COMPUTING HOMOLOGY: 0 b lk. CS 468 Lecture 7 11/6/2. Afra Zomorodian CS 468 Lecture 7 - Page 1

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1 COMPUTING HOMOLOGY: b b lk 0 0 CS 468 Lecture 7 11/6/2 Afra Zomorodian CS 468 Lecture 7 - Page 1

2 TIDBITS Lecture 8 is on Tuesday, November 12 me about projects! Projects will be November 27th and December 4th. November 20th? Triangulation of example Afra Zomorodian CS 468 Lecture 7 - Page 2

3 (LAST TIME) EULER-POINCARÉ chain complex C : k+1... C k+1 C k k Ck 1... Homology functors H H (C ) is a chain complex: k+1... H k+1 H k What is its Euler characteristic? (Theorem) χ(k) = χ(c ) = χ(h (C )). k Hk 1... i ( 1)i s i = i ( 1)i rank(h i ) = i ( 1)i β i Afra Zomorodian CS 468 Lecture 7 - Page 3

4 OVERVIEW Dualities Data structures Quad-Edge Edge-Facet Computing Homology Reduction Algorithm Incremental Algorithm Afra Zomorodian CS 468 Lecture 7 - Page 4

5 DUALITY φ a c abc abd acd bcd b ac ab bc ad cd bd a b c d d φ (a) Tetrahedron (b) Poset Afra Zomorodian CS 468 Lecture 7 - Page 5

6 PLATONIC SOLIDS solid vertices edges faces tetrahedron cube octahedron dodecahedron icosahedron Afra Zomorodian CS 468 Lecture 7 - Page 6

7 ORIENTATION The Optiverse [Sullivan 98] Afra Zomorodian CS 468 Lecture 7 - Page 7

8 BACKGROUND-FOREGROUND Afra Zomorodian CS 468 Lecture 7 - Page 8

9 COMPLEMENTARITY Afra Zomorodian CS 468 Lecture 7 - Page 9

10 TIME REVERSAL Afra Zomorodian CS 468 Lecture 7 - Page 10

11 DUALITY Orientation: inside vs. outside Structure: primal vs. dual (poset vs. upside-down poset) Complementarity: background vs. foreground Time reversal: forward vs. backward Afra Zomorodian CS 468 Lecture 7 - Page 11

12 DIRECTED EDGE Left Right An edge e has two vertices A directed edge goes from Org (e) to Dest (e) An edge separates two cells Sym (e) goes in the opposite direction Afra Zomorodian CS 468 Lecture 7 - Page 12

13 QUAD-EDGE Left Right Rot(e) gives you the dual edge (clockwise) and Tor(e) (counter) Edge e stores its number and the next clockwise edge with the same origin: Onext (e) A Quad-Edge is Edge[4]: edge, rot, sym, tor All operations O(1) Afra Zomorodian CS 468 Lecture 7 - Page 13

14 OPERATIONS Left Right Oprev (e) = (Rot Onext Rot)(e) Dnext (e) = (Sym Onext Sym)(e) Lnext (e) = (Tor Onext Rot)(e) Afra Zomorodian CS 468 Lecture 7 - Page 14

15 ORIENTED TRIANGLES a b Enext Enext Enext abc bca cab Sym Sym Sym bac cba Enext Enext acb c Enext Sym does one transposition (changes orientation) Enext does two transpositions (rotate by 60 degrees clockwise) Array of six edge-facets for each triangle All operations O(1) Afra Zomorodian CS 468 Lecture 7 - Page 15

16 FNEXT c a d e b Fnext (bac) = bad Fnext (abd) = abc Fnext (abc) = abe Each store its Fnext Afra Zomorodian CS 468 Lecture 7 - Page 16

17 FACES OF A TETRAHEDRON c c a d a d b b (a) f = bac (b) Fnext (f) = bad c c a d a d b b (c) Sym (Fnext (f)) = abd (d) Sym (Fnext (Enext (f))) = cad Afra Zomorodian CS 468 Lecture 7 - Page 17

18 HOMOLOGY The kth homology group is H k = Z k /B k = ker k /im k+1. Compute a basis for ker k Compute a basis for im k + 1 Ck+1 δ k+1 C k δ k C k 1 Z k+1 Z k Z k 1 B k+1 B k B k Afra Zomorodian CS 468 Lecture 7 - Page 18

19 MATRIX REPRESENTATION Boundary homomorphism is linear, so it has a matrix k : C k C k 1 Use oriented simplices as bases for domain and codomain! M k is the standard matrix representation for k Afra Zomorodian CS 468 Lecture 7 - Page 19

20 EXAMPLE a b d c M 1 = ab bc cd ad ac a b c d Afra Zomorodian CS 468 Lecture 7 - Page 20

21 ELEMENTARY OPERATIONS The elementary row operations on M k are 1. exchange row i and row j, 2. multiply row i by 1, 3. replace row i by (row i) + q(row j), where q is an integer and j i. Similar elementary column operations on columns Effect: change of bases Afra Zomorodian CS 468 Lecture 7 - Page 21

22 DESCRIPTION Homology groups are finitely generated abelian. (Theorem) Every finitely generated abelian group is isomorphic to product of cyclic groups of the form β k = β(h k ) Torsion coefficients Z m1 Z m2... Z mr Z Z... Z, Afra Zomorodian CS 468 Lecture 7 - Page 22

23 INTUITION How do we find cycles? How do we find boundaries? What does a free generator correspond to? How does a torsional element appear? Afra Zomorodian CS 468 Lecture 7 - Page 23

24 REDUCTION ALGORITHM Like Gaussian elimination, we keep changing the basis to get to the (Smith) normal form: M k = b b lk 0 0 l k = rank M k = rank M k, b i 1 b i b i+1 for all 1 i < l k Afra Zomorodian CS 468 Lecture 7 - Page 24

25 REDUCED EXAMPLE a b d c M 1 = cd bc ab z 1 z 2 d c c b b a a z 1 = ad bc cd ab and z 2 = ac bc ab form a basis for Z 1 {d c, c b, b a} is a basis for B 0 Afra Zomorodian CS 468 Lecture 7 - Page 25

26 REDUCED EXAMPLE M 2 = abc acd ac 1 1 ad 0 1 cd 0 1 bc 1 0 ab 1 0 M 2 = abc acd + abc ac bc ab 1 0 ad cd bc ab 0 1 cd 0 0 bc 0 0 ab 0 0 Afra Zomorodian CS 468 Lecture 7 - Page 25-1

27 NORMAL FORM M k = b b lk 0 0 Description: 1. the torsion coefficients of H k 1 are b i 1 2. {e i l k + 1 i m k } is a basis for Z k. Therefore, rank Z k = m k l k. 3. {b i ê i 1 i l k } is a basis for B k 1. Equivalently, rank B k = rank M k+1 = l k+1. β k = rank Z k rank B k = m k l k l k+1 Afra Zomorodian CS 468 Lecture 7 - Page 26

28 IN S 3 Algorithm takes O(m 3 ) operations, but integers can get large Subcomplexes are torsion-free, so we don t need the force! k-chain: c = i n i[σ i ], n i Z, σ i K Different view, n i are coefficients We can multiply, but not divide (in Z) We can also change to other coefficients, such as R, Q, etc. Z 2 Homology restrict to 0,1, so unoriented simplices σ = σ Addition is symmetric sum: c + d = (c d) (c d). Afra Zomorodian CS 468 Lecture 7 - Page 27

29 FILTRATION a b a b a b a b a b a b d c d c d c d c d c 0 a, b 1 c, d, ab, bc 2 cd, ad 3 ac 4 abc 5 acd A filtration of a complex K is = K 0 K 1... K m = K. A filtration is a partial ordering Sort according to dimension Break other ties arbitrarily Algorithm for K = S 3 Afra Zomorodian CS 468 Lecture 7 - Page 28

30 ALEXANDER DUALITY Alexander Duality: β 0 measures the number of components of the complex. β 1 is the rank of a basis for the tunnels. β 2 counts the number of voids in the complex. An incremental approach: add each simplex in turn check to see if we form a new cycle class or destroy one. Afra Zomorodian CS 468 Lecture 7 - Page 29

31 VERTICES Vertices always add a new component, so β Why? Union-find data-structure: MAKESET: initializes a set with an item FIND: finds the set an element belongs to UNION: forms the union of two sets Very simple to implement O(n) space Amortized α(m) FIND, UNION MAKESET for each vertex Afra Zomorodian CS 468 Lecture 7 - Page 30

32 EDGES (a) β 0 (b) β 1 ++ (a) Two FINDs, one UNION (b) Two FINDs Afra Zomorodian CS 468 Lecture 7 - Page 31

33 TRIANGLES AND TETRAHEDRA (a) β 1 (b) β 2 ++ Tetrahedra always fill voids, so β 2 Afra Zomorodian CS 468 Lecture 7 - Page 32

34 COMPUTING VOIDS tri 1 void complement 1 component dual 1 vertex 2 voids tri 2 components edge 2 vertices tet 1 void tet 1 component vertex 1 vertex tet 1 component complement tet vertex 1 void dual We can always look at the complement in S 3 Dualize to get vertices and edges Reverse time to get to Union-Find Afra Zomorodian CS 468 Lecture 7 - Page 33

35 INCREMENTAL ALGORITHM Three passes: One pass to identify negative edges One reverse dual pass on the complement space to get positive triangles One pass to compute them all (the Betti numbers) O(mα(m)) dim Afra Zomorodian CS 468 Lecture 7 - Page 34