Further Connections between Contract-Scheduling and Ray-Searching - - PowerPoint PPT Presentation

Further Connections between Contract-Scheduling and Ray-Searching Problems

Spyros Angelopoulos CNRS and University Pierre and Marie Curie

Friday, October 23, 15

Outline and motivation

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain)

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain) The second problem: how to obtain effjcient anytime algorithms

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain) The second problem: how to obtain effjcient anytime algorithms Objective: address variants of these problems with a common approach

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain) The second problem: how to obtain effjcient anytime algorithms Objective: address variants of these problems with a common approach

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain) The second problem: how to obtain effjcient anytime algorithms Objective: address variants of these problems with a common approach

Friday, October 23, 15

Outline and motivation

Two well-studied problems:

• ne from OR/TCS, the other from AI

The first problem: searching on a star (unbounded domain) The second problem: how to obtain effjcient anytime algorithms Objective: address variants of these problems with a common approach Robots in this presentation are benign!*

*certain conditions may apply Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Friday, October 23, 15

The first problem: Ray searching

Performance evaluation: Competitive ratio

Friday, October 23, 15

The first problem: Ray searching

Performance evaluation: Competitive ratio

α = sup total distance to find distance of from origin

Friday, October 23, 15

The first problem: Ray searching

Performance evaluation: Competitive ratio

α = sup total distance to find distance of from origin

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

. . . . .

time

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

interruption at time t

. . . . .

time

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

interruption at time t

. . . . .

time

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

interruption at time t

. . . . .

time

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

interruption at time t

contract execution

• f worst progress

. . . . .

time

Friday, October 23, 15

The second problem: Scheduling contract algorithms

Contract algorithms execution time given as input

may return non-meaningful solutions if interrupted

Interruptible algorithms may be interrupted at will

always return meaningful solutions (improving with time)

[Russell and Zilberstein 1991]: Interruptible algorithms via schedules of executions of contract algorithms

interruption at time t

contract execution

• f worst progress

Performance evaluation: Acceleration ratio

. . . . .

time

β = sup

t

interruption time t length of

Friday, October 23, 15

Some related previous work

Ray searching m rays Contract scheduling

n problems Early work by Bellman, Beck and Newman for

m = 2

Optimal strategies by [Gal 74] Re-discovered in CS context [Baeza-Yates et al. 93] Several other settings: Randomization [Kao et al. 96] Multiple searchers [Lopez- Ortiz and Schuierer 04] Turn cost [Demaine et al. 06], [A. et al. 14+] New measures [Kirkpatrick 09] [Russell and Zilberstein 91]: n = 1 [Zilberstein et al. 03]: , multiple processors

n = 1

[Bernstein et al. 02]: general n [Bernstein et al. 03], [Lopez-Ortiz et al. 06] general n, multiple processors [A. and Lopez-Ortiz 08]: soft interruptions [A. and Lopez-Ortiz 09]: new measures

Friday, October 23, 15

Some related previous work

Ray searching m rays Contract scheduling

n problems Early work by Bellman, Beck and Newman for

m = 2

Optimal strategies by [Gal 74] Re-discovered in CS context [Baeza-Yates et al. 93] Several other settings: Randomization [Kao et al. 96] Multiple searchers [Lopez- Ortiz and Schuierer 04] Turn cost [Demaine et al. 06], [A. et al. 14+] New measures [Kirkpatrick 09] [Russell and Zilberstein 91]: n = 1 [Zilberstein et al. 03]: , multiple processors

n = 1

[Bernstein et al. 02]: general n [Bernstein et al. 03], [Lopez-Ortiz et al. 06] general n, multiple processors [A. and Lopez-Ortiz 08]: soft interruptions [A. and Lopez-Ortiz 09]: new measures

[Bernstein et al. 03]: Connections between cyclic strategies for the two problems

Friday, October 23, 15

Results

Friday, October 23, 15

Results

Ray searching m rays Contract scheduling

n problems

Friday, October 23, 15

Results

Ray searching m rays Contract scheduling

n problems

Stochastic setting

Setting: Target detection with probability Results: Strategy with competitive ratio + No strategy is better than

• competitive

p Θ(m/p2)

Setting: Contracts are Monte Carlo algorithms with success prob. Results: Schedule with

• accel. ratio

+ No schedule better than

• acceleration

p

n/p (e(n + 1))/p

m/(2p)

Friday, October 23, 15

Results

Ray searching m rays Contract scheduling

n problems

Stochastic setting Fault tolerance / redundancy

Setting: Target detection with probability Results: Strategy with competitive ratio + No strategy is better than

• competitive

p Θ(m/p2)

Setting: Contracts are Monte Carlo algorithms with success prob. Results: Schedule with

• accel. ratio

+ No schedule better than

• acceleration

p

n/p (e(n + 1))/p

Setting: Target detection

• n the -th visit

r

Results: Strategy with competitive ratio + no strategy better than

• competitive

r(n + 1) ✓ 1 + 1 rn ◆rn

Setting: Output the

• th smallest contract

r

Results: Strategy with acceleration ratio + no strategy better than

• competitive

r(m − 1) ✓ m m − 1 ◆m + 2 − r

rm/2 rn m/(2p)

Friday, October 23, 15

Results

Ray searching m rays Contract scheduling

n problems

Stochastic setting Fault tolerance / redundancy Methodology

Setting: Target detection with probability Results: Strategy with competitive ratio + No strategy is better than

• competitive

p Θ(m/p2)

Setting: Contracts are Monte Carlo algorithms with success prob. Results: Schedule with

• accel. ratio

+ No schedule better than

• acceleration

p

n/p (e(n + 1))/p

Setting: Target detection

• n the -th visit

r

Results: Strategy with competitive ratio + no strategy better than

• competitive

r(n + 1) ✓ 1 + 1 rn ◆rn

Setting: Output the

• th smallest contract

r

Results: Strategy with acceleration ratio + no strategy better than

• competitive

r(m − 1) ✓ m m − 1 ◆m + 2 − r

rm/2 rn Non-trivial analysis

• f cyclic strategies

Non-cyclic strategies that improve upon the best cyclic ones m/(2p)

Friday, October 23, 15

Results (continued)

Friday, October 23, 15

Results (continued)

Ray searching m rays Contract scheduling

n problems

Randomized scheduling

Known: Randomization helps improve the competitive ratio [Kao et al. 96] Result: Randomized schedule of acceleration ratio about 0.6 times the deterministic acceleration ratio

Friday, October 23, 15

Results (continued)

Ray searching m rays Contract scheduling

n problems

Randomized scheduling Trade ofgs between performance and turns / executions

Known: Randomization helps improve the competitive ratio [Kao et al. 96] Result: Randomized schedule of acceleration ratio about 0.6 times the deterministic acceleration ratio Setting: we are interested in the # of turns Results: Optimal trade-

• fgs between competitive

ratio and # of turns Setting: we are interested in the # of executions Results: Optimal trade-

• fgs between acceleration

ratio and # of executions

Friday, October 23, 15

Results (continued)

Ray searching m rays Contract scheduling

n problems

Randomized scheduling Trade ofgs between performance and turns / executions Methodology

Known: Randomization helps improve the competitive ratio [Kao et al. 96] Result: Randomized schedule of acceleration ratio about 0.6 times the deterministic acceleration ratio Setting: we are interested in the # of turns Results: Optimal trade-

• fgs between competitive

ratio and # of turns Setting: we are interested in the # of executions Results: Optimal trade-

• fgs between acceleration

ratio and # of executions Similar strategies but difgerent analysis (no closed form in the case of contract scheduling) Combination of uniform and exponentially increasing strategies

Friday, October 23, 15

The stochastic setting

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p)

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p) We analyze a round-robing strategy: search with lengths x1, x2, . . .

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p) We analyze a round-robing strategy: search with lengths x1, x2, . . .

• 1. Upper bound the expected cost of the strategy

(we get an expression that contains the ’s and xi p

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p) We analyze a round-robing strategy: search with lengths x1, x2, . . .

• 1. Upper bound the expected cost of the strategy

(we get an expression that contains the ’s and

• 2. Assume exponential strategies: for fixed

In the resulting expression, there is a term of the form xi p xi = bi b > 1

∞

X

j=2

((bm(1 − p))j−1

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p) We analyze a round-robing strategy: search with lengths x1, x2, . . .

• 1. Upper bound the expected cost of the strategy

(we get an expression that contains the ’s and

• 2. Assume exponential strategies: for fixed

In the resulting expression, there is a term of the form xi p xi = bi b > 1

∞

X

j=2

((bm(1 − p))j−1

• 3. Setting and requiring that we obtain

λ := bm(1 − p) λ < 1 α ≤ 1 + 2 bm bm − 1 · 1 1 − λ

Friday, October 23, 15

The stochastic setting

Easy bound of Ω(m/p) We analyze a round-robing strategy: search with lengths x1, x2, . . .

• 1. Upper bound the expected cost of the strategy

(we get an expression that contains the ’s and

• 2. Assume exponential strategies: for fixed

In the resulting expression, there is a term of the form xi p xi = bi b > 1

∞

X

j=2

((bm(1 − p))j−1

• 3. Setting and requiring that we obtain

λ := bm(1 − p) λ < 1 α ≤ 1 + 2 bm bm − 1 · 1 1 − λ

• 4. Applying some calculus we show that α ≤ 1 + 8m/p2

Friday, October 23, 15

Randomized scheduling of contract algorithms

Friday, October 23, 15

Randomized scheduling of contract algorithms

Algorithm

• 1. Choose a random permutation of

the n problems and random ✏ ∈ (0, 1)

• 2. In the i-th step schedule a contract for

problem and of length i mod n bi+✏

Friday, October 23, 15

Randomized scheduling of contract algorithms

Algorithm

• 1. Choose a random permutation of

the n problems and random ✏ ∈ (0, 1)

• 2. In the i-th step schedule a contract for

problem and of length i mod n bi+✏

Inspired by the randomized ray-searching algorithm of [Kao et al. 95]

Friday, October 23, 15

Randomized scheduling of contract algorithms

Algorithm

• 1. Choose a random permutation of

the n problems and random ✏ ∈ (0, 1)

• 2. In the i-th step schedule a contract for

problem and of length i mod n bi+✏

Inspired by the randomized ray-searching algorithm of [Kao et al. 95]

20 40 60 80 100 120 140 10 20 30 40 50 60 70 80 randomized deterministic

140 120 100 80 60 40 20 20 40 60 80 70 50 30 10 Randomized Deterministic

no

0 0

1521

n β

Friday, October 23, 15

Randomized scheduling of contract algorithms

Algorithm

• 1. Choose a random permutation of

the n problems and random ✏ ∈ (0, 1)

• 2. In the i-th step schedule a contract for

problem and of length i mod n bi+✏

Inspired by the randomized ray-searching algorithm of [Kao et al. 95]

20 40 60 80 100 120 140 10 20 30 40 50 60 70 80 randomized deterministic

140 120 100 80 60 40 20 20 40 60 80 70 50 30 10 Randomized Deterministic

no

0 0

1521

A closed formula does not appear to exist We can give analytical bounds for n → ∞ n β

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

“Given a target at distance d what is the minimum number of turns required to guarantee a certain competitive ratio?” “Given an interruption at time t what is the minimum number of contracts required to guarantee a certain acceleration ratio?”

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

“Given a target at distance d what is the minimum number of turns required to guarantee a certain competitive ratio?” “Given an interruption at time t what is the minimum number of contracts required to guarantee a certain acceleration ratio?”

pathwise search expanding search

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

“Given a target at distance d what is the minimum number of turns required to guarantee a certain competitive ratio?” “Given an interruption at time t what is the minimum number of contracts required to guarantee a certain acceleration ratio?”

pathwise search expanding search contract scheduling scheduling with preemptions

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

“Given a target at distance d what is the minimum number of turns required to guarantee a certain competitive ratio?” “Given an interruption at time t what is the minimum number of contracts required to guarantee a certain acceleration ratio?”

pathwise search expanding search contract scheduling scheduling with preemptions

Friday, October 23, 15

Tradeofgs between performance and turns/executions

Ray searching m rays Contract scheduling

n problems

“Given a target at distance d what is the minimum number of turns required to guarantee a certain competitive ratio?” “Given an interruption at time t what is the minimum number of contracts required to guarantee a certain acceleration ratio?”

pathwise search expanding search contract scheduling scheduling with preemptions Theorem: For any strategy with acceleration ratio , for and there exists t such that the # of preemptions up to t is at least . Furthermore, there is a strategy with acceleration ratio and at most preemptions. n(b + 1) − ✏ b > 1 ✏ > 0 n logb ✓t(b − 1) n + 1 ◆ − n n logb ✓t(b − 1) n + 1 ◆ + n n(b + 1)

Friday, October 23, 15

Conclusions and outlook

Friday, October 23, 15

Conclusions and outlook

Connections between two well-studied problems Similarities in settings, common algorithmic approach The two problems are similar, but also have certain difgerences

Friday, October 23, 15

Conclusions and outlook

Connections between two well-studied problems Similarities in settings, common algorithmic approach The two problems are similar, but also have certain difgerences More work needed on the stochastic setting (tight bounds for probabilistic searching, more elaborate study of stochastic contract scheduling) Heterogeneous environments? (e.g., every ray has its own probability of target location) Connections with other problems? (older work: speedup of Las Vegas algorithms [Luby et al. 93]; newer work: progressive algorithms [Alewijnse 15] )

Friday, October 23, 15

Conclusions and outlook

Connections between two well-studied problems Similarities in settings, common algorithmic approach The two problems are similar, but also have certain difgerences More work needed on the stochastic setting (tight bounds for probabilistic searching, more elaborate study of stochastic contract scheduling) Heterogeneous environments? (e.g., every ray has its own probability of target location) Connections with other problems? (older work: speedup of Las Vegas algorithms [Luby et al. 93]; newer work: progressive algorithms [Alewijnse 15] ) Full version of the paper available at www.arxiv.org or at www.lip6.fr/Spyros.Angelopoulos

Friday, October 23, 15

Conclusions and outlook

Connections between two well-studied problems Similarities in settings, common algorithmic approach The two problems are similar, but also have certain difgerences More work needed on the stochastic setting (tight bounds for probabilistic searching, more elaborate study of stochastic contract scheduling) Heterogeneous environments? (e.g., every ray has its own probability of target location) Connections with other problems? (older work: speedup of Las Vegas algorithms [Luby et al. 93]; newer work: progressive algorithms [Alewijnse 15] )

Thank you!

Full version of the paper available at www.arxiv.org or at www.lip6.fr/Spyros.Angelopoulos

Friday, October 23, 15