Lower bound: Special example

Online Motion Planning MA-INF 1314 General rays! Searching in streets!

Elmar Langetepe University of Bonn

WH: General rays, spiral search

• Lemma Worst-case: Ray is a tangent

• Lemma Ratio C(β, α) = C(α) sin(β) + cos(β)

• Ratio depends on α (angle γ(α))

π −α

a

β

γ(α) q0

p

q

s

WH: Minimize worst-case for s = q

• Lemma Solve Equation: sin(α−γ(α))^sinα = E^{cotα(2π+γ(α))}

• Optimize:

C(α) = ^|Π

p

a|+|pq⁰|

|aq⁰| :=

1

sinα·cosα + ^sinγ(α)

sin² α

E^{b(2π+γ(α))} + cot α

• Maximize: g(β) := C(α_min) sin β + cos β!

γ(α)

a

β α

q p

q0

s

Optimizing the spiral: Theorem

• Ratio: C(α) for s = q⁰ minimal for α = 1.4575 . . .

• C(α) = 22.4908 . . .Theo.

• Adversary choose β for max.

D(β, α) = C(α) sin(β) + cos(β)

• For α = 1.4575 . . .

choose β = 1.526363 . . .

D(β, C(α)) = 22.51306056 . . . Kor.

γ(α)

a

β α

δ q p

q0

s

Lower bound: Special example

• General problem of lower bounds

• Special example: Searching for a special ray in the plane

• Cross R and detect s

• Special case: No move to s

• Alpern/Gal (Spiral search):

Ratio C = 17.289 . . .

• Best ratio among monotone and periodic strategies

• ”Complicated task”: There is an optimal periodic/monotone strategy

p

s R

a

Lower bound construction

• n known rays emanating from a

• Angle α = ^2π_n

• Find s on one of the rays

• Also non-periodic and non- monotone strategies

• Stragegy S: Visits the rays in some order

• Hit x_k, leave β_kx_k (β_k ≥ 1)

xk+2 βk+1xk+1

xk+1 xk

βkxk

α

Lower bound construction

• Find s on a ray visited up to β_kx_k at the last time, now at x_Jk

• Note: Any order is possible

• Worst-case, s close to β_kx_k

• Ratio: C(S)

P_Jk⁻¹

i=1

q

(β_ix_i)²−2β_ix_ix_i+1 cosγ_i,i+1+x²_i+1+(β_ix_i−x_i) β_kx_k

• Monotone/Periodic, Funktional??

xJ_k

xk+2 βk+1xk+1

xk+1 α

xk βkxk s

a

Lower bound construction

• Ratio: C(S)

P_Jk⁻¹

i=1

q(β_ix_i)²−2β_ix_ix_i+1 cosγ_i,i+1+x²_i+1+(β_ix_i−x_i)

β_kx_k

• Shortest distance to next ray:

β_ix_i sin ^2π_n

• Lower bound for

q

(β_ix_i)² − 2β_ix_ix_i+1 cosγ_i,i+1 + x²_i+1

• Lower bound: C(S) ≥ sin 2π

n

PJ_k−1

i=1 β_ix_i β_kx_k

xi+1

βixi xi

2πn βJ_kxJ_k

xJ_k

xk βkxk

Lower bound construction

• Lower bound

P_Jk⁻¹

i=1 f_i f_k

• Equals functional of standard m- ray search

• Optimal strategy:

monotone/periodic (Alpern/Gal)

• f_i =

n

n−1

ⁱ

, ratio:(n − 1)

n

n−1

ⁿ

• C(S) ≥ sin ^2π_n (n − 1)

n

n−1

ⁿ

xi+1

βixi xi

2πn βJ_kxJ_k

xJ_k

xk βkxk

Lower bound construction

• C(S) ≥ sin ^2π_n (n − 1)

n

n−1

ⁿ

n→∞lim (n − 1)

n

n − 1

ⁿ

sin 2π n = 2π e = 17.079 . . .

• Lower bound: Theorem

xi+1

βixi xi 2πn

βJkxJk

xJk

xk βkxk

Summary

• The Window-Shopper-Problem

• Optimal strategy C = 1.059 . . .: Theorem

• Interesting design technique

• Rays in general

• Lower C ≥ 2π e = 17.079 . . . (Theorem) and upper bound C = 22.51 . . . (Theorem)

• Lower bound construction

• Also a lower bound for special case with C = 17.289 . . .

Searching in street polygons

• Searching in simple ploygons, visibility

• Subclass: Streets

• Start- and target

• Target t unknown, search for t!

• Compare to shortest path to t! Comp. factor!!

π_Opt P_L s

P_R t

Formal definition

Def. Let P be a simple polygon with t and s on the boundary of P. Let P_L und P_R denote the left and right boundary chain from s to t.

P is denoted as a street, if P_L and P_R are weakly visible, i.e., for any point p ∈ P_L there is at least one point q ∈ P_R that is visible, and vice versa.

π_Opt P_L s

P_R t

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

Reasonable movements: Struktural property!

• Inner wedge is important: Between . . .

• Rightmost left reflex vertex, leftmost right reflex Vertex

• By contradiction: Assume that the goal is not there!No street!

q

P_R

v_r=v_r¹ v_l=v_l¹

v_r² t

P⁰ t

v_l e_l

v⁰_l

P_L

E(e_l)

P_R

s P_L

u

φ v_l³

v_l²

Reasonable strategies

• Always into wedge of c, v_l and v_r

• Goal visible, move directly toward it

• Cave behind v_{`</sub> or v<sub>r</sub> fully visible, no target as for q (v<sub>`} or v_r vanishes), agent moves directly to the opposite vertex

q

P_R

v_r=v_r¹ v_l=v_l¹

v²_r t

P⁰ t

v_l e_l

v_l⁰

P_L

E(e_l)

P_R P_L

u

φ v_l³

v_l²

Reasonable strategies

• Always into wedge of c, v_l and v_r

• Another vertex (for example) v_{`</sub><sup>2</sup> appears behind v<sub>`}. Change to the wedge c, v_l² and v_r

v_r=v¹_r v_l=v_l¹

v_r² t

P_L u

p

φ v_l³

v_l²

q

P_R

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_{`</sub> and t<sub>r</sub> respectively and which are finally connected by a line segment t<sub>`}t_r is called a funnel (polygon).

t_`

v_`

φ

v_r

φ₀

t_r

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. Beweis:

`· sin ^φ₂

`

`· cos ^φ₂ m

φ s

t_` t_r

Detour at least: _|π^|πS|

Opt| = ^`cos

φ

2+` sin ^φ₂

` = √

1 + sin φ.

Opt. strat. opening angle 0 ≤ ϕ

≤ π !

• 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

• Here: Start from s with angle φ₀ ≥ π/2.

• Remaining case: Exercise!

`

`· cos ^φ₂

`· sin ^φ₂

φ

t_` t_r

m

Opt. strat. opening angle π ≥ ϕ

≥ π/2!

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

• K_π = 1 and K_π-competitive opt. trategy 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

`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

`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 t_l

P_L

l₀

r₀

P_R

φ₀ p₁ φ₁

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 t_l

P_L

l₀

r₀

P_R

φ₀ p₁ φ₁

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 t_l

P_L

l₀

r₀

P_R φ₁