1. Deterministic Distributed Resource Discovery
Shay Kutten Technion
David Peleg Weizmann Inst.
Uzi Vishkin Univ. of Maryland & Technion

2. 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

4. 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)

5. 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

6. Think Distributed Mr. and Ms. B 132.12.16.64 IP street Interland A?
a pointer used by
the algorithm
Mr. and Ms. B
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”)

7. When the Algorithm is startedC 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”)

8. 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.

9. 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

10. 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

11. Algorithm’s result
pointers form star, knowledge graph is strongly connected,
Plus, possibly, additional links (possibly clique) .

12. 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)

13. Motivation (2) Gnutella
Napster server (mp3 music files)
I have “Madonna”
I want “Love Song”
I have “Love Song”

14. Motivation (2) Gnutella
Napster server
Ask A A
I want Love Song
I have Love Song

15. Motivation (2) Gnutella
Napster server
Ask A A
I want Love Song

16. Motivation (2) Gnutella
Napster server
Ask A A
Love Song

17. Motivation (2) Gnutella vs. Napster
Napster server
If the court closes
The service collapsed

18. Motivation (2) Gnutella vs. Napster
If the court closes
The service collapsed
Napster server

19. Motivation (2) Gnutella vs. Napster
If the court closes
the service collapses
Napster server

20. 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

21. 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.

22. 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.

23. Motivation: “Peer to Peer” Applications (P2P)
Akamai
Gnutella (originally Nullsoft (Winamp)) 2000
Gpulp: general Purpose Location Protocol
(4) Genny
(5) Freenet
(6) JXTA (Sun Microsystems)
April 2001
(7) Retsina: Discovery of Infrastructure in Multi-Agent Systems (CMU)

24. 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

25. (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])

26. Example of a technique taken from [SV82] (Handling tall trees in “pointer graph”)
Shortcuts [SV82] C B A

27. 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

28. Handing tall trees in “pointer graph” (cont.)
Shortcuts D E F G H I C B A

29. 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

30. Algorithm (2) handling active star roots
join 5 12
join
join 24 3
Star roots ask to connect

31. 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.

32. Algorithm (2) handling active star roots5 12 24
-An undirected subgraph is created by join
Messages.
-Each root connects to local minima (ID) root
On undirected subgraph. 3

33. Algorithm (2) handling active star roots12
join 24 7
(note differences from [SV82])
Non-stars do not join 9 20

34. 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

35. Algorithm (2) handling active star roots
join 5 12
join
join 16
join 24 7 3 9 20

36. Algorithm (2) handling active star roots
join 5 12
join
join 16
join 24 7 3 9 20

37. Algorithm (2) handling active star roots
join 5 12
join 16
join
join 24 7 3 9 20

38. Algorithm handling passive star roots5
join 24 9
Node 9 is “passive” since it does not know any
Node  it cannot initiate any joining

39. Algorithm handling passive star roots8
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

40. Algorithm handling passive star roots8
join
join 5 12
join 24 7 3
Node 12 joins its lowest “suitor” 3.
Suitors of 12 join 3 too. 9 20

41. 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

42. (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

43. (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

44. (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

45. (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

46. Lemma: always some active star
Correctness
Lemma: No cycles are created (hook on smaller +
non-star does not connect)
Assured by a new technique:
37
Possible because of
Weak connectivity!! 14 3 7 50 30
200
No connection
 No cycle
Lemma: always some active star

47. (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.

48. 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).