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Congruence Testing in 4 Dimensions G¨ unter Rote joint work with Heuna Kim Freie Universit¨ at Berlin A B ? ∼ = G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Congruence Testing in 4 Dimensions G¨ unter Rote joint work with Heuna Kim Freie Universit¨ at Berlin A B ? ∼ = A G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Overview • 1 dimension • 2 dimensions • 3 dimensions • 4 dimensions • d dimensions G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Overview  • 1 dimension   • 2 dimensions O ( n log n ) time • 3 dimensions   • 4 dimensions NEW, joint work with Heuna Kim O ( n ⌈ d/ 3 ⌉ log n ) time [Brass and Knauer 2002] • d dimensions O ( n ⌊ d/ 2 ⌋ / 2 log n ) time Monte Carlo[Akutsu 1998/Matouˇ sek] G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Overview  • 1 dimension   • 2 dimensions O ( n log n ) time • 3 dimensions   • 4 dimensions NEW, joint work with Heuna Kim O ( n ⌈ d/ 3 ⌉ log n ) time [Brass and Knauer 2002] • d dimensions O ( n ⌊ d/ 2 ⌋ / 2 log n ) time Monte Carlo[Akutsu 1998/Matouˇ sek] • Problem statement and variations • Dimension reduction as in [Alt, Mehlhorn, Wagener, Welzl] • Atkinson’s reduction (pruning/condensation) • (Planar) graph isomorphism • Hopf fibrations • Pl¨ ucker coordinates • Coxeter groups G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Rotation or Rotation+Reflection? We only need to consider proper congruence (orientation-preserving congruence, of determinant +1 ). If mirror-congruence is also desired, repeat the test twice, for B and its mirror image B ′ . A B ? ∼ = B ′ ? ∼ = G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Congruence = Rotation + Translation Translation is easy to determine: The centroid of A must coincide with the centroid of B . A B ? ∼ = → from now on: All point sets are centered at the origin 0 : � � a = b = 0 a ∈ A b ∈ B We need to find a rotation around the origin (orthogonal matrix T with determinant +1 ) which maps A to B : TA = B G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Exact Arithmetic The proper setting for this (mathematical) problem requires real numbers as inputs and exact arithmetic. → the Real RAM model (RAM = random access machine): One elementary operation with real numbers ( + , ÷ , √ , sin ) is counted as one step. A regular 5-gon, 7-gon, 8-gon, . . . with rational coordinates does not exist in any dimension. G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Applications Congruence testing is the basic problem for many pattern matching tasks • computer vision • star matching • brain matching • . . . The proper setting for this applied problem requires tolerances, partial matchings, and other extensions. G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Arbitrary Dimension A, B ⊂ R d , | A | = | B | = n . We consider the problem for fixed dimension d . When d is unrestricted, the problem is equivalent to graph isomorphism: G = ( V, E ) , V = { 1 , 2 , . . . , n } ∪{ e i + e j | ij ∈ E } ⊂ R n �→ A = { e 1 , . . . , e n } 2 � �� � e i = (0 , . . . , 0 , 1 , 0 , . . . , 0) regular simplex CONJECTURE: Congruence can be tested in O ( n log n ) time for every fixed dimension d . Current best bound: O ( n ⌈ d/ 3 ⌉ log n ) time G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Arbitrary Dimension A, B ⊂ R d , | A | = | B | = n . We consider the problem for fixed dimension d . When d is unrestricted, the problem is equivalent to graph isomorphism: G = ( V, E ) , V = { 1 , 2 , . . . , n } ∪{ e i + e j | ij ∈ E } ⊂ R n �→ A = { e 1 , . . . , e n } 2 � �� � e i = (0 , . . . , 0 , 1 , 0 , . . . , 0) regular simplex CONJECTURE: Congruence can be tested in O ( n log n ) time for every fixed dimension d . Current best bound: O ( n ⌈ d/ 3 ⌉ log n ) time G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Two dimensions Can be done by string matching. [ Manacher 1976 ] Sort points around the origin. Encode alternating sequence of distances r i and angles ϕ i . ( r 1 , ϕ 1 , r 2 , ϕ 2 , . . . , r n , ϕ n ) r 3 r 2 ϕ 1 Check whether the r 1 corresponding sequence of B is a cyclic shift. → O ( n log n ) + O ( n ) time. G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Two dimensions Can be done by string matching. [ Manacher 1976 ] Sort points around the origin. Encode alternating sequence of distances r i and angles ϕ i . Even more CANONICAL DIRECTIONS can be done: A c ( A ) The canonical set c ( A ) : [Akutsu 1992] A ∼ = B ⇐ ⇒ c ( A ) = c ( B ) → searching in a database G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Three dimensions [ Sugihara 1984; Alt, Mehlhorn, Wagener, Welzl 1988 ] Project points to the unit sphere, 1 and keep distances as labels . 1 1 Compute the convex hulls P ( A ) and P ( B ) , in O ( n log n ) time. Check isomorphism between the corresponding LABELED planar graphs. Vertex labels: from the radial projection Edge labels: dihedral angles and face angles. In O ( n ) time, [ Hopcroft and Wong 1974 ] or in O ( n log n ) time. [ Hopcroft and Tarjan 1973 ] G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A Apply some criterion that distinguishes points (distance from the center, number of closest neighbors, . . . ) G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A Apply some criterion that distinguishes points (distance from the center, number of closest neighbors, . . . ) A ′ | A ′ | ≤ | A | 2 Throw away all but the smallest resulting class, and repeat. Simultaneously apply this procedure to B . A ′ and B ′ may have more congruences! G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A Make some construction (midpoints of closest-pair edges, . . . ) Simultaneously apply this procedure to B . A ′ and B ′ may have more congruences! G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A Make some construction (midpoints of closest-pair edges, . . . ) Simultaneously apply this procedure to B . A ′ and B ′ may have more congruences! G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017

Pruning/Condensing A Make some construction (midpoints of closest-pair edges, . . . ) A ′ | A ′ | ≤ | A | 2 Simultaneously apply this procedure to B . A ′ and B ′ may have more congruences! G¨ unter Rote, Freie Universit¨ at Berlin Congruence Testing in 4 Dimensions CALDAM 2017 Pre-Conference School on Algorithms and Combinatorics, BITS, Goa, February 13–14, 2017