Is 1 different from 12?Is 1 different from 12?

Eli GafniUCLA

Eli GafniUCLA

OROR

•Is theoretical distributed computing is a brunch of Mathematics or Engineering?

•What is theoretical distributed computing ``on one leg?’’

•An algorithm has a geometric representation and a ``code’’ representation. Which should I use?

•Is theoretical distributed computing is a brunch of Mathematics or Engineering?

•What is theoretical distributed computing ``on one leg?’’

•An algorithm has a geometric representation and a ``code’’ representation. Which should I use?

Consensus TaskConsensus Task

•Two versions:

•0-1: procs input is 0 or 1

•All output o or all output 1

•if output i then some proc input is i

•election: proc input is its id

•all output the same participating id

•Two versions:

•0-1: procs input is 0 or 1

•All output o or all output 1

•if output i then some proc input is i

•election: proc input is its id

•all output the same participating id

k-set Cons (Chaudhuri 90)

k-set Cons (Chaudhuri 90)

•Two versions

•k-value: Proc’s input and output from {0,1,...,k}

•At most k distinct output, if i is output then some proc input is i

•k-Election: Proc input is its id

•All output together at most k participating ids

•Two versions

•k-value: Proc’s input and output from {0,1,...,k}

•At most k distinct output, if i is output then some proc input is i

•k-Election: Proc input is its id

•All output together at most k participating ids

1=?k1=?k

•Can any property of cons has it analogue to k-set cons?

•TST= 2 procs cons ?

•0-1 cons = election ?

•State machine approach to distributed computing ?

•Can any property of cons has it analogue to k-set cons?

•TST= 2 procs cons ?

•0-1 cons = election ?

•State machine approach to distributed computing ?

What about n Processors with m<n consensus?

What about n Processors with m<n consensus?

•With 2-cons can do Test-and-Set•With 2-cons can do Test-and-Set

TST (Test-and-Set)TST (Test-and-Set)•TST: One proc output 0 all the rest 1 (2,1)

•k-TST: At most k at least one proc output 0 all the rest 1 (k+1,k)

•TST implementable for 2 proc cons =(2,1)

•k-TST implementable from (k+1,k) set cons.

•TST: One proc output 0 all the rest 1 (2,1)

•k-TST: At most k at least one proc output 0 all the rest 1 (k+1,k)

•TST implementable for 2 proc cons =(2,1)

•k-TST implementable from (k+1,k) set cons.

nn33

2211

But what about m=3But what about m=3

•(3,1), (k+2,k)

•...[Afek, Weisberger]

•(3,1), (k+2,k)

•...[Afek, Weisberger]

Is View of k-set a la Chahudhuri is the most

convenient?

Is View of k-set a la Chahudhuri is the most

convenient?

•What will be the analogue of binary consensus?

•With consensus b.c.=election, how do I use at most k values to elect at most k leaders?

•What will be the analogue of binary consensus?

•With consensus b.c.=election, how do I use at most k values to elect at most k leaders?

All of DC ``on one foot!’’All of DC ``on one foot!’’

•Geometric approach

• Agreement:

•Geometric approach

• Agreement:

Digression: DC= -Agreement

Digression: DC= -Agreement

11 11 11

11 11

11

000000

0000

00 00 00 00

Center of GravityCenter of Gravity

MWMR RegistersMWMR RegistersWhile total weight < While total weight < WW

make next 0 into 1 make next 0 into 1 at your cornerat your cornerReturn CGReturn CG

AlgorithmicallyAlgorithmically

•All processors return vertices from same triangle restricted to the face of participation

•All processors return vertices from same triangle restricted to the face of participation

aa

bbcc

d...d...

How to converge on ``abstract’’ subdivided

simplex

How to converge on ``abstract’’ subdivided

simplex

•Add ``weight’’ until the ``longest path’’

•If not max add 1 and step one in shortest path to max

•Cannot converge without ``adding’’ unless n=2.

•Add ``weight’’ until the ``longest path’’

•If not max add 1 and step one in shortest path to max

•Cannot converge without ``adding’’ unless n=2.

Ahm...but 2 procs can do 2 0-1 independent cons with at least one output!

Ahm...but 2 procs can do 2 0-1 independent cons with at least one output!

•Can do - agreement at least one agreement no proc will output ?

•But can do - agreement for any number of processors if I can group them in two groups

•With 2-set cons can solve at least one of 2 independent cons!

•Can do - agreement at least one agreement no proc will output ?

•But can do - agreement for any number of processors if I can group them in two groups

•With 2-set cons can solve at least one of 2 independent cons!

11

332200

22

1100

33??

??

o-1 = election (k?)o-1 = election (k?)

•What is the meaning of 0-1 when it comes to k-set???

•What is the meaning of 0-1 when it comes to k-set???

In General In General •3 procs

•Cons i=1,2,3

•Say cons 2

• -agreement on the line perpendicular to the base

•Only 2 lines intersect at a point

•3 procs

•Cons i=1,2,3

•Say cons 2

• -agreement on the line perpendicular to the base

•Only 2 lines intersect at a point

1(2)1(2)0(2)0(2)

0(2)0(2)

11

00

What if 2-set and 3 ind cons?

What if 2-set and 3 ind cons?

•2 out of the 3 cons will return decision

•With k set cons and m>k ind cons then m-k+1 cons will return

•2 out of the 3 cons will return decision

•With k set cons and m>k ind cons then m-k+1 cons will return

TextText

How to do Multi-ValuedHow to do Multi-Valued

121233

123123

123123123123

121233

112323

112233

112233

121233

112233

AlgorithmicallyAlgorithmically

•2 set = 2 ind cons at least on return

•Choose two vectors (a1,a2) (b1,b2)

•post vector proposal for cons on 1st entry

•if alone return for first entry

•else proceed return b2 for 2nd entry

•2 set = 2 ind cons at least on return

•Choose two vectors (a1,a2) (b1,b2)

•post vector proposal for cons on 1st entry

•if alone return for first entry

•else proceed return b2 for 2nd entry

AlgorithmicallyAlgorithmically•2 set cons with 3 ind cons return from at

least 2 con’s

•will have 2 3-entry vectors (a1,a2,a3) (b1,b2,b3)

•post for 1st, if alone return from 1st, post chosen value, proceed to 2nd

•if not alone proceed with B vector to second

•...

•Return all posted values

•2 set cons with 3 ind cons return from at least 2 con’s

•will have 2 3-entry vectors (a1,a2,a3) (b1,b2,b3)

•post for 1st, if alone return from 1st, post chosen value, proceed to 2nd

•if not alone proceed with B vector to second

•...

•Return all posted values

How to do election from set binary-cons

How to do election from set binary-cons

•Will see ``prefix’’ of id for jth cons and propose a bit continuation for a leader name

•But how I know who return what from were: While I assume nobody returned in fact ``she just was late’’

•Will see ``prefix’’ of id for jth cons and propose a bit continuation for a leader name

•But how I know who return what from were: While I assume nobody returned in fact ``she just was late’’

How to do Multi-ValuedHow to do Multi-Valued

•If take small enough if any proc commit all commit or adopt same

•If take small enough if any proc commit all commit or adopt same

adopt 0adopt 0

adopt 1adopt 1

commit 1commit 1 commit 0commit 0

1-State-Machine1-State-Machine

•Want to emulate:

•Repeat:

•All send command to the ``center’’

•Center chooses one and send it to all

•Implement in rounds in SM using consensus in each round

•Want to emulate:

•Repeat:

•All send command to the ``center’’

•Center chooses one and send it to all

•Implement in rounds in SM using consensus in each round

2-State-Machine2-State-Machine

•Want to emaulate

•Repeat synchronously

•All send to ``center1’’ and ``center2’’ at least one of which is omission non-faulty at the round

•Center(s) choose and send back

•Can emulate with 2-set cons?

•Want to emaulate

•Repeat synchronously

•All send to ``center1’’ and ``center2’’ at least one of which is omission non-faulty at the round

•Center(s) choose and send back

•Can emulate with 2-set cons?

Solution 2-State-Machine with 2-set cons

Solution 2-State-Machine with 2-set cons

•Do 2 independents cons

•If get ``adopt v’’ at cons[i] the next command to cons[i] is the adopted value

•If the adopted value was commited all will propose same and commit

•If collid with the new value just commit the adopted

•Do 2 independents cons

•If get ``adopt v’’ at cons[i] the next command to cons[i] is the adopted value

•If the adopted value was commited all will propose same and commit

•If collid with the new value just commit the adopted

Too Numerous RamificationsToo Numerous Ramifications•Subsumes BG

•What is BG?

•If cannot answer 1st-cons both participate, will proceed to second, if collide in 2nd-cons, see exact positions in 1st-cons and can decide it

•Subsumes BG

•What is BG?

•If cannot answer 1st-cons both participate, will proceed to second, if collide in 2nd-cons, see exact positions in 1st-cons and can decide it

??2211

??

ConclusionsConclusions

•It is........ MATH.....•It is........ MATH.....