1 r. johnsonbaugh, discrete mathematics chapter 4 algorithms

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R. Johnsonbaugh,

Discrete Mathematics

Chapter 4

Algorithms

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4.1 Introduction An algorithm is a finite set of instructions with

the following characteristics: Precision: steps are precisely stated

Deterministic: Results of each step of execution are uniquely defined. They depend only on inputs and results of preceding steps

Finiteness: the algorithm stops after finitely many steps.

Correctness: The output produced is correct.

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More characteristics of algorithms

Input: the algorithm receives input

Output: the algorithm produces output

Generality: the algorithm applies to various sets of inputs

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Example: a simple algorithmAlgorithm to find the largest of three

numbers a, b, c:Assignment operator

s := k means “copy the value of k into s”

1. x:= a

2. If b > x then x:= b

3. If c > x then x:= c

A trace is a check of the algorithm for specific values of a, b and c

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4.2 Examples of algorithms

Pseudocode:

Instructions given in a generic language

similar to a computer language such as C++

or Pascal. procedure if-then, action if-then-else begin

return while loop for loop end

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Variable declarationinteger x, y; real x;

orx : integer;boolean a; char c, d;datatype x;

Note: We shall prefer not to give a complete declaration when the context of variables is obvious.

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Assignment statementsx := expression;

or x = expression;or x expression;

eg. x 1 + 3 *2 y := a * y + 2;

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Control structures if

if <condition> then<a sequence of statements>

[else<a sequence of statements>]

endif

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Control structures (cont.)

While

while <condition> do

<statements>;

endwhile

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Control structures (cont.) loop-until

loop <statements>;

until < condition>

Note: In comparison to while statement, the loop-until guarantees that the <statements> will be executed at least once.

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Control structures (cont.) for

for i = <n1> to <n2> [step d]

<statements>

endfor

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Control structures (cont.) Case

case : <condition1> : <statements1>;

<condition2> : <statements2>;

: <conditionn> : <statementsn>;

[default : <statements>] endcase

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I/O statements

read(<argument list>);

print(<argument list>);

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Exit statementExample

while condition1 do

while condition2 do while condition3 do

if…then exit (exit from the outmost loop)

endwhile endwhile endwhile

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Functions and procedures function name(parameter list) begin declarations statements; return(value); endname

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Functions and proceduresprocedure name(parameter list) begin declarations statements; endnameNote: Procedures are the similar to functions

but they have no return statement.

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Examples Procedure swap( x, y) /* in this case x, y are inout*/

begintemp x

x y y temp endswap

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Examples: Find the maximum value of 3 numbersInput: a, b, cOutput: large (the largest of a, b, and c)procedure max(a,b,c) { large = a if (b > large) large = b if (c > large) large = c return large}

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Examples: Find the maximum value in a sequence S1, S2, S3,…, Sn

Input: S, nOutput: large (the largest value in the

sequence S)procedure max(S,n) { large = S1

for i = 2 to n if (Si > large)

large = Si

return large}

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Examples: Print the even numbers

2, 4, 6,…, 10

Input: NoneOutput: even numbers 2, 4, 6,…, 10 procedure printEven() { for i = 2 to 10 print i;}

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Examples: Read 10 integer values, sort them in ascending order and print the even numbers

Input: NoneOutput: Sorted 10 integer values in ascending

order procedure simpleSort() { integer numbers[10] for i = 1 to 10 read numbers[i];

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Examples: Read 10 integer values, sort them in ascending order and print the even numbers for i = 1 to 10 { for j = i+1 to 10 { if (numbers[i] > numbers[j] ) then { temp = numbers[i] numbers[i] = numbers[j] numbers[j] = temp } } }

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Examples: Read 10 integer values, sort them in ascending order and print the even numbers

for i = 1 to 10 print numbers[i];}

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4.3 Analysis of algorithms Big O notationDefinition: f(n) = O(g(n)) iff there exist two

positive constants c and n0 such that

|f(n)| <= c |g(n)| for all n >= n0.

Theorem If A(n) = am nm +…+ a1n + a0 is a polynomial of degree m then A(n) = O(nm).

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Analysis of algorithms

Ex.f(n) = 3 n2

f(n) = O(n2) Ex.

f(n) = 3 n2 + 5n + 4 f(n) = O(n2)

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4.4 Recursive algorithms

A recursive procedure is a procedure that invokes itself Example: given a positive integer n, factorial of n is

defined as the product of n by all numbers less than n and greater than 0. Notation: n! = n(n-1)(n-2)…3.2.1

Observe that n! = n(n-1)! = n(n-1)(n-2)!, etc. A recursive algorithm is an algorithm that contains

a recursive procedure

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Fibonacci sequence Leonardo Fibonacci (Pisa, Italy, ca. 1170-1250) Fibonacci sequence f1, f2,… defined recursively

as follows:

f1 = 1

f2 = 2

fn = fn-1 + fn-2 for n > 3 First terms of the sequence are: 1, 2, 3, 5, 8, 13,

21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,…

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How to write a recursion algorithm

Function a(input)begin basis steps; /* for minimum size input */ call

a(smaller input); /* could be a number of recursive calls*/ combine the sub-solution of the recursive call;end a

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Recursion (cont.)

function max( A[i : j])begin

if i = j then return( A[i]); else m1 = max ( A[ i : (i+j)/2] );

m2 = max ( A[ (i+j)/2 + 1 : j] ); if m1 > m2 then return( m1) else return(m2) endif endifendmax

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Recursion: 4 Basic Rules Base cases can be solved without

recursion. Making progress toward a base case for

cases that are to be solved recursively.

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Recursion: 4 Basic Rules Design rule. Assume that all recursive

calls work. Compound interest rule. Never

duplicate work by solving the same instance of a problem in separate recursive calls.

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Recursion: A poor use of recursionfib (int : n)begin if ( n <= 1) then return 1; else return fib(n-1) + fib(n-2) /*redundant

work*/ endifend fib

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4.5 Complexity of algorithms

Complexity: the amount of time and/or space needed to execute the algorithm.

Complexity depends on many factors: data representation type, kind of computer, computer language used, etc.

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Types of complexity

Best-case time = minimum time needed to execute the algorithm for inputs of size n

Worst-case time = maximum time needed to execute the algorithm for inputs of size n

Average-case time = average time needed

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Order of an algorithmLet f and g be functions with domain ZZ+ = {1, 2, 3,…} f(n) = O(g(n)): f(n) is of order at most g(n)

if there exists a positive constant C1 such that |f(n)| < C1|g(n)| for all but finitely many n

f(n) = (g(n)): f(n) is of order at least g(n) if there exists a positive constant C2 such that |f(n)| > C2|g(n)|

for all but finitely many n

f(n) = (g(n)): f(n) is or order g(n) if it is O(g(n)) and (g(n)).

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Review of Big O

Example:3n + 2 = O(n) as 3n + 2 <= 4n for all n >= 2.3n + 3 = O(n) as 3n + 3 <= 4n for all n >= 3.100n + 6 = O(n) as 100n + 6 <= 101n for all n >= 10.10n2 + 4n + 2 = O(n2) as 10n2 + 4n + 2 <= 11 n2 for n>= 5.3n + 2 is not O(1) as 3n + 2 is not less than or

equal to c or any constant c for all n.

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Review of Big O O-notation is a means for describing an

algorithm’s performance. O-notation is used to express an upper

bound on the value of f(n). O-notation doesn’t say how good the

bound is.

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Review of Big O Notice that n = O(n2), n = O(n3), n = O(2n) etc. In order for the statement

f(n) = O(n) to be informative, g(n) should be the as small a function of n as one can come up. We shall never say 3n + n = O(n2) even though it’s correct.

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Review of Big O f(n) = O(g(n)) is not the same as O(g(n)) =

f(n).

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Review of 3n + 2 = (n) as 3n + 2 >= 3n for all n >= 1.3n + 3 = (n) as 3n + 3 >= 3n for all n >= 1.100n + 6 = (n) as100n + 6 >= 100n for all n >= 1.10n2 + 4n + 2 = (n2) as 10n2 + 4n + 2 >= n2 for n>= 1.3n + 2 is not O(1) as 3n + 2 is not less than or equal to c or

any constant c for all n.

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Review of 3n + 2 = (n) as 3n + 2 >= 3n for all n >= 1 and 3n + 2 <= 4n for all n >= 2 so c1 = 3 and c2 = 4 and n0 = 2

Note: The theta is more precise than both the big O and omega notations.

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Review of notationsNotice that the coefficients in all the g(n)

used have been 1. This is accordance with practice. We shall never find ourselves saying that 3n + 3 = O(4n), or that 10 = O(100) even though each of these statements is true.