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Difference / order nlog(n) VS. n^2 algorithm.

What is the difference between an order nlog(n) and n^2 algorithm.  Please give a detailed explanation and examples of each in VB5.  I believe this relates to asymtotic boundaries and sorting heaps and such, but need a definitive answer that an MIS dude can understand.  Email to me if you have the time:

jhowell@cyberhighway.net

Thanks!!!
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jwhowell
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1 Solution

Commented:
MsgBox 10 ^ 2   '>> it is like 10 times 10
MsgBox 10 * Log(10) ' >> it is like 10 times log(base 10)of 10
End
End Sub

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Author Commented:
a111a111a111,

Thank you for the math lesson on natural logrithms, but I was hoping for a a bit more.  This relates to sorting algorithms and I need examples of each equation where I can generate a large random array or something and time the sorting of each algorithm based on the volume of the sort.  Do you have any code for timing sorting like I describe?

jwhowell
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Commented:
The "speed" of a sort algorythm is measured based on the number of operations to be done, compared to the number of elements to sort.

For example, the simple "bubble sort" is of order n^2, because the number of times you go through the loop and the test increases by the square of the number of values to sort:

For i = 1 to NbElement - 1
For j = i to NbElement
if strArray(i) > strArray(j) then
strTemp = strArray(i)
strArray(i) = strArray(j)
strArray(j) = strTemp
end if
Next j
Next i

If NbElement is 100, you need about 5,000 iterations.
If NbElements is 1,000, you need about 500,000 iterations, 100 times more while the number of elements is only 10 times greater.

There is one sort algorithm which is of order n*ln(n): it is called QuickSort. Here is the routine:

Sub QuickSort_VB(iLow As Long, iHigh As Long)
' QuickSort works by picking a random "pivot" element in lInput, then
' moving every element that is bigger to one side of the pivot, and every
' element that is smaller to the other side.  QuickSort is then called
' recursively with the two subdivisions created by the pivot.  Once the
' number of elements in a subdivision reaches two, the recursive calls end
' and the array is sorted.

Dim i As Long
Dim j As Long
Dim iRand As Long
Dim lPartition As Long
Dim lTemp As Long

If iLow < iHigh Then

' Only two elements in this subdivision; swap them if they are out of
' order, then end recursive calls:
If iHigh - iLow = 1 Then
If bt(iLow) > bt(iHigh) Then
lTemp = bt(iLow)
bt(iLow) = bt(iHigh)
bt(iHigh) = lTemp
End If
Else

' Pick a pivot element at random, then move it to the end:
iRand = iLow + Int(Rnd * (iHigh - iLow + 1))
lTemp = bt(iHigh)
bt(iHigh) = bt(iRand)
bt(iRand) = lTemp
lPartition = bt(iHigh)
Do

' Move in from both sides towards the pivot element:
i = iLow
j = iHigh
Do While (i < j) And (bt(i) <= lPartition)
i = i + 1
Loop
Do While (j > i) And (bt(j) >= lPartition)
j = j - 1
Loop

' If we haven't reached the pivot element, it means that two
' elements on either side are out of order, so swap them:
If i < j Then
lTemp = bt(i)
bt(i) = bt(j)
bt(j) = lTemp
End If
Loop While i < j

' Move the pivot element back to its proper place in the array:
lTemp = bt(i)
bt(i) = bt(iHigh)
bt(iHigh) = lTemp

' Recursively call the QuickSort procedure (pass the smaller
' subdivision first to use less stack space):
If (i - iLow) < (iHigh - i) Then
QuickSort_VB iLow, i - 1
QuickSort_VB i + 1, iHigh
Else
QuickSort_VB i + 1, iHigh
QuickSort_VB iLow, i - 1
End If
End If
End If
End Sub

The procedure above sorts the array bt().

I think some mathematicians have demonstrated that this sort is much faster, although it is impossible to exactly predict hw much time it will take to sort an array. It depends on the "disorder' of the initial array.

Hope this help,
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Author Commented:
Steve06,

You are the man!  Thanks a bunch.  I may want to Email you about the small comparison app I'll build using your algorithms.  Could you send me your Email address?  I also ordered a book called 'Ready-To-Run Visual Basic Algorithms.'  Do you know this book?

Here are the points!

Jason W. Howell
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Commented:
Jason,

I don't know the book you mention. May be it contains sort algorithms as well.

I am happy that I could answer your question. My e-mail is
steve06@infonie.be

Steve.
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