1. The Efficiency of Algorithms
Chapter 4
Adapted from Pearson Education, Inc.

2. Contents Motivation Measuring an Algorithm’s Efficiency
Counting Basic Operations
Best, Worst, and Average Cases
Big Oh Notation
The Complexities of Program Constructs
Picturing Efficiency
The Efficiency of Implementations of the ADT Bag
An Array-Based Implementation
A Linked Implementation
Comparing the Implementations
Adapted from Pearson Education, Inc.

3. Objectives Assess efficiency of given algorithm
Compare expected execution times of two methods
Given efficiencies of algorithms
Adapted from Pearson Education, Inc.

4. Motivation - Even a simple program can be inefficient
Three algorithms for computing ∑ i
Trace each with n=5 and time yourself
Trace each with n=10 and time yourself
What do you notice after doubling n?
When doubling n
The time remained fixed
When doubling n
The time was doubled
When doubling n
The time was quadrupled
Adapted from Pearson Education, Inc.

5. Measuring an Algorithm’s Efficiency
Complexity
Space requirements
Time requirements
Other issues for best solution
Generality of algorithm
Programming effort
Problem size – number of items program will handle
Growth-rate function
Adapted from Pearson Education, Inc.

6. Counting Basic Operations
Linearly related to n
Quadratically
related to n
Independent
of n
Adapted from Pearson Education, Inc.

7. Counting Basic Operations
Linearly related to n
Quadratically
related to n
Independent
of n
Adapted from Pearson Education, Inc.

8. (a) Typical growth-rate functions – (b) Effect of doubling the size of n – (c) An example
time to process
106 1MIPS
(b)
(c)
Adapted from Pearson Education, Inc.

9. Best, Worst, and Average Cases
Some algorithms depend only on size of data set
Other algorithms depend on nature of the data
Best case search when item at beginning
Worst case when item at end
Average case somewhere between
Adapted from Pearson Education, Inc.

10. Big Oh Notation A function f(n) is of order at most g(n) that is,
We can say f(n) = (n2+n)/2
is of order g(n) = n2
f(n) has a Big Oh of n2  O(n2)
This means, with c=1 and N=1
(n2+n)/2 ≤ 1∙ n2 for all n ≥ 1
A function f(n) is of order at most g(n) that is,
f(n) is O(g(n)) — if :
A positive real number c
and positive integer N exist
such that f(n) ≤ c ∙ g(n)
for all n ≥ N.
That is, c ∙ g(n) is an upper bound on f(n) when n is sufficiently large.
Adapted from Pearson Education, Inc.

11. Big Oh Identities O(k g(n)) = O(g(n)) for a constant k
O(g1(n)) + O(g2(n)) = O(g1(n) + g2(n))
O(g1(n)) x O(g2(n)) = O(g1(n) x g2(n))
O ( g1(n) + g2(n) gm(n) ) = O ( max[ g1(n), g2(n), . . ., gm(n) ] ) =
max [ O(g1(n)), O(g2(n)), , O(gm(n)) ]
Adapted from Pearson Education, Inc.

12. Big Oh Identities O(k g(n)) = O(g(n)) for a constant k
O(g1(n)) + O(g2(n)) =
O(g1(n) g2(n))
O(g1(n)) x O(g2(n)) =
O( g1(n) x g2(n) )
O ( g1(n) + g2(n) gm(n) ) = O (max[ g1(n), g2(n), , gm(n) ] ) =
max[O(g1(n)), O(g2(n)), , O(gm(n)) ]
O(3 g(n)) = O(g(n))
O(n) + O(3n2) =
O(n + 3n2)
O(n) x O(3n2) =
O(n x 3n2)
O (n + 3n2 + 5logn+n ) = O ( max[ n, 3n2, 5logn+n ] ) =
max [ O(n), O(3n2), O(5logn+n) ]
= O(n+n2)
= O( max(n, n2))
= O(n2)
= O(3n3)
= O(n3)
= O(2n+3n3+5logn)
= max(O(n),O(n3),O(logn))
= O(n3)
Adapted from Pearson Education, Inc.

13. Figure 4-6 An O(n) algorithm
Adapted from Pearson Education, Inc.

14. Figure 4-7 An O(n2) algorithm
Adapted from Pearson Education, Inc.

15. Figure 4-8 Another O(n2) algorithm
Adapted from Pearson Education, Inc.

16. Complexities of program constructs
Time Complexity
Consecutive segments
If that choses between if- part and else-part
A loop that iterates m times
Add growth-rates  max
O(cond) +
max(O(if-part),O(else-part))
m * (O(cond) + O(body))
Adapted from Pearson Education, Inc.

17. Array Based Implementation (fixed size)
Adding an entry to a bag, an O(1) method,
Adapted from Pearson Education, Inc.

18. Array Based Implementation (fixed size)
Adding an entry to a bag, an O(1) method,
O(1)
O(1)
+O(1)
= O(1)
O(1)
Max(O(1),O(1)) = O(1)
O(1)
O(1)
O(1)
O(1) +O(1) +O(1)
= O(1)
Adapted from Pearson Education, Inc.

19. Array Based Implementation
Searching for an entry, O(1) best case, O(n) worst or average case  O(n) method overall
Adapted from Pearson Education, Inc.

20. Array Based Implementation
Searching for an entry, O(1) best case, O(n) worst or average case  O(n) method overall
The size of the bag, is the size of the problem
 numberOfEntries is our n
This loop executes numberOfEntries times, i.e. n times
O(1)
n * (O(1) + O(1))
= n * O(1)
= O(n)
O(1)
O(1)
+O(1)
=O(1)
O(1)
O(1)
O(1)
O(1) +O(n) +O(1)
= O(n)
Adapted from Pearson Education, Inc.

21. A Linked Implementation
Adding an entry to a bag, an O(1) method,
O(1)
+O(1)
= O(1)
Adapted from Pearson Education, Inc.

22. A Linked Implementation
Searching a bag for a given entry, O(1) best case, O(n) worst case  O(n) overall
Adapted from Pearson Education, Inc.

23. A Linked Implementation
Searching a bag for a given entry, O(1) best case, O(n) worst case  O(n) overall
The size of the bag, is the size of the problem
This loop executes n times
O(1)
n * (O(1)+O(1))
= n * O(1)
= O(n)
O(1)
O(1)
O(1)
O(1) +O(n) +O(1)
= O(n)
Adapted from Pearson Education, Inc.

24. Time efficiencies of the ADT bag operations
Adapted from Pearson Education, Inc.

25. End
Chapter 4
Adapted from Pearson Education, Inc.