HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Algorithms for Radio Networks Exercise 11 Stefan Rührup

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 2 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/s 2 = a max and assume a time interval of ∆ = 1 sec. –Draw the location-velocity-diagram of the scenario.

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 3 50m Exercise 22 distance d = 50 m 20 sensors acceleration: g ≈ 10m/s 2 = a max time interval ∆ = 1 s

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 4 Acceleration bound a max Positions u,v and speed vectors u’,v’ known Maximum distance after time interval ∆ (  transmission range): Vehicular Model uncertainty due to acceleration uw velocity

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 5 Exercise 22 Location-Velocity-Diagram: y vyvy

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 6 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 INSTITUTE University of Paderborn Algorithms and Complexity 7 Given the positions u,w and the velocity bound v max 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 uw

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 8 Exercise 23 The relation between the crowdedness and the maximum number of pedestrians per square –Consider the radius 2v max ∆ for v max = 1/2 m/s and ∆ = 1 s. –Crowdedness is linear in the maximum number of pedestrians per square.

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 9 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 INSTITUTE University of Paderborn Algorithms and Complexity 10 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/n c ) 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/n c.

HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 11 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 INSTITUTE University of Paderborn Algorithms and Complexity 12 Balls into Bins

HEINZ NIXDORF INSTITUTE University 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 INSTITUTE University of Paderborn Algorithms and Complexity 14 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 o(1)