On Unfolding Trees and Polygons on Various Lattices

Transcription

1 On Unfolding Trees and Polygons on Various Lattices Author: Sheung-Hung Poon, Presented by: Elena Mumford Technical University of Eindhoven (TU/e), The Netherlands The 19th Canadian Conference on Computational Geometry (CCCG), August 20-22, 2007 Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 1

2 Introduction Introduction Rigid edges can rotate around vertex w/o crossing other edges. Straightening = move rigid edges to lie on a straight line. r r Convexifying = move rigid edges until polygon becomes convex. locking = cannot be straightened or convexified. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 2

3 Introduction Introduction Rigid edges can rotate around vertex w/o crossing other edges. Straightening = move rigid edges to lie on a straight line. r r Convexifying = move rigid edges until polygon becomes convex. locking = cannot be straightened or convexified. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 2

4 Introduction Introduction Rigid edges can rotate around vertex w/o crossing other edges. Straightening = move rigid edges to lie on a straight line. r r Convexifying = move rigid edges until polygon becomes convex. locking = cannot be straightened or convexified. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 2

5 Introduction Introduction Rigid edges can rotate around vertex w/o crossing other edges. Straightening = move rigid edges to lie on a straight line. r r Convexifying = move rigid edges until polygon becomes convex. locking = cannot be straightened or convexified. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 2

6 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

7 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

8 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

9 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

10 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

11 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

12 Introduction Applications: movements of robot arms, molecular conformation, wire bending, rigidity & knot theory, (Borrowed from Univ. College London) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 3

13 Definition of Lattice Polygons/Trees Definition of Lattice Polygons/Trees In this talk, we mainly consider lattice polygons/trees. A unit polygon/tree = with all its edges of unit-length. A lattice polygon/tree = with all its edges from a lattice. Square lattice Hexagonal lattice Triangular lattice tree tree tree Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 4

14 Definition of Lattice Polygons/Trees Definition of Lattice Polygons/Trees In this talk, we mainly consider lattice polygons/trees. A unit polygon/tree = with all its edges of unit-length. A lattice polygon/tree = with all its edges from a lattice. Square lattice Hexagonal lattice Triangular lattice tree tree tree Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 4

15 Definition of Lattice Polygons/Trees Previous Work Carpenter s Rule Conjecture (solved): Chains (polygons) in 2D can be straightened (convexified). [Connelly, Demaine and Rote 00][Strienu 00] Trees (polygons) in 4D+ can be straightened (convexified). [Cocan and O Rourke 01] A tree in 2D & a 5-chain in 3D can lock. [Biedl et al 01] (a) A locked tree in 2D. (b) A locked 5-chain in 3D Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 5

16 Definition of Lattice Polygons/Trees Previous Work Carpenter s Rule Conjecture (solved): Chains (polygons) in 2D can be straightened (convexified). [Connelly, Demaine and Rote 00][Strienu 00] Trees (polygons) in 4D+ can be straightened (convexified). [Cocan and O Rourke 01] A tree in 2D & a 5-chain in 3D can lock. [Biedl et al 01] (a) A locked tree in 2D. (b) A locked 5-chain in 3D Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 5

17 Definition of Lattice Polygons/Trees Previous Work Carpenter s Rule Conjecture (solved): Chains (polygons) in 2D can be straightened (convexified). [Connelly, Demaine and Rote 00][Strienu 00] Trees (polygons) in 4D+ can be straightened (convexified). [Cocan and O Rourke 01] A tree in 2D & a 5-chain in 3D can lock. [Biedl et al 01] (a) A locked tree in 2D. (b) A locked 5-chain in 3D Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 5

18 Definition of Lattice Polygons/Trees PSPACE-complete: to reconfigure 2D-trees or 3D-chains. [Alt et al. 03] A unit tree of diameter 4 can always be straightened. [Poon 05] A 2D/3D lattice tree can always be straightened, and A 2D lattice polygon can always be convexified. [Poon 06] Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 6

19 Definition of Lattice Polygons/Trees PSPACE-complete: to reconfigure 2D-trees or 3D-chains. [Alt et al. 03] A unit tree of diameter 4 can always be straightened. [Poon 05] A 2D/3D lattice tree can always be straightened, and A 2D lattice polygon can always be convexified. [Poon 06] Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 6

20 Definition of Lattice Polygons/Trees PSPACE-complete: to reconfigure 2D-trees or 3D-chains. [Alt et al. 03] A unit tree of diameter 4 can always be straightened. [Poon 05] A 2D/3D lattice tree can always be straightened, and A 2D lattice polygon can always be convexified. [Poon 06] Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 6

21 Our Results Our Results A move = a monotonic increase/decrease of angle at a vertex. Theorem A hexagonal/triangular lattice chain can be straightened in O(n) moves and time (n = no. of edges). Theorem A hexagonal/triangular lattice tree can be straightened in O(n 2 ) moves and time. Theorem A hexagonal/triangular lattice polygon can be convexified in O(n 2 ) moves and time. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 7

22 Our Results Our Results A move = a monotonic increase/decrease of angle at a vertex. Theorem A hexagonal/triangular lattice chain can be straightened in O(n) moves and time (n = no. of edges). Theorem A hexagonal/triangular lattice tree can be straightened in O(n 2 ) moves and time. Theorem A hexagonal/triangular lattice polygon can be convexified in O(n 2 ) moves and time. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 7

23 Our Results Our Results A move = a monotonic increase/decrease of angle at a vertex. Theorem A hexagonal/triangular lattice chain can be straightened in O(n) moves and time (n = no. of edges). Theorem A hexagonal/triangular lattice tree can be straightened in O(n 2 ) moves and time. Theorem A hexagonal/triangular lattice polygon can be convexified in O(n 2 ) moves and time. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 7

24 Hexagonal Lattice Hexagonal Lattice Chains Hexagonal Lattice Chains Algorithm: Fold up the end edges of the chain iteratively. Unfold the final folded spring/zig-zag path. PS. The algorithm is similar to that for square lattice. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 8

25 Hexagonal Lattice Hexagonal Lattice Trees Hexagonal Lattice Trees Given a hexagonal lattice tree P. Let r be root = the leftmost vertex of P. Algorithm: Our algorithm proceeds by pulling P to the left successively until the whole tree is straightened. In each pulling step: Each vertex v is pulled along its edge connecting to its parent; The motion of v stops when v is coincident with its parent in the previous step. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 9

26 Hexagonal Lattice Hexagonal Lattice Trees In each pulling step: Each vertex v is pulled along its edge connecting to its parent; The motion of v stops when v is coincident with its parent in the previous step. r r r r Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 10

27 Hexagonal Lattice Hexagonal Lattice Trees The pulling step is repeated n times so that, finally, tree P is straightened. r r r P 2 P P 1 r P 5 r P 4 r P 3 Each pulling step takes O(n) moves. Thus n pulling steps take O(n 2 ) moves in total. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 11

28 Hexagonal Lattice Hexagonal Lattice Polygons Hexagonal Lattice Polygons Use similar technique of block-collapsing as square lattice. New definition: A block = a hexagonal cell. Algorithm: Collapse leftmost collapsible block iteratively. Case 1a Case 1b Case 2 Case 3 Observe: operation in Case 3, no edges are reduced. After at most O(n) operations of Case 3, we reach one Case 1 or Case 2. Thus whole algorithm takes O(n 2 ) moves and time. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 12

29 Triangular Lattice Triangular Lattice Chains Triangular Lattice Chains Use similar technique as for square & hexagonal lattice. Algorithm: Fold up the end edges of the chain iteratively. Unfold the final folded spring/zig-zag path. Cases to handle: 1 When angle α % end edge & its adjacent edge is π/3: e e e e e or e e (a) (b) 2 When angle α = 2π/3 or π, we can handle similarly. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 13

30 Triangular Lattice Triangular Lattice Trees Triangular Lattice Trees Given a triangular lattice tree P. Let r be root = the leftmost vertex of P. Algorithm: Our algorithm proceeds by pulling P to the left successively until the whole tree is straightened. In each pulling step: Each vertex v is pulled along its edge (or its extension) connecting to its parent; The motion of v stops when v is coincident with its parent in the previous step. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 14

31 Triangular Lattice Triangular Lattice Trees In each pulling step: Each vertex v is pulled along its edge connecting to its parent; The motion of v stops when v is coincident with its parent in the previous step. r r r r During motion, a vertex never exceed middle point of an extension edge(dashed blue). So no edge crossings occur. Each pulling step takes O(n) moves, and thus the whole algorithm takes O(n 2 ) moves and time. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 15

32 Triangular Lattice Triangular Lattice Polygons Triangular Lattice Polygons We extend block-collapsing technique for square/hexagonal lattice. New definition: A block = a (a) parallelogram, (b) trapezoidal or (c) triangular block. (a) (b) (c) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 16

33 Triangular Lattice Triangular Lattice Polygons Algorithm: Collapse leftmost collapsible block iteratively. Cases to handle: (a) Collapse a parallelogram block; (b) Collapse two trapezoidal/triangular blocks; and (c) Collapse an extended triangular/trapezoidal block. (a) (b) (c) Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 17

34 Conclusions Conclusions We obtain results for chains/trees/polygons on hexagonal or triangular lattice can be straightened/covexified. Conjecture: A unit tree in two dimensions can always be straightened. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 18

35 Conclusions Conclusions We obtain results for chains/trees/polygons on hexagonal or triangular lattice can be straightened/covexified. Conjecture: A unit tree in two dimensions can always be straightened. Author: Sheung-Hung Poon, Presented by: Elena Mumford On Unfolding Trees and Polygons on Various Lattices 18