Linear time 8 / 3 -approx. of r -star guards in simple orthogonal - PowerPoint PPT Presentation

Linear time 8 / 3 -approx. of r -star guards in simple orthogonal art galleries obert Mezei 1 Ervin Gy˝ ori & Tam´ as R´ ICGT 2018, July 9, Lyon 1 Alfr´ ed R´ enyi Institute of Mathematics, Hungarian Academy of Sciences 1/17

The Art Gallery problem - for Orthogonal Polygons • Art gallery: P ⊂ R 2 , a simple orthogonal polygon • Point guard: fixed point g ∈ P , has 360 ◦ line of sight vision • Objective: place guards in the gallery so that any point in P is seen by at least one of the guards P g 2/17

The art gallery theorem for orthogonal polygons Theorem (Kahn, Klawe and Kleitman, 1980) � n � guards are sometimes necessary and always sufficient to cover 4 the interior of a simple orthogonal polygon of n vertices. 3/17

The art gallery theorem for orthogonal polygons Theorem (Kahn, Klawe and Kleitman, 1980) � n � guards are sometimes necessary and always sufficient to cover 4 the interior of a simple orthogonal polygon of n vertices. Theorem (Schuchardt and Hecker, 1995) Finding a minimum size point guard system is NP -hard in simple orthogonal polygons 3/17

Rectangular or r -vision Rectangular vision: two points x , y ∈ P have r -vision of each other if there is an axis-parallel rectangle inside P , containing x and y . x y 4/17

Rectangular or r -vision Rectangular vision: two points x , y ∈ P have r -vision of each other if there is an axis-parallel rectangle inside P , containing x and y . x z x and z have unrestricted vision of each other 4/17

Rectangular or r -vision Rectangular vision: two points x , y ∈ P have r -vision of each other if there is an axis-parallel rectangle inside P , containing x and y . x z x and z do not have rectangular vision of each other 4/17

Rectangular or r -vision Rectangular vision: two points x , y ∈ P have r -vision of each other if there is an axis-parallel rectangle inside P , containing x and y . r -star: an orthogonal polygon that can be covered by one guard equipped with r -vision 4/17

Rectangular or r -vision Rectangular vision: two points x , y ∈ P have r -vision of each other if there is an axis-parallel rectangle inside P , containing x and y . During the rest of the talk, vision means r -vision 4/17

Complexity results for r -vision Theorem (Worman and Keil, 2007) There is an ˜ O ( n 17 ) time algorithm that computes the minimum size set of point guards equipped with r -vision covering an n -vertex simple orthogonal polygon. 5/17

Complexity results for r -vision Theorem (Worman and Keil, 2007) There is an ˜ O ( n 17 ) time algorithm that computes the minimum size set of point guards equipped with r -vision covering an n -vertex simple orthogonal polygon. Theorem (Lingas, Wasylewicz, ˙ Zyli´ nski, 2012) There is a linear time 3-approximation algorithm for minimum size point guard system with rectangular vision. 5/17

Complexity results for r -vision Theorem (Worman and Keil, 2007) There is an ˜ O ( n 17 ) time algorithm that computes the minimum size set of point guards equipped with r -vision covering an n -vertex simple orthogonal polygon. Theorem (Lingas, Wasylewicz, ˙ Zyli´ nski, 2012) There is a linear time 3-approximation algorithm for minimum size point guard system with rectangular vision. The novelty of our algorithm is not so much the lower approximation ratio, but the extremal style of the result (we will see this) 5/17

Mobile guards in orthogonal polygons A mobile guard is an axis-parallel line segment L inside the gallery. The guard sees a point x ∈ P iff there is a point y ∈ L such that x is visible from y . P L y x Sliding camera (introduced by Katz and Morgenstern, 2011): a mobile guard whose line segment is maximal, equipped with r -vision 6/17

Results on the complexity of sliding camera problems Theorem (Gy˝ ori and M, 2016) There is a linear time algorithm that finds a covering set of mobile � 3 n +4 � guards of cardinality at most , even if the patrols are 16 required to be pairwise disjoint. The complexity of the optimization problem is open. 7/17

Our result p : minimum number of point guards required to cover P m V : min. number of vertical sliding cameras required to cover P m H : min. number of horizontal sliding cameras required to cover P 8/17

Our result p : minimum number of point guards required to cover P m V : min. number of vertical sliding cameras required to cover P m H : min. number of horizontal sliding cameras required to cover P Theorem For any simple orthogonal polygon there is a linear time algorithm which finds a point guard of size at most 4 3 ( m V + m H − 1) . 8/17

Our result p : minimum number of point guards required to cover P m V : min. number of vertical sliding cameras required to cover P m H : min. number of horizontal sliding cameras required to cover P Theorem For any simple orthogonal polygon there is a linear time algorithm which finds a point guard of size at most 4 3 ( m V + m H − 1) . Since m V , m H ≤ p , we have 4 3( m V + m H − 1) ≤ 8 3 p , so the algorithm provides an 8 3 -approximation solution. 8/17

High level description of the algorithm

Horizontal and vertical R -trees ortho. poly. P tree T H h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 0 h 1 h 2 Cut horizontally at each reflex vertex, join touching slices by an edge Gy˝ ori et. al. (1995) drafts that T H can be computed in linear time 9/17

Horizontal and vertical R -trees ortho. poly. P tree T V h 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Do the same for vertical slices 9/17

Horizontal and vertical R -trees ortho. poly. P bipartite G h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Join two slices iff their interiors intersect G may have Ω( n 2 ) edges 9/17

Horizontal and vertical R -trees ortho. poly. P h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V v 4 Each neighborhood in G forms a path in the appropriate R -tree Observation 1: by storing only the ends of the path formed by the neighborhood of each vertex of G , the graph can be described in O ( n ) space 9/17

Working with the sparse representation of G • Choose arbitrary roots in the R -trees • The lowest common ancestors algorithm (for eg. the one due to Gabow and Tarjan, 1985) requires linear time 10/17

Working with the sparse representation of G • Choose arbitrary roots in the R -trees • The lowest common ancestors algorithm (for eg. the one due to Gabow and Tarjan, 1985) requires linear time • Observation 2: for any v 1 , v 2 ∈ S V , the ends of the path formed by N G ( v 1 ) ∩ N G ( v 2 ) in T H can be computed using 6 LCA queries 10/17

Working with the sparse representation of G • Choose arbitrary roots in the R -trees • The lowest common ancestors algorithm (for eg. the one due to Gabow and Tarjan, 1985) requires linear time • Observation 2: for any v 1 , v 2 ∈ S V , the ends of the path formed by N G ( v 1 ) ∩ N G ( v 2 ) in T H can be computed using 6 LCA queries • Observation 3: if G is 2-connected then for any v 1 v 2 ∈ E ( T V ) we have | N G ( v 1 ) ∩ N G ( v 2 ) | ≥ 2 10/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V The intersection graph structure in connection with mobile guards lafiejski, and ˙ has been studied by Kosowski, Ma� Zyli´ nski (2007) With respect to rectangular vision, it is enough to know the pixels containing the points. 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Point guard ↔ Edge 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V v 3 Sliding camera ↔ Vertex 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 4 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Sliding camera ↔ Vertex 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Rectangular vision ( e 1 ∩ e 2 � = ∅ ) 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 3 h 4 h 5 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V v 3 v 7 Rectangular vision ( G [ e 1 ∪ e 2 ] ∼ = C 4 ) 11/17

Pixelation graph P G h 0 h 1 h 1 h 2 h 3 h 3 h 4 h 5 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V v 3 v 5 v 7 G is chordal bipartite: any cycle of length at least 6 has a chord (eg.: h 5 v 5 ) 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V v 1 v 4 v 7 Covering set of vert. sliding cameras ↔ M V ⊆ S V dominating S H 11/17

Pixelation graph P G h 0 h 1 h 2 h 3 h 3 h 4 h 4 h 5 h 6 S H h 6 h 5 h 4 h 3 h 0 h 1 h 2 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 S V Covering set of horiz. sliding cameras ↔ M H ⊆ S H dominating S V 11/17