The Searchlight Problem for Road Networks Dariusz Dereniowski a - PowerPoint PPT Presentation

The Searchlight Problem for Road Networks Dariusz Dereniowski a Hirotaka Ono a Ichiro Suzuki a Łukasz Wrona a Masafumi Yamashita a Paweł Żyliński a GRASTA-MAC 2015, Montreal

Problem definition The searchlight problem in a road network What is the worst-case number s ( n, g ) of searchlights, each placed at one of the g guard positions, required to successfully search a given road network of n lines/line segments? → A mobile intruder capable of moving continuously and arbitrarily fast is hiding. → The objective of the guards is to detect the intruder using the rays. → The intruder is considered detected at the moment he is illuminated by one of the rays → or he reaches a position where a guard is located. a a b b undetected ◮ Sugihara, Suzuki and Yamashita. (1990): The searchlight scheduling problem ◮ Yen and Tang (1995): The searchlight guarding problem on weighted trees 1/13

Problem definition The searchlight problem in a road network What is the worst-case number s ( n, g ) of searchlights, each placed at one of the g guard positions, required to successfully search a given road network of n lines/line segments? → A mobile intruder capable of moving continuously and arbitrarily fast is hiding. → The objective of the guards is to detect the intruder using the rays. → The intruder is considered detected at the moment he is illuminated by one of the rays → or he reaches a position where a guard is located. F ′ (a) (b) (c) F F F ′′ F ′′ F ′ v 0 v 0 v 0 v 1 v 1 v 1 · · · F ′′ F F ′ A sample search strategy for an ( n, 2) -arrangement, n ≥ 4 . 1/13

Lines Arrangements of lines : a lower bound of 2 g − 1 A g A 2 · · · v g − 1 v 0 v 1 v 0 v 1 v 2 (2 g, g ) -arrangement A g = ( L g , { v 0 , . . . , v g − 1 } ) , g ≥ 2 (4 , 2) -arrangement A 2 = ( L 2 , { v 0 , v 1 } ) that requires at least three searchlights s ( n, g ) ≥ 2 g − 1 2/13

Lines Arrangements of lines : a lower bound of 2 g − 1 v g − 1 A ′ g v k · · · · · · v k − 1 v 0 v 1 v 2  2 g − 1 if 1 ≤ g ≤ n 2 ;      n − 2 if n For 1 ≤ g ≤ n − 1 , s ( n, g ) ≥ 2 < g ≤ n − 2;    n − 1 if g = n − 1 .   2/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 s ( n, g ) ≤ 3 g 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 recursion this searchlight is fixed s ( n, g ) ≤ 3 g 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 remains clean this searchlight is fixed recursion s ( n, g ) ≤ 3 g 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 remains clean this searchlight is fixed remains clean recursion s ( n, g ) ≤ 3 g 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 6 v 5 v 0 v 3 v 0 v 1 v 1 v 4 x no intersection points between v 1 and x v 0 is incident to v 1 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 2 v 6 v 5 x v 3 v 0 v 3 v 1 v 4 no intersection points between v 3 and x v 2 is incident to v 3 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 6 v 5 v 0 v 3 v 0 v 3 v 1 v 4 there is a ‘free’ cycle around v 3 v 0 is not incident to v 3 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 6 v 5 v 3 v 0 v 3 v 1 v 4 any arborescence: 3 searchlights at the root v 0 2 searchlights at any other vertex ordering: v 0 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 2 v 6 v 5 v 0 v 0 v 3 v 0 v 1 v 1 v 3 v 1 v 3 v 4 any arborescence: 3 searchlights at the root v 0 2 searchlights at any other vertex ordering: v 0 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 wrong order: correct order: v 3 is handled before handling v 2 v 3 is handled after handling v 2 s ( n, g ) ≤ 7 g 3 − 1 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 v 2 v 2 v 6 v 5 v 0 v 0 v 3 v 0 v 1 v 1 v 3 v 1 v 3 v 4 any arborescence: 3 searchlights at the root v 0 2 searchlights at any other vertex ordering: v 0 , v 1 , v 2 , v 3 , v 4 , v 5 , v 6 wrong order: correct order: v 3 is handled before handling v 2 v 3 is handled after handling v 2 s ( A ) ≤ 2 g + ( h − 1) . 3/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 s ( n, g ) ≤ 7 g 3 − 1 4/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 s ( n, g ) ≤ 7 g 3 − 1 4/13

Lines Arrangements of lines : an upper bound of 7 g 3 − 1 s ( n, g ) ≤ 7 g 3 − 1 4/13

Line segments Arrangements of line segments : a lower bound of Ω( g log n g ) A ✷ 3 ¯ A ✷ 2 ˆ ˇ A ✷ A ✷ 2 2 A ✷ 2 A 3 A 2 s ( n, 2) = Ω(log n ) 5/13

Line segments Arrangements of line segments : a lower bound of Ω( g log n g ) ¯ A ✷ 2 A ✷ 2 ˆ ˇ A ✷ A ✷ 2 2 A ✷ 2 A 2 A 2 s ( n, 2) = Ω(log n ) 5/13

Line segments Arrangements of line segments : a lower bound of Ω( g log n g ) A ✷ 3 ¯ A ✷ 2 ˆ ˇ A ✷ A ✷ 2 2 A ✷ 2 A 3 A 2 s ( n, 2) = Ω(log n ) 5/13

Line segments Arrangements of line segments : a lower bound of Ω( g log n g ) A ✷ k ¯ A ✷ k − 1 ˆ ˇ A ✷ A ✷ k − 1 k − 1 A ✷ k − 1 A k A k − 1 s ( n, 2) = Ω(log n ) 5/13

Line segments Arrangements of line segments : a lower bound of Ω( g log n g ) A k A k A k A k s ( n, g ) = Ω( g log n g ) 5/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights nice arrangement all guards are “outside” 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights not nice arrangement some guards are “inside” 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights not nice arrangement some guards are “inside” 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights not nice arrangement some guards are “inside” 6/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights 7/13

Line segments Arrangements of line segments : an upper bound of O ( g 2 log n ) ◮ partitioning into nice arrangements: O ( g ) searchlights (remain fixed) ◮ recursive searching of nice arrangements (divide-and-conquer) → depth of the recursion with respect to a guard v : O (log n ) → divide-and-conquer: O (1) searchlights per each guard → the total number of O ( g 2 log n ) searchlights splitter S balanced partition 7/13