Objective: To get a feel for big-oh notation by analyzing algorithms as well as timing them (Part A), and gain
some experience writing Python classes.
Informal Big-oh (and Big-Theta) Definition: As the size of a computational problem grows (i.e., more data),
we expect our program to run longer, but this run-time growth is not necessarily linear. Big-oh notation gives us
an idea how our program’s run-time will grow with respect to its problem size on larger data.
This might seem like a lot of mathematical mumbo-jumbo, but knowing an algorithms big-oh notation can help us
predict its run-time on large problem sizes. While running a large size problem, we might want to know if we
have time for a quick lunch, a long lunch, a long nap, go home for the day, take a week of vacation, pack-up the
desk because the boss will fire you for a slow algorithm, etc.
For example, consider the following algorithm:
result = 0
for r in range(n): ← loops n times
for c in range(n): ← executes a total of n*n times
for d in range(n//2): ← executes a total of n*n*n/2 times
result = result + d ← executes a total of n*n*n/2 times
# end for
# end for
# end for
Clearly, the body of the inner-most loop (the “result = result + d” statement) will execute n3 /2 times, so this
algorithm is “big-oh” of n-cubed, O( n3 ). Thus, the execution-time formula with-respect-to n is:
T(n) = c n3 + (slower growing terms).
For large values of n, T(n) c n3, where c is the constant of proportionality on the fastest growing term (the
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machine dependent time related to how long it takes to execute the inner-most loop once) and T(n) represents
execution time as a function of n. If we know that T(10,000) = 1 second = c x 10,0003 , then we can predict what
T(1,000,000). First approximate c as c T(n) / n3 = 1 second / 10,0003 = 1 second / 1012 = 10-12 seconds. Since
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we are running the algorithm on the same machine, c is unchanged for the larger problem. Thus, T(1,000,000) c
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1,000,0003 = c 1018 = 10-12 seconds * 1018 = 106 seconds or about 11.6 days. (A couple weeks of vacation is
appropriate!)
To start the lab: Download and unzip the file lab2.zip from
Part A: In the folder lab2, open the timeStuff.py program in IDLE. (Right-click on timeStuff.py | Edit with
IDLE) Start it running in IDLE by selecting Run | Run Module from the menu. While it is running, answer
the following questions about each of the algorithms in timeStuff.py.
a) What is the big-oh of Algorithm 0?
Algorithm 0:
result = 0
for i in range(10000000):
result = result + i
b) What is the big-oh of Algorithm 1?
Algorithm 1:
result = 0
for i in range(n):
result = result + i
# end for
Data Structures (CS 1520) Lab 2 Name:___________________________
Lab 2 - 1
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