CS6100: Topics in Design and Analysis of Algorithms

Transcription

1 CS6100: Topics in Design and Analysis of Algorithms Guarding and Triangulating Polygons John Augustine CS6100 (Even 2012): Guarding and Triangulating Polygons

2 The Art Gallery Problem A simple polygon is a region enclosed by a single closed polygonal chain that does not intersect itself. Models 2D spaces such as floor plans. Given a simple polygon P with n vertices, place a small number of guards inside P so that it is completely monitored. I.e., for any point p inside P, there must be a guard positioned at g such that line segment pq is completely inside P. Computing the optimum (i.e., minimum) is NP-hard. CS6100 (Even 2012): Guarding and Triangulating Polygons 1

3 Triangulation A convex polygon only needs one guard! So, if we can decompose P into convex pieces, we need one per convex piece. A diagonal of P is a line segment that connects two vertices of P and is contained in the interior of P. A triangulation is a decomposition of P into triangles be a maximal number of non-intersecting diagonals. Theorem 1. Any simple polygon P with n vertices admits a triangulation and any triangulation has (n 2) triangles. CS6100 (Even 2012): Guarding and Triangulating Polygons 2

4 w w v v v u u Proof Sketch. First we show that a diagonal always exists. (See figures above.) Then, we can always cut along such a diagonal, thereby decomposing P into two smaller polygons that we can induct on to show both claims. Assume for now that a triangulation of P is given. Do we really need n 2 guards? Can we do better? CS6100 (Even 2012): Guarding and Triangulating Polygons 3

5 Back to the Art Gallery Problem A 3-colouring of a triangulated polygon P is an assignment of one of three distinct colours to each vertex such that any two vertices connected by an edge or a diagonal are assigned different colours. Note. The three vertices of any triangle will be assigned different colours. How can we show that n/3 guards suffice if 3-colouring? The dual graph of a triangulated polygon P is a graph G = (V, E) such that V is the set of (n 2) triangles and edges connect two triangles that share an edge. We use DFS on the dual graph to 3-colour P. µ ν? CS6100 (Even 2012): Guarding and Triangulating Polygons 4

6 Invariant. Vertices of all triangles corresponding to dual graph nodes traversed are 3-colored. Start DFS from any node. corresponding triangle. 3-Colour the vertices of Consider any subsequent edge traversed. It goes through a diagonal d and enters a new triangle. The ends of d are already coloured and the third vertex of the newly entered triangle can only be coloured by one colour. The good news is that the invariant is maintained. Theorem 2 (Art Gallery Theorem). To guard a simple polygon with n vertices, n/3 guards are sufficient and sometimes necessary. n/3 prongs CS6100 (Even 2012): Guarding and Triangulating Polygons 5

7 Overview of Approach Theorem 3. We can compute the positions of the n/3 guards for a simple polygon P with n vertices in O(n log n) time. The birds eye view of the algorithm is as follows. 1. Decompose P into y-monotone pieces. (Definition pending.) 2. Triangulate each y-monotone piece. This gives a triangulation of P colour the vertices of P using the triangulation of P. 4. Find the colour used least and place guards on vertices of that colour. CS6100 (Even 2012): Guarding and Triangulating Polygons 6

8 Decomposing P into y-monotone pieces A y-monotone polygon is a simple polygon such that its intersection with any line perpendicular to the y-axis is connected. y-axis To triangulate P, we first partition P into y-monotone sub-polygons (in O(n log n) time and then triangulate the y-monotone pieces (again, in O(n log n) time. CS6100 (Even 2012): Guarding and Triangulating Polygons 7

9 Classification of Vertices A vertex at which, as we walk along the edges, the direction of the edges change from downward to upward or vice versa is called a turn vertex. A vertex at which the direction does not change is called a regular vertex. The following figure illustrates 4 types of turn vertices. v 10 v 5 v 6 e5 e 4 e 6 v 9 v 7 e 8 v 8 e 7 e 9 e 10 e 11 v 12 v 11 v 3 v 4 e 3 e 2 v 2 e 1 v 1 e 15 e 14 v 14 v 15 e 13 e 12 v 13 = start vertex = end vertex = regular vertex = split vertex = merge vertex Lemma 1. A simple polygon is y-monotone if it does not contain any split or merge vertex. The proof is left as an exercise. While this lemma is easy to see, a formal proof needs a bit of care. CS6100 (Even 2012): Guarding and Triangulating Polygons 8

10 Plane Sweep Method Let v 1 be the rightmost vertex in P. Traverse the edges of P starting from v 1 so that the polygon is always to your left. In so doing, denote the ith vertex you encounter as v i for all 1 < i n. Denote edge v i v i+1 as e i for 1 i < n. Denote edge v n v 1 as e n. Let Left be the set of edges of P such that the polygon lies to its right. We sweep a line l from top to bottom. The status of the sweep line l at any given time is the edges in Left that l intersects. They are stored in L R order in a balanced binary tree. The events are the vertices of P and since they are known up front, they can be stored in an array sorted according to the descending order of y coordinates. CS6100 (Even 2012): Guarding and Triangulating Polygons 9

11 Let e be an edge in the status of l. We define the helper of e, denoted helper(e), as the lowest vertex above l such that the horizontal line segment connecting the vertex to e lies inside P. Note: helper(e) may be the upper endpoint of e. (The figure below shows the helper of an edge e j.) e j helper(e j ) e k v i e i 1 e i l Given Lemma 1, we can focus our efforts on eliminating merge and split vertices. CS6100 (Even 2012): Guarding and Triangulating Polygons 10

12 Pseudocode from BCKO 1 Algorithm MAKEMONOTONE(P) Input. A simple polygon P stored in a doubly-connected edge list D. Output. A partitioning of P into monotone subpolygons, stored in D. 1. Construct a priority queue Q on the vertices of P, using their y-coordinates as priority. If two points have the same y-coordinate, the one with smaller x-coordinate has higher priority. 2. Initialize an empty binary search tree T. 3. while Q is not empty 4. do Remove the vertex v i with the highest priority from Q. 5. Call the appropriate procedure to handle the vertex, depending on its type. HANDLESTARTVERTEX(v i ) 1. Insert e i in T and set helper(e i ) to v i. HANDLEENDVERTEX(v i ) 1. if helper(e i 1 ) is a merge vertex 2. then Insert the diagonal connecting v i to helper(e i 1 ) in D. 3. Delete e i 1 from T. HANDLESPLITVERTEX(v i ) 1. Search in T to find the edge e j directly left of v i. 2. Insert the diagonal connecting v i to helper(e j ) in D. 3. helper(e j ) v i 4. Insert e i in T and set helper(e i ) to v i. 1 Computational Geometry: Algorithms and Applications, by de Berg, Cheong, van Krevald, and Overmars. CS6100 (Even 2012): Guarding and Triangulating Polygons 11

13 HANDLEMERGEVERTEX(v i ) 1. if helper(e i 1 ) is a merge vertex 2. then Insert the diagonal connecting v i to helper(e i 1 ) in D. 3. Delete e i 1 from T. 4. Search in T to find the edge e j directly left of v i. 5. if helper(e j ) is a merge vertex 6. then Insert the diagonal connecting v i to helper(e j ) in D. 7. helper(e j ) v i HANDLEREGULARVERTEX(v i ) 1. if the interior of P lies to the right of v i 2. then if helper(e i 1 ) is a merge vertex 3. then Insert the diagonal connecting v i to helper(e i 1 ) in D. 4. Delete e i 1 from T. 5. Insert e i in T and set helper(e i ) to v i. 6. else Search in T to find the edge e j directly left of v i. 7. if helper(e j ) is a merge vertex 8. then Insert the diagonal connecting v i to helper(e j ) in D. 9. helper(e j ) v i Lemma 2. Algorithm MakeMonotone correctly adds a set of non-intersecting diagonals that partitions P into monotone subpolygons. Theorem 4. A simple polygon with n vertices can be partioned into y-monotone polygons in O(n log n) time with an algorithm that uses O(n) storage. Proofs are left as exercises. CS6100 (Even 2012): Guarding and Triangulating Polygons 12

14 e j v i e k v m diagonal will be added when the sweep line reaches v m v 5 e5 e 4 v 6 e 6 v 9 v 7 e v 7 8 e 8 e 9 v 10 e 10 e11 e 13 v 3 v 4 e 3 e 2 v 2 e 1 e 15 e 14 v 14 v 15 v 1 v 11 v 12 e 12 v 13 CS6100 (Even 2012): Guarding and Triangulating Polygons 13

15 Triangulating a Monotone Polygon Given a y-monotone polygon P, we want to triangulate it. We assume that P is given as a sequence of vertices ordered anticlockwise. From the given sequence, we can order the vertices in decreasing order of their y-coordinates. How? Once ordered, we pass through the vertices top bottom. As the algorithm progresses, a (connected) portion of P will be untriangulated, while the rest is triangulated. Even though we might have passed some of the vertices, they may still be on the boundary of the untriangulated part of P. CS6100 (Even 2012): Guarding and Triangulating Polygons 14

16 triangles split off not yet triangulated As can be seen above, the untriangulated, but passed vertices form a geometric shape that has an inverted funnel-like structure with the following properties. Straight side. This is a part of an edge. Concave side. The other side forms a sequence of reflex vertices (i.e., vertices whose internal angle is > 180 ). This side might have triangulated pieces sticking to it. CS6100 (Even 2012): Guarding and Triangulating Polygons 15

17 We will need a stack S, which is initially empty. At any time, it stores the vertices of the untriangulated part. The vertices are stored in S such that if popped one after another, we will get the lowest hightest sequence. If next vertex v j is on the straight side: We pop vertices one-by-one and add diagonals from each popped vertex to v. The last vertex popped will be the other side of the straight edge, so we don t add a diagonal between the last vertex and v. popped pushed e v j popped and pushed CS6100 (Even 2012): Guarding and Triangulating Polygons 16

18 If next vertex v j is on the concave side: We again pop vertices from the stack and add diagonals as long as (i) we can add diagonals that don t intersect 2 edges of P and (ii) S is not empty. If the diagonal to top vertex v in S intersects edges of P, then do not pop v. Furthermore, push the vertex that was popped immediately prior to v (if it exists) back into the stack. Finally, push v j into the stack as well. popped and pushed popped popped and pushed v j pushed v j pushed 2 How can this be tested in O(1) time? CS6100 (Even 2012): Guarding and Triangulating Polygons 17

19 Algorithm TRIANGULATEMONOTONEPOLYGON(P) Input. A strictly y-monotone polygon P stored in a doubly-connected edge list D. Output. A triangulation of P stored in the doubly-connected edge list D. 1. Merge the vertices on the left chain and the vertices on the right chain of P into one sequence, sorted on decreasing y-coordinate. If two vertices have the same y-coordinate, then the leftmost one comes first. Let u 1,...,u n denote the sorted sequence. 2. Initialize an empty stack S, and push u 1 and u 2 onto it. 3. for j 3 to n 1 4. do if u j and the vertex on top of S are on different chains 5. then Pop all vertices from S. 6. Insert into D a diagonal from u j to each popped vertex, except the last one. 7. Push u j 1 and u j onto S. 8. else Pop one vertex from S. 9. Pop the other vertices from S as long as the diagonals from u j to them are inside P. Insert these diagonals into D. Push the last vertex that has been popped back onto S. 10. Push u j onto S. 11. Add diagonals from u n to all stack vertices except the first and the last one. Theorem 5. A strictly y-monotone polygon with n vertices can be triangulated in O(n) time. Theorem 6. A simple polygon with n vertices can be triangulated in O(n log n) time. Theorem 7. A planar subdivision with n vertices can be triangulated in O(n log n) time. CS6100 (Even 2012): Guarding and Triangulating Polygons 18