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

2. This talk is about... 0 00 1 1 122 2 1 0 2 Symmetric and chromatic subdivision 0 2 1 0 21 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?

3. This talk is about... Impossible for dimension 1 01 0 0 1 1 w.l.o.g. Since the map must be symmetric The map does not exist for any subdivision ?

4. 0 00 1 1 122 2 1 0 2 0 2 1 0 21 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

5. 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

6. MODEL OF COMPUTATION

7. n+1 asynchronous processors with id’s 0, 1,... n... 01n

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

9. 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

10. 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

11. WEAK SYMMETRY BREAKING (WSB)

12. WSBWSB output values:, input values: id’s

13. WSBWSB output values:, input values: id’s

14. WSBWSB output values:, input values: id’s

15. 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

16. 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 ???

17. 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!!

18. Results n n+1 For some exceptional values of n there is an algorithm for WSB for n+1 processors n+1 1... 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, 14... New upper and lower bounds for renaming

19. TOPOLOGICAL REPRESENTATION ALGORITHM FOR WSB

20. 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

21. 0 00 0 1 11 1 22 2 2 The complex is a chromatic and binary colored subdivision of a proper colored simplex. 0 12 Initial state of the system All possible executions

22. 0 00 0 1 11 1 22 2 2 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

23. solo executions All processors participate Two processors participate 0 00 0 1 11 1 22 2 2 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

24. 0 00 0 1 11 1 22 2 2 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

25. 0 00 0 1 11 1 22 2 2 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

26. [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

27. 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

28. DERIVING ALGORITHMS FOR WSB

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

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

31. 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

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

33. The Chromatic Cone 1. Assume a symmetric boundary 2. Put a red monochromatic triangle at the center 3. Connect them 2 0 1 2 22 2 2 0 0 0 1 11 1 1 0 0 4. Each simplexes on bd with carrier of same dim, is connected to the face of the center that completes its id’s

34. Every corner produces a triangle Every edge produces a triangle If red monochromatic then red monochromatic Only has red monochromatic n-simplexes 2 0 1 2 22 2 2 0 0 0 1 11 1 1 0 0 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)

35. 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

36. 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

37. STEP 2: CANCELING SIMPLEXES +1 WITH –1

38. #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

39. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

40. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

41. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

42. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

43. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

44. +1 K Cancel out a simplex of K counted as +1 with a simplex counted as –1 by subdividing a path which connects them

45. 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

46. The easiest case is when simplexes are adjacent 0 0 1 2

47. 0 0 1 2 2 1

48. We did not modify the boundary 0 0 1 2 2 1

49. An example of a path of size 4 0 0 0 11 2

50. 0 0 0 11 2 2 0

51. 0 0 0 11 2 2 0 2 1

52. 0 0 0 11 2 2 0 2 1 2 1

53. The boundary is the same 0 0 0 11 2 2 0 2 1 2 1

54. A path of size 6

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

60. 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 0 21 0 21 0 0 1 1 2 2

61. 0 0 1 1 2 2 Example A 0 0 1 2

62. 0 0 1 1 2 2 0 0 1 2 2 1 0 0 1 2

63. 0 0 1 1 2 2 0 0 1 2 2 1 0 0 1 2

64. 0 0 1 2 2 1 0 0 1 2 0 0 1 1 2 2

65. Example B 2 0 0 1 12 00 2 22 1

66. 2 0 0 1 12 00 2 22 1 00 2 22 1 2 1

67. 2 0 0 1 12 00 2 22 1 00 2 22 1 2 1

68. 2 0 0 1 12 00 2 22 1 00 2 22 1 2 1

69. 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)

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

71. +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:

72. 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 +1 +1 The Algorithm:

73. 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 +1 +1 The Algorithm:

74. 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 +1 +1 The Algorithm:

75. 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 +1 +1 +1 The Algorithm:

76. 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 +1 +1 The Algorithm:

77. 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:

78. 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

79. 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

80. 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

82. Same size as the input

91. 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

92. 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

93. The end