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Compact Routing and Locality in Peer-to-Peer Systems. Ittai Abraham School of Computer Science and Engineering Hebrew University of Jerusalem. Internet Activity. After 1992 – Dominated by the web browser Client Server paradigm Clients are Lightweight and Transient

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compact routing and locality in peer to peer systems

Compact Routing and Locality in Peer-to-Peer Systems

Ittai Abraham

School of Computer Science and Engineering

Hebrew University of Jerusalem

internet activity
Internet Activity
  • After 1992 – Dominated by the web browser
  • Client Server paradigm
  • Clients are Lightweight and Transient
    • Low bandwidth, computational power and storage requirements
  • Most servers are relatively simple and static (HTTP)
  • Distributed Computing community mostly focused on robustifying servers for load balancing and high availability, targeting mostly at clusters of dozens of servers at the most
  • How will Internet Activity look like in the future ?
tomorrow s internet
Tomorrow’s Internet ?
  • Internet peers will be stateful and will have a persistent connection
    • Have high bandwidth, computational power and storage capabilities
  • Peers are capable of acting both as a server and as a client
  • Will have an active network presence
  • Symmetric, decentralized, self organizing paradigm
peer to peer systems
Peer-to-peer Systems
  • Group of peers wish to maintain a shared information data structure
  • Complications
    • Large group
    • Enormous amounts of information
    • Group is spatially distributed
    • Dynamically changing
    • Heterogeneous
    • Selfish/faulty/malicious participants
  • Challenge: provide efficient access to the shared information data structure
distributed hash tables
Distributed Hash Tables
  • A universe U of object ids. A hash function h maps U into a smaller set S spreading ids without many collisions
  • A set V of nodes that forms a distributed system
  • The set S is partitioned in to |V| parts, and each node in V maintains its relevant part of the hash table
  • Basic operations are lookup and store: Given an object id A, find the hash value h(A), route to the node that maintains the key h(A), and read/write the object A
dht lookup example
DHT Lookup Example
  • Hash function h(x) = x mod 1009
  • For this example each node with id i maintains all the objects whose has hash value is in [1009*i/6,1009*(i+1)/6)

0

5

6

Hash value is 1, so need to route to node with id 0

3

2

Node 2 wants to find the value of the object with id 1009001

4

1

traditional complexity measures of distributed hash tables
Traditional Complexity Measures of Distributed Hash Tables
  • Degree (local memory) of the overlay network

O(1), O(logk n), O(n1/k)

  • Number of hops from source node to target node

O(log n), O(log n/loglog n), O(1)

  • But counting hops does not take into account a weighted communication network
low stretch routing
Low Stretch Routing
  • Two models:
    • Peers communicate through a weighted network G=<V,E,ω>
    • The cost of communication d(s,t) between peers in V induces a metric space M=<V,d>I
      • General metric
      • Growth bounded metric, Euclidean metric, Doubling metric
  • In either case the stretch of a routing scheme RS is the maximal ratio over all pairs of dRS(s,t)/d(s,t)
routing on a weighted graph
Routing on a weighted graph
  • Devise a distributed routing scheme such that: a node that knows the label of a target node can send a message that will be routed to the target node
  • Main complexity measures:
    • Stretch: the ratio between the cost of the path taken by the routing protocol and the cost of a minimum cost path from source to destination.
    • Memory: the number of bits stored in each node.
  • A solution is compact if memory is o(n)
routing on a weighted graph1
Routing on a weighted graph
  • Lower bounds:
    • Stretch < 3 requires Ω(n) bits per node [Gavoille & Gengler 01]
    • Stretch < 5 requires Ω (√n) bits per node [Thorup & Zwick 01]
    • Stretch < 2k-1 requires Ω (n1/k) bits per node [Thorup & Zwick 01] Under the Erdosh conjecture
  • Two main variants:
    • Labeled routing: designer can choose the labels of nodes
    • Name Independent routing: node labels are given by an adversary
  • Labeled routing:
    • Stretch 3 with Õ(n2/3) bits [Cowen 99]
    • Stretch 3 with Õ(√n) bits [Thorup & Zwick 01]
    • Stretch 4k-1 with Õ(n1/k) bits [Thorup & Zwick 01]
name independent routing
Name Independent Routing
  • Awerbuch, Bar-Noy, Linial & Peleg 89
    • With Õ(n1/k) bits – stretch O(k29k)
    • With Õ(n2/3) bits – stretch 468
    • With Õ(√n) bits – stretch 2593
  • Awerbuch & Peleg 90 (Sparse Partitions)
    • For diameters that are polynomial in n
    • With Õ(n1/k) bits – stretch O(k2)
    • With Õ(n2/3) bits – stretch 624
    • With Õ(√n) bits – stretch 1088
  • Arias, Cowen, Laing, Rajaraman & Taka 03
    • With Õ(√n) bits – stretch 5

[A, Gavoille, Malkhi], DISC 04, Stretch O(k)

[A, Gavoille, Malkhi, Nisam, Thorup], SPAA 04, Stretch 3

compact name independent routing with minimum stretch a gavoille malkhi nisan and thorup spaa 2004
Compact Name-Independent Routing with Minimum Stretch[A, Gavoille, Malkhi, Nisan, and Thorup SPAA 2004]
  • Optimal stretch 3 with Õ(√n) bits
  • Construction in polynomial time
  • Routing decisions performed in constant time
  • Surprisingly, with Õ(√n) bits allowing the designer to label the nodes does not improve the stretch factor compared to the task when node labels are predetermined by an adversary.
the recipe
The Recipe
  • Ingredients
    • Vicinity routing
    • Random coloring to √n colors
    • Hash labels to colors
    • Labeled routing on trees
    • Landmarks
    • Partial shortest path trees
vicinity routing

w

v

Vicinity Routing
  • Let B(u) denote the (√n log n)-closest nodes to u (ties broken consistently)
  • For all vB(u), node u stores the next hop of a minimum cost path from u to v
  • Simple property [ABLP 89]: If vB(u) and w is on a minimum cost path from u to v, then vB(w)

u

B(u)

random coloring to n colors
Random Coloring to √n Colors
  • Every node u chooses a random color c(u)
  • With high probability
    • Every color set has O(√n) nodes
    • Every node has in its vicinity at least one node from every color set
  • Polynomial number of tests
  • Each test can be done in logspace
  • Derandomization using the pseudo random generator of Nisan
hash labels to colors
Hash Labels to Colors
  • Label u is hashed to a color h(u){1… √n}
  • At most O(√n log n) hashed to same color
  • Trivial if node labels are a permutation of 1…n
  • Otherwise can collision free hash to n2.5 and then use the techniques of Tarjan and Yao to hash to √n in constant time. Can be deradomized similarly to deterministic dictionaries of Hagerup, Milerstein, & Pagh
labeled routing on trees
Labeled Routing on Trees
  • Based on DFS Interval Routing, improved by Thorup & Zwick 01 and Frainiaud & Gavoille 01
  • A node is heavy if its sub-tree contains more than half of the nodes of its parent’s sub-tree
  • Each node stores its DFS interval and the DFS interval of its heavy child (if it has one)
    • Storage is O(log n) bits
  • A node’s label consists of the names of the non-heavy nodes on the path from the root
    • Labels require O(log2 n) bits
  • Routing to v on node u:
    • If v is not in u’s interval then send to parent
    • If v is in u’s heavy child interval then send to heavy
    • Otherwise u’s label contains the appropriate child
landmarks
Landmarks
  • Let R(T,v) be the routing information stored at node v for routing on tree T
  • Let T(u) be the minimum cost tree rooted at u
  • One color is designated as special
  • Let L be the set of all nodes l such that c(l)=special color
  • Every node u maintains R(T(l),u) for all lL
  • This requires Õ(√n) bits
  • For node u, let l(u) be a landmark in B(u)
partial shortest path trees
Partial Shortest Path Trees
  • Every node v stores R(T(u),v) for all uB(v)
  • Requires Õ(√n) bits
  • Let L(T,u) be the label of u on tree T
  • Simple property: If xB(y) then given L(T(x),y), node x can route to node y along a minimum cost path

x

w

y

B(y)

case 1 inside b u
Case 1: Inside B(u)
  • Use vicinity routing

u

v

case 2 b u and b v are close

w

Case 2: B(u) and B(v) are close
  • Any node on any minimal path from u to v is either in B(u) or in B(v)
  • Vicinity route to wB(u) s.t. c(w)=h(v)
  • Node w stores L(T(w),u),x,(xy),L(T(y),v)
  • Partial tree route to u on T(w)
  • Vicinity routing to y
  • Partial tree route to v on T(y)

u

y

x

v

case 3 b u and b v are far

l(v)

Case 3: B(u) and B(v) are far
  • Any minimal path from u to v contains a node that is not in B(u) or in B(v)
  • Vicinity route to wB(u) s.t. c(w)=h(v)
  • Node w stores L(T(l(v)),l(v)) and L(T(l(v)),v)
  • Tree route on T(l(v)) to l(v) and then to v

u

w

v

storage on node u
Storage on node u
  • Routing information and colors in B(u) [Vicinity]
  • R(T(l),u) for all lL [Landmarks]
  • R(T(v),u) for all vB(u) [Partial Trees]
  • For all v such that c(u)=h(v) minimum of
    • Path to l(v) and then to v. Store <L(T(l(v)), l(v)) , L(T(l(v)), v)>
    • Let P(u,w,v) be a path from u to v composed of an MCP from u to w, and of an MCP from w to v, such that:
      • u \in B(w),
      • there exists an edge (x  y) along the minimum path from w to v such that x  B(w) and y  B(v)

Among all the these paths choose the lowest cost path P(u,w,v) and store <L(T(u), w), x, (x  y), L(T(y),v)>

routing from u to v
Routing from u to v
  • If vB(u) use vicinity routing
  • If vL use tree routing on T(v)
  • Otherwise vicinity route to wB(u) such that c(w)=h(v)
  • Node w stores either
    • <R(T(l(v)), l(v)) , R(T(l(v)), v)>, and routing proceeds to l(v) and then to v
    • <R(T(u), w), x, (x  y), R(T(y),v)>, and routing proceeds to w then x to y and finally to v
better than stretch 3
Better than stretch 3 ?
  • There are worst case metrics in which stretch 3 is the best possible (many edges and high girth)
  • But do these metrics depict real world distances ?
  • Studies show that many networks have a bounded expansion ratio. Density changes are somewhat gradual
  • [Plaxton, Rajaraman & Rica 1997]:
    • Required
    • Expected (large) constant stretch
    • Deployments: Tapestry (Berkeley), Pastry (MS UK)