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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 nonnegative 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 Now your task is to 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 

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
The code fails for this input set:
Code: x1 = 1 x2 = 1 n = 2 

28 Jul 2017, 14:02 

l4m2
Quote:


28 Jul 2017, 14:04 

l4m2
another prob comes


29 Jul 2017, 15:35 

Furs
That's just a very inefficient way to do Radix Sort.


29 Jul 2017, 17:16 

l4m2
Furs wrote: That's just a very inefficient way to do Radix Sort. 

30 Jul 2017, 13:39 

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<<(ds),w)*sh(1<<s,(w<<(ds))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
l4m2 wrote: Why Radix, not Counting? It does the whole sort in one run 

30 Jul 2017, 13:52 

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 modulotaking are all O(1). Comparisons (less than, greater than, equal to) are all O(1). Shiftleft and shiftright 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 logtaking 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 noninitialized, 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 

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