Solved–Lab 3– Solution

Description

1. The connection of the road network is speci ed as a matrix M ( lename is \road.txt”). There are 10 sites indexed as (0; 1; : : : ; 9). Entry M(i; j) denotes the length of the road between site i

and j in kms (the roads are uni-directional). The roads are uni-directional (i.e., there could be a road from A to B, but not from B to A). If the entry is 0, it means no connection. Vehicles depart from a source and then take a path ( lename \vehicle.txt”). The time of departure (in minutes) is given in lename \time.txt”, n^th vehicle departs at time instance n. The speed (in kmph) of the vehicle in a road is given by the formula e^0:5x=(1 + e^0:5x) + 15=(1 + e^0:5x), where x is the number of vehicles ahead in a given road. If there are n vehicles in the road, then for the rst vehicle x = 0, second vehicle x = 1, and for the last vehicle x = n 1. The time taken

to travel the road is dist/speed. Implement a function that takes time as input and reports

the position of various vehicles at that given time. [20]

2. There is an under-powered car stuck in the bottom of a 1-dim valley. It needs to nd its way to the top. The car has three actions namely A = f 1; 0; +1g which means accelerate backward, no acceleration and accelerate forward, respectively. A chosen action is applied for a xed duration of time, after which a new action is chosen. The ranges for position and velocity are

• 1:2; 0:5] and [ 0:07; 0:07], respectively. The car is needs to reach the top on the right, i.e., position of 0:5. The dynamics is according to the equations:

v_n+1 = v_n + 0:001a_n 0:0025 cos(3p_n)

pn+1 = pn + vn
(a) Implement a function which takes in the current position and action and outputs the
position and velocity at the next step.	[25]

2. Implement a random agent which at each time instant chooses one of the three actions uniformly at random. Plot f(p_n; v_n); n 1g produced by the random agent. Take the

initial position to be 0:5, and initial velocity to be 0. [15]

(c) Implement another agent which accelerates in the direction of the current velocity. Plot f(p_n; v_n); n 1g produced by this agent. Take the initial position to be 0:5, and initial

velocity to be 0. [10]

3. Read the text le \speeches.txt”

(a)	Compute the next word probabilities.	[10]
(b)	Based on the next word probabilities, write down a code that will produce random text
	of length 5000.	[10]
(c)	Compute the next word probability from the previous task and show that it converges to
	the next word probabilities calculated from the original text.	[10]

End of paper