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

Spyros Angelopoulos CNRS-UPMC

Searching with turn cost and related problems

Joint work with Alex Lopez-Ortiz

• Univ. of Waterloo

Diogo Arsenio CNRS Christoph Dürr CNRS

Monday, April 7, 14

Outline of the presentation

Monday, April 7, 14

Outline of the presentation

Setting : A searcher that must locate a fixed target Starting point: environment = set of rays Objective : As quickly as possible -> performance guarantees Variant : Turning direction incurs a fixed cost

Monday, April 7, 14

Outline of the presentation

Setting : A searcher that must locate a fixed target Starting point: environment = set of rays Objective : As quickly as possible -> performance guarantees Variant : Turning direction incurs a fixed cost Main result :

Monday, April 7, 14

Outline of the presentation

Setting : A searcher that must locate a fixed target Starting point: environment = set of rays Objective : As quickly as possible -> performance guarantees Variant : Turning direction incurs a fixed cost Main result : ★ Tight bounds on the performance measures

Monday, April 7, 14

Outline of the presentation

Setting : A searcher that must locate a fixed target Starting point: environment = set of rays Objective : As quickly as possible -> performance guarantees Variant : Turning direction incurs a fixed cost Main result : ★ Tight bounds on the performance measures ★ Explore the role of infinite LPs + duality

Monday, April 7, 14

Outline of the presentation

Setting : A searcher that must locate a fixed target Starting point: environment = set of rays Objective : As quickly as possible -> performance guarantees Variant : Turning direction incurs a fixed cost Main result : ★ Tight bounds on the performance measures ★ Explore the role of infinite LPs + duality ★ Connections with problems in AI

Monday, April 7, 14

Warm-up : The cow-path problem

Monday, April 7, 14

Warm-up : The cow-path problem

Monday, April 7, 14

Warm-up : The cow-path problem

Monday, April 7, 14

Warm-up : The cow-path problem

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Optimal strategy : Geometric search

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Optimal strategy : Geometric search

b

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Optimal strategy : Geometric search

b b2

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Optimal strategy : Geometric search

b b2 b3

Monday, April 7, 14

Warm-up : The cow-path problem

λ

Search strategy : an (infinite) sequence of turn points Competitive ratio = sup total distance of searcher

λ

Optimal strategy : Geometric search

b b2 b3 b4

Monday, April 7, 14

A long history of previous research

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem [Beck& Newman 70] Yet more on the linear search problem

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem [Beck& Newman 70] Yet more on the linear search problem [Beck& Warren 73] The return of the linear search problem

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem [Beck& Newman 70] Yet more on the linear search problem [Beck& Warren 73] The return of the linear search problem [Beck& Beck 84] Son of the linear search problem

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem [Beck& Newman 70] Yet more on the linear search problem [Beck& Warren 73] The return of the linear search problem [Beck& Beck 84] Son of the linear search problem [Beck& Beck 86] The linear search problem rides again

Monday, April 7, 14

A long history of previous research

Initially proposed by Bellman and Beck (1963) in a Bayesian context First solved by Beck and Newman : optimal competitive ratio of 9

[Beck 64] On the linear search problem [Beck 65] More on the linear search problem [Beck& Newman 70] Yet more on the linear search problem [Beck& Warren 73] The return of the linear search problem [Beck& Beck 84] Son of the linear search problem [Beck& Beck 86] The linear search problem rides again [Beck& Beck 92] Revenge of the linear search problem

Monday, April 7, 14

A generalization : Star search or ray search

Monday, April 7, 14

A generalization : Star search or ray search

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

b

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

b b2

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

b b2 b3

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

Multiple searchers [Lopez-Ortiz and Schuierer 2002]

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

Multiple searchers [Lopez-Ortiz and Schuierer 2002] Turn cost [Demaine et al. 2004]

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

Multiple searchers [Lopez-Ortiz and Schuierer 2002] Turn cost [Demaine et al. 2004] Randomized strategies [Kao et al. 1996]

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

Multiple searchers [Lopez-Ortiz and Schuierer 2002] Turn cost [Demaine et al. 2004] Randomized strategies [Kao et al. 1996] Searching on the plane [Gal 1980 and Langetepe 2010]

b b2 b3 b4

Monday, April 7, 14

A generalization : Star search or ray search

m infinite rays, one target Competitive ratio = sup (search cost) /λ Optimal strategy : geometric search [Gal 72]

Many variants:

Multiple searchers [Lopez-Ortiz and Schuierer 2002] 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

b b2 b3 b4

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., 2006 & 2009] [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

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 and interruptible algorithms [A. et al., 2006 & 2009] [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 In preparation : More connections between searching

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 Competitive ratio = sup

total search cost λ

Monday, April 7, 14

Online searching with turn cost

+d +d +d

Total cost = Distance traversed + overall turn cost Competitive ratio = sup

total search cost 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 Competitive ratio = sup

total search cost 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.)

x1

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4 2x1 + 2x2 + x1 + 2d ≤ 9x1 + B

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4 2x1 + 2x2 + 2x3 + x2 + 3d ≤ 9x2 + B

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4 2x1 + 2x2 + 2x3 + 2x4 + x3 + 4d ≤ 9x3 + B

Monday, April 7, 14

LP formulations of the problem (Demaine et al.)

x1 x2 x3 x4 2x1 + 2x2 + 2x3 + 2x4 + x3 + 4d ≤ 9x3 + 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

x1

Monday, April 7, 14

Ray searching with turn cost

x1 x2

Monday, April 7, 14

Ray searching with turn cost

x1 x2 x3

Monday, April 7, 14

Ray searching with turn cost

x1 x2 x3 x4

Monday, April 7, 14

Ray searching with turn cost

x1 x2 x3 x4

• comp. ratio = 1 + 2M

where M =

bm b−1 and b = m m−1

Monday, April 7, 14

Ray searching with turn cost

x1 x2 x3 x4

• comp. ratio = 1 + 2M

Using the strategy xi = d

2(bi − 1) yields

B = (M − m)d where M =

bm b−1 and b = m m−1

Monday, April 7, 14

Extension to ray searching (Demaine et al.)

Monday, April 7, 14

min B (P1) s.t. 2 Xm−1

j=1 xj − B 6 −d(m − 1)

2 Xm+i

j=1 xj − 2Mxi+1 − B 6 −d(m + i)

∀i = 0 . . . k B, x1, . . . , xm+k > 0,

We can obtain an infinite family of LP formulations

Extension to ray searching (Demaine et al.)

Monday, April 7, 14

min B (P1) s.t. 2 Xm−1

j=1 xj − B 6 −d(m − 1)

2 Xm+i

j=1 xj − 2Mxi+1 − B 6 −d(m + i)

∀i = 0 . . . k B, x1, . . . , xm+k > 0,

with corresponding dual LPs We can obtain an infinite family of LP formulations

max ✓ (m − 1)z + Xk

i=0 yi(m + i)

◆ d (D1) s.t. z + Xk

i=0 yi 6 1

⇢ z , j ≤ m − 1 0 , otherwise

• +

Xk

i=max(0,j−m) yi − Myj−1 > 0

∀j = 1 . . . m + k z, y0, . . . , yk > 0

Extension to ray searching (Demaine et al.)

Monday, April 7, 14

Monday, April 7, 14

Dealing with the infinite LP

Monday, April 7, 14

Demaine et al. focus on the infinite dual LP

Dealing with the infinite LP

max ⇣ (m − 1)z + X∞

i=0 yi(m + i)

⌘ d (D∞

1 )

s.t. z + X∞

i=0 yi 6 1

⇢ z , j ≤ m − 1 0 , otherwise

• +

X∞

i=max(0,j−m) yi − Myi > 0

∀j = 1, 2, . . . z, y0, y1, . . . > 0

Monday, April 7, 14

Demaine et al. focus on the infinite dual LP

Dealing with the infinite LP

max ⇣ (m − 1)z + X∞

i=0 yi(m + i)

⌘ d (D∞

1 )

s.t. z + X∞

i=0 yi 6 1

⇢ z , j ≤ m − 1 0 , otherwise

• +

X∞

i=max(0,j−m) yi − Myi > 0

∀j = 1, 2, . . . z, y0, y1, . . . > 0

and argue that the following is a feasible solution to the infinite dual LP z = m

M

y0 = y1 = ym−2 = . . . =

1 M , ym−1 = 1 M (1 − z)

and yi = yi−1 −

1 M yi−m, for all i ≥ m,

Monday, April 7, 14

Demaine et al. focus on the infinite dual LP

Dealing with the infinite LP

max ⇣ (m − 1)z + X∞

i=0 yi(m + i)

⌘ d (D∞

1 )

s.t. z + X∞

i=0 yi 6 1

⇢ z , j ≤ m − 1 0 , otherwise

• +

X∞

i=max(0,j−m) yi − Myi > 0

∀j = 1, 2, . . . z, y0, y1, . . . > 0

and argue that the following is a feasible solution to the infinite dual LP z = m

M

y0 = y1 = ym−2 = . . . =

1 M , ym−1 = 1 M (1 − z)

and yi = yi−1 −

1 M yi−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 This means we cannot trust the infinite dual LP Instead we should work on finite LPs (and obtain the best bound at the limit) Md > (M − m)d = upper bound

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 This means we cannot trust the infinite dual LP Instead we should work on finite LPs (and obtain the best bound at the limit) Md > (M − m)d = upper bound 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 Contract algorithms: Prespecified amount of execution time given as part of the algorithm’ s input (+) flexible! (-) complicated, no performance guarantees

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 (+) flexible! (-) complicated, no performance guarantees (-) less flexible (+) easier to program, analyze

Monday, April 7, 14

From contract to interruptible algorithms

Main goal: “Black-box” techniques for turning every contract algorithm to its interruptible version Establish measures of how good this simulation is Find efficient simulations in this measure Survey: “Using anytime algorithms in intelligent systems” (Zilberstein)

“Plan synthesis must have anytime, incremental

• characteristics. It should be possible to stop a plan

synthesis algorithm at any time during its execution and expect useful results. One should expect the “quality” of the results to improve continuously as a function of time.” John Bressina and Mark Drummond NASA Ames Research Center

Monday, April 7, 14

From contract to interruptible algorithms

Main goal: “Black-box” techniques for turning every contract algorithm to its interruptible version Establish measures of how good this simulation is Find efficient simulations in this measure Survey: “Using anytime algorithms in intelligent systems” (Zilberstein)

“Plan synthesis must have anytime, incremental

• characteristics. It should be possible to stop a plan

synthesis algorithm at any time during its execution and expect useful results. One should expect the “quality” of the results to improve continuously as a function of time.” John Bressina and Mark Drummond NASA Ames Research Center

Monday, April 7, 14

A quick example

Suppose we are given a contract algorithm Run the algorithm for 1 step, then for 2 steps, then for 4 steps and so forth

Monday, April 7, 14

A quick example

Suppose we are given a contract algorithm Run the algorithm for 1 step, then for 2 steps, then for 4 steps and so forth 1 2 4 8

Monday, April 7, 14

A quick example

Suppose we are given a contract algorithm Run the algorithm for 1 step, then for 2 steps, then for 4 steps and so forth 1 2 4 8 interruption at t=10

Monday, April 7, 14

A quick example

Suppose we are given a contract algorithm Run the algorithm for 1 step, then for 2 steps, then for 4 steps and so forth 1 2 4 8 interruption at t=10 In hindsight we could have run the algorithm for 10 units, but the best we achieved is a running time of 4 Inefficiency = max

t

t longest contract finished by t

Monday, April 7, 14

Conclusion and outlook

Monday, April 7, 14

Conclusion and outlook

We study online search problems with turn cost using infinite LP formulations Caveats of duality in infinite LPs Further applications : Search problems in unbounded domains Resource allocation problems with infinite horizon Many other variants of ray searching remain open

Monday, April 7, 14

Conclusion and outlook

We study online search problems with turn cost using infinite LP formulations Caveats of duality in infinite LPs Further applications : Search problems in unbounded domains Resource allocation problems with infinite horizon Many other variants of ray searching remain open

Thank you!

Monday, April 7, 14