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

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

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**Deriving an Algorithm for the Weak Symmetry Breaking Task**Armando Castañeda Sergio Rajsbaum Universidad Nacional Autónoma de México**Symmetric and chromatic subdivision**Chromatic and binary sphere 0 2 1 symmetric map that no maps on mono?? 2 0 0 1 0 1 2 1 2 2 1 0 0 1 2 This talk is about ... Symmetric map: Faces same dim => mapped same binary colors Exists subdivision s.t. map exists? Symmetric: Faces same dim => same subdivision All possible assignments of binary values**0**1 0 1 ? 1 0 This talk is about ... Impossible for dimension 1 w.l.o.g. Since the map must be symmetric The map does not exist for any subdivision**0**2 1 2 0 0 1 0 1 2 1 2 2 1 0 0 1 2 This talk is about ... Impossible for dimension 2 Does not exist for any subdivision Impossible for dimension 3, 4 Possible for dim n iff #faces of n-simple are relatively prime Possible for dimension 5 Impossible for dimension 6, 7, 8 Possible for dimension 9**This talk is about ...**The relation with distributed computing: If the subdivision exists for dimension nthen 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**0**1 n . . . • n+1 asynchronous processors with id’s 0, 1, ... n**0**1 n . . . atomic snapshot write . . . • n+1 asynchronous processors with id’s 0, 1, ... n • shared memory with n+1 atomic registers**. . .**0 1 n . . . • 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**. . .**NO restriction on relative speeds 0 1 n . . . • 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 Many possible schedulings: order processes’ operations**output values: ,**input values:id’s W S B**output values: ,**input values:id’s W S B**output values: ,**input values:id’s W S B**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**2**0 1 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 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 It has to decide the same!! ??? zzz 2 0 1**n+1**1 n+1 2 n+1 n ... • Exceptional n = • are relatively prime • Results • For some exceptional values of n there is an algorithm for WSB for n+1 processors New upper and lower bounds for renaming n= 5, 9, 11, 13, 14 ... • For the other values of n there is no algorithm for WSB for n+1 processors**In 1993 it was discovered the deep relationship between**topology and distributed computing [Borowsky & Gafni 93] [Herlihy & Shavit93,99] [Saks & Zaharoglou93, 00] Here we focus on WSB • Represent the global state of an execution of an algorithm as a simplex • All executions are represented by a complex**0**2 1 0 1 2 0 0 2 1 0 2 1 2 1 The complex is a chromatic and binary colored subdivision of a proper colored simplex. All possible executions Initial state of the system**0**2 1 0 1 2 0 0 2 1 0 2 1 2 1 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 All possible executions Initial state of the system Simplex proper colored with id’s procs participate Binary coloring = output value**0**2 1 0 1 2 0 0 2 1 0 solo executions Two processors participate All processors participate 2 1 2 1 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**2 1 0 1 2 0 0 2 1 0 2 1 2 1 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 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**0**2 1 0 1 2 0 0 2 1 0 2 1 2 1 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 NO monochromatic simplexes of dimension n Representation WSB algorithm: chromatic subdivision with a symmetric binary coloring and no monochromatic n-simplexes**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 [Borowsky & Gafni93, 97] [Herlihy & Shavit93, 99][Saks & Zaharoglou93, 00] [Attiya & Rajsbaum02] Impossibility for WSB:for somen, 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 exceptionaln, there are subdivision with symmetry and no monochromatic**n+1**1 n+1 2 n+1 n ... • Exceptionaln= • are relatively prime • Goal: For exceptional n, construct a subdivisionK • chromatic • binary coloring • symmetric on the boundary • no monochromatic n-simplexes**n+1**1 n+1 2 n+1 n + 1 = 0 + ... + + k1 k0 kn-1 Key: there exist integers ki‘s which satisfy the equation if and only if n is exceptional**The construction in two steps:**• Use these ki’s to construct a symmetric subdivision K with 0 monochromatic n-simplexes counted by orientation: x counted as +1 and x counted as –1 • Cancel out the simplexes counted as +1 with the simplexes counted as –1 without modifying the boundary of K**0**2 1 0 0 1 2 1 2 0 0 0 2 1 1 2 1 2 The ChromaticCone 1. Assume a symmetric boundary 2. Put a red monochromatic triangle at the center 3. Connect them 4. Each simplexes on bd with carrier of same dim, is connected to the face of the center that completes its id’s**0**2 1 0 0 1 2 1 2 0 0 0 2 1 1 2 1 2 The ChromaticCone Every corner produces a triangle Every edge produces a triangle If red monochromatic then red monochromatic 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) Only has red monochromatic n-simplexes**Step 1:**bd(K) S • Construct K by dimension: each proper i-face is appropriate subdivided such that ithas ki red-mono i-simplexes. All i-faces have thesame subdivision (binary coloring is symmetric) • 2. Once the boundary bd(K) is done, do a chromatic cone with a red-mono simplex at the center Not any subdivision with ki red mono i-simplexes works Every ki, it is possible to construct the appropriate subdivision There is a restriction for k0 but it is not a big problem**By construction**n+1 i +1 n - 1 i = 0 = 0 #mono = 1 +sum ki n-simplexes produced by one sub(i-face) simplex at the center # i-faces For a subdivision with a symmetric a binary coloring#mono is • Orient K such that simplex at the center is counted as +1 • Count the number of monochromatic n-simplexes: The boundary induces the number of monochromatic simplexes!! Using Index Lemma => for any pseudomanifold, the boundaryinduces #mono**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 monocounted as –1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1 +1 -1 +1 -1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1 +1 -1 +1 -1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1 +1 -1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1 +1 -1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them -1 +1**Cancel out a simplex of K counted as +1 with a simplex**counted as –1 by subdividing a path which connects them n exceptional => subdivision K with no monochromatic => algorithm for WSB The algorithm works for any dimension n >= 2**1**0 0 2 The easiest case is when simplexes are adjacent**1**2 0 0 1 2 The easiest case is when simplexes are adjacent**1**2 0 0 1 2 The easiest case is when simplexes are adjacent We didnot modify the boundary**An example of a path of size 4**0 0 2 1 1 0**An example of a path of size 4**0 0 2 1 0 1 2 0