COMP252, Winter 2022. Jan 13, 2022. Due Jan 20, 2022, 4pm.

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Exercise 1. A question related to Fibonacci sequences. A Fibonacci-like sequence of integers
is defined as follows: x0 = · · · = x9 = 1, xn = xn−5 + xn−10, n ≥ 10.
(i) Show that xn = Θ(c
n) for some positive constant c and determine c.
(ii) Give an algorithm that, under the ram model, computes xn in time O(log n).
Exercise 2. Bit complexity. Give an efficient bit-level algorithm for computing 73
n
, and determine
a good upper bound for its bit complexity in O(·) format.
Exercise 3. Algorithm design: Finding the polygon that contains the origin.
Consider n lines in the plane all described by equations of the form aix+biy = 1, 1 ≤ i ≤ n. For simplicity, assume that all coefficients are strictly positive and different. Assume also that no three lines
meet in one point. The n lines partition the positive quadrant in the plane plane into a quadratic
number of polygonal regions. The goal is to find
the collection of lines that determine the polygon
that contains the origin. These are all the lines
whose union covers the border of the polygon. Assuming a ram model of computation and an oracle
for finding the intersection point of two lines, give
a recursive O(n log n) algorithm for solving this
problem. Hint: Refer to the convex hull problem
seen in class.

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COMP252, Winter 2022. Jan 13, 2022. Due Jan 20, 2022, 4pm.

COMP252, Winter 2022. Jan 13, 2022. Due Jan 20, 2022, 4pm.相关推荐

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