed-degree graphs =-=[25], -=-[42]. Let Nmax be the maximum possible number of nodes in the system (such as 10 10 ). Organize the space of all possible nodes between [0, Nmax − 1] into a modulo-Nmax number field. Now notice that... |