Deriving an Algorithm for the Weak Symmetry Breaking Task Armando Castañeda Sergio Rajsbaum...

Deriving an Algorithm for the Weak Symmetry Breaking Task

Armando Castañeda

Sergio Rajsbaum

Universidad Nacional Autónoma de México

This talk is about ...

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Symmetric and chromatic subdivision

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Chromatic and binary sphere

symmetric map that no mapno mapss on mono??

Symmetric: Faces same dim => same subdivision

All possible assignments of binary values

Symmetric map: Faces same dim => mapped

same binary colors

Exists subdivision s.t. map exists?


Impossible for dimension 1

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w.l.o.g.

Since the map must be symmetric

The map does not exist for any subdivision

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Impossible for dimension 2

Impossible for dimension 3, 4

Possible for dimension 5

Impossible for dimension 6, 7, 8

Possible for dimension 9

Possible for dim ndim n iff #faces#faces of nn-simple are

relatively prime

Does not exist for any subdivision


The relation with distributed computing:

If the subdivision exists for dimension nn then

There exists a distributed algorithm for n+1n+1 processors for the Weak Symmetry Breaking task

Does not exists for 2, 3, 4, 5 processors

Exists for 6 processors

MODEL OF COMPUTATION

• n+1 asynchronous processors with id’s 0, 1, ... n

. . .0 1 n


• shared memory with n+1 atomic registers

. . .

. . .0 1 n

writeatomic snapshot



• at most n processors can fail by crashing

. . .

. . .0 1 n



• at most n processors can fail by crashing

• wait-free algorithms: a correct processor cannot wait forever

. . .

. . .0 1 n

NO restriction on relative speeds

Many possible schedulings: order processes’ operations

WEAK SYMMETRY BREAKING (WSB)

WS B

output values: ,input values: id’s

Trivial algorithm: processors with even id decide and processors with odd id decide

Avoiding trivial solutions. Each processors can only do comparisons A > B? A = B? It does not know its id!!

This requirement implies symmetry on the outputs of executions with similar scheduling




0 1 2zzz???




0 1 2zzz ???

It has to decide the same!!

Results

• For some exceptional values of nn there is an algorithm for WSB for n+1n+1 processors

n+1

1...

n+1

2

n+1

nExceptional nn =

are relatively prime

• For the other values of nn there is no algorithm for WSB for n+1n+1 processors

nn = 5, 9, 11, 13, 14 ...

New upper and lower bounds

for renaming

TOPOLOGICAL REPRESENTATION ALGORITHM

FOR WSB

In 1993 it was discovered the deep relationship between topology and distributed computing

[Borowsky & Gafni 93][Herlihy & Shavit 93, 99]

[Saks & Zaharoglou 93, 00]

• Represent the global state of an execution of an algorithm as a simplex

• All executions are represented by a complex

Here we focus on WSB

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The complex is a chromatic and binary colored subdivision of a proper colored simplex.

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Initial state of the system

All possible executions

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The complex is a chromatic and binary colored subdivision of a proper colored simplex. The more steps processors execute, the more fine the subdivision is

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Initial state of the system

All possible executions

Simplex proper colored with id’s procs participate

Binary coloring = output value

solo executions

All processors participate

Two processors participate

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Comparison requirement =>

symmetry on the boundary

For two i-faces s1, s2, there is a simplicial bijection from sub(s1) to sub(s2) that

preserves id coloring and binary coloring

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NO monochromatic

simplexes of dimension n

Representation WSB algorithm: chromatic subdivision with a symmetric binary coloring

and no monochromatic n-simplexes

[Borowsky & Gafni 93, 97] [Herlihy & Shavit 93, 99] [Saks & Zaharoglou 93, 00] [Attiya & Rajsbaum 02]

If there exists an algorithm for WSB for n+1 processors then there exists a chromatic subdivision of dim n with a symmetric binary coloring and no monochromatic n-simplexes

Impossibility for WSB: for some n, symmetry => any such a subdivision contains monochromatic

If there exists a chromatic subdivision of dim n with a symmetric binary coloring and no monochromatic n-simplexes then there exists an algorithm for WSB for n+1 processors

Asynchronous Computability Theorem [Herlihy & Shavit 93, 99],Simplex Convergence Algorithm [Borowsky & Gafni 97]

Algorithm for WSB: for exceptional n, there are subdivision with symmetry and no monochromatic

DERIVING ALGORITHMS FOR WSB

Goal: For exceptional nn, construct a subdivision KK• chromatic • binary coloring• symmetric on the boundary• no monochromatic n-simplexes

n+1

1...

n+1

2

n+1

nExceptional nn =

are relatively prime

Key:

there exist integers ki‘s which satisfy the equation if and only if nn is exceptional

n+1

1+ ... +

n+1

2

n+1

n+ + 1 = 0k0 k1 kn-1

The construction in two steps:

1. Use these kkii’s to construct a symmetric subdivision KK with 0 monochromatic n-simplexes counted by orientation: x counted as +1 and x counted as –1

2. Cancel out the simplexes counted as +1 with the simplexes counted as –1 without modifying the boundary of KK

STEP 1: A SUBDIVISION WITH #mono#mono=0

The Chromatic Cone

1. Assume a symmetric boundary

2. Put a red monochromatic triangle at the center

3. Connect them

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4. Each simplexes on bd with carrier of same dim, is connected to the face of the center that completes its id’s

Every corner produces a triangle

Every edge produces a triangle

If red monochromatic then red monochromatic

Only has red monochromatic n-simplexes

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The Chromatic Cone

for i-faces s1, s2 => n-simplexes produced by isomorphic i-simplexes of sub(s1) and sub(s2) are counted in the same way (by orientation)

1. Construct KK by dimension: each proper i-face is appropriate subdivided such that it has kkii red-mono i-simplexes. All i-faces have the same subdivision (binary coloring is symmetric)

bd(bd(KK))

S

Step 1:

2. Once the boundary bd(bd(KK)) is done, do a chromatic cone with a red-mono simplex at the center

Not any subdivision with ki red mono i-simplexes worksEvery ki, it is possible to

construct the appropriate subdivision

There is a restriction for k0 but it is not a big problem

3. Orient KK such that simplex at the center is counted as +1

4. Count the number of monochromatic n-simplexes:

n-simplexes produced by one

sub(i-face)

# i-facessimplex at the

center

n+1

i +1i = 0

n - 1

#mono = 1 + sum ki = 0

By construction

The boundary induces the number of monochromatic simplexes!!Using Index Lemma => for any

pseudomanifold, the boundary induces #mono#monoFor a subdivision with a symmetric a binary coloring #mono #mono is

STEP 2: CANCELING SIMPLEXES +1 WITH –1

From step 1: symmetric subdivision K with #mono= 0#mono= 0 n-simplexes, counting by orientation

Goal: subdivision of K with NO mono n-simplexes andthe same boundary (to preserve symmetry)

Idea: algorithm to cancel out each mono counted as +1with a mono counted as –1

-1

-1

-1 +1

+1+1

Cancel out a simplex of KK counted as +1 with a simplex counted as –1 by subdividing a path which connects them

-1

-1 +1

+1


-1

+1


The algorithm works for any dimension n >= 2


nn exceptional => subdivision K with no monochromatic => algorithm for WSB

The easiest case is when simplexes are adjacent

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We did not modify the boundary

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An example of a path of size 4

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The boundary is the same

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A path of size 6

A path of size 6

The algorithm takes the path and stretches it on the chromatic and binary sphere

The chromatic and binary sphere

Contains a proper colored n-simplex for every possible assignment of n+1n+1 binary values to the n+1n+1 colors

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Example A

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Example A

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Example A

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Example B

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Example B

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For a input path PP, #mono#mono is 0

The algorithm does not touch bd(P)bd(P), therefore #mono#mono ofsub(P)sub(P) is 0

Always exists a subdivision of PP that is mapped exactly0 times to the mono simplexes of the chromatic andbinary sphere BB

It makes a continuous transformation from PP to sub(P)sub(P)

The Algorithm:

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-1

1. Inspect shared (n-1)-faces from the beginning to find a subdividing point

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2. Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1

The Algorithm:



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The Algorithm:



3. Produce two paths of size smaller than or equal the size of original path

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-1

-1 +1

The Algorithm:




4. Proceed recursively on resulting paths

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-1

-1 +1

The Algorithm:





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-1

-1 +1 -1-1 +1+1

The Algorithm:





-1

+1 -1 +1+1+1 -1

-1

The Algorithm:





The Algorithm:

Subdividing point:Notation for a path

S0 – S1 – S2 – ... – Sq-1 – Sq

Red-mono counted as +1 and -1

No mono

For Si – Si+1, Si,i+1 is the (n-1)-face shared by Si and Si+1

The subdividing point is the smallest m such that #red(Sm+1,m+2) >= n+1-m

The subdividing point m is like the middle of the path

+1

-1

P1

+1

P1

-1

P2Shortest path P2

that completes P1, | P1 | = | P2 |

In the middle we can produce paths of

size smaller than or equal original

• Once the algorithm finds the subdividing point, there are 6 cases

• Each case is tailor-made subdivided

• For 4 cases algorithm produces paths of size smaller than the original path

• For 2 cases algorithm produces a path of size equal than the original

• When a resulting paths is of size equal to the input, paths of size smaller on the next recursively invocation

Same size as the input

Conclusions

1. WSB task: processors decide red or blue. If all processors participate, not all decide the same value. Comparison based algorithms

2. Relation distributed computing and topology => there is a chromatic subdivision of an n-simplex with a symmetric binary coloring and no monochromatic n-simplexes iff there is an algorithm for WSB for n+1 processors

Conclusions

3. For non-exceptional nnon-exceptional n, there is no algorithm for WSB for n+1n+1 processors

4. For exceptional nexceptional n, there exists an algorithm for WSB for n+1n+1 processor

a) chromatic subdivision KK with a symmetric binary coloring and #mono#mono = 0

b) Subdivision of KK with the same boundary and no monochromatic n-simplexes

The end

Deriving an Algorithm for the Weak Symmetry Breaking Task Armando Castañeda Sergio Rajsbaum...

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