Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Degree–Optimal Routing for P2P Systems Giovanni Chiola, Gennaro Cordasco, Luisa

Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006

Degree–Optimal Routing for P2P

Systems

Giovanni Chiola, Gennaro Cordasco, Luisa Gargano, Mikael

Hammar, Alberto Negro, and Vittorio Scarano

Meeting WEBMINDS 2006Genova, 27/28 ottobre

Università di Salerno - GL7


OutlineOutline

• Peer-to-Peer• Distribute Hash Table• Overlay Network• Uniform Networks vs. Randomized Networks• Greedy Routing vs. Neighbor of Neighbor Routing

• Our Proposals• F-Chord (2004) ----> Generalize Base-k (2006)• H-Networks (2005) ----> Overlay network with class

(2006)

• Conclusions


• Distributed version of a hash table data structure• Stores (key, value) pairs

• The key is like a filename• The value can be file contents

• Goal: Efficiently insert/lookup/delete (key, value) pairs

• Each peer stores a subset of (key, value) pairs in the system

• Core operation: Find node responsible for a key• Map key to node• Efficiently route insert/lookup/delete requests to this

node

Distributed Hash Table (DHT)Distributed Hash Table (DHT)


Overlay networkOverlay network

• [I. Stoica] “An overlay network is a virtual network of nodes and logical links that is built on top of an existing network with the purpose to implement a network service that is not available in the existing network.In particular an overlay network:• adds an additional layer of

indirection/virtualization• changes properties in one or more areas of

underlying network”


Overlay Network: PerformanceOverlay Network: Performance

• Routing table size (degree)• Storage cost• Measure the cost of self-stabilization for

adapting to node joins/leaves • Diameter and Average path length

• Time cost• Fault tolerance

• Congestion• Clustering


Chord Chord • Chord uses a one-dimensional circular key space (ring) of

N=2m identifiers• The node responsible for the key is the node whose identifier

most closely follows the key• Chord maintains two sets of neighbors:

• A successor list of r nodes that immediately follows it in the key space

• A finger list of m = log N nodes spaced exponentially around the key space

• Routing consists in forwarding to the node closest, but not past, the key

• Performance:• Diameter: log N (O(log n) whp) where n denote the number of

nodes present in the network• Routing table size: log N (O(log n) whp)• Average path length: ½ log N

Routing correctness

Routing efficiency


An Example: Chord An Example: Chord

ID Resp.

8+1=9 14

8+2=11 14

8+8=16 21

8+16=24 24

8+32=40 42

8+4=12 14

m=6

indice Nodo

1 14

2 21

4 32

5 38

6 42

3 24

Successors

Predecessor

Nodo 1


Uniform Routing AlgorithmUniform Routing Algorithm• We consider a ring of N identifiers labeled from 0

to N-1• A routing algorithm is uniform if for each identifier

v, v is connected to u iff v+w is connected to u+w (i.e. : all the connection are symmetric).Advantages

+Easy to implement+Greedy algorithm

+ Simple – to understand and implement+ Local – routing occurs inside the portion of ring that is

delimited by source and destination+ In some cases – (Hypercube, Chord) – the best we can do

+No node congestion+Quick Bootstrap

Drawback– Less powerful (De Bruijn Graph and Neighbor of

Neighbor Greedy routing are more powerful)


t

Greedy Routing: move to the neighbor that minimizes the distance to the target.

s

Greedy Routing Greedy Routing


Fib-Chord (2004)Fib-Chord (2004)

• FormallyLet N (Fib(m-1), Fib(m)]. The scheme uses m-2 jumps of size Fib(i) for i = 2,3, … , m-1

• Fib-Chord• Diameter :

• Degree :

123 5 8 13 21 34 55 89

Fib-ChordFib-Chord

1log 0.72021 log

2N N

log 1.44042 logN N

618.12

51


F-Chord(F-Chord() (2004)) (2004)

• F-Chord()Fib(2i), for i = 1,2, …,(1-)(m-2)Fib(i), for i = 2 (1-)(m-2) +2, …, m-1

• F-Chord() uses (m-2) jumps

2 5 13 34 89

Fib-ChordFib-Chord

1 3 8 21 55

even index jumps

all jumps

[1/2,1]


Property of F-ChordProperty of F-Chord• Diameter:

Theorem [CGHSN04] For any value of , the diameter of F-

Chord() is m/2 0.72021 log n

• Average Path Length:Theorem [CGHSN04] The average path length of the F-Chord()

scheme is bounded by 0.39812 log n + (1-)0.24805 log n

• Degree: F-Chord() use (m-2) jumps


0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

Degree

Diameter

Average Path Length

Chord APL

Chord Degree &Diameter

hops

x

log

n

F-Chord (2004)F-Chord (2004)

Lower is

better


Generalized Base-Generalized Base-kk• We generalize the property of F-Chord(1/2)

to bases greater than 2• Let R(d) the size of the greatest ring with

diameter d (using Greedy routing).• Generalized Base-k:

J(0)=1, R(0)=1,for each l 0

for each i=1,…,k-1J((k-1)l+i)=J((k-1)l)+iR(l)R(l+1)= J((k-1)l)+kR(l)


Generalized Base-Generalized Base-k: An Examplek: An Example• Generalized Base-k: J(0)=1, R(0)=1,

for each l 0 for each i=1,…,k-1 J((k-1)l+i)=J((k-1)l)+iR(l) R(l+1)= J((k-1)l)+kR(l)

Example k=3 and n=R(2)=15 J(0)=1, R(0)=1 J(1)=2, J(2)=3,R(1)=4 J(3)=7, J(4)=11,

Generalized Base-3Generalized Base-3

1 2 3 7 11


Generalized Base-Generalized Base-kk• For k=2 the sequence of jump is

1,2,5,13,34,… (F-Chord(1/2) jumps)

Generalized Base-2 = F-Chord(1/2)

Generalized Base-k is the proper generalization of F-

Chord(1/2) to base k

12 5 13 34 89


Property of Property of Generalized Base-Generalized Base-kk

• Diameter:Let d the diameter of Generalized Base-k scheme

with n=R(d) nodes, d<logk+((k-1)/k)(n+1)

• Average Path Length: The average path length of Generalized Base-k scheme

with n=R(d) nodes is bounded by ((k+2)/(k+3))d

• Degree:Let the node degree of Generalized Base-k scheme with n=R(d) nodes, = (k-1)d+1


Simulation Results: DegreeSimulation Results: DegreeB-k: Base-kB-k: Base-k

G-k: Gen. Base-kG-k: Gen. Base-k

Fib: F-Chord Fib: F-Chord

F-k: Ext. Fib.F-k: Ext. Fib.

G-(k+1)B-k

B-3B-3

G-3G-3

B-2B-2


Simulation Results: APLSimulation Results: APLB-k: Base-kB-k: Base-k

G-k: Gen. Base-kG-k: Gen. Base-k

Fib: F-Chord Fib: F-Chord

F-k: Ext. Fib.F-k: Ext. Fib.

G-kB-k for k>2


Randomized NetworksRandomized Networks

Networks whose nodes are connected to each other randomly+ Small Average Path Length (APL) Small Clustering The global structure of the network cannot be a global information Routing is not local Optimal trade-off can be reached only through a estimation of the

number of participants in the system The optimal trade-off will only occur for a range of n ?Quick Bootstrap? ?Congestion?

• Examples:• Viceroy Network

• Degree: O(1)• APL: O(log N)

• Mariposa • Degree: O()• APL: O(log N)


t

NoN Greedy routing NoN Greedy routing

s


• Benefits• Efficient• Local Routing• Estimation of n not required

• Drawbacks• Neighbor lists should be maintained• No quick bootstrap

Neighbor of Neighbor RoutingNeighbor of Neighbor Routing


• Consider a deterministic network U having n nodes and degree

• Let j1, j2, . . . , j all the jumps of U (ordered by their size). (i.e.: For each node v and for each i=1,…, ,

v is connected with v+ji mod n)

• Let H() denote a good hash function, that maps an id on the interval [0,1)

• The network H-U is obtained from U as follows:• For each i=1,…, , node v is connected to the nodes

v+ji+(ji+1− ji)H(v) mod n (where j+1 = n)

H-Networks (2005)H-Networks (2005)

jiiji+1i+1


0

1

2

3

4

5

6

7

8

9

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

log N

hops

Chord

R-ChordGreedyR-Chord NoN

R-Chord SP

H-ChordGreedyH-Chord NoN

H-Chord SP

ChordChordLower is

better


H-F-ChordH-F-Chord vsvs. . H-ChordH-Chord

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

0 50000 100000 150000 200000 250000 300000 350000n

hops

H-F-Chord(1/ 2) H-Chord H-F-Chord(0.69424) H-F-Chord(1)

Lower is better


Overlay Network with classOverlay Network with class• Uniform Networks

• Greedy routing is optimal: log N hops with log N fingers per node

• Greedy routing is easy to program• Uniformity allows quick bootstrap for new peers joining

an already established DHT• Congstion-free and high Clustered

• Non-Uniform Networks• Greedy routing is not optimal: other strategies such as

“Neighbor of Neighbor” (NoN) yield log N / log log N hops with log N degree per node

• NoN routing is similar to Greedy routing in terms of implementation

• Non-Uniformity prevents quick bootstrap for new peers joining an already established DHT

• ?Congestion and Clustering?


• Split nodes into a c classes• Routing tables are uniform within the same class• Different classes follow different patterns

• Each peer chooses its class pseudo-randomly (based on Hash digest of its identifier)

• Quick bootstrap is obtained when finding a neighbor belonging to the same class

Overlay Network with classOverlay Network with class


• Let U be uniform overlay network having N nodes and degree .

• Let j1, j2, . . . , j all the jumps of D (ordered by their size).

• Let c be any given positive integer in the interval [1, 2m] and H() be a cryptographic hash function that maps an id on the interval [0, 1).

• Consider c real numbers 0, 1,…, c-1 in the interval

[0, 1) such that 0 = 0<1<…< c-1 < 1• The network Hc-U is obtained from U as follows:

• For each i = 0,…, -1, node v is connected by an edge to the nodes

v + ji + cv(ji+1- ji),

where cv = cH(v).



• TheoremTheorem Let 1 < c Let 1 < c log n/ log log n, the average path log n/ log log n, the average path

length is O(loglength is O(logcc n) hops for the NoN Greedy n) hops for the NoN Greedy algorithm on Halgorithm on Hcc-Chord with n = 2-Chord with n = 2mm peers peers

• CorollaryCorollary When c = log n/ log log n, the average path When c = log n/ log log n, the average path

length islength is

O(log n/ log log n) hops and the number of hops O(log n/ log log n) hops and the number of hops for the completion of the join operation are for the completion of the join operation are O(log n log log n) (w.h.p.) O(log n log log n) (w.h.p.)

n<2m

Chernoff Bound



• TheoremTheorem Both the diameter and the average Both the diameter and the average

(shortest) path length of an (shortest) path length of an Hc-Network with degree O(log n) and c classes are with degree O(log n) and c classes are (log(logcc n) n)

Hc-Chord is

asymptotically optimal



• Extension to other networks:• Symphony −> Hc-Symphony

• Hypercube −> Hc-Hypercube

• Small-world -> …

How Much Independent Should

Individual Contacts be to Form a Small-

World?



Simulation ResultsSimulation Results


• Hc-Networks:• NoN routing on few Classes gives almost

the same performance as NoN routing on truly randomized systems

• Few Classes guarantee a quick bootstrap with very high probability

• 2 Classes are sufficient for DHTs with up to hundred thousand peers

• 4 Classes are sufficient for DHTs with billions of peers

• The result applies to other network topologies



ReferencesReferences• G. Cordasco, L. Gargano, M.Hammar, A. Negro and V. Scarano,

“F-Chord: Improved Uniform Routing on Chord”. In Proc. of 11th Colloquium on Structural Information and Communication Complexity (Sirocco) Smolenice Castle, Slovakia June 21-23, 2004.

• G. Cordasco, L. Gargano, M.Hammar and V. Scarano, “Degree-Optimal Deterministic Routing for P2P Systems”. In Proc. of 10th IEEE Symposium on computers and communications (ISCC 2005) La Manga del Mar Menor, Cartagena, SPAIN June 27-30, 2005. (see also Twenty-Third Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2004) (Brief Announcement) St. John's, Newfoundland, Canada July 25-28, 2004.)

• G. Cordasco, L. Gargano, M.Hammar, A. Negro and V. Scarano, “Non-uniform deterministic routing on F-Chord(α)”. In Proc. of First International Workshop on Hot Topics in Peer-to-Peer Systems (Hot-P2P04) Volendam (The Netherland) October 08, 2004.

• G. Chiola, G. Cordasco, L. Gargano, A. Negro and V. Scarano, “Overlay networks with class”. In Proc. of 8th International Symposium on Parallel Architectures, Algorithms, and Networks (I-SPAN 2005) Las Vegas, Nevada, USA. December 7-9, 2005.

• G. Chiola, G. Cordasco, L. Gargano, A. Negro and V. Scarano, “Optimizing the finger table in Chord-like DHTs”. In Proc. of Third International Workshop on Hot Topics in Peer-to-Peer Systems (HotP2P 2006) (Co-Located with IPDPS 2006) Rhodes Island, Greece. April 29, 2006.

• G. Cordasco and L. Gargano, “How Much Independent Should Individual Contacts be to Form a Small-World?”. In Proc. of The 17th International Symposium on Algorithms and Computation (ISAAC 2006) December 18-20, 2006 - Kolkata, India. (To Appear)


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Gennaro Cordasco - Degree–Optimal Routing for P2P Systems - 28/10/2006 Degree–Optimal Routing for P2P Systems Giovanni Chiola, Gennaro Cordasco, Luisa