Post on 08-Jul-2020
transcript
MA 252: Data Structures and Algorithms Lecture 3
http://www.iitg.ernet.in/psm/indexing_ma252/y12/index.html
Partha Sarathi Mandal
Dept. of Mathematics, IIT Guwahati
Asymptotic Notation
• Convenient for describing worst-case running time function T(n) • Big-Oh
– f(n) is O(g(n)) if f(n) is asymptotically less than or equal to g(n)
• big-Omega – f(n) is W(g(n)) if f(n) is asymptotically greater than or equal to g(n)
• big-Theta – f(n) is Q(g(n)) if f(n) is asymptotically equal to g(n)
• little-oh – f(n) is o(g(n)) if f(n) is asymptotically strictly less than g(n)
• little-omega – f(n) is w(g(n)) if is asymptotically strictly greater than g(n)
P. S. Mandal, IITG
big-oh (O)-notation
big-Oh (O)-notation:
f(n) is O(g(n)) if there is a constant c > 0
and an integer constant n0>=1 such
that 0=<f(n) =< c g(n) for n >= n0
• an upper bound that is asymptotically tight.
P. S. Mandal, IITG
big-Omega (W)-notation
big-Omega (W)-notation:
f(n) is W(g(n)) if there is a constant c > 0
and an integer constant n0>=1 such that
f(n) >= c g(n) >=0 for n >= n0
• a lower bound and that is asymptotically
tight. P. S. Mandal, IITG
big-Theta (Q)-notation
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Example Show that:
P. S. Mandal, IITG
Little-oh & Little-omeha • little-oh (o)-notation
– f(n) is o(g(n)) if, for any constant c > 0, there is an integer constant n0 >= 0 such that 0=<f(n) < c g(n) for n >= n0
– an upper bound that is not asymptotically tight.
• little-omega (w)-notation – f(n) is w(g(n)) if, for any constant
c > 0, there is an integer constant n0 >= 0 such that f(n) > c g(n) >=0 for n >= n0
– a lower bound that is not asymptotically tight.
P. S. Mandal, IITG
The family of Bachmann–Landau notations
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Comparison of Functions
• Transitivity:
• Reflexivity:
• Symmetry:
• Transpose Symmetry:
P. S. Mandal, IITG
Exercise1: Give a big-Oh characterization
Algorithm Ex1(A, n)
Input an array X of n integers
Output the sum of the elements in A
s A[0]
for i 0 to n 1 do
s s + A[i]
return s
P. S. Mandal, IITG
Exercise2: Give a big-Oh characterization
Algorithm Ex2(A, n)
Input an array X of n integers
Output the sum of the elements at even cells in A
s A[0]
for i 2 to n 1 by increments of 2 do
s s + A[i]
return s
P. S. Mandal, IITG
Exercise3: Give a big-Oh characterization
Algorithm Ex3(A, n)
Input an array X of n integers
Output the sum of the prefix sums A s 0
for i 0 to n 1 do
s s + A[0]
for j 1 to i do
s s + A[j]
return s
P. S. Mandal, IITG
Merge sort
• divide-and-conquer paradigm:
– Split the list in two parts, sort both, then merge the sorted results to get an overall sorted list.
P. S. Mandal, IITG
Merge sort Algorithm
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Merging two sorted arrays
P. S. Mandal, IITG
Analyzing merge sort
P. S. Mandal, IITG
Recurrence for merge sort
• We shall usually omit stating the base case when T(n) = Θ(1) for sufficiently small n, but only when it has no effect on the asymptotic solution to the recurrence.
P. S. Mandal, IITG
How to solve Recurrence equation ?
• Solve T(n) = 2T(n/2) + cn, where c > 0 is constant.
P. S. Mandal, IITG