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Algorithmic Cost and Complexity
Algorithms can be analyzed from two different points of view:
time
or
space
.
For the most part we are more interested in time than space.
o
Time: each instruction takes time.
How many instructions are
executed by the algorithm?
How fast does the algorithm run?
What
affects the run-time of the algorithm?
o
Space: data structures require space.
What kind of data structures
are used? How does the choice of data structure affect the run-
time?
Measures of work:
Worst case performance, best case performance,
and average performance.
o
What situations will produce worst case performance?
What
situations will produce best case performance?
What is average
performance?
How is average performance determined?
Algorithm Analysis
In some of the labs this term and in class we have looked at techniques where
we determined, essentially the number of times a statement in a loop is
executed.
Remember, that baring some type of looping statement, the code in
a program (algorithm) is executed sequentially.
This means that without loops,
each statement is executed exactly one time and the running time of the
algorithm is very easy to establish.
Loops cause iteration and iteration
increases the running time depending on how much iteration occurs.
Therefore,
we need to know, for the statements inside the loop, how many times they are
executed.
Consider the two code segments shown below:
What is the number of addition operations performed in these two code
segments?
Computational Complexity -
1
Computational Complexity
Segment #1:
grandtotal = 0;
for (k = 0; k < n – 1; ++k) {
rows[k] = 0;
for (j = 0; j < n – 1; ++j)
rows[k] = rows[k] + matrix[k][j];
grandtotal = grandtotal + matrix[k][j];
}
}

This
** preview**
has intentionally

For segment #1 the number of addition operations is 2n
2
while for segment
#2 the value is n
2
+ n.
Assume that we are working with a hypothetical computer that requires 1
microsecond (10
-6
) seconds to perform an addition.
If the value of
n
= 1000
the segment #1 would require just over 2 seconds to execute
[(2*(1000)
2
)inst] * 10
-6
sec/inst = 2 seconds.
On the other hand, segment
#2 would require just over 1 second [(1000)
2
+ 1000] inst * 10
-6
sec/inst =
1.001 seconds.
If the value of
n
is increased to 100,000 then code
segment #1 would require about 6 hours and code segment #2 would
require about 3 hours.
The table shown below gives the Big-Oh complexity

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