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Chapter 3
3.3 Complexity of Algorithms
– Time Complexity
– Understanding the complexity of Algorithms
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Complexity of Algorithm
• Computational Complexity (of the Algorithm)
• Time Complexity: Analysis of the time required.
• Space Complexity: Analysis of the memory
required.
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Time Complexity
• Example 1: Describe the time complexity of
Algorithm 1 of section 3.1 for finding the maximum
element in a set (in terms of number of comparisons).
• Algorithm 1: Finding the maximum element in a
finite sequence.
procedure max(a1, a2, . . . ,an: integers)
max := a1
for i: =2 to n
if max < ai then max := ai
{max is the largest element}
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• Example 2: Describe the time complexity of the
linear search algorithm.
• Algorithm 2 : the linear search algorithm
procedure linear search (x: integer, a1, a2, …,an: distinct integers)
i :=1;
while ( i ≤n and x ≠ ai)
i := i + 1
If i ≤ n then location := i
Else location := 0
{location is the subscript of the term that equals x , or is 0 if x is
not found}
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• Example 3: Describe the time complexity of the binary search
algorithm in terms of the number of comparisons used .
(and ignoring the time required to compute m= (i  j ) / 2 in
each iteration of the loop in the algorithm)
• Algorithm 3: the binary search algorithm
Procedure binary search (x: integer, a1, a2, …,an: increasing integers)
i :=1 { i is left endpoint of search interval}
j :=n { j is right endpoint of search interval}
While i < j
begin
m := (i  j ) / 2
if x > am then i := m+1
else j := m
end
If x = ai then location := I
else location :=0
{location is the subscript of the term equal to x, or 0 if x is not found}
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• Example 4: Describe the average-case performance
of the linear search algorithm, assuming that the
element x is in the list.
• Example 5: What is the worst-case complexity of
the bubble sort in terms of the number of
comparisons made?
• ALGORITHM 4: The Bubble Sort
procedure bubble sort (a1, a2, …,an: real numbers with n ≥2)
for i := 1 to n-1
for j := 1 to n- i
if aj > aj+1 then interchange aj and aj+1
{a1, a2, …,an is in increasing order}
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• Example 6: What is the worst-case complexity of the
insertion sort in terms of the number of comparisons
made?
• Algorithm 5: The Insertion Sort
procedure insertion sort (a1, a2, …,an: real numbers with n ≥2)
for j := 2 to n
begin
i := 1
while aj > ai
i := i + 1
m := aj
for k :=0 to j-i-1
aj-k := a j-k-1
ai := m
end {a1, a2, …,an are sorted}
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Understanding the complexity of
Algorithms
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• Solvable (in polynomial time, or in exponential time)
• Tractable: A problem that is solvable using an
algorithm with polynomial worst-case complexity.
• Intractable: The situation is much worse for
problems that cannot be solved using an algorithm
with worst-case polynomial time complexity. The
problems are called intractable.
• NP problem.
• NP-complete problem.
• Unsolvable problem: no algorithm to solve them.
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• Big-O estimate on the time complexity of an algorithm provides
an upper, but not a lower, bound on the worst-case time
required for the algorithm as a function of the input size.
• Table 2 displays the time needed to solve problems of various
sizes with an algorithm using the indicated number of bit
operations. Every bit operation takes nanosecond. Times of
more than 10100 years are indicated with an asterisk.
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