For small input instances, n 30, you may output your results to the command window. Larger input instances must be written to a file. In either case, you may need to produce an output file for trace runs or asymptotic performance analysis.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

 

For small input instances, n 30, you may output your results to the command window. Larger input instances must be written to a file. In either case, you may need to produce an output file for trace runs or asymptotic performance analysis.

1. Closest Pairs [30 points]

There are several practical applications of this algorithm, such as, moving objects in a game, classification algorithms, computational biology, computational finance, genetic algorithms, and N -body simulations.

• [5 points] Construct a brute-force algorithm for finding the closest pair of points in a set of n points in a two-dimensional plane and show the worst-case big-O estimate for the number of operations used by the algorithm. Hint: First determine the distance between every pair of points and then find the points with the closest

 

• [5 points] Implement the brute-force algorithm from part (a).

 

• [10 points] Given a set of n points (x1, y1), ..., (xn, yn) in a two dimensional plan, where the distance be- tween two points (xi, yi) and (xj, yj) is measured by using the Euclidean distance (xi xj)2  + (yi      yj)2, design an algorithm to find the closest pair of points that can be found in an efficient way and the worst- case big-O

 

• [10 points] Implement the algorithm from part (c) with the following methods:
  1. [5 points] The distance of the closest pair of points. Return the distance between closest single pair

• [5 points] The distance of the closest m pairs of points, where m (n-1). Return the distances between the closest m pairs of points.

2. Deterministic Turing Machine (DTM) [70 points/10 available extra credit points]

In this problem you will implement four variants of a DTM. A DTM consists of the following:

• A finite state control,
• A infinitely long tape divided into tape squares, and
• A read-write
  • [30 points] Design and code software implementing a generic DTM. Your generic DTM should have a finite state control mechanism that can be tailored for each parts (b) - (e). Your implementation must address each of the DTM components: a finite state control, an infinitely long tape divided into tape squares, and a read-write
  • [10 points] Tailor your DTM from (a) to implement
    1. [5 points] Using the input  tape contents produce trace runs for your
    2. [5 points] Define at least four other input tapes and produce trace runs for your
  • [15 points] Tailor your DTM from (a) to implement binary addition - The tape will need to handle variable length binary inputs, g., tape values of 101100101 add 101 or add 101101101.

 

Table 2: Binary Addition Rules

 

Operation x + xt	Result	Carry
0 + 0	0	0
0 + 1	1	0
1 + 0	1	0
1 + 1	0	1

 

• [15 points] Tailor your DTM from (a) to implement binary subtraction - The tape will need to handle variable length binary inputs, g., tape values of 101100101 subtract 101 or subtract 101101.

 

Table 3: Binary Subtraction Rules

 

Operation x − xt	Result	Borrow
0 − 0	0	0
0 − 1	1	1
1 − 0	1	0
1 − 1	0	0

 

• Extra Credit [10 points] Tailor your DTM from (a) to implement binary multiplication - The tape will need to handle variable length binary inputs, g., tape values of 101100101 multiply 101 or multiply 1011.

 

Table 4: Binary Multiplication Rules

 

Operation x × xt	Result
0 × 0	0
0 × 1	0
1 × 0	0
1 × 1	1

 

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