Searching with turn cost and related problems Spyros Angelopoulos - PowerPoint PPT Presentation

A generalization : Star search or ray search m infinite rays, one target Competitive ratio = sup (search cost) / λ b 2 Optimal strategy : geometric search [Gal 72] b 4 b 3 b Multiple searchers [Lopez-Ortiz and Schuierer 2002] Many variants: Turn cost [Demaine et al. 2004] Randomized strategies [Kao et al. 1996] Searching on the plane [Gal 1980 and Langetepe 2010] Monday, April 7, 14

A generalization : Star search or ray search m infinite rays, one target Competitive ratio = sup (search cost) / λ b 2 Optimal strategy : geometric search [Gal 72] b 4 b 3 b Multiple searchers [Lopez-Ortiz and Schuierer 2002] Many variants: Turn cost [Demaine et al. 2004] Randomized strategies [Kao et al. 1996] Searching on the plane [Gal 1980 and Langetepe 2010] Textbook by Alpern and Gal: The Theory of Search Games and Rendezvous Monday, April 7, 14

More recent related work Monday, April 7, 14

More recent related work [Kirkpatrick 2009] and [Mc Gregor et al. 2009] : New measures for analysis: OPT does not have the complete picture of the instance Monday, April 7, 14

More recent related work [Kirkpatrick 2009] and [Mc Gregor et al. 2009] : New measures for analysis: OPT does not have the complete picture of the instance [A. et al. 2011] Multi-target search [Tseng and Kirkpatrick 2011] Input-thrifty algorithms Monday, April 7, 14

More recent related work [Kirkpatrick 2009] and [Mc Gregor et al. 2009] : New measures for analysis: OPT does not have the complete picture of the instance [A. et al. 2011] Multi-target search [Tseng and Kirkpatrick 2011] Input-thrifty algorithms [Bose et al., 2013] Linear search with distance bounds Monday, April 7, 14

More recent related work [Kirkpatrick 2009] and [Mc Gregor et al. 2009] : New measures for analysis: OPT does not have the complete picture of the instance [A. et al. 2011] Multi-target search [Tseng and Kirkpatrick 2011] Input-thrifty algorithms [Bose et al., 2013] Linear search with distance bounds [Bernrstein et al. 2003] Ray search Interruptible algorithms [A. et al., 2006 & 2009] Monday, April 7, 14

More recent related work [Kirkpatrick 2009] and [Mc Gregor et al. 2009] : New measures for analysis: OPT does not have the complete picture of the instance [A. et al. 2011] Multi-target search [Tseng and Kirkpatrick 2011] Input-thrifty algorithms [Bose et al., 2013] Linear search with distance bounds [Bernrstein et al. 2003] Ray search Interruptible algorithms [A. et al., 2006 & 2009] In preparation : More connections between searching and interruptible algorithms Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost λ Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost Monday, April 7, 14

Online searching with turn cost +d Monday, April 7, 14

Online searching with turn cost +d +d Monday, April 7, 14

Online searching with turn cost +d +d +d Monday, April 7, 14

Online searching with turn cost +d +d +d Total cost = Distance traversed + overall turn cost Monday, April 7, 14

Online searching with turn cost +d +d +d Total cost = Distance traversed + overall turn cost total search cost Competitive ratio = sup λ Monday, April 7, 14

Online searching with turn cost +d +d +d Total cost = Distance traversed + overall turn cost total search cost Competitive ratio = sup λ Objective : Find the smallest B such that Total cost of searching ( Comp. ratio ) λ + Β ≤ · Monday, April 7, 14

Online searching with turn cost +d +d +d Total cost = Distance traversed + overall turn cost total search cost Competitive ratio = sup λ Objective : Find the smallest B such that Total cost of searching ( Comp. ratio ) λ + Β ≤ · competitive ratio = 9 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) Monday, April 7, 14

LP formulations of the problem (Demaine et al.) Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 2 x 1 + 2 x 2 + x 1 + 2 d ≤ 9 x 1 + B Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 2 x 1 + 2 x 2 + 2 x 3 + x 2 + 3 d ≤ 9 x 2 + B Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 2 x 1 + 2 x 2 + 2 x 3 + 2 x 4 + x 3 + 4 d ≤ 9 x 3 + B Monday, April 7, 14

LP formulations of the problem (Demaine et al.) x 1 x 2 x 3 x 4 2 x 1 + 2 x 2 + 2 x 3 + 2 x 4 + x 3 + 4 d ≤ 9 x 3 + B Infinite LP : Min B subject to an infinite number of constraints Monday, April 7, 14

Ray searching with turn cost Monday, April 7, 14

Ray searching with turn cost Monday, April 7, 14

Ray searching with turn cost x 1 Monday, April 7, 14

Ray searching with turn cost x 2 x 1 Monday, April 7, 14

Ray searching with turn cost x 2 x 3 x 1 Monday, April 7, 14

Ray searching with turn cost x 2 x 3 x 4 x 1 Monday, April 7, 14

Ray searching with turn cost x 2 x 3 x 4 x 1 comp. ratio = 1 + 2 M b m m where M = b − 1 and b = m − 1 Monday, April 7, 14

Ray searching with turn cost x 2 x 3 x 4 x 1 comp. ratio = 1 + 2 M b m m where M = b − 1 and b = m − 1 x i = d 2 ( b i − 1) yields Using the strategy B = ( M − m ) d Monday, April 7, 14

Extension to ray searching (Demaine et al.) Monday, April 7, 14

Extension to ray searching (Demaine et al.) We can obtain an infinite family of LP formulations min ( P 1 ) B X m − 1 s.t. 2 j =1 x j − B 6 − d ( m − 1) X m + i 2 j =1 x j − 2 Mx i +1 − B 6 − d ( m + i ) ∀ i = 0 . . . k B, x 1 , . . . , x m + k > 0 , Monday, April 7, 14

Extension to ray searching (Demaine et al.) We can obtain an infinite family of LP formulations min ( P 1 ) B X m − 1 s.t. 2 j =1 x j − B 6 − d ( m − 1) X m + i 2 j =1 x j − 2 Mx i +1 − B 6 − d ( m + i ) ∀ i = 0 . . . k B, x 1 , . . . , x m + k > 0 , with corresponding dual LPs ✓ ◆ X k max ( m − 1) z + i =0 y i ( m + i ) ( D 1 ) d X k s.t. z + i =0 y i 6 1 ⇢ z , j ≤ m − 1 � X k + i =max(0 ,j − m ) y i − My j − 1 > 0 ∀ j = 1 . . . m + k 0 , otherwise z, y 0 , . . . , y k > 0 Monday, April 7, 14

Monday, April 7, 14

Dealing with the infinite LP Monday, April 7, 14

Dealing with the infinite LP Demaine et al. focus on the infinite dual LP ⇣ X ∞ ⌘ max ( m − 1) z + i =0 y i ( m + i ) ( D ∞ 1 ) d X ∞ s.t. z + i =0 y i 6 1 ⇢ z , j ≤ m − 1 � X ∞ + i =max(0 ,j − m ) y i − My i > 0 ∀ j = 1 , 2 , . . . 0 , otherwise z, y 0 , y 1 , . . . > 0 Monday, April 7, 14

Dealing with the infinite LP Demaine et al. focus on the infinite dual LP ⇣ X ∞ ⌘ max ( m − 1) z + i =0 y i ( m + i ) ( D ∞ 1 ) d X ∞ s.t. z + i =0 y i 6 1 ⇢ z , j ≤ m − 1 � X ∞ + i =max(0 ,j − m ) y i − My i > 0 ∀ j = 1 , 2 , . . . 0 , otherwise z, y 0 , y 1 , . . . > 0 and argue that the following is a feasible solution to the infinite dual LP z = m 1 1 y 0 = y 1 = y m − 2 = . . . = M , y m − 1 = M (1 − z ) M 1 and y i = y i − 1 − M y i − m , for all i ≥ m, Monday, April 7, 14

Dealing with the infinite LP Demaine et al. focus on the infinite dual LP ⇣ X ∞ ⌘ max ( m − 1) z + i =0 y i ( m + i ) ( D ∞ 1 ) d X ∞ s.t. z + i =0 y i 6 1 ⇢ z , j ≤ m − 1 � X ∞ + i =max(0 ,j − m ) y i − My i > 0 ∀ j = 1 , 2 , . . . 0 , otherwise z, y 0 , y 1 , . . . > 0 and argue that the following is a feasible solution to the infinite dual LP z = m 1 1 y 0 = y 1 = y m − 2 = . . . = M , y m − 1 = M (1 − z ) M 1 and y i = y i − 1 − M y i − m , for all i ≥ m, which yields an objective value of (M-m)d (which is optimal) Monday, April 7, 14

but there is a problem..... Monday, April 7, 14

but there is a problem..... One can show that a different feasible solution to the infinite dual LP yields an objective of Md > ( M − m ) d = upper bound This means we cannot trust the infinite dual LP Instead we should work on finite LPs (and obtain the best bound at the limit) Monday, April 7, 14

but there is a problem..... One can show that a different feasible solution to the infinite dual LP yields an objective of Md > ( M − m ) d = upper bound This means we cannot trust the infinite dual LP Instead we should work on finite LPs (and obtain the best bound at the limit) Finding the best dual solution : establishing some properties of the linear recurrence y More precisely: for which initial data does y become eventually negative? Monday, April 7, 14

A related problem in AI : Anytime algorithms Monday, April 7, 14

A related problem in AI : Anytime algorithms Quality of output improves as a function of time [Dean and Boddy 1987], [Russell and Zilberstein 1991] Interruptible algorithms: Can be interrupted at any time, must be able to produce a solution Contract algorithms: Prespecified amount of execution time given as part of the algorithm’ s input Monday, April 7, 14

A related problem in AI : Anytime algorithms Quality of output improves as a function of time [Dean and Boddy 1987], [Russell and Zilberstein 1991] Interruptible algorithms: Can be interrupted at any time, must be able to produce a solution (+) flexible! (-) complicated, no performance guarantees Contract algorithms: Prespecified amount of execution time given as part of the algorithm’ s input Monday, April 7, 14

A related problem in AI : Anytime algorithms Quality of output improves as a function of time [Dean and Boddy 1987], [Russell and Zilberstein 1991] Interruptible algorithms: Can be interrupted at any time, must be able to produce a solution (+) flexible! (-) complicated, no performance guarantees Contract algorithms: Prespecified amount of execution time given as part of the algorithm’ s input (-) less flexible (+) easier to program, analyze Monday, April 7, 14