Rep.: Lower Bound!

Online Motion Planning MA-INF 1314 Searching in streets!

Elmar Langetepe University of Bonn

Rep.: Street

Def. Polygonal boundary chains P_L and P_R of P between s and t weakly visible.

Task: Start at s, find t!

π_Opt P_L s

P_R t

Rep.: Lower Bound!

Theorem: No strategy attains a ratio better than √

2 versus the length of the shortest path.

Beweis:

m t?

s

π 2

1 1

1

√2 √

2 t?

t_` t_r

Detour of factor at least √ 2

Rep.: Reasonable strategies!

• Rightmost left reflex vertex, leftmost right reflex vertex!

• Move into the wedge of c, v_l and v_r

• One side-candidate vanishes, move directly to the other

• Extreme vertices change over time

v_r=v¹_r v_l=v_l¹

v_r² t

(i) P_L

s u

p

φ v_l³

v_l²

q

P_R

Rep.: Funnel polygons!

• It is sufficient to analyse special streets

• Def. Polygon, conve vertex s, two opening convex polygonal chains P_L and P_R starting in s ending at t_` and t_r, respectively.

Segment t_`t_r closes the funnel (polygon).

t_`

v_`

φ

v_r

s φ₀

t_r

Rep.: Generalized LB for funnels

• Lemma LB for funnel of opening angle φ: K_φ := √

1 + sin φ.

• Strongly increasing: 0 ≤ φ ≤ π/2, Interval [1, √ 2]

• Strongly decreasing: π/2 ≤ φ ≤ π, Interval [√

2, 1]

• Subdivide: Strategy up to φ₀ = π/2, Strategy from φ₀ = π/2

`· sin ^φ₂

`

`· cos ^φ₂ m

φ s

t_` t_r

Rep.: Opt. strat. for angles π ≥ ϕ

≥ π/2!

• Backward analysis: For ϕ_n := π optimal strategy.

• K_π = 1 and K_π-competitive opt. strategy with path l_n or r_n!

• Assumption: Opt. strategy for some φ₂ with factor K_φ2 ex.

• How to prolong for φ₁ with factor K_φ1 where ^π₂ ≤ φ₁ < φ₂?

• We have K_φ1 > K_φ2

v_`

φ₂

p₁

`₁

p_n r_n

p₂ φ₁ w

`_n

r₁

`₂ r₂

v_r

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

• Situation: Opt. strategy for φ₂ with ratio K_φ2

• How to get opt. strategy for K_φ1?

• Conditions for the path w? Design!

• Goal behing v_l, path: |w| + K_φ2 · `₂, optimal: l₁

• Goal behind v_r, path: |w| + K_φ2 · r₂, optimal: r₁

• Means: ^|w|+K_l ^φ2·`2

1 ≤ K_φ1 and ^|w|+K_r ^φ2·r2

1 ≤ K_φ1

v_`

φ₂

p₁

`₁

p_n r_n

p₂ φ₁ w

`_n

r₁

`₂ r₂

v_r

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

• Guarantee: ^|w|+K_l ^φ2·`2

1 ≤ K_φ1 and ^|w|+K_r ^φ2·r2

1 ≤ K_φ1

• Combine, single condition for w

• |w| ≤ min{K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ }

• Change of a vertex at p₂? Remains guilty!

v_`

φ₂

p₁

`₁

p_n r_n

p₂ φ₁ w

`_n

r₁

`₂ r₂

v_r

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

• Change left hand: Condition

|w| ≤ min{K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ }

• There is opt. strategy for φ₂

• Show: ^{|w|+Kφ2·(`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

s t_l

P_L

l₀

r₀

P_R

φ₀ p₁ φ₁

w

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

|w| ≤ K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂

= K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ + K_φ2`<sup>0</sup><sub>2</sub> − K<sub>φ2</sub>`⁰₂

≤ K_φ1(`<sub>1</sub> + `⁰₂) − K_φ2(`<sub>2</sub> + `⁰₂)

v_l

φ₂ p₂

l₂ r₂

l₁ l⁰₂

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

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

Lemma Let S be a strategy for funnels with opening angles φ₂ ≥ ^π₂ and competitive ratio K_φ2. We can extend this strategy to a

strategy with ratio K_φ1 for funnels with opening angles φ₁ where φ₂ > φ₁ ≥ ^π₂, if we guarantee

|w| ≤ min{ K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ }

for the path w from p₁ (opening angle φ₁) to p₂ (opening angle φ₂).

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

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

• If |w| ≤ min{K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ } holds, then

• |W| ≤ min{ K_φ0 · |P_L| − K_π`_End , K_φ0 · |P_R| − K_πr_End }.

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

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

• |w| ≤ min{K_φ1`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ }

• How to fulfil this?

• Equality for both sides: K_φ2(`<sub>2</sub> − r<sub>2</sub>) = K<sub>φ1</sub>(`₁ − r₁)

• Good choice for both sides!

• Defines a curve!

• We start with A = K_φ0(`₀ − r₀)

• Parametrisation!

A = K

(`

− r

)

• Hyperbola: ^X_a2² − ^Y_b2² = 1, l − r = 2a, 2c, a² + b² = c²

• Circle: X² + (Y − x)² = z², r = z, (0, x)

(0,0)

1 2

z x

l(p)

v_r v_l

φ p

r(p)

φ 2

π −φ

Intersection with circle and hyperbola

• 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)

v_r v_l

φ p

r(p)

φ 2

π−φ

Opt. strat. for opening angle π ≥ ϕ

≥ π/2!

Intersection: Verification by insertion!

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

Opt. strat. for opening angle π ≥ ϕ

≥ 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

Opt. strat. for opening angle π ≥ ϕ

≥ 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`<sub>1</sub> − K<sub>φ2</sub>`₂ , K_φ1r₁ − K_φ2r₂ }

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

Experiementally!

Opt. strat. opening angle 0 ≤ ϕ

≤ π/2!

• The same approach

• But independent from the angle

• Dominated by factor K_π/2 = √ 2

• Require: w ≤ min{ √

2(`<sub>1</sub> − `₂) , √

2(r₁ − r₂) }.

• Equality: `<sub>1</sub> − `₂ = r₁ − r₂

• Current angular bisector: Hyberbola!

p₁

φ₂

r₁ p₂ w

`₂

r₂ v_`

v_r

φ₁

`₁

Opt. strat. opening angle 0 ≤ ϕ

≤ 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.

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_{`</sub><sup>0</sup> or v<sub>r</sub><sup>0</sup> instead of v<sub>`} 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.