Download presentation

Presentation is loading. Please wait.

Published byManuel Brake Modified over 3 years ago

1
Bigamy Eli Gafni UCLA GETCO 2010

2
Outline Models, tasks, and solvability What is SM? r/w w.f. asynch computability Sperner Lemma as a consequence of the impossibility of set consensus Impossibility of 0-1 coloring 1111… Simple algorithmic reduction of t-resiliency to wait-free (characterizing t-resiliency) – Consensus contingent on a resolver t-resiliency by r/w reduction Algorithmic Asynch Computability

3
(Inputless) Tasks Point to set map from a chromatic simplex to a chromatic labeled complex that preserve dimension and colors Eg two proc cons between p0 and p1 p0p1 p0p1p0 0011

4
Tasks cont’d A task is ``invoked’’ by a proc and returns a value to the invoker. If just p0 invoked, the task will return 0 If just p1 invoked, the task will return 1 If both invoked, it will either return 0 to both or 1 to both

5
A model A task whose labels also identify a single consistent participating set for each label To make 2 cons a task we have 4 outputs – ((P0,0,p0),(p1,0,{p0,p1})) – ((P1,1,p1),(p0,1,{p0,p1})) – ((P0,0,{p0,p1}),(p1,0,{p0,p1})) – ((P1,1,{p0,p1}),(p0,1,{p0,p1})) coloroutput Participating set

6
A model cont’ed A task/model solves a task if when iterating the model/task enough times to get a task A then there exists a chromatic simplicial map ``boundary preserving’’ from A to the task. SM wait-free model/task is the immediate snaps task

7
What is SM? The task implemented operationally as follows: All interleaving of w(i), followed by any permutation of of r(i,j) j=1,…,n An r(I,j) ``reads’’ j for pi iff w(i) precedes it. pi returns all the ids it read What is the (succinct) ``spec’’ of the possible simplexes returned?

8
SM spec There must be at least one proc who returns all (the proc to write last). Take all the procs that return the set n and eliminate their ids from smaller returned sets, ie move their write up in the sequence… Continue inductively. The returned sets left is an ``immediate snapshot’’ simplex SM is a ``fat’’ immediate snapshot n iterations of SM/task maps to immediate snapshot

9
Set Cons Task (n) If a set P, |P|=k<n invoke the task each proc returns an id from P |P|=n one of the ids is not returned by anybody.

10
Sperner Lemma Sperner coloring of a subdivided simplex: Each 0-face distinct color. A vertex gets the color of one of the 0-faces carrying it Lemma: Any Sperner coloring forces a fully colored n-simplex

11
Asynch Computability Any r/w w.f. algorithm for set cons implies a Sperner coloring of a chromatic subdivided simplex Any Sperner coloring of a chromaic subdivided simplex without fully colored simplex implies a r/w w.f. algorithm for set cons Conclusion: Sperner holds for chromatic subdivided simplexes Will later show a property that holds only for chromatic, so general Sperner is not ``automatic’’

12
General Sperner from Chromatic Sperner Let A be a subdivided simplex and let A be Sperner colored, then there exists a refinement r(A) of A which is chromatic and moreover, If A has no fully colored simplex so does r(A). This is not a ``disguised derivation’’ of Sperner as no ``combinatorics’’ were relied on.

13
Impossibility of 0-1 1111…task Want to prove the following Lemma: A 0-1 coloring of a chromatic subdivided simplex such that each boundary simplex has at least one vertex colored 0, forces a mono-chromatic full simplex Want to prove the Lemma from the impossibility of set cons.

14
Proof The counter of the lemma implies a r/w w.f. algorithm on n procs such that of n processors at least one returns 0 and at least one returns 1. A proc that return 0 now chooses it name, at proc that returns 1 writes it in SM and reads and return a participating id that did not write it returned 1. The first to write 1 will not be returned by anybody = set cons.

15
R/W Reduction What’s a researcher in Distributed Algorithms to do after the discovery of the connection to topology? – Take topo 101 in order to remain ``viable’’? Analogue thing occurred with NP-completeness. Some do ``algorithms’’ some do ``complexity’’ and suddenly a big implication from complexity on algorithm: – NP-completeness reduction Here: Do r/w reductions Characterization of t-resilient by reduction to w.f.:

16
t-resiliency Model/Task implementation: – w(i) followed by any permutation of r(i,j) j=1,…,n – Interleave on I – T-resiliency: The first read is preceded by at least n-t w’s No ``fat’’ ``t-resilient immediate snapshot’’! Can be shown: 2 iterations of immediate snaps with ``views’’ that ``see’’ less than n-t removed Will show equivalent to: Wait once until see n-t then proceed w.f.

17
t-res by r/w reduction Consensus contingent on a ``resolver’’ – If all propose same all output same – Else if resolver is alive all output its value – Epsilon ¼ agreement – If not ? Then output else wait for resolver 01 0?0? 1?1?

18
t-res cont For simplicity t=1 Wait until n-1, or n In Round Robin service the ``reads’’ by resolver consensus, with the resolver for he code of pi is pi If stuck with no output then from there on al agree on the ``reads’’ until the stuck is resolved and jumps to another code.

Similar presentations

OK

R/W Reductions Eli Gafni UCLA ICDCN06 12/30. Outline Tasks and r/w impossible task: –2 cons –3 cons NP-completeness R/W reduction “Weakest Unsolvable.

R/W Reductions Eli Gafni UCLA ICDCN06 12/30. Outline Tasks and r/w impossible task: –2 cons –3 cons NP-completeness R/W reduction “Weakest Unsolvable.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google