Rep.: Funnel situation!

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Online Motion Planning MA-INF 1314 Alternative cost measures!

Elmar Langetepe University of Bonn

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Rep: Searching for target of a street

π_Opt P_L s

P_R t

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Rep: Lower Bound

Theorem No strategy can achieve a path length smaller than

√2 × πOpt.

Proof:

m t?

s

π 2

1 1

1

√2 √

2 t?

t_` t_r

Detour with ratio√ 2

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Rep.: Funnel situation!

• It is sufficient to consider special streets only!

• Combine them piecewise!

• Def. A polygon that start with a convex vertex s and consists of two opening convex chains ending at t_` and t_r respectively and which are finally connected by a line segment t_`t_r is called a funnel (polygon).

t_`

v_`

φ

v_r

s φ₀

t_r

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Rep.: Generalized Lower Bound

Lemma For a funnel with opening angle φ ≤ π no strategy can guarantee a path length smaller than K_φ · |Opt| where

K_φ := √

1 + sin φ. Proof:

`· sin ^φ₂

`

`· cos ^φ₂ m

φ s

t_` t_r

Detour at least: _|π^|π^S^|

Opt| = ^`^cos

φ

2+` sin ^φ₂

` = √

1 + sin φ.

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Rep.: Opening angle π ≥ ϕ

₀

≥ π/2!

• Backward analysis!

• ^|w^|+K_l ^φ²^·`²

1 ≤ K_φ₁ and ^|w^|+K_r ^φ²^·r²

1 ≤ K_φ₁

• Combine to one condition for w

• |w| ≤ min{K_φ₁`₁ − K_φ₂`₂ , K_φ₁r₁ − K_φ₂r₂ }

v_`

φ₂

p₁

`₁

p_n r_n

p₂ φ₁ w

`_n

r₁

`₂ r₂

v_r

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Rep.: Opening angle π ≥ ϕ

₀

≥ π/2!

• Change of the reflex vertices! Sufficient!

• Change left side! Condition:

|w| ≤ min{K_φ₁`₁ − K_φ₂`₂ , K_φ₁r₁ − K_φ₂r₂ }

• And also: ^|w^|+K^φ²^·(`²^+`

0 2)

(l₁+`⁰₂) ≤ K_φ₁

v_l

φ₂ p₂

l₂ r₂

l₁ l⁰₂

v_l⁰ t_r

p_end W

r₁

rend l_end

v_r t_l

P_L

l₀

r₀

P_R

φ₀ p₁ φ₁

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Rep.: Opening angle π ≥ ϕ

₀

≥ π/2!

Lemma: Let S be a strategy, that searches for the target in a funnel with opening angle φ₂ for φ₂ ≥ ^π₂ with competitive ratio K_φ₂. This strategy can be extended to a strategy of ratio K_φ₁ and opening angle φ₁ for φ₂ > φ₁ ≥ ^π₂, if we guarantee

|w| ≤ min{ K_φ₁`₁ − K_φ₂`₂ , K_φ₁r₁ − K_φ₂r₂ } for the corresponding connecting path.

v_l

φ₂ p₂

l₂ r₂

l₁ l₂⁰

v_l⁰ t_r

p_end W

r₁

rend l_end

v_r

s t_l

P_L

l₀

r₀

P_R

φ₀ p₁ φ₁

w

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Rep.: Opening angle π ≥ ϕ

₀

≥ π/2!

• Equality: K_φ₂(`₂ − r₂) = K_φ₁(`₁ − r₁), A := K_φ₀(`₀ − r₀)

• Hyperbola: ^X²

A 2Kφ

2 − ^Y ²

(¹₂)²⁻

A 2Kφ

2 = 1

• Circle: X² +

Y + ^cot₂ ^φ ²

= ¹

4 sin² φ

(0,0)

1 2

z x

l(p)

vr

v_l

φ p

r(p)

φ 2

π−φ

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Rep.: Opening angle π ≥ ϕ

₀

≥ π/2!

Intersection Hyp. Circle: Curve!!

X(φ) = A

2 · cot ^φ₂

1 + sin φ · s

1 + tan φ 2

2

− A² Y (φ) = 1

2 · cot φ 2 ·

A²

1 + sin φ − 1

where A = K_φ₀(`₀ − r₀)

-0.5 0

Y

-0.5 X 0.5

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Opt. strat. for opening angle π ≥ ϕ

₀

≥ π/2!

X(φ) = A

2 · cot ^φ₂

1 + sin φ · s

1 + tan φ 2

²

− A² Y (φ) = 1

2 · cot φ 2 ·

A²

1 + sin φ − 1

Change of the boundary points. A also changes, new piece of curve!

v_l

φ₂ p₂

l₂ r₂

l₁ l⁰₂

v_l⁰ t_r

p_end

r₁

rend l_end

v_r

s P_L

l₀

r₀

P_R

φ₀ p₁

w₁ t_l

w₂ w₃

φ₁

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Opt. strat. for opening angle π ≥ ϕ

₀

≥ π/2!

Theorem: The goal of a funnel with opening angle φ₀ > ^π₂ can be found with ratio K_φ₀.

Proof: Show that the curves fulfil:

|w| ≤ min{ K_φ₁`₁ − K_φ₂`₂ , K_φ₁r₁ − K_φ₂r₂ }

For any small piece w of the curve. Analytically, lengthy proof!

Experimentally!

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Opt. strat. opening angle 0 ≤ ϕ

₀

≤ π/2!

• The same approach

• But independent from the angle

• Dominated by factor K_π/2 = √ 2

• Require: w ≤ min{ √

2(`₁ − `₂) , √

2(r₁ − r₂) }.

• Equality: `₁ − `₂ = r₁ − r₂

• Current angular bisector: Hyberbola!

φ₂

r₁ p₂ w

`₂

r₂ v_`

v_r

φ₁

`₁

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Opt. strat. opening angle 0 ≤ ϕ

₀

≤ π !

Combine strategy 1 and strategy 2

t

s

Theorem: In an unknown street-polygon beginning from the source s we can find the target t with an optimal online strategy with

competitive ratio √ 2.

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Optimal strategy “Worst-Case-Aware”

As long as target t is not visible:

Compute current v_` and v_r.

If only one exists: Move directly toward the other.

Otherwise. Repeat:

New reflex vertex v_`⁰ or v_r⁰ is detected:

Use v_`⁰ or v_r⁰ instead of v_` or v_r.

Let φ be the angle between v_`, the current poyition and v_r. If φ ≤ ^π₂: Follow the current angular bisctor!

If φ > ^π₂: Follow the curve (X(φ), Y (φ)).

Until either v_` or v_r is explored.

Move toward the non-explored vertex.

Move toward the goal.

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Opening angle π ≥ ϕ

₀

≥ π/2!

Theorem: Searching for the target in a street polygon can be realized within a competitive ratio of √

2.

• From ϕ ≥ π/2 curve fulfils w condition, analysis/experiments!

• For smaller angles: √

2 substitute for all K_φ

t

s

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Optimal searchpath

• We have seen:

Searching for a goal (polygon) in general not competitive

• Question: What is a good searchpath (for polygons)?

• Searching: Target point unknown!

• Offline-Searching: Environment is known

• Online-Searching: Environment unknown

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Quality measures!

• Competitive ratio of search strategy A in polygons:

C := sup

P

sup

p∈P

|A(s, p)|

|sp(s, p)|

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Optimal search path in polygons

• Competitive analysis:

• Agent with visibiltly

• Adversary forces any strategy to visit any corridor

• Optimal path is short

• ⇒ Any strategy fails

(not constant competitive)

goal goal

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Optimal search path in polygons

• Strat1:

fully visit any corridor

• Strat2:

visit all corr. depth d = 1 visit all corr. depth d = 2

visit all corr. depth d = 4 etc.

• Strat2 seems to be better:

close targets s are visited earlier

• Can we give a measure?

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Search ratio for polygons

π: Searchpath, quality for π: SR(π, P) = max

p∈P

|π_s^p⁰| + |p⁰p|

|sp(s, p)|

sp(s, p) s

π p⁰

p

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Search ratio in general

Given: Environment E, Set of goals G ⊆ E Graphs G = (V, E): Vertices G = V

Geometric Search G = V ∪ E (Requirement: ∀p ∈ E : |sp(s, p)| = |sp(p, s)|)

Search ratio of a search strategy A for E: SR(A, E) := sup

p∈G

|A(s, p)|

|sp(s, p)|

Optimal search ratio:

SR_OPT(E) := inf

A SR(A, E)

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Path with optimal search ratio

• Graphs (offline): NP-hard

• Polygons (offline): ???

• Online: Approximation is possible

⇒ Goal: Approximate the path with opt. search ratio

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Search ratio approximation

• Competitive ratio : C := sup

E

sup

p∈G

|A(s, p)|

|sp(s, p)|

• Search ratio: SR(A, E) := sup

p∈G

|A(s, p)|

|sp(s, p)|

• Optimal search ratio: SR_OPT(E) := inf

A SR(A, E)

• Approximation: A search-competitiv C_s := sup

E

SR(A, E) SR_OPT(E)

• Comparison not against SP, but against best possible SR

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Depth-restricted exploration

Def. Exploration-Strategy Expl for E is called depth-restrictable, if we can derive a strategy Expl (d) such that:

• Expl (d) explores E up to depth d ≥ 1

• return to s after the exploration

• Expl (d) is C-competitive, i.e.,∃C ≥ 1 : ∀E:

|Expl (d)| ≤ C · |Expl_OPT(d)| .

P(d) d

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Searchpath approximation

Algorithm

• Explore E by increasing depth: Expl (2ⁱ) f¨ur i = 1, 2, . . . Lemma:

• Agent without vision system

• Environment E

• Expl_ONL: C-competitive, depth-restrictable, online exploration strategy for E

(d. h. |Expl (d)| ≤ C · |Expl_OPT(d)|)

⇒ Algorithm gives 4C-Approximation of optimal search path!

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Searchpath approximation proof

|Π_Expl_opt_(d)| ≤ d · (SR(Π_opt) + 1) (1)

SR(Π) ≤

j+1

P

i=1

|Π_Expl(2i)| 2^j + ε

≤

(Ass.⁾

C 2^j

j+1

X

i=1

|Π_Expl

opt(2ⁱ)| ≤₍₁₎ C 2^j

j+1

X

i=1

2ⁱ · (SR(Π_opt) + 1)

≤ C ·

2^j⁺² − 1 2^j

· (SR(Π_opt) + 1) ≤ 4C · (SR(Π_opt) + 1)

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Applications

• Corollay:

Trees: Exploration by DFS (C = 1) Online or Offline, depth restricted, simple

⇒ Searchpath approximation of factor 4

• Graphs: Online and Offline!

CFS (C = 4 + _α⁸) depth-restrictable!!

• But: Factor depends on rope length (1 + α)r by depth r

• CFS sometimes explores more than d (precisely (1 + α)d)

⇒ Expl (d) not comparable to Expl_OPT(d)

• Workaround: Compare Expl (d) with Expl_OPT(β · d)

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β -depth restricted exploration

Def. Exploration strategy Expl for E is denoted as β-depth restrictable, if we can derive a strategy Expl (d) such that:

• Expl (d) explores E only up to depth d ≥ 1

• returns to the start s

• Expl (d) is C_β-competitiv, i.e., ∃C_β ≥ 1, β > 0 : ∀E:

|Expl (d)| ≤ C_β · |Expl_OPT(β · d)| .

P(d) d

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Searchpath approximation

Theorem:

• Agent without vision

• Environment E

• Expl : C_β-competitive, β-depth restrictable, online exploration strategy for E, (i.e., |Expl (d)| ≤ C_β · |Expl_OPT(β · d)|)

⇒ Algorithm (exploration/double depth) gives a 4βC_β-approximation of the optimal searchpath

Corollary: Unknown graphs, Algorithm with CFS is 4(1 + α)(4 + _α⁸)-approximation of optimal searchpath