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The Computational Geometry of Congruence Testing, Part II G¨ unter Rote Freie Universit¨ at Berlin A B ? ∼ = G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

The Computational Geometry of Congruence Testing, Part II G¨ unter Rote Freie Universit¨ at Berlin A B ? ∼ = A G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Overview • 1 dimension   • 2 dimensions  O ( n log n ) time • 3 dimensions   • 4 dimensions today (joint work with Heuna Kim) O ( n ⌈ d/ 3 ⌉ log n ) time [Brass and Knauer 2002] • d dimensions O ( n ⌊ ( d +2) / 2 ⌋ / 2 log n ) Monte Carlo [Akutsu 1998/Matouˇ sek] ↓ O ( n ⌊ ( d +1) / 2 ⌋ / 2 log n ) time G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Overview • 1 dimension   • 2 dimensions  O ( n log n ) time • 3 dimensions   • 4 dimensions today (joint work with Heuna Kim) O ( n ⌈ d/ 3 ⌉ log n ) time [Brass and Knauer 2002] • d dimensions O ( n ⌊ ( d +2) / 2 ⌋ / 2 log n ) Monte Carlo [Akutsu 1998/Matouˇ sek] ↓ O ( n ⌊ ( d +1) / 2 ⌋ / 2 log n ) time • Rotations in 4-space • Pl¨ ucker coordinates for 2-planes in 4-space • The Hopf fibration of S 3 • Closest pair graph • 2+2 dimension reduction • Coxeter classification of reflection groups G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

4 Dimensions: Algorithm Overview joint work with Heuna Kim ≤ 100 |P| markers lower-dimensional components Mirror ≤ n mirror Marking and 2 symmetry Case Condensing |P| ≤ n/ 200 Iterative Great Circles Generating edge- Pruning planes transitive Orbit Cycles ≤ n 2 2+2 Dimension Reduction n = | A | is bounded 1+3 Dimension Reduction G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

4 Dimensions: Algorithm Overview joint work with Heuna Kim ≤ 100 |P| markers lower-dimensional components Mirror ≤ n mirror Marking and 2 symmetry Case Condensing |P| ≤ n/ 200 Iterative Great Circles Generating edge- Pruning planes transitive Orbit Cycles ≤ n 2 2+2 Dimension Reduction n = | A | is bounded 1+3 Dimension Reduction G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Initialization: Closest-Pair Graph 1) PRUNE by distance from the origin. ⇒ we can assume that A lies on the 3-sphere S 3 . • = 2) Compute the closest pair graph G ( A ) = ( A, { uv : � u − v � = δ } ) where δ := the distance of the closest pair, in O ( n log n ) time. [ Bentley and Shamos, STOC 1976 ] • We can assume that δ is SMALL: δ ≤ δ 0 := 0 . 0005 . (Otherwise, | A | ≤ n 0 , by a packing argument.) G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Everything looks the same! By the PRUNING principle, we can assume that all points look locally the same: • All points have congruent neighborhoods in G ( A ) . (The neighbors of u lie on a 2-sphere in S 3 ; 5 2 δ There are at most K 3 = 12 neighbors.) 6 3 1 u 7 8 4 G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Everything looks the same! By the PRUNING principle, we can assume that all points look locally the same: • All points have congruent neighborhoods in G ( A ) . (The neighbors of u lie on a 2-sphere in S 3 ; 5 2 δ There are at most K 3 = 12 neighbors.) 6 3 1 u 7 • Make a directed graph D from G ( A ) 8 4 and PRUNE its arcs uv by the joint neighborhood of u and v . v u G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Everything looks the same! By the PRUNING principle, we can assume that all points look locally the same: • All points have congruent neighborhoods in G ( A ) . (The neighbors of u lie on a 2-sphere in S 3 ; 5 2 δ There are at most K 3 = 12 neighbors.) 6 3 1 u 7 • Make a directed graph D from G ( A ) 8 4 and PRUNE its arcs uv by the joint neighborhood of u and v . v u G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Everything looks the same! By the PRUNING principle, we can assume that all points look locally the same: • All points have congruent neighborhoods in G ( A ) . (The neighbors of u lie on a 2-sphere in S 3 ; 5 2 δ There are at most K 3 = 12 neighbors.) 6 3 1 u 7 • Make a directed graph D from G ( A ) 8 4 and PRUNE its arcs uv by the joint neighborhood of u and v . v u G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Everything looks the same! By the PRUNING principle, we can assume that all points look locally the same: • All points have congruent neighborhoods in G ( A ) . (The neighbors of u lie on a 2-sphere in S 3 ; 5 2 δ There are at most K 3 = 12 neighbors.) 6 3 1 u 7 • Make a directed graph D from G ( A ) 8 4 and PRUNE its arcs uv by the joint neighborhood of u and v . v u • . . . until all arcs uv look the same. G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Algorithm Overview ≤ 100 |P| markers lower-dimensional components Mirror ≤ n mirror Marking and 2 symmetry Case Condensing |P| ≤ n/ 200 Iterative Great Circles Generating edge- Pruning planes transitive Orbit Cycles ≤ n 2 2+2 Dimension Reduction n = | A | is bounded 1+3 Dimension Reduction G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Algorithm Overview ≤ 100 |P| markers lower-dimensional components Mirror ≤ n mirror Marking and 2 symmetry Case Condensing |P| ≤ n/ 200 Iterative Great Circles Generating edge- Pruning planes transitive Orbit Cycles ≤ n 2 2+2 Dimension Reduction n = | A | is bounded 1+3 Dimension Reduction G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Algorithm Overview ≤ 100 |P| markers lower-dimensional components Mirror ≤ n mirror Marking and 2 symmetry Case Condensing |P| ≤ n/ 200 Iterative Great Circles Generating edge- Pruning planes transitive Orbit Cycles ≤ n 2 2+2 Dimension Reduction n = | A | is bounded 1+3 Dimension Reduction G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Predecessor-Successor Figure Pick some α . s ( uv ) := { vw : vw ∈ E, ∠ uvw = α } w s ( uv ) v u α p ( uv ) t G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Predecessor-Successor Figure Pick some α . s ( uv ) := { vw : vw ∈ E, ∠ uvw = α } w ′ w s ( uv ) τ t ′ v w ′ torsion angle τ t ′ u α p ( uv ) t G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Predecessor-Successor Figure Pick some α . s ( uv ) := { vw : vw ∈ E, ∠ uvw = α } w ′ w s ( uv ) τ t ′ v w ′ torsion angle τ t ′ u α p ( uv ) t G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Predecessor-Successor Figure Pick some α . s ( uv ) := { vw : vw ∈ E, ∠ uvw = α } canonical directions w ′ w s ( uv ) τ t ′ v w ′ torsion angle τ can PRUNE arcs t ′ from s ( u, v ) u α p ( uv ) t G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017

Predecessor-Successor Figure Pick some α . s ( uv ) := { vw : vw ∈ E, ∠ uvw = α } canonical directions w ′ w s ( uv ) τ t ′ v w ′ torsion angle τ can PRUNE arcs t ′ from s ( u, v ) u α For every path tuv p ( uv ) τ 0 with ∠ tuv = α , ∃ vw with ∠ uvw = α t and torsion angle τ 0 . G¨ unter Rote, Freie Universit¨ at Berlin The Computational Geometry of Congruence Testing Journ´ ees de G´ eometrie Algorithmique, Aussois, December 11–15, 2017