problem - question we seek an answer for, e.g., "what is the largest item in an array?"

parameters - variables with unspecified values

instance - assignment of values to parameters, i.e., the specific input to the problem

algorithm - step-by-step procedure for producing a solution

Text Notation: text uses a C++ pseudo code

Data Type	Meaning
keytype	items from an ordered set (we can compare two items for =, <, >, etc.)
index	integer used as an array index
number	either integer or real numbers
bool	a Boolean that can be assigned either "true" or "false"

Comparison Operator	C++ syntax
x = y	(x == y)
x y	(x != y)
x y	(x <= y)
x y	(x >= y)

Algorithm 1.1 Sequential Search

Problem: Is the key x in the array S of n keys?

Inputs (parameters): positive integer n, array of keys S indexed from 1 to n, and a key x.

Outputs: location, the location of x in S (0 if x is not in S).

void seqsearch ( int n,

const keytype S[ ],

keytype x,

index & location )

{

location = 1;

while (location <= n && S[location] != x)

location++;

if (location > n)

location = 0;

}

Algorithm 1.2 Add Array Members

Problem: Add all the numbers in the array S of n numbers.

Inputs: positive integer n, array of numbers S indexed from 1 to n.

Outputs: sum, the sum of the numbers in S.

number sum ( int n, const keytype S[ ] )

{

index i;

number result;

result = 0;

for (i = 1; i <= n; i++)

result = result + S [ i ];

return result;

}

basic operation - fundamental operation in the algorithm (i.e., operation done the most)

We want to derive a function for the number of times that the basic operation is performed related to the problem size.

problem size - input size. For algorithms involving arrays, the problem size is the number of elements.

For Add Array Members algorithm, what is the basic operation?

What is the problem size?

Every-case time complexity analysis, T(n) = c₁ + c₂ n = 100 + 10 n

What determines the length of c₁ and c₂?

When n is "sufficiently large," the 10 n term dominates. For what value of n, does 10 n = 100?

Big-oh Definition - asymptotic upper bound

For a given complexity function f(n), O( f(n) ) is the set of complexity functions g(n) for which there exists some positive real constant c and some nonnegative integer N such that for all n N,

T(n) = c₁ + c₂ n = 100 + 10 n is O( n ).

"Proof": Pick c = 110 and N = 1, then 100 + 10 n 110 n for all n 1.

100 + 10 n 110 n

100 100 n

1 n

Problem with big-oh:

If T(n) is O(n), then it is also O(n²), O(n³), O(n³), O(2ⁿ), .... since these are also upper bounds.

Omega Definition - asymptotic lower bound

For a given complexity function f(n), ( f(n) ) is the set of complexity functions g(n) for which there exists some positive real constant c and some nonnegative integer N such that for all n N,

T(n) = c₁ + c₂ n = 100 + 10 n is ( n ).

"Proof": We need to find a c and N so that the definition is satisfied, i.e.,

100 + 10 n c n for all n N.

What c and N will work?

Theta Definition - asymptotic upper and lower bound, i.e., a "tight" bound or "best" big-oh

For a given complexity function f(n), ( f(n) ) is the set of complexity functions g(n) for which there exists some positive real constants c and d and some nonnegative integer N such that for all n N,

T(n) = c₁ + c₂ n = 100 + 10 n is ( n ).