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Algorithms

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Algorithm. A structured method of solving a problem. Assertions ... Represents the amount of work an algorithm does in order to process n elements. Example: ... – PowerPoint PPT presentation

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Title: Algorithms


1
Algorithms
  • Introductory Discrete Mathematics (CS/MAT165)

2
Algorithm
  • A structured method of solving a problem
  • Assertions
  • Pre-condition
  • Initial state represented as a predicate
  • Post-condition
  • Final state represented as a predicate

3
Statements
  • Assignment
  • Decision
  • if-then-else
  • Loops
  • while
  • repeat
  • for

4
Assignment
  • Associates a value with a variable
  • Example
  • X 5
  • X is assigned the value 5

5
Assertion
  • A predicate associated with a specific point in
    an algorithm
  • Example
  • X 10
  • x 10
  • X X 5
  • x 15

6
Decisions
  • Also called a branch
  • Indicates one or two statements to execute based
    on the value of a condition
  • If condition then then-statement else
    else-statement
  • Example
  • If X lt 5 then Y 1 else Y 2

7
Decisions
  • if Score lt 60
  • then Grade NP
  • else Grade P
  • What must be true for the then-statement to be
    executed?
  • The else-statement?

8
Try this
  • What if the grade breakdown is as follows
  • A 90 100B 80 89F lt80
  • What would the decision statement look like?

9
Try this
  • If Score lt 80 then Grade F else if Score lt
    90 then Grade B else Grade A

10
Loops
  • While
  • Repeat
  • For

11
While Loop
  • while (guard) body end while
  • Guard Boolean expression
  • Sometimes called a condition
  • Body is repeated (iterated) as long as the guard
    is true

12
While Loop
  • X 1
  • while x lt 10
  • x x 1
  • end while
  • Q What is the value of X after the loop has
    terminated?

13
Loop Invariant
  • Predicate with domain a set of integers
  • For each iteration of the loop,if the predicate
    is true before the iteration,then it is true
    after the iteration

14
Loop Invariant
  • If the following are true, then the loop is
    correct with respect to its pre- and
    post-conditions
  • It is true before the first iteration
  • If the loop terminates after a finite number of
    iterations, the truth of the loop invariant
    insures the truth of the post-condition

15
Try this
  • Prove the correctness of the following
  • Pre-condition A and B are integers,A gt B ? 0,
    a A, b B, r B
  • while (b ? 0) r a mod b a b b rend
    while
  • Post-condition a gcd(A,B)

Loop invariant?
16
For Loop
  • Every for-loop can be re-written as a while-loop

for x 1 to 5 ltbodygtend for
x 1while (x ? 5) ltbodygt x x 1end while
17
Timing Function
  • f(n)
  • Represents the amount of work an algorithm does
    in order to process n elements
  • Example
  • for x 1 to n ltbodygtend for
  • f(n) Cn

18
Big-?
  • Upper bound
  • A function that g(n) that approximates a timing
    function f(n)
  • f(n) ?(g(n)) ??N, ?C such that f(n) ? Cg(n),
    ?n ? N

19
Big-?
  • g(n) is a function that hugs f(n) just above for
    all n ? N

20
Try this
  • Given f(n) n2
  • Which of the following could be g(n)?
  • n
  • n3
  • 1/n

21
Try this
  • Find a suitable big-?, g(n), for each of the
    following
  • f(n) n n2 n3
  • f(n) 1/n
  • f(n) ln(n) n
  • General ruleGiven a function consisting of the
    sum of terms, a good g(n) candidate is the term
    that grows the quickest

22
Proving Big-?
  • Given f(n) n n2
  • Try g(n) n2
  • Proof
  • N 1
  • f(n) ? Cg(n)n n2 ? (C)(n2)1/n1 ? C1/1
    1 ? C2 ? C


23
One more time
  • Given f(n) n2
  • Which of the following could be g(n)?
  • n
  • n3
  • 1/n

Prove it!
24
Common Big-?s
  • ?(1) constant
  • ?(log n) logarithmic
  • ?(n) linear
  • ?(n log n) linearithmic
  • ?(nc), c gt 1 polynomial
  • ?(cn), c gt 1 exponential
  • ?(n!) factorial

25
Power Functions
  • f(n) nc
  • Polynomials are made from power functions

c gt 1
c lt 0
c 0
0 lt c lt 1
c 1
26
Nested Loops
  • Q How many times will the body execute in the
    following algorithm?
  • while 1 ? a ? n ltbodygtend while
  • A n times

27
Nested Loops
  • Q How many times will the body execute in the
    following algorithm?
  • while 1 ? a ? n while 1 ? b ? n ltbodygt end
    whileend while
  • A n2 times

28
Nested Loops
  • Q How many times will the body execute in the
    following algorithm?
  • while 1 ? a ? n while 1 ? b ? a ltbodygt end
    whileend while
  • Loop invariant?

29
Nested Loops
  • Q How many times will the body execute in the
    following algorithm?
  • while 1 ? a ? n while 1 ? b ? 5 ltbodygt end
    whileend while

30
Big-?
  • Lower bound
  • A function that g(n) that approximates a timing
    function f(n)
  • f(n) ?(g(n)) ??N, ?C such that f(n) ? Cg(n),
    ?n ? N

31
Big-?
  • A function that g(n) that approximates a timing
    function f(n)
  • f(n) ?(g(n)) ??N, ?A ,?B such that Bg(n) ?
    f(n) ? Ag(n), ?n ? N

32
Properties
  • If f(n) is O(g(n)) and g(n) is O(h(n)),then f(n)
    is O(h(n)) (transitivity)
  • If f(n) is O(h(n)) and g(n) is O(h(n)),then
    f(n)g(n) is O(h(n))
  • The function ank is O(nk)
  • The function nk is O(nkj) for any positive j

33
Properties
  • If f(n) cg(n), then f(n) is O(g(n))
  • The function loga n is O(logb n) for any positive
    number a and b ? 1
  • loga n is O(lg n) for any positive a ? 1,where
    lg n log2 n

34
Insertion Sort
  • Simple to implement
  • Efficient on (quite) small data sets
  • Efficient on data sets which are already
    substantially sorted it runs in ?(n d) time,
    where d is the number of inversions
  • Otherwise ?(n2)
  • Why?

35
Insertion Sort
  • subroutine insert (a, length, value)i length
    1while (i ? 0 ? ai gt value) ai 1
    ai i i 1end whileai 1 value
  • subroutine insertionSort (a, length)i 0while
    (i lt length) insert(a, i, ai) i i 1end
    while

36
Merge Sort
  • Divide the unsorted list into two sublists of
    about half the size
  • Sort each of the two sublists recursively until
    we have list sizes of length 1, in which case the
    list itself is returned
  • Merge the two sorted sublists back into one
    sorted list

37
Merge Sort
  • function mergesort(m) var list left, right,
    result if length(m) 1 return m else
    middle length(m) / 2 for each x in m up to
    middle add x to left for each x in m after
    middle add x to right left mergesort(left)
    right mergesort(right) result merge(left,
    right) return result

38
Merge Sort
  • function merge(left,right) while (length(left))
    gt 0 ? (length(right) gt 0) if first(left)
    first(right) append first(left) to
    result left rest(left) else append
    first(right) to result right rest(right) end
    ifend while if length(left) gt 0 append
    rest(left) to resultend if if length(right) gt
    0 append rest(right) to resultend if return
    result

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