Algorithms for Radio Networks Exercise 11

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Algorithms for Radio Networks

Exercise 11

Stefan Rührupsr@upb.de

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 22

• Consider a multistory building of height 50 m. At each floor of height 2.5 m a sensor node is attached to the wall. Now, every 1 second a sensor is dropped from the top of the building.

–Calculate the transmission radius of the falling sensors which is needed to maintain a connection to the static nodes. Use the acceleration bounded (vehicular) mobility model with acceleration g ≈ 10m/s2 = amax and assume a

time interval of ∆ = 1 sec.

–Draw the location-velocity-diagram of the scenario.

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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50m

Exercise 22

distance d = 50 m20 sensors

acceleration: g ≈ 10m/s2 = amax time interval ∆ = 1 s

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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•Acceleration bound amax

•Positions u,v and speed vectors u’,v’ known

•Maximum distance after time interval ∆ ( transmission range):

Vehicular Model

uncertainty due to acceleration

u w

velocity

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 22

• Location-Velocity-Diagram:

y

vy

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 23

• Consider a quadratic area which is divided into n squares of size 1m x 1m. Now, n pedestrians are placed randomly and uniformly in this area.

–What is the expected number of pedestrians per square?

–What is the relation between the crowdedness and the maximum number of pedestrians per square?

–What is the probability that exactly k pedestrians are in one square?

–What is the probability at least k pedestrians are in one square?

–For which k is this probability smaller than 1/n?

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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•Given the positions u,w and the velocity bound vmax

•Maximum distance after time interval ∆ ( transmission range):

•Crowdedness: Maximum number of nodes that can collide with a given node in time span [0,Δ]:

Velocity bounded (pedestrian) model

uncertainty

u w

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 23

• The relation between the crowdedness and the maximum number of pedestrians per square

–Consider the radius 2vmax ∆

for vmax = 1/2 m/s and ∆ = 1 s.

–Crowdedness is linear in the maximum number of pedestrians per square.

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Exercise 23

• Random placement:

–What is the probability that at least k pedestrians are in one square?

–For which k is this probability smaller than 1/n?

• Balls into Bins:

–Assume n balls are thrown sequentially into n bins (randomly and uniformly distributed)

–What is the maximum nuber of balls per bin?

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Balls into Bins

Theorem: The probability that at least t log n/log log n balls fall

into a single bin is at most O(1/nc) for constants t and c.

With high probability (P = 1 - 1/n(1)) at most O(log n/log log n)

balls fall into one bin.

Proof: • Determine the Probability (generally) that at least k out of n

Balls fall into a certain bin.• Consider the case that at least k out of n balls fall into any

of the n bins• Choose k such that this holds with probability 1/nc.

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Balls into Bins

Probability that exactly k balls fall into a certain bin:

Probability that at least k of n balls fall into a certain bin:

follows from Sterling´s formula:

We use

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Balls into Bins

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

13

Balls into Bins

Probability that at least k of n balls fall into a certain bin:

For which k is the probability

For which k holds ?

We only consider the dominant terms:

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

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Balls into Bins

For which k holds ? Inverse of k ln k?

So, we choose k as follows:

...

Probability that at least k of n balls fall into a certain bin:

Probability that at least k of n balls fall into any of the n bins:

i.e. for a constant c = t - 1 + o(1)