10-Complexity - What Makes a Good Algorithm CS/ENGRD 2110...

6/15/2011
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CS/ENGRD 2110
Object-Oriented Programming
and Data Structures
Spring 2011
Thorsten Joachims
Lecture 10: Asymptotic
Complexity and
O
What Makes a Good Algorithm?
• Suppose you have two possible algorithms or
data structures that basically do the same thing;
which is better?
• Well… what do we mean by better?
– Faster?
– Less space?
– Easier to code?
– Easier to maintain?
– Required for homework?
• How do we measure time and space for an
algorithm?
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Sample Problem: Searching
static boolean find(int[] a, int item) {
for (int i = 0; i < a.length; i++) {
if (a[i] == item) return true;
}
return false;
}
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 Determine if a sorted array of integers contains a given integer
 First solution: Linear Search (check each element)
static boolean find(int[] a, int item) {
for ( int x : a ) {
if (x == item) return true;
}
return false;
}
Sample Problem: Searching
static boolean find (int[] a, int item) {
int low = 0;
int high = a.length - 1;
while (low <= high) {
int mid = (low + high)/2;
if (a[mid] < item)
low = mid + 1;
else if (a[mid] > item)
high = mid - 1;
else return true;
}
return false;
}
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Second solution:
Binary Search
Linear Search vs Binary Search
• Which one is better?
– Linear Search is easier to
program
– But Binary Search is
faster… isn’t it?
• How do we measure to
show that one is faster
than the other
– Experiment
– Proof
• Which inputs do we use?
 Simplifying assumption #1:
 Use the size of the input
rather than the input itself
 For our sample search
problem, the input size is
n+1 where n is the array size
 Simplifying assumption #2:
 Count the number of “ basic
steps ” rather than
computing exact times
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One Basic Step = One Time Unit
• Basic step:
– input or output of a scalar
value
– accessing the value of a scalar
variable, array element, or
field of an object
– assignment to a variable,
array element, or field of an
object
– a single arithmetic or logical
operation
– method invocation (not
counting argument evaluation
and execution of the method
body)
• For a conditional, count
number of basic steps on
the branch that is executed
•