1. CMSC 203, Section 0401 Discrete Structures Fall 2004 Matt Gaston

2. Algorithms Ch

3. Definition Book definition: Better definition (?):
Algorithm: a finite set of precise instructions for performing a computation or for solving a problem
Better definition (?):
Algorithm: An algorithm is a finite set of unambiguous, executable instructions that directs a terminating activity.

4. Algorithm Features Input Output Definiteness Correctness Finiteness
Effectiveness
Generality

5. Linear Search
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
How many steps? Worse case? On average?

6. Binary Search
procedure binary search (x:integer, a1, a2, , an: increasing integers)
i := 1
j := n
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

7. Growth of Functions Big-O Notation: C and k are called “witnesses
Let f and g be functions from the set of integers to the set
of integers or the set of real numbers to the set of real
numbers. We say that f(x) is O(g(x)) if there are constants
C and k such that
| f(x) |  C | g(x) |
whenever x > k.

8. Pictorial: Big-O Notation
C g(x)
f(x)
g(x) k

9. Example
7x2 = O(x3)

10. Combinations of Functions
f(x) = anxn + an-1xn a1x + a0
f(x) = O(xn)
(f1 + f2)(x) = O(max( g1(x), g2(x) ))
(f1f2)(x) = O(g1(x)g2(x))

11. Big-Omega Notation
Let f and g be functions from the set of integers to the set
of integers or the set of real numbers to the set of real
numbers. We say that f(x) is (g(x)) if there are constants
C and k such that
| f(x) |  C | g(x) |
whenever x > k.

12. Big-Theta Notation f(x) is of order g(x)
Let f and g be functions from the set of integers to the set
of integers or the set of real numbers to the set of real
numbers. We say that f(x) is (g(x)) if f(x) is O(g(x)) and
f(x) is (g(x)).

13. Bubble Sort procedure bubble sort (a1, a2, . . . , an)
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 }

14. 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 := aj-k-1
ai := m
end {a1, a2, , an are sorted }

15. Make Change!

16. Understanding Complexity
n!, 2n, n2, n log(n), n, log(n), 1
Tractable/Intractable
Solvable/Unsolvable – Halting Problem
Class P, Class NP
NP-Complete