Deriving an Algorithm for the Weak Symmetry Breaking Task Armando Castañeda Sergio Rajsbaum Universidad Nacional Autónoma de México

This talk is about Symmetric and chromatic subdivision Chromatic and binary sphere no maps symmetric map that no maps 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?

This talk is about... Impossible for dimension w.l.o.g. Since the map must be symmetric The map does not exist for any subdivision ?

This talk is about... Impossible for dimension 2 Impossible for dimension 3, 4 Possible for dimension 5 Impossible for dimension 6, 7, 8 Possible for dimension 9 dim n #facesn Possible for dim n iff #faces of n-simple are relatively prime Does not exist for any subdivision

This talk is about... The relation with distributed computing: n If the subdivision exists for dimension n then n+1 There exists a distributed algorithm for n+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... 01n

n+1 asynchronous processors with id’s 0, 1,... n shared memory with n+1 atomic registers... 01n write atomic snapshot

n+1 asynchronous processors with id’s 0, 1,... n shared memory with n+1 atomic registers at most n processors can fail by crashing... 01n

n+1 asynchronous processors with id’s 0, 1,... n shared memory with n+1 atomic registers at most n processors can fail by crashing wait-free algorithms: a correct processor cannot wait forever... 01n NO restriction on relative speeds Many possible schedulings: order processes’ operations

WEAK SYMMETRY BREAKING (WSB)

WSBWSB output values:, input values: id’s

WSBWSB output values:, input values: id’s

WSBWSB 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

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 01 2 zzz ???

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 01 2 zzz ??? It has to decide the same!!

Results n n+1 For some exceptional values of n there is an algorithm for WSB for n+1 processors n n+1 2 n+1 n n Exceptional n = are relatively prime n n+1 For the other values of n there is no algorithm for WSB for n+1 processors n n = 5, 9, 11, 13, 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

The complex is a chromatic and binary colored subdivision of a proper colored simplex Initial state of the system All possible executions

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 0 12 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 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 0 12

Comparison requirement => symmetry on the boundary For two i-faces s 1, s 2, there is a simplicial bijection from sub(s 1 ) to sub(s 2 ) that preserves id coloring and binary coloring

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 0 12 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

n K Goal: For exceptional n, construct a subdivision K chromatic binary coloring symmetric on the boundary no monochromatic n-simplexes n n+1 2 n+1 n n Exceptional n = are relatively prime

Key: there exist integers k i ‘s which satisfy the equation n if and only if n is exceptional n n+1 2 n+1 n = 0 k0k0 k1k1 k n-1

The construction in two steps: k i K 1.Use these k i ’s to construct a symmetric subdivision K with 0 monochromatic n-simplexes counted by orientation: x counted as +1 and x counted as –1 K 2.Cancel out the simplexes counted as +1 with the simplexes counted as –1 without modifying the boundary of K

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

The Chromatic Cone 1. Assume a symmetric boundary 2. Put a red monochromatic triangle at the center 3. Connect them 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 The Chromatic Cone for i-faces s 1, s 2 => n-simplexes produced by isomorphic i-simplexes of sub(s 1 ) and sub(s 2 ) are counted in the same way (by orientation)

K k i 1.Construct K by dimension: each proper i-face is appropriate subdivided such that it has k i red-mono i- simplexes. All i-faces have the same subdivision (binary coloring is symmetric) bd( K ) S Step 1: bd( K ) 2. Once the boundary bd( K ) is done, do a chromatic cone with a red-mono simplex at the center Not any subdivision with k i red mono i-simplexes works Every k i, it is possible to construct the appropriate subdivision There is a restriction for k 0 but it is not a big problem

K 3.Orient K 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-faces simplex at the center n+1 i +1 i = 0 n - 1 #mono = 1 + sum k i = 0 By construction The boundary induces the number of monochromatic simplexes!! #mono Using Index Lemma => for any pseudomanifold, the boundary induces #mono #mono For a subdivision with a symmetric a binary coloring #mono is

STEP 2: CANCELING SIMPLEXES +1 WITH –1

#mono= 0 From step 1: symmetric subdivision K with #mono= 0 n-simplexes, counting by orientation Goal: subdivision of K with NO mono n-simplexes and the same boundary (to preserve symmetry) Idea: algorithm to cancel out each mono counted as +1 with a mono counted as –1

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

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

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

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

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

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

The algorithm works for any dimension n >= 2 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them n n exceptional => subdivision K with no monochromatic => algorithm for WSB

The easiest case is when simplexes are adjacent

We did not modify the boundary

An example of a path of size

The boundary is the same

A path of size 6

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

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

Example A

Example B

P#mono For a input path P, #mono is 0 bd(P)#mono The algorithm does not touch bd(P), therefore #mono of sub(P) sub(P) is 0 P Always exists a subdivision of P that is mapped exactly 0 times to the mono simplexes of the chromatic and B binary sphere B P It makes a continuous transformation from P tosub(P)

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

+1 1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and – The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 3.Produce two paths of size smaller than or equal the size of original path The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 3.Produce two paths of size smaller than or equal the size of original path 4.Proceed recursively on resulting paths The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 3.Produce two paths of size smaller than or equal the size of original path 4.Proceed recursively on resulting paths The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 3.Produce two paths of size smaller than or equal the size of original path 4.Proceed recursively on resulting paths The Algorithm:

1.Inspect shared (n-1)-faces from the beginning to find a subdividing point 2.Subdivide a shared (n-1)-face to produce two red-mono n-simplexes counted as +1 and –1 3.Produce two paths of size smaller than or equal the size of original path 4.Proceed recursively on resulting paths The Algorithm:

Subdividing point: Notation for a path S 0 – S 1 – S 2 –... – S q-1 – S q Red-mono counted as +1 and -1 No mono For S i – S i+1, S i,i+1 is the (n-1)-face shared by S i and S i+1 The subdividing point is the smallest m such that #red(S m+1,m+2 ) >= n+1-m

The subdividing point m is like the middle of the path +1 P1P1 +1 P1P1 P2P2 Shortest 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 non-exceptional n n+1 3.For non-exceptional n, there is no algorithm for WSB for n+1 processors exceptional n n+1 4.For exceptional n, there exists an algorithm for WSB for n+1 processor K #mono a) chromatic subdivision K with a symmetric binary coloring and #mono = 0 K b) Subdivision of K with the same boundary and no monochromatic n-simplexes

The end