Deterministic Distributed Resource Discovery Shay Kutten Technion David Peleg Weizmann Inst. Uzi Vishkin Univ. of Maryland & Technion
Distributed Resource Discovery [HLL99] router B C A pointer The corresponding directed graph (on nodes A,B,C): (assumption: weakly connected) A B C Problem: compute the connected component
Distributed Resource Discovery- [Harcol-Balter, Leighton, Lewin 1999] (“Names Dropper” protocol) Licensed to AKAMAI, a leading Web caching provider The resource discovery task: the set of servers changes Find the current set (towards deciding which Akamai server to assign to each user)
Akamai servers need to learn how to reach each other via the Internet (towards deciding which Akamai server is “close” to the user). user CNN Internet Akamai Server
Think Distributed Mr. and Ms. B 132.12.16.64 IP street Interland A? a pointer used by the algorithm Mr. and Ms. B 132.12.16.64 IP street Interland A? A B a pointer used by the algorithm (one per node, initially pointing at the node itself) A Address “known” to Node A (“knowledge graph”)
When the Algorithm is started C A D B Weakly connected directed graph a pointer used by the algorithm (one per node, initially pointing at the node itself) A Address “known” to Node A (“knowledge graph”)
Algorithm Actions C A D C B Node A learns C’s addr. From B, thus, directed edge (A C) is added to knowledge graph.
Algorithm Actions C A D C B Node A learns C’s addr. From B, thus, directed edge (A C) is added to knowledge graph. (2) Node D who knows of C, change PTR to point at C, thus, pointer graph changes. D
Algorithm Actions change C A D C B Node A learns C’s addr. From B, thus, directed edge (A C) is added to knowledge graph. (2) Node D who knows of C, change PTR to point at C, thus, pointer graph changes. D
Algorithm’s result pointers form star, knowledge graph is strongly connected, Plus, possibly, additional links (possibly clique) .
Model [HLL99] - synchronous (not necessary for our alg) - simultaneously start (for our alg: just “close”) - complete comm. Graph Complexity: #Msgs (connections [HLL99]), #bits (#pointers*log n), time (#rounds)
Motivation (2) Gnutella Napster server (mp3 music files) I have “Madonna” I want “Love Song” I have “Love Song”
Motivation (2) Gnutella Napster server Ask A A I want Love Song I have Love Song
Motivation (2) Gnutella Napster server Ask A A I want Love Song
Motivation (2) Gnutella Napster server Ask A A Love Song
Motivation (2) Gnutella vs. Napster Napster server If the court closes The service collapsed
Motivation (2) Gnutella vs. Napster If the court closes The service collapsed Napster server
Motivation (2) Gnutella vs. Napster If the court closes the service collapses Napster server
Motivation (2) Gnutella vs. Napster Napster server If the court closes the service collapses Gnutella: attempts to solve by having every client=server Same problem: user knows some others find more increase connectivity to withstand disconnections
Motivation(2) Gnutella Bob Users get disconnected often Alice Carol David Initially Alice knows only Bob & Carol (personally, or from ICQ, or from Google…) and can connect via them To others they know.
Motivation(2) gnutella Users get disconnected often Bob Alice Carol David Initially Alice knows only Bob & Carol (personally, or from ICQ, or from Google…) Alice learned from Carol how to reach David, so that when Bob & Carol are not online Alice is still connected.
Motivation: “Peer to Peer” Applications (P2P) Akamai http://www.akamai.com 1999 Gnutella http://gnutella.wego.com (originally Nullsoft (Winamp)) 2000 Gpulp: general Purpose Location Protocol http://www.gpulp.com (4) Genny http://www.gnutelladev.com (5) Freenet http://freenet.sourceforge.net 2000 (6) JXTA http://www.jxta.org (Sun Microsystems) April 2001 (7) Retsina: Discovery of Infrastructure in Multi-Agent Systems (CMU)
This paper [HLL99] deterministic randomized termination detection no detection O(log n) rounds O(log n) O(n log n) messages O(n log n) O(E log n) bits O(n log n) 2 3
(1) Shrink tall trees to make them stars [SV82] Algorithm strategies (1) Shrink tall trees to make them stars [SV82] D E F G H I (2) Merge stars to get one tree (changed from [SV82]) 12 5 (3) Carefully connect weakly connected (explained later; one of the differences from [Shiloach, Vishkin 82])
Example of a technique taken from [SV82] (Handling tall trees in “pointer graph”) Shortcuts [SV82] C B A
the pointer to its grandparent Handing tall trees in “pointer graph” (cont.) Shortcuts ([SV82] D E F G H I C B A Each child learns the pointer to its grandparent
Handing tall trees in “pointer graph” (cont.) Shortcuts D E F G H I C B A
Add edges to knowledge graph: Algorithm: finding who to merge with [SV82] D A E,E,F,G,H, I,C E B F C C G H Add edges to knowledge graph: node I, on becoming a child of root D, tells D about C I
Algorithm (2) handling active star roots join 5 12 join join 24 3 Star roots ask to connect
Algorithm (2) handling active star roots join 5 12 join join 24 3 -An undirected subgraph is created by join Messages. -Each root connects to local minima (ID) root On undirected subgraph.
Algorithm (2) handling active star roots 5 12 24 -An undirected subgraph is created by join Messages. -Each root connects to local minima (ID) root On undirected subgraph. 3
Algorithm (2) handling active star roots 12 join 24 7 (note differences from [SV82]) Non-stars do not join 9 20
Algorithm (2) handling active star roots Join smallest id- prevents cycles 12 5 join 5 joins 12!!! : smallest id neighbor join join 24 7 3 (only stars join, as opposed to [SV82]) 9 20
Algorithm (2) handling active star roots join 5 12 join join 16 join 24 7 3 9 20
Algorithm (2) handling active star roots join 5 12 join join 16 join 24 7 3 9 20
Algorithm (2) handling active star roots join 5 12 join 16 join join 24 7 3 9 20
Algorithm handling passive star roots 5 join 24 9 Node 9 is “passive” since it does not know any Node it cannot initiate any joining
Algorithm handling passive star roots 8 join join 5 12 Node 12 is passive: It does not know Any non-child. It cannot Initiate any progress. join 24 7 3 9 20
Algorithm handling passive star roots 8 join join 5 12 join 24 7 3 Node 12 joins its lowest “suitor” 3. Suitors of 12 join 3 too. 9 20
Algorithm: An Example of a Technique different than [SV82] (weakly connected directed graph) Without an additional technique, may take (n) time: Worst case (only A is active, its star grows by 1 per phase): Phase 1 A
(weakly connected diagraph) Without an additional technique, may take (n) time: Worst case (only A is active, its star grows by 1 per phase): A phase 2
(weakly connected diagraph) Without an additional technique, may take (n) time: Worst case (only A is active, its star grows by 1 per phase): phase n-1 A
(weakly connected diagraph) Prevent (n) time, and still do not send too many messages: i Idea: star root A picks 2 neighbors (not just 1) in phase i. (In phase i there are only n/2 stars) i A Example: i=2 2 =4 i
(weakly connected diagraph) Prevent (n) time: Idea: A picks 2 neighbors (not just 1) in phase i. i A Some delicate points: resolving collisions, mixed topologies prove convergence in O(log n) phases
Lemma: always some active star Correctness Lemma: No cycles are created (hook on smaller + non-star does not connect) Assured by a new technique: 37 Possible because of Weak connectivity!! 14 3 7 50 30 200 No connection No cycle Lemma: always some active star
(2) Active star roots merge. Complexity Two kinds of progress in each phase: A tall tree gets shallower by a constant factor. (2) Active star roots merge. Obstacles: # of star trees may grow because of (1) Trees may get taller because of (2) But (with the right combination) there is a progress in the combination.
Conclusion improved complexity in all measures (verifies [HLL99] conjecture, forecasting a simple algorithm with these complexities). Deterministic. Terminates (answers an open problem of Lipton). Further research Adaptive algorithms (vs. “one shot”) Lower bounds? (in [CGK95] O(n) msgs for undirected case).