Chapter 9
Efficiency of Algorithms
9.3
Efficiency of Algorithms
Efficiency of Algorithms
• Two aspects of algorithm efficiency are important:
– 1. the amount of time required for execution
– 2. amount of memory space needed when it runs.
• Algorithms are usually analyzed in terms of their best case,
worst case, and average.
• How can the time efficiency of an algorithm be calculated?
– Have to factor in the input size to the algorithm
– Nature of the input data
Efficiency of Algorithms
• Because the time efficiency can be influenced by many
physical parameters of a computing device, i.e.
processor speed, memory size, multi-core, etc., a
method of analysis must be used that is not a factor of
the processing platform.
• Evaluation of algorithms can be based on the number
of elementary operations required by the algorithm.
– Elementary operations are addition, subtraction,
multiplication, division and comparison.
– All elementary ops are viewed as taking 1 time unit on any
system.
Example
• Consider algorithms A & B designed to accomplish the
same task. For input size of n
– A requires 10n to 20n elementary operations.
– B requires 2n2 to 4n2 elementary operations.
• Which algorithm is more efficient?
– for n ≤ 10, 2n2 < 20n, and hence, B is better
– for n > 10, 20n < 2n2, and in this case A is better
• The answer is dependent on the size of the input. For a
small n B is better, but for a larger inputs A wins. It is
important to understand the constraints on the order.
Definition
• Let A be an algorithm
1. Suppose the number of elementary ops performed when
A is executed for an input of size n depends on n alone
and not on the nature of the input data; say it equals
f(n). If f(n) is Θ(g(n)) then, A is of order g(n).
2. Suppose the number of elementary operations
performed when A is executed for an input of size n
depends on the nature of the input data as well as on n.
1.
2.
Let b(n) be the minimum number of elementary operations
required to execute A for all possible input sets of size n. If b(n)
is Θ(g(n)), we say A has a best case order of g(n).
Let w(n) be the maximum number of elementary operations
required to execute A for all possible input sets of size n. If w(n)
is Θ(g(n)), we say A has a worst case order of g(n).
Time Comparisons of Algorithm Orders
Example
• Consider the following algorithm segment
p =0, x=2
for i = 2 to n
p = (p + i) * x
next I
• Compute the actual number of elementary ops?
–
–
–
–
1 multi, 1 add for each iteration.
2 ops per iteration.
num of iteration = n – 2 + 1 = n-1.
num of elementary ops = 2(n-1)
• Find an order from among the set of power functions
– by theorem on polynomial orders, 2n – 2 is Θ(n)
– and thus, segment is Θ(n).
Example
• Consider a segment with a nested loop (loop inside of a loop)
s=0
for i=1 to n
for j= 1 to i
s=s + j * (i-j+1)
next j
next i
• Compute the number of elementary ops.
–
–
–
–
2 adds, 1 multi, and 1 minus (4 ops) for each iteration
inside loop (j) iterates i=1 1, i=2 2, i=3 3 … i=n n times
inside loop: 1 + 2 + 3 + … + n = n(n+1)/2
num of ops = 4 * n(n+1)/2 = 2*n(n+1) = 2n2 + 2n
• Find the order among the set of power functions
– 2n(n+1) = 2n2 + 2n is Θ(n2),
– hence, segment is Θ(n2)
Example
• Consider the segment where the floor function is used.
for i = ⎣n/2⎦ to n
a=n–I
next I
• Compute the actual number of subtractions performed.
– 1 subtraction for each iteration.
– loop iterates n - ⎣n/2⎦ + 1 times.
•
•
•
•
n is even: ⎣n/2⎦ = n/2
n – n/2 + 1 = (2n – n + 2)/2 = (n+2)/2
n is odd: ⎣n/2⎦ = (n-1)/2
n – (n-1)/2 + 1 = [2n – (n-1) + 2]/2 = (n+3)/2
• Find an order for this segment
– (n+2)/2 is Θ(n) and hence, segment is Θ(n)
Sequential Search
• Sequential search occurs when every element
in the list is compared to a particular x until
either a match is found or the end of the list.
Example
• Find the best and worst case orders for
sequential search.
– Best case: the best case occurs when the first
element of the list is the item that is being
pursued (searched for). The search takes 1
comparison. Θ(1)
– Worst case: the element being searched for is the
last element or not in the last (these two
situations are equal). The search takes n
comparisons. Θ(n)