1. Viceroy: A scalable and dynamic emulation of the Butterfly Presented in CS294-4 by Sailesh Krishnamurthy Sep 22, 2003

2. Viceroy Goals: An overlay routing network with Logarithmic path lengths Constant join/leave cost Balanced congestion (~ log(n)/n) Keys, servers mapped to unit ring [0,1) In Chord: each node has all log(n) links In Viceroy: Each node has one log(n) link A link to 1/2 k distance points to a node with a link to 1/2 (k+1) distance

3. Viceroy Topology Each node has a level between 1..log(n) A `level-k’ node has “right child”: A long-range link to distance 1/2 k (approx.), to a level-(k+1)’ node “left child”: A local link to `level-(k+1)’ node Level ring links (pred,succ of the same level) “up”: A local link to ‘level-(k-1)’ node Ring links (pred,succ on the ring)

4. Viceroy Topology 01 Level 1 Level 2 Level 3 x y

5. Distributed level selection Select Level (s) Let n0 = 1/d(s,succ(s)) Select a level among [1 … log(n0)] uniformly at random Sanity When n servers present, then w.h.p. every server estimates: log(n/2logn)<=log(n0)<=3logn Any level l <log(n/2logn) is “sane”

6. Lookup target in Viceroy Three phases: Proceed to root: up-links to a `level-1’ node for level k = 1..log(n) If distance < 1/2 k use down-left (short link) If distance >= 1/2 k use down-right (long link) If reqd down link doesn’t exist (or if you overshoot target) break to next phase Traverse the ring (pred/succ links, whichever is closer)

7. Analysis Constant out-degree Expected constant in-degree log(n ) w.h.p. O(log(n )) lookup steps w.h.p log(n ) to `level-1’ node log(n ) for binary search log(n ) for final local search Congestion: Expected log(n )/n lookup load O(log 2 (n )/n) w.h.p.

8. Improving Lookup Simple lookup - third phase may be too long May be log 2 n links to traverse :-( Fancy lookup Use a combination of global and level rings to get a dilation of log(n) w.h.p. Greedy approach - use the level links if you are still too far away from the target.

9. What happens on node leave? Problem: we have constant out-degree, but even with a sparse network we could have log(n) in-degree on average Solution: “buckets” - extra background process Idea: improve identity/level-selection so that we have constant number of nodes in each stretch of (log(n)/n) nodes Maintain ~n buckets of ~log(n) contiguous non-overlapping nodes.

10. Bucket properties Size: always θ(log(n)) Merge with neighbouring bucket when size falls below log(n) Split bucket when size grows above clog(n) Diversity in bucket Each level in [1..log(n)] is represented by (1,c) nodes. Claim: this limits the indegree to 2c - how ?

11. Some comparisons SkipNets log(n) log(n) ?? SkipList

12. Some questions How important are const degree networks ? Dilation and congestion same as chord Depends on bucket mechanism What about fault tolerance ?