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Author
 Thread  l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
Usually when we say complexity, we have some assumptions. Michael Brand's assumptions, however, have some bugs. This code can sort an array of non-negative integers(add -min if not sure) in O(n):
Code:
```; Fasm G
; Init
x1 = 13
x2 = 7
x3 = 11
x4 = 3
n = 4
; Sort
; Assuming any two numbers differ.
; If not sure, multiply them by n and plus its index
; Also assuming n is at least 2
a = 0
rept n i
a = a + (1 shl (x##i * n))
end rept
rept n i
rept 1 t:(a shr (x##i * n)) mod ((1 shl n) - 1)
y##t = x##i
end rept
end rept
; Output
rept n i
rept 1 t:y##i
display `t, 13, 10
end rept
end rept    ```

1. sort an array of real in O(n).
Note: Real constants are unnecessary, but allowed as a step.

2. factorize a positive integer in o(lg^c n), where c is constant
real numbers are banned

Last edited by l4m2 on 29 Jul 2017, 15:34; edited 3 times in total 28 Jul 2017, 13:24  revolution When all else fails, read the source Joined: 24 Aug 2004 Posts: 17278 Location: In your JS exploiting you and your system revolution What is the average run complexity? The worst case complexity? Because I expect when you claim O(n) you are suggesting the best case, right? I don't think anyone has yet discovered a sorting algorithm in the O(n) complexity class yet, except in the best case where the list is already sorted. Quite a few algorithms can do O(n) in the best case, but they can't keep such a low complexity when sorting non-ordered lists. 28 Jul 2017, 13:39
l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
I meant the worst case complexity.
In reality it is not O(n), but here we assumed any number, no matter how large, goes calculated in O(1), thus something weird happened. 28 Jul 2017, 13:46  revolution When all else fails, read the source Joined: 24 Aug 2004 Posts: 17278 Location: In your JS exploiting you and your system revolution The code fails for this input set: Code:```x1 = 1 x2 = 1 n = 2 ``` 28 Jul 2017, 14:02
l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
Quote:
Code:
```; Sort
; Assuming any two numbers differ.
; If not sure, multiply them by n and plus its index     ``` 28 Jul 2017, 14:04  l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
another prob comes 29 Jul 2017, 15:35  Furs

Joined: 04 Mar 2016
Posts: 1471
Furs
That's just a very inefficient way to do Radix Sort. 29 Jul 2017, 17:16  l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
Furs wrote:
That's just a very inefficient way to do Radix Sort.
Why Radix, not Counting? It does the whole sort in one run 30 Jul 2017, 13:39  l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2
Answer to the 1st problem in Python 2:
Code:
```import math

def sh(n,a):
return ((1<<(a*n))-1)/((1<<a)-1)

def sortr(a):
#;Assuming all values are in [0,1)
n=len(a) #;length
s=l=0 #;s means [0,1) is cutted into 2**s equal part, l is the list to store indices
m=y=int(math.log(n,2)+1) #;every part costs y bits, but only the last m bits mean
for i in range(1,n):
r=int((1<<s)*a[i])
t=(l>>(y*r))%(1<<m)
if (int((1<<s)*a[t])==r) & (a[t]!=a[i]): #;Another value is in the same part
d=int(math.floor(-math.log(abs(a[t]-a[i]),2))) #; New s that is enough. easily proved
w=(y<<s)+y #; Old l's length plus y, so the next code won't overlap
l=l*sh(1<<(d-s),w)*sh(1<<s,(w<<(d-s))-y)
#; abcd -> Abcd0AbcdaBcd0aBcdabCd0abCdabcD0abcD if s=2 and d=3
s=d
y=w
r=int((1<<s)*a[i]) #; The old r won't work, so recomputing is necessary
l=l-(l>>(y*r))%(1<<y)*(1<<(y*r))+(i<<(y*r)) #; Set the value there to the new one
#;sort ints, every number mult by 2**s
x=[int(a[i]*(1<<s))*n+i for i in range(0,n)]
b=sum([1<<(x[i]*n) for i in range(0,n)])
y=range(0,n)
for i in range(0,n):
y[b%(1<<(x[i]*n))%((1<<n)-1)] = a[i] #; Count those who are smaller than the current item
return y

sortr([.67, .233, .2328])    ```

Last edited by l4m2 on 01 Aug 2017, 03:52; edited 2 times in total 30 Jul 2017, 13:41  Furs

Joined: 04 Mar 2016
Posts: 1471
Furs
l4m2 wrote:
Why Radix, not Counting? It does the whole sort in one run
To me it looked like counting based on a large array (well in your case, just stuffed into an integer, but that's just semantics...) which works in your specific case since the numbers are small, but otherwise is inefficient. I'm not very familiar with counting sort, but doesn't it have 3 steps? (iterating through the range) 30 Jul 2017, 13:52  l4m2

Joined: 15 Jan 2015
Posts: 648
l4m2

The Usual Complexity Assumptions
"The usual complexity assumptions" is a term refering to a set of conventions employed so often in complexity calculations that they are usually taken for granted.

For example, one often refers to Bubble sorting as an O(n2)-time operation, when, in fact, it is not. It merely requires O(n2) comparison operations. If we were to compare two integers, a and b using (for example) a Turing machine, we would, in the general case, need to go over all their bits in order to find a difference, so this operation may take as long as O(log(min(|a|,|b|))), rather than O(1).

Similarly, we casually refer to storing real numbers and performing operations on them, when, in fact, it is impossible to store a full representation of a general real number on any computer or any Turing machine.

"The usual complexity assumptions" therefore refers to the most commonly used (and least commonly explicitly specified) computational model. It defines "integers", "reals" and "arrays" as abstract data types, and also defines an execution model that uses them. This, in turn, means that certain operations are assumed to be atomic O(1)-time operations.

The following are the O(1) operations assumed:

Regarding integers

An integer can be stored in O(1) space.
The four basic arithmetic operations (addition, subtraction, multiplication and division) as well as modulo-taking are all O(1).
Comparisons (less than, greater than, equal to) are all O(1).
Shift-left and shift-right by n bits, for any integer n, are O(1). (The result of this is the same as multiplication/division by 2n.)
Conversion to real numbers is O(1).

Note that bitwise operations other than shifts are not considered O(1), unless explicitly stated.
Regarding real numbers

A real number can be stored in O(1) space.
The four basic arithmetic operations (addition, subtraction, multiplication and division) are all O(1).
Comparisons (less than, greater than, equal to) are all O(1).
All trigonometric and inverse trigonometric functions are O(1).
Exponentiation and log-taking are both O(1).
Conversion to integers by either "floor" or "ceil" is O(1).

Regarding arrays

Allocating/deallocating an array (of reals or integers) of any size is O(1) time. (An array stores a single integer/real in each of its cells. Cell indices are integers. Arrays start off as non-initialized, meaning that if an array cell is read before it was ever written to, the value read may be any integer or real. Arrays cannot change their sizes once allocated.)
Accessing a given array position for either reading or writing is O(1). (Accessing an array position outside the array bounds [either by use of an index with a negative value or one larger than the array's maximum] leads to undefined results. Array indices are zero based.)
Input and output of the program are special cases of array reading and array writing (respectively), and follow the same complexity.

Note that use of an integer/real variable is modeled as the use of an array of size 1. Integer/real constants (i.e. program literals) are like variables, but unlike regular arrays/variables do get initialized and cannot be overwritten with new values.
Regarding flow control

Jumps are performed in O(1) time.
Conditional jumps are perfomed in O(1) time, in addition to the time it takes to evaluate the condition.

Though anybody who has done any complexity calculation has probably been exposed to this computational model, its assumptions are rarely stated explicitly. They are repeated here because in riddles they are often stretched ad absurdum, so it is useful to keep them in mind. 01 Aug 2017, 15:20  Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First  Jump to: Select a forum Official----------------AssemblyPeripheria General----------------MainDOSWindowsLinuxUnixMenuetOS Specific----------------MacroinstructionsCompiler InternalsIDE DevelopmentOS ConstructionNon-x86 architecturesHigh Level LanguagesProgramming Language DesignProjects and IdeasExamples and Tutorials Other----------------FeedbackHeapTest Area

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