Algorithms worksheet
Work in groups or alone and use the time today to answer all of the questions below.
You may use your books or notes from class, and do not turn in this assignment: it is yours
to study for the exam next week. Recall that:
Defn: An algorithm consists of three stages: input, procedure, and output. All arith-
metic operations (+,−,×,÷) and comparisons (≤, <, =) must occur in the procedure
stage. The complexity of the algorithm is the total number of arithmetic operations and/or
comparisons used for the algorithm to run for an input in the worst case.
Defn: A recurrence relation an=ran−1+san−2has solution
an=(c1xn
1+c2xn
2if x16=x2,
c1xn
0+c2nxn
0if x1=x2= 0,
where x1, x2are the roots of the characteristic polynomial
x2−rx −s= (x−x1)(x−x2)=0,
and c1, c2are unknown constants which are solved for using the initial conditions of the
recursively defined sequence.
Defn: Alist of elements of a set Ais an ordered n-tuple, or element of An=A×A× · · · × A
| {z }
ntimes
.
1. Find a recursive formula and a closed formula for the sequence -1, 17, 11, 113, 179, 857 . . .
2. Write an algorithm that outputs the nth term of the sequence above using the recursive
definition of the sequence. Now write an algorithm that outputs the nth term of the
sequence using the closed formula using only arithmetic operations. You may not use
”exponents” but only multiplication ×. Find the complexity of each algorithm.
3. Write an algorithm takes takes as input a list of real numbers L= (a1, . . . , an) and outputs
(1) the number of times the number 5 appears in the list, and (2) the number of times an
entry is between 0 and 5, non-inclusive.
4. Describe the relationship between the input and the output of the algorithm below. In
other words, what does the algorithm do? What is the maximum value of sif d=n= 5?
What is the complexity of the algorithm for fixed d? for fixed n?
Input: A natural number dand a list L= (a1, . . . , an) of real numbers.
Procedure: Initialize: s=0.
Step 1:
For i= 1, . . . ,n
For j= 1, . . . ,d
If ai=j, then set sto s+ 1.
Output: s.