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Viceroy: Scalable Emulation of Butterfly Networks For Distributed Hash Tables By: Dahlia Malkhi, Moni Naor & David Ratajzcak Nov. 11, 2003 Presented by Zhenlei Jia Nov. 11, 2004

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Acknowledgments Some of the following slides are adapted from the slides created by the authors of the paper

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Outline DHT Properties Viceroy Structure Routing Algorithm Join/Leave Bounding In-degree: Bucket Solution Fault Tolerance Summary

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DHT What ’ s DHT Store (key, value) pairs Lookup Join/Leave Examples CAN, Pastry, Tapestry, Chord etc.

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DHT Properties Dilation Efficient lookup, usually O(log(n)) Maintenance cost Support dynamic environment Control messages, affected servers Degree Number of opened connections Servers impacted by node join/leave Heartbeat, graceful leave

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DHT Properties (cont.) Congestion: Peers should share the routing load evenly Load (of a node): the probability that it is on a route with random source and destination. If path length = O(log(n)) then on average, each node is on n 2 x O(log(n))/n = O(nlog(n)) routes. Average load = O(nlogn)/n 2 = O(log(n))/n

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Previous Works

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Intuition Route is a combination of links of appropriate size Chord: Each node has ALL log(n) links Viceroy Each node has ONE of the long-range links A link of length 1/2 k points to a node has link of length 1/2 k+1 Chrod

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Level 1 Level 2 Level 3 Level 4 A Butterfly Network Each node has ONE of the long-range links A link of length 1/2 k points to a node has link of length 1/2 k+1 Nodes “ share ” each other ’ s long link Routing 1.Route to root 2.Route to right group 3.Route to right level Path: O(log(n)) Degree: O(1)

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A Viceroy network Level 1 Level 2 Level 3 Ideally, there should be log(n) levels There is not a global counter Later, we will see how a node can estimate log(n) locally

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Structure: Nodes Node Id: 128 bits binary string, u Level: positive integer, u.level Order of ids b 1 b 2 … b k ∑ i=1 … k b i /2 i Each node has a SUCCESSOR and a PREDECESSOR SUCC(u), PRED(u) Node u stores the keys k such that u≤k< SUCC (u)

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Structure: Nodes 01 x SUCC (x) PRED (x) Keys stored on x Lemma 2.1 Let n 0 = 1/d(x, SUCC (x)), then w.h.p. (i.e. p>1-1/n 1+e ) that log(n)-log(log(n))-O(1)

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Structure: Links A node u in level k has six out links 2 x Short: SUCCESSOR,PREDECESSOR 2 x Medium: (left) closest level-(k+1) node whose id matches u.id[k] and is smaller than u.id. 1 x Long: the closest level-(k+1) node with prefix u 1 … u k-1 (1-u k )(?) u 1 … u k-1 (1-u k )u k+1 … u w * where w=log( n0)-log(log(n0)) 1 x Parent: closest level-(k-1) node Also keeps track of in-bound links

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Structure: Links Level 1 Level 2 Level Short link Parent link, to level k-1 Medium link Matches x[k]0* Long link, cross over about 1/2 k Matches u[w] except k th bit. (11*) Matches 1* Wrong!

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Routing: Algorithm LOOKUP (x, y): Initialization: set cur to x Proceed to root: while cur.level > 1: cur = cur.parent Greedy search: if cur.id ≤ y < SUCC (cur).id, return cur. Otherwise, choose m from links of cur that minimize d(m, y), move to m and repeat. Demo:

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Routing: Example Level 1 Level 2 Level y x

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One Observation Level 1 Level 2 Level

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Routing: Analysis (1) Level 1 Level 2 Level y x

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Routing: Analysis (2) Expected path length = O(log(n)) log(n ) to `level-1’ node log(n ) for traveling among clusters log(n ) for final local search

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Routing: Theorems Theorem 4.4 The path length from x to y is O(log(n)) w.h.p. Proof is based on several lemmas Lemma 4.1 For every node u with a level u.level < log(n)-log(log(n)), the number of nodes between u and u.Medium-left (Medium-right), if it exists, is at most 6log 2 (n) w.h.p.

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Routing: Theorems (2) Lemma 4.2 In the greedy search phase of a lookup of value Y from node x, let the j ’ th greedy step v j, for 1 ≤ j ≤ m, be such that v j is more than O(log 2 (n)) nodes away from y. Then w.h.p. node v j is reached over a Medium or Long link, and hence satisfies v j.level = j and v j [j] = Y[j]. m = log(n)-2loglog(n)-log(3+e) W.h.p. within m steps, we are n/2 m = 6log 2 (n) nodes away from the destination

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Routing: Theorems (3) Lemma 4.3 Let v be a node that is O(log 2 (n)) nodes away from the target y. Then w.h.p., within O(log(n)) greedy steps that target y is reached from v. Theorem 4.4 The total length of a route from x to y is O(log(n)) w.h.p. Theorem 4.6 Expected load on every node is O(log(n)/n). The load on every node is log 2 (n)/n w.h.p. Theorem 4.7 Every node u has in-degree O(log(n)) w.h.p.

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Join: Algorithm 1. Choose identifier: select a random 128 bits x 1 x 2 … x Setup short links: invoke LOOKUP (x), let x ’ be the result node. Insert x between x ’ and x ’. SUUCESSOR. 3. Choose level: let k be the maximal number of matching prefix bits between x and either SUCC (x) or PRED (x), choose level from 1 … k. 4. Set parent link: If SUCC (x) has level x.level-1, set x.parent to it. Otherwise, move to SUCC (x) and repeat. 5. Set long link: p = x 1 … x k-1 (1-x k )x k+1 … x w Invoke LOOKUP(p), stop after a node at level x.level+1 and matches p is reached.

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Join: Algorithm (cont.) 6. Set medium links: Denote p = x 1 x 2 … x x.level. If SUCC (x) has prefix p and level x.level+1, set x.Medium-right link to it. Otherwise, move the SUCC(x) and repeat. 7. Set inbound links: Denote p = x 1 x 2 … x x.level. Set inbound Medium links: Following SUCC links, so long as successor y has a prefix p and a level different from x.level, if y.level = x.level-1, set y.Medium-left to x. Set inbound long links: Following SUCC links, find y that has a prefix matches p and has level x.level. Take any inbound links that is closer to x than y. Set inbound parent links: Following PRED link, find y such that y.level = x.level+1. Repeat until meet a node in same level as x.

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Join: Example Level 1 Level 2 Level Lookup(x) 0111 Set Medium link: O(lg 2 n) w.h.p p = x 1 x 2 … x k (01) If y[k] != p: stop If y[k]=p and y.level=k+1: set Medium link Otherwise, move to succ(y) STOP Set Parent link: Following SUCC link, find a node has level k-1. Set long link P = x 1 … x k-1 (1-x k ) … x w stop at level k+1? In this case, find 00* Set inbound long links: Following short links, find y such that y[k]=x[k] and y.level = x.level, check y ’ s inbound links. X

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Join: Analysis LOOKUP takes O(log(n)) messages w.h.p. Travels on short links during link setting phase is O(lg 2 n) w.h.p. A Medium link is within 6log 2 (n) nodes from x w.h.p. Similar for others Theorem 5.1: A JOIN operation by a new node x incurs expected O(log(n)) number of messages, and O(log 2 (n)) messages w.h.p. The expected number of nodes that change their state as a result of x ’ s join is constant, and w.h.p is O(log(n)). Because node x has O(log(n)) in-degrees w.h.p. Similar results holds for LEAVE.

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Bounding In-degrees Theorem 4.7 Every node has expected constant in-degree, and has O(log(n)) in-degree w.h.p. In-degree=# of servers affected by join/leave How to guarantee constant in-degree? Bucket solution A background process to balance the assignment of levels

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Bucket Solution: Intuition Level k-1 Level k Node x has log(n) in-degree, assuming Medium Right ~log(n) x Too many nodes at level k-1; Improve the level selection procedure Too few nodes at level k

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Bucket Solution The name space is divided into non- overlapped buckets. A bucket contains m nodes, where log(n) ≤m ≤ clog(n), for c>2. In a buckets, levels are NOT assigned randomly For each 1≤j≤log(n), there are 1 … c nodes at level j in each bucket In(x) < 7c (?? 2c)

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Maintaining Bucket Size n can be accurately estimated When bucket size exceeds clog(n), the bucket is split into two equal size buckets. When bucket size drops below log(n), it is merged with a neighbor bucket. Further more, if the merged bucket is greater than log(n) x (2c+2)/3, the new bucket is split into two buckets. (c+1)/3 > 1 since c>2 Buckets are organized into a ring, which can be merged or split with O(1) message.

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Maintain Level Property Node join/leave without merging or splitting O(1) Join: size < clog(n), choose a level that has less that c nodes Leave: If it is the only node in its level, find another level that has two nodes, reassign level j to one of them. Bucket merge or split may result in a reassignment of the levels to all nodes in the bucket(s) O(log(n)) Merging/splitting are expensive, but they do not happen very often After a merging or splitting of buckets, at least log(n) (c-2)/3 JOIN/LEAVE must happen in this bucket until another merging or splitting of this bucket is performed Amortized Overhead = c/((c-2)/3) = O(1) for c>2

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Amortized analysis Log(n) clog(n) d1d1 d2d2 d 1, d 2 > (c-2)/3 New bucket size Max bucket size min(c/2lgn, (c+1)/3lgn)max(c/2lgn, (2c+2)/3lgn)

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Viceroy has no built in support for fault tolerance Viceroy requires graceful leave Leaves are NOT the same as failures Performance is sensitive to failure External techniques: Thickening Edges State Machine Replication Fault Tolerance

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State Machine Replication Old New SMR Super nodeViceroy nodes

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Related Works De Bruijn Graph Based Network Distance halving D2B Koorde Others Symphony (Small world model) Ulysses (ButterFly, log(n), log(n)/loglogn)

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Summary Constant out-degree Expected constant in-degree O(log(n )) w.h.p. O(1) with bucket solution O(log(n )) path length w.h.p Expected log(n )/n load: O(log 2 (n)/n) w.h.p. Weakness/improvements: Not Locality Aware No Fault Tolerance Support Due to the lack of flexibility of ButterFly network

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Question Photo by Peter J. Bryant

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