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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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018

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 Workshop on Geometric Computation and Applications, Trinity College, Dublin, June 17–21, 2018