Upper Bound on the Complexity of Solving Renaming Ami Paz, Technion - PowerPoint PPT Presentation

Upper Bound on the Complexity of Solving Renaming Ami Paz, Technion Joint work with: Hagit Attiya, Technion Armando Castañeda, Technion Maurice Herlihy, Brown PODC 2013 Best Student Paper Award

Introduction 2

The Model ... p 1 p 2 p 3 p n Atomic Read/Write  n asynchronous processes.  At most n – 1 processes can crash.  Wait-free algorithms: each nonfaulty process produces an output.  Full information. 3

Iterated Atomic Snapshot  Execution induced by a sequence of blocks:  Write together;  Read together.  Fresh copy of the memory every time.  Implemented in 𝑃 𝑜 2 overhead [Borowsky and Gafni 97] . 4

Comparison Based Algorithms  Processes only compare their identifiers.  Execution by P 1 , P 2 , P 3 looks like execution by P 1 , P 2 , P 4 . p 1 p 3 p 1 p 2 p 3 p 1 p 4 p 1 p 2 p 4 5

M -Renaming [Attiya et al. 90] 5 8 2 n processes Outputs: 1 ,…, M Unique values With identifiers Processes are only allowed to compare their identifiers 6

Weak Symmetry Breaking (WSB) [Gafni et al. 06] 1 1 0 n processes Outputs: 0/1 If all output: not all the same With identifiers Processes are only allowed to compare their identifiers 7

M -Renaming Bounds n +1,...,2 n -2 n +1,...,2 n -2 2 n -1,... 2 n -1,... M : 1 ,...,n 1 ,...,n WSB WSB Several Papers [Attiya et al. 90] [Attiya et al. 90] 8

M -Renaming Bounds [Castañeda and Rajsbaum 10]: Lower bounds are wrong. n +1,... n +1,... 2 n -1,... 2 n -1,... M : 1 ,...,n 1 ,...,n 2 n -2 2 n -2 n WSB WSB Prime Power Non ? Prime Power 9

Renaming Bounds [Castañeda and Rajsbaum 10]: Lower bounds are wrong.  Existential proof.  No bounds on steps complexity. 10

Our Results  n -process algorithm for WSB and (2𝑜 − 2) -renaming, when n is not a prime power.  Bounded step complexity: 𝑃(𝑜 𝑟+5 ) , where q is the largest prime power dividing n. 11

Topology & Distributed Computing 12

Simplexes  Sets of objects.  Represented as convex hulls of points. 𝑦 𝑦, 𝑧 𝑦, 𝑧, 𝑨 𝑦, 𝑧, 𝑨, 𝑥 13

Simplicial Complexes  “ Gluings ” of simplexes.  Some complexes are called subdivisions of others. 14

Topology & Distributed Computing [Borowsky and Gafni 93]; [Herlihy and Shavit 93,99]; [Saks and Zaharoglou 93,00]; [Herlihy and Rajsbaum 94,00].  Simplicial complexes represent states of the system. y a x x z z 15

Topology & Distributed Computing [Borowsky and Gafni 93]; [Herlihy and Shavit 93,99]; [Saks and Zaharoglou 93,00]; [Herlihy and Rajsbaum 94,00].  Simplicial complexes represent states of the system. (x, y, z) (x, a, z)  Colored. 16

Topology & Distributed Computing  An execution. x x,y x,y x,y 17

Topology & Distributed Computing  An execution. x x,y x,y x,y x,y y  All 1-step interleaving. x x,y x,y y 18

Subdivision Implies Algorithm  Simplicial approximation: processes converge on a simplex. 19

Subdivision Implies Algorithm  Execution: 20

Subdivision Implies Algorithm  Execution: 21

Subdivision Implies Algorithm  Execution: 22

Subdivision Implies Algorithm  Execution: 23

Subdivision Implies Algorithm  Execution: 24

Subdivision Implies Algorithm  Execution: 25

Outputs  Each vertex has double coloring:  Process id  Output value 26

Subdivision Implies Algorithm  Simplicial approximation 0 0 1 0 1 0 0 1 1 1 0 0 27

Chromatic Subdivisions  Chromatic subdivision: can assign a process to each vertex.  An algorithm is induced by a specific subdivision:  Standard chromatic subdivision. Simplex Standard Subdivision Second subdivision: S ... Std K ( S ) Std( S ) Std 2 ( S ) 28

Topological Notions  Simplicial complex  Subdivision  Chromatic Subdivision  Standard chromatic Subdivision 29

Topology & Distributed Computing Chromatic Distributed Subdivision Algorithm 30

From Subdivision to Algorithm Standard Chromatic Distributed Chromatic Subdivision Algorithm Subdivision simulated ← 0 Write(initialState i ) to R i while true do r ← Scan (R 0 ,...,R n − 1 ) if r contains all then return simulated simulated ← 1 Execute Local A (r) if A returns v then return the same value v Write ( r) to R 31 …

Colored Simplicial Approximation [Herlihy and Shavit 99] Standard Chromatic Chromatic Subdivision Subdivision  Colored simplicial approximation theorem: any chromatic subdivided simplex can be “approximated” by a standard chromatic subdivision std K ( S ) …  …for large enough K.  Yields no bound on K . 32

Subdivision Implies Algorithm Standard Distributed Chromatic Algorithm Subdivision  We count subdivisions, to get the step complexity. 33

Solving WSB Properties of the desired solution 34

Recall: WSB [Gafni et al. 06] 1 1 0 n Processes Outputs: 0/1 If all output: not all the same With identifiers Processes are only allowed to compare their identifiers 35

Binary Outputs  All output values are binary. 1 0 0 0 0 1 0 1 1 1 1 0 36

Monochromatic Simplexes  Represent executions with a single output.  Forbidden! 37

Comparison Based Algorithms  Processes only compare their values.  Execution by P 1 , P 2 , P 3 looks like execution by P 1 , P 2 , P 4 . p 1 p 3 p 1 p 2 p 3 p 1 p 4 p 1 p 2 p 4  Topology: implies symmetry on the boundary. 38

Who is Bigger? 39

Symmetric Output Coloring 40

Three Steps to Solution 41

Our Goal Construct a subdivided simplex & coloring, s.t.:  Symmetric coloring on the boundry.  Without monochromatic simplexes.  Standerd chromatic subdivision. 42

Three Step Plan  Step 1: find a symmetric subdivision with only good monochromatic simplexes.  Step 2: eliminate mono. simplexes, while preserving symmetry.  Step 3: get a mapping from standard subdivision, yielding a WSB coloring and algorithm. 43

Step One: Symmetric Boundary 44

1. Create Boundary  Start by creating a symmetric boundary.  Each i -face is subdivided and colored:  Create 𝑙 𝑗 0-mono. simplexes, for some integer 𝑙 𝑗 .  Number of i -faces = 𝑜 𝑗 . 45

1. Fill in the Interior  Add internal 0-mono. simplex.  More 0-mono. simplexes are created.  Total number of mono.: 𝑜−1 1 + 𝑜 𝑙 𝑗 𝑗 𝑗=1 46

1. Counting Mono. Simplexes  Each 𝑙 𝑗 has a sign.  We want: 𝑜−1 1 + 𝑜 𝑙 𝑗 = 0 𝑗 𝑗=1 47

1. Creating the Boundary  We want: 1 + 𝑜 𝑗 𝑙 𝑗 = 0 .  Subdivide boundaries simultaneously.  𝑃(1) subdivisions. 48

Step Two: Eliminating Mono. Simplexes 49

Eliminating Monochromatic Simplexes  Use subdivisions to eliminate monochromatic simplexes.  While preserving symmetry on the boundary.  Adjacent case. 50

Eliminating Monochromatic Simplexes  Use subdivisions to eliminate monochromatic simplexes.  Non Adjacent case. 51

Eliminating Monochromatic Simplexes  We can use subdivisions to eliminate monochromatic simplexes.  Similar constructions for longer paths.  𝑃(ℓ) subdivisions for ℓ -length path. 52

Odd Paths  Eliminate odd length paths?  Impossible!  We can eliminate only simplexes of even distance. 53

Signs  Give each maximal simplex a sign.  Can eliminate only opposite signs.  Count monochromatic simplexes by their sign.  This is an invariant. 54

2. Create Path  Choose mono. simplexes of opposite signs.  Find a connecting path. 55

2. Eliminate  Choose mono. simplexes of opposite signs.  Find a connecting path  Eliminate. 56

2. Longer Paths  Path between simplexes of opposite signs.  The longer the path, more subdivisions are needed. 57

2. Longer Paths  Path between simplexes of opposite signs.  The longer the path, more subdivisions are needed.  Solution:  Break into short paths.  Many n -length paths, subdivided simultaneously in 𝑃 𝑜 . 58

2. Eliminate Paths  Match all simplexes in pairs.  Eliminate pairs.  Cannot be done simultaneously. 59

2. Number of paths  Number of paths:  Half the number of mono. simplexes: 𝑜−1 1 1 + 𝑜 ∈ 𝑃 𝑜 𝑟+2 𝑙 𝑗 2 𝑗 𝑗=1  q is the largest prime power dividing n . 60

2. Number of Subdivisions  The “expensive” part:  A simplex shared by many paths is subdivided many times.  𝑃 𝑜 𝑟+3 subdivisions. 61

Step Three: The Output Map 62

Cone Subdivision Simplex S Cone subdivision Second cone subdivision L -cone ... 63 subdivision

Cone Subdivision Simplex S Cone subdivision Second cone subdivision ... L -cone 64 subdivision

Constructing Subdivisions 1. Pick simplexes and an integer L . 2. L -cone (in parallel) these simplexes. 3. Extend to all simplexes. 65