Hello all this is my very first question here. I am new to datastructure and algorithms my teacher asked me to compare time complexity of different algorithms including: merge sort, heap sort, insertion sort, and quick sort. I search over internet and find out that quick sort is the fastest of all but my version of quick sort is the slowest of all (it sort 100 random integers in almost 1 second while my other sorting algorithms took almost 0 second). I tweak my quick sort logic many times (taking first value as pivot than tried to take middle value as pivot but in vain) I finally search the code over internet and there was not much difference in my code and code on internet. Now I really am confused that if this is behaviour of quick sort is natural (I mean whatever your logic is you are gonna get same results.) or there are some specific situations where you should use quick sort. In the end I know my question is not clear (I don't know how to ask besides my english is also not very good.) I hope someone can help me I really wanted to attach picture of awkward result I am having but I can't (reputation < 10).
Theoretically, quicksort is supposed to be the fastest algorithm for sorting, with a runtime of O(nlogn). It's worst case would be O(n^2), but only occurs if there are repeated values are equal to the pivot.
In your situation, I can only assume that your pivot value is not ideal in your data array, but is still able to sort the values using that pivot. Otherwise, your quicksort implementation is unfortunately incorrect.
Quicksort has O(n^2) worst-case runtime and O(nlogn) average case runtime. A good reason why Quicksort is so fast in practice compared to most other O(nlogn) algorithms such as Heapsort, is because it is relatively cache-efficient. Its running time is actually O(n/Blog(n/B)), where B is the block size. Heapsort, on the other hand, doesn't have any such speedup: it's not at all accessing memory cache-efficiently.
The value you choose as pivot may not be appropriate hence your sorting may be taking some time.You can avoid quicksort’s worst-case run time of O(n^2) almost entirely by using an appropriate choice of the pivot – such as picking it at random.
Also , the best and worst case often are extremes rarely occurring in practice.But any average case analysis assume some distribution of inputs. For sorting, the typical choice is the random permutation model (as assumed on Wikipedia).
I'm writing a program which can significantly lessen the number of collisions that occur while using hash functions like 'key mod table_size'. For this I would like to use Genetic Programming/Algorithm. But I don't know much about it. Even after reading many articles and examples I don't know that in my case (as in program definition) what would be the fitness function, target (target is usually the required result), what would pose as the population/individuals and parents, etc.
Please help me in identifying the above and with a few codes/pseudo-codes snippets if possible as this is my project.
Its not necessary to be using genetic programming/algorithm, it can be anything using evolutionary programming/algorithm.
thanks..
My advice would be: don't do this that way. The literature on hash functions is vast and we more or less understand what makes a good hash function. We know enough mathematics not to look for them blindly.
If you need a hash function to use, there is plenty to choose from.
However, if this is your uni project and you cannot possibly change the subject or steer it in a more manageable direction, then as you noticed there will be complex issues of getting fitness function and mutation operators right. As far as I can tell off the top of my head, there are no obvious candidates.
You may look up e.g. 'strict avalanche criterion' and try to see if you can reason about it in terms of fitness and mutations.
Another question is how do you want to represent your function? Just a boolean expression? Something built from word operations like AND, XOR, NOT, ROT ?
Depending on your constraints (or rather, assumptions) the question of fitness and mutation will be different.
Broadly fitness is clearly minimize the number of collisions in your 'hash modulo table-size' model.
The obvious part is to take a suitably large and (very important) representative distribution of keys and chuck them through your 'candidate' function.
Then you might pass them through 'hash modulo table-size' for one or more values of table-size and evaluate some measure of 'niceness' of the arising distribution(s).
So what that boils down to is what table-sizes to try and what niceness measure to apply.
Niceness is context dependent.
You might measure 'fullest bucket' as a measure of 'worst case' insert/find time.
You might measure sum of squares of bucket sizes as a measure of 'average' insert/find time based on uniform distribution of amongst the keys look-up.
Finally you would need to decide what table-size (or sizes) to test at.
Conventional wisdom often uses primes because hash modulo prime tends to be nicely volatile to all the bits in hash where as something like hash modulo 2^n only involves the lower n-1 bits.
To keep computation down you might consider the series of next prime larger than each power of two. 5(>2^2) 11 (>2^3), 17 (>2^4) , etc. up to and including the first power of 2 greater than your 'sample' size.
There are other ways of considering fitness but without a practical application the question is (of course) ill-defined.
If your 'space' of potential hash functions don't all have the same execution time you should also factor in 'cost'.
It's fairly easy to define very good hash functions but execution time can be a significant factor.
I have a hardware related question that I debated with a friend.
Consider two processors from two different manufacturers with the same amount of gigaflops put into the same computers (i.e. RAM and such as are the same for both computers).
Now given a simple program will there be any difference in execution times between the two computers with the same processors. I.e. will the two computers handle the code differently (for-loops, while-loops, if-statements and such)?
And if, is that difference noticably or can one say that the computers would approximately perform the same?
Short answer: Yes, they will be different, possibly very much so.
Flops are just about floating point operations, so it is a very very crude measure of CPU performance. It is, in general, a decent proxy for performance for scientific computations of certain kinds, but not for general performance.
There are CPUs which are strong in FLOPS - the Alpha is a historical example - but which have more moderate performance in integer computations. This means that an alpha and an x86 CPU with similar FLOPS would have very different MIPS-performance.
The truth is that it is very hard to make a good generic benchmark, though many have tried.
Another critical factor in comparing the performance of two processors with the same FLOP measure is the rate at which they can move data between CPU and RAM. Add memory cache into your thinking to complicate matters further.
I'm trying to figure out the best programming language for an analytical model I'm building. Primary consideration is speed at which it will run FOR loops.
Some detail:
The model needs to perform numerous (~30 per entry, over 12 cycles) operations on a set of elements from an array -- there are ~300k rows, and ~150 columns in the array. Most of these operations are logical in nature, e.g., if place(i) = 1, then j(i) = 2.
I've built an earlier version of this model using Octave -- to run it takes ~55 hours on an Amazon EC2 m2.xlarge instance (and it uses ~10 GB of memory, but I'm perfectly happy to throw more memory at it). Octave/Matlab won't do elementwise logical operations, so a large number of for loops are needed -- I'm relatively certain that I've vectorized as much as possible -- the loops that remain are necessary. I've gotten octave-multicore to work with this code, which makes some improvement (~30% speed reduction when I get it running on 8 EC2 cores), but ends up being unstable with file locking, etc.
+I'm really looking for a step change in run-time -- I know that actually using Matlab might get me as much as a 50% improvement from looking at some benchmarks, but that is cost-prohibitive. The original plan when starting this was to actually run a Monte Carlo with this, but at 55 hours a run, that's completely impractical.
The next version of this is going to be a complete rebuild from the ground up (for IP reasons I won't get in to if nothing else), so I'm completely open to any programming language. I'm most familiar with Octave/Matlab, but have dabbled in R, C, C++, Java. I'm also proficient w/ SQL if the solution involves storing the data in a database. I'll learn any language for this -- these aren't complicated functionality we're looking for, no interfacing with other programs, etc., so not too concerned about learning curve.
So with all that said, what's the fastest programming language specifically for FOR loops? From a search of SO and Google, Fortran and C bubble to the top, but looking for some more advice before diving in to one or the other.
Thanks!
This for loop looks no more complex than this when it hits the CPU:
for(int i = 0; i != 1024; i++) translates to
mov r0, 0 ;;start the counter
top:
;;some processing
add r0, r0, 1 ;;increment the counter by 1
jne top: r0, 1024 ;;jump to the loop top if we havn't hit the top of the for loop (1024 elements)
;;continue on
As you can tell, this is sufficiently simple you can't really optimize it very well[1]... Refocus towards the algorithm level.
The first cut at the problem is to look at cache locality. Look up the classic example of matrix multiplication and swapping the i and j indexes.
edit: As a second cut, I would suggest evaluating the algorithm for data-dependencies between iterations and data dependency between localities in your 'matrix' of data. It may be a good candidate for parallelization.
[1] There are some micro-optimizations possible, but those will not produce the speedsups you're looking for.
~300k * ~150 * ~30 * ~12 = ~16G iterations, right?
This number of primitive operations should complete in a matter of minutes (if not seconds) in any compiled language on any decent CPU.
Fortran, C/C++ should do it almost equally well. Even managed languages, such as Java and C#, would only fall behind by a small margin (if at all).
If you have a problem of ~16G iterations running 55 hours, this means that they are very far from being primitive (80k per second? this is ridiculous), so maybe we should know more. (as was already suggested, is disk access limiting performance? is it network access?)
As #Rotsor said, 16G operations / 55 hours is about 80,000 operations per second, or one operation every 12.5 microseconds. That's a lot of time per operation.
That means your loops are not the cause of poor performance, it's what's in the innermost loop that's taking the time. And Octave is an interpreted language. That alone means an order of magnitude slowdown.
If you want speed, you first need to be in a compiled language. Then you need to do performance tuning (aka profiling) or, just single step it in a debugger at the instruction level. That will tell you what it is actually doing in its heart of hearts. Once you've got it to where it's not wasting cycles, fancier hardware, cores, CUDA, etc. will give you a parallelism speedup. But it's silly to do that if your code is taking unnecessarily many cycles. (Here's an example of performance tuning - a 43x speedup just by trimming the fat.)
I can't believe the number of responders talking about matlab, APL, and other vectorized languages. Those are interpreters. They give you concise source code, which is not at all the same thing as fast execution. When it comes down to the bare metal, they are stuck with the same hardware as every other language.
Added: to show you what I mean, I just ran this C++ code, which does 16G operations, on this moldy old laptop, and it took 94 seconds, or about 6ns per iteration. (I can't believe you baby-sat that thing for 2 whole days.)
void doit(){
double sum = 0;
for (int i = 0; i < 1000; i++){
for (int j = 0; j < 16000000; j++){
sum += j * 3.1415926;
}
}
}
In terms of absolute speed, probably Fortran followed by C, followed by C++. In practical application, well written code in any of the three, compiled with a descent compiler should be plenty fast.
Edit- generally you are going to see much better performance with any kind of looped or forking (e.g.- if statements) code with a compiled language, versus an interpreted language.
To give an example, on a recent project I was working on, the data sizes and operations were about 3/4 the size of what you're talking about here, but like your code, had very little room for vectorization, and required significant looping. After converting the code from matlab to C++, runtimes went from 16-18 hours, down to around 25 minutes.
For what you're discussing, Fortran is probably your first choice. The closest second place is probably C++. Some C++ libraries use "expression templates" to gain some speed over C for this kind of task. It's not entirely certain those will do you any good, but C++ can be at least as fast as C, and possibly somewhat faster.
At least in theory, there's no reason Ada couldn't be competitive as well, but it's been so long since I used it for anything like this that I hesitate to recommend it -- not because it isn't good, but because I just haven't kept track of current Ada compilers well enough to comment on them intelligently.
Any compiled language should perform the loop itself on roughly equal terms.
If you can formulate your problem in its terms, you might want to look at CUDA or OpenCL and run your matrix code on the GPU - though this is less good for code with lots of conditionals.
If you want to stay on conventional CPUs, you may be able to formulate your problem in terms of SSE scatter/gather and bitmask operations.
Probably the assembly language for whatever your platform is. But compilers (especially special-purpose ones that specifically target a single platform (e.g., Analog Devices or TI DSPs)) are often as good as or better than humans. Also, compilers often know about tricks that you don't. For example, the aforementioned DSPs support zero-overhead loops and the compiler will know how to optimize code to use those loops.
Matlab will do element-wise logical operations and they are generally quite fast.
Here is a quick example on my computer (AMD Athalon 2.3GHz w/3GB) :
d=rand(300000,150);
d=floor(d*10);
>> numel(d(d==1))
ans =
4501524
>> tic;d(d==1)=10;toc;
Elapsed time is 0.754711 seconds.
>> numel(d(d==1))
ans =
0
>> numel(d(d==10))
ans =
4501524
In general I've found matlab's operators are very speedy, you just have to find ways to express your algorithms directly in terms of matrix operators.
C++ is not fast when doing matrixy things with for loops. C is, in fact, specifically bad at it. See good math bad math.
I hear that C99 has __restrict pointers that help, but don't know much about it.
Fortran is still the goto language for numerical computing.
How is the data stored? Your execution time is probably more effected by I/O (especially disk or worse, network) than by your language.
Assuming operations on rows are orthogonal, I would go with C# and use PLINQ to exploit all the parallelism I could.
Might you not be best with a hand-coded assembler insert? Assuming, of course, that you don't need portability.
That and an optimized algorithm should help (and perhaps restructuring the data?).
You might also want to try multiple algorithms and profile them.
APL.
Even though it's interpreted, its primitive operators all operate on arrays natively, therefore you rarely need any explicit loops. You write the same code, whether the data is scalar or array, and the interpreter takes care of any looping needed internally, and thus with the minimum overhead - the loops themselves are in a compiled language, and will have been heavily optimised for the specific architecture of the CPU it's running on.
Here's an example of the simplicity of array handling in APL:
A <- 2 3 4 5 6 8 10
((2|A)/A) <- 0
A
2 0 4 0 6 8 10
The first line sets A to a vector of numbers.
The second line replaces all the odd numbers in the vector with zeroes.
The third line queries the new values of A, and the fourth line is the resulting output.
Note that no explicit looping was required, as scalar operators such as '|' (remainder) automatically extend to arrays as required. APL also has built-in primitives for searching and sorting, which will probably be faster than writing your own loops for these operations.
Wikipedia has a good article on APL, which also provides links to suppliers such as IBM and Dyalog.
Any modern compiled or JITted language is going to render down to pretty much the same machine language code, giving a loop overhead of 10 nano seconds or less, per iteration, on modern processors.
Quoting #Rotsor:
If you have a problem of ~16G iterations running 55 hours, this means that they are very far from being primitive (80k per second? this is ridiculous), so maybe we should know more.
80k operations per second is around 12.5 microseconds each - a factor of 1000 greater than the loop overhead you'd expect.
Assuming your 55 hour runtime is single threaded, and if your per item operations are as simple as suggested, you should be able to (conservatively) achieve a speedup of 100x and cut it down to under an hour very easily.
If you want to run faster still, you'll want to look at writing multi-threaded solution, in which case a language that provides good support would be essential. I'd lean towards PLINQ and C# 4.0, but that's because I already know C# - YMMV.
what about a lazy loading language like clojure. it is a lisp so like most lisp dialects lacks a for loop but has many other forms that operate more idiomatically for list processing. It might help your scaling issues as well because the operations are thread safe and because the language is functional it has fewer side effects. If you wanted to find all the items in the list that were 'i' values to operate on them you might do something like this.
(def mylist ["i" "j" "i" "i" "j" "i"])
(map #(= "i" %) mylist)
result
(true false true true false true)
Does anyone know if there's a real benefit regarding decreasing collision probability by combining hash functions? I especially need to know this regarding 32 bit hashing, namely combining Adler32 and CRC32.
Basically, will adler32(crc32(data)) yield a smaller collision probability than crc32(data)?
The last comment here gives some test results in favor of combining, but no source is mentioned.
For my purpose, collision is not critical (i.e. the task does not involve security), but I'd rather minimize the probability anyway, if possible.
PS: I'm just starting in the wonderful world of hashing, doing a lot of reading about it. Sorry if I asked a silly question, I haven't even acquired the proper "hash dialect" yet, probably my Google searches regarding this were also poorly formed.
Thanks.
This doesn't make sense combining them in series like that. You are hashing one 32-bit space to another 32-bit space.
In the case of a crc32 collision in the first step, the final result is still a collision. Then you add on any potential collisions in the adler32 step. So it can not get any better, and can only be the same or worse.
To reduce collisions, you might try something like using the two hashes independently to create a 64-bit output space:
adler32(data) << 32 | crc32(data)
Whether there is significant benefit in doing that, I'm not sure.
Note that the original comment you referred to was storing the hashes independently:
Whichever algorithm you use there is
going to be some chance of false
positives. However, you can reduce
these chances by a considerable margin
by using two different hashing
algorithms. If you were to calculate
and store both the CRC32 and the
Alder32 for each url, the odds of a
simultaneous collision for both hashes
for any given pair of urls is vastly
reduced.
Of course that means storing twice as
much information which is a part of
your original problem. However, there
is a way of storing both sets of hash
data such that it requires minimal
memory (10kb or so) whilst giving
almost the same lookup performance (15
microsecs/lookup compared to 5
microsecs) as Perl's hashes.