1. Compact Name-Independent Routing with Minimum Stretch
Ittai Abraham (HUJI)
Cyril Gavoille (LABRI)
Dahlia Malkhi (HUJI)
Noam Nisan (HUJI)
Mikkel Thorup (ATT)

2. 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 ?

3. 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

4. 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

5. 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

6. 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) 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 4 1

7. 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

8. Locality Awareness x t s

9. Compact Low Stretch Routing
Peers communicate through a weighted network G=<V,E,ω>
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 maximal ratio over all pairs of dRS(s,t)/d(s,t)
Memory: the number of bits stored in each node.

10. 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]

11. 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

12. Compact Name-Independent Routing with Minimum Stretch
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.

13. The Recipe Ingredients Vicinity routing Random coloring to √n colors
Hash labels to colors
Labeled routing on trees
Landmarks
Partial shortest path trees

14. 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 v w
B(u)

15. 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

16. 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

17. Labeled Routing on Trees
Based on DFS Interval Routing, improved by Frainiaud & Gavoille 01 and Thorup & Zwick 01
Storage O(log n) bits, Label O(log2 n) bits, Stretch 1
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

18. 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)

19. 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)

20. Case 1: Inside B(u)
Use vicinity routing u v

21. 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 x y v w

22. 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
l(v) v
b(u)+d(w,l)+b(v) <=
b(u)+b(u)+d(u,v)+b(v)+b(v)
<= 3 d(u,v)

23. 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)>

24. 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

25. Questions

26. Thank you