I keep getting wrong outcome, while I try to sum big arrays. I have isolated problem to the following code sample (its not an sum of a big array, but I believe this is The problem)
Compilable Sample:
template <typename T>
void cpu_sum(const unsigned int size, T & od_value) {
od_value = 0;
for (unsigned int i = 0; i < size; i++) {
od_value += 1;
}
}
int main() {
typedef float Data;
const unsigned int size = 800000000;
Data sum;
cpu_sum(size, sum);
cout << setprecision(35) << sum << endl; // prints: 16777216 // ERROR !!!
getchar();
}
Environment:
OS: Windows 8.1 x64 home
IDE: Microsoft Visual Studio 2015
Error Description:
While my outcome should obviously be sum == 800000000 I keep getting sum == 16777216.
That is very weird for me, because float max value is far above this one, and yet it looks like sum variable reach its limit.
What did I miss??
It is a well known problem. Gradually your sum becomes so big that next summand becomes comparable with an epsilon (about 10^-14) of it. At that moment you start loosing precision.
Standard solution is to change summation tactics: when array larger than, say, 100 elements, split it in halves and sum each half separately. It goes on recursively and tend to keep precision much better.
As pointed out by Michael Simbursky, this is a well-known problem when trying to sum a large number of limited-precision numbers. The solution he provides works well, and is quite efficient.
For the curious reader, a slower method is to sort your array before computing the sum, assuring that the values are added in order of increasing absolute value. This ensures that the least-significant values make their contributions to the overall sum before being overwhelmed by the other values. This technique is used more as an example/illustration than in any serious programming ventures.
For serious scientific programming where the overall size of the data can vary a great deal, another algorithm to be aware of is the Kahan Summation Algorithm, and the referenced wikipedia page provides a nice description of it.
Related
The task is:
Given is an array containing N numbers, A[0],A[1],...A[N-1]. Compute the array B of length N, such that B[i]=A[0]*A[1]*...A[i-1]*A[i+1]...*A[N-1]. You shouldn't use division and both time and memory complexity should be O(N).
#include <iostream>
#include <math.h>
#include <conio.h>
#include <time.h>
long int find_solution(int result, int multiplier, int increment)
{
int safety=1;
long int solution=0;
while(safety==1)
{
if (solution*multiplier==result)
{
safety=0;
return solution;
}
else if((solution+increment)*multiplier>result)
{
safety=0;
return find_solution(result,multiplier, increment*0.1);
}
solution=solution+increment;
}
}
int main()
{
double time_spent=0.0;
int A[20] = {2, 3, 4, 5, 6,1,2,3,4,5,1,1,1,2,3,1,1,1,1,1};
clock_t begin=clock();
int i;
long int main_result=1;
for(i=0;i<20;i++)
{
main_result=main_result*A[i];
}
printf("\n%d\n",main_result);
long int B[20];
for(i=0;i<20;i++)
{
B[i]=find_solution(main_result, A[i],100000000);
}
for(i=0;i<20;i++)
{
printf("%d\n",B[i]);
}
clock_t end=clock();
time_spent=(double)(end-begin)/CLOCKS_PER_SEC;
printf("Time elapsed: %.20f\n",time_spent);
getch();
return 0;
}
I analyzed this function myself and it seems OK. When I calculate the time on the computer, it's fine if the input is 5,10,15 (I made 21 measures for each amount of input). When it's 20, the time starts growing more than it probably should.
5-0.01333333
10-0.02276
15-0.03757
20-0.07066
What might be the reason? I didn't use any nested loops. Is my function OK?
I would suggest generating much larger inputs to test time complexity, for example N1=100000, N2=200000 ... . For small inputs and small timeframes compilers, caches and other processes may impact the time you measure, and shouldnt be part of your analysis.
Your algorithm looks to be O(N) to me, as find_solution() is constant time and is not called more than some constant times. A very cool and creative solution to the problem, however, you might just be reinventing division as you are answering the question "what multiplied by the denominator equals the numerator".
My solution after thinking a little is to for each index calculate the product of all elements to the right, this can be done in O(N) time. Do the same for the other side and multiply the sides together and you have B, right?
EDIT: Also, when discussing time complexity, measurements are fine but they do not "prove" the time complexity, that has to be done manually by understanding the time complexity of different parts of the algorithm and analysing how many times something is ran.
O(N) means that the algorithm should take N operations, which is limited to the input array size of A in your case.
At main function the loop iterates 20 times but within "find_solution" method loop the complexity duplicates.
I think you are a bit confused, your task requires just the multiplication of each item of A for all B's whatever the i of B is:
B[i]=A[0]A[1]...A[i-1]*A[i+1]...*A[N-1].
So multiply each item of A once, and put it into all B's within a for loop of N is O(N).
Problem Statement is to find prime number below 2 billion in timeframe < 20 sec.
I followed below approaches.
Divide the number n with list of number k ( k < sqrt(n)) - took 20 sec
Divide the number n with list of prime number below sqrt(n).In this scenario I stored prime numbers in std::list - took more than 180 sec
Can someone help me understand why did 2nd approach take longtime even though we reduced no of divisions by 50%(approx)? or Did I choose wrong Data Structure?
Approach 1:
#include <iostream>
#include<list>
#include <ctime>
using namespace std;
list<long long> primeno;
void ListPrimeNumber();
int main()
{
clock_t time_req = clock();
ListPrimeNumber();
time_req = clock() - time_req;
cout << "time taken " << static_cast<float>(time_req) / CLOCKS_PER_SEC << " seconds" << endl;
return 0;
}
void check_prime(int i);
void ListPrimeNumber()
{
primeno.push_back(2);
primeno.push_back(3);
primeno.push_back(5);
for (long long i = 6; i <= 20000000; i++)
{
check_prime(i);
}
}
void check_prime(int i)
{
try
{
int j = 0;
int limit = sqrt(i);
for (j = 2 ; j <= limit;j++)
{
if(i % j == 0)
{
break;
}
}
if( j > limit)
{
primeno.push_back(i);
}
}
catch (exception ex)
{
std::cout << "Message";
}
}
Approach 2 :
#include <iostream>
#include<list>
#include <ctime>
using namespace std;
list<long long> primeno;
int noofdiv = 0;
void ListPrimeNumber();
int main()
{
clock_t time_req = clock();
ListPrimeNumber();
time_req = clock() - time_req;
cout << "time taken " << static_cast<float>(time_req) / CLOCKS_PER_SEC << " seconds" << endl;
cout << "No of divisions : " << noofdiv;
return 0;
}
void check_prime(int i);
void ListPrimeNumber()
{
primeno.push_back(2);
primeno.push_back(3);
primeno.push_back(5);
for (long long i = 6; i <= 10000; i++)
{
check_prime(i);
}
}
void check_prime(int i)
{
try
{
int limit = sqrt(i);
for (int iter : primeno)
{
noofdiv++;
if (iter <= limit && (i%iter) == 0)
{
break;
}
else if (iter > limit)
{
primeno.push_back(i);
break;
}
}
}
catch (exception ex)
{
std::cout << "Message";
}
}
The reason your second example takes longer is you're iterating a std::list.
A std::list in C++ is a linked list, which means it doesn't use contiguous memory. This is bad because to iterate the list you must jump from node to node in a (to the CPU/prefetcher) unpredictable way. Also, You're most likely only "using" a few bytes of each cacheline. RAM is slow. Fetching a byte from RAM takes a lot longer than fetching it from L1. CPUs are fast these days, so your program is most of the time not doing anything and waiting for memory to arrive.
Use a std::vector instead. It stores all values one after the other and iterating is very cheap. Since you're iterating forward in memory without jumping, you're using the full cacheline and your prefetcher will be able to fetch further pages before you need them because your access of memory is predictable.
It has been proven by numerous people, including Bjarne Stroustrup, that std::vector is in a lot of cases faster than std::list, even in cases where the std::list has "theoretically" better complexity (random insert, delete, ...) just because caching helps a lot. So always use std::vector as your default. And if you think a linked list would be faster in your case, measure it and be surprised that - most of the time - std::vector dominates.
Edit: as others have noted, your method of finding primes isn't very efficient. I just played around a bit and implemented a Sieve of Eratosthenes using a bitset.
constexpr int max_prime = 1000000000;
std::bitset<max_prime> *bitset = new std::bitset<max_prime>{};
// Note: Bit SET means NO prime
bitset->set(0);
bitset->set(1);
for(int i = 4; i < max_prime ; i += 2)
bitset->set(i); // set all even numbers
int max = sqrt(max_prime);
for(int i = 3; i < max; i += 2) { // No point testing even numbers as they can't be prime
if(!bitset->test(i)) { // If i is prime
for(int j = i * 2; j < no_primes; j += i)
bitset->set(j); // set all multiples of i to non-prime
}
}
This takes between 4.2 and 4.5 seconds 30 seconds (not sure why it changed that much after slight modifications... must be an optimization I'm not hitting anymore) to find all primes below one Billion (1,000,000,000) on my machine. Your approach took way too long even for 100 million. I cancelled the 1 Billion search after about two minutes.
Comparison for 100 million:
time taken: 63.515 seconds
time taken bitset: 1.874 seconds
No of divisions : 1975961174
No of primes found: 5761455
No of primes found bitset: 5761455
I'm not a mathematician so I'm pretty sure there are still ways to optimize it further, I only optimize for even numbers.
The first thing to do is make sure you are compiling with optimisations enabled. The c++ standard library template classes tend to perform very poorly with unoptimised code as they generate lots of function calls. The optimiser inlines most of these function calls which makes them much cheaper.
std::list is a linked list. Its is mostly useful where you want to insert or remove elements randomly (i.e. not from the end).
For the case where you are only appending to the end of a list std::list has the following issues:
Iterating through the list is relatively expensive as the code has to follow node pointers and then retrieve the data
The list uses quite a lot more memory, each element needs a pointer to the previous and next nodes in addition to the actual data. On a 64-bit system this equates to 20 bytes per element rather than 4 for a list of int
As the elements in the list are not contiguous in memory the compiler can't perform as many SIMD optimisations and you will suffer more from CPU cache misses
A std::vector would solve all of the above as its memory is contiguous and iterating through it is basically just a case of incrementing an array index. You do need to make sure that you call reserve on your vector at the beginning with a sufficiently large value so that appending to the vector doesn't cause the whole array to be copied to a new larger array.
A bigger optimisation than the above would be to use the Sieve of Eratosthenes to calculate your primes. As generating this light require random deletions (depending on your exact implementation) std::list might perform better than std::vector though even in this case the overheads of std::list might not outweigh its costs.
A test at Ideone (the OP code with few superficial alterations) completely contradicts the claims made in this question:
/* check_prime__list:
time taken No of divisions No of primes
10M: 0.873 seconds 286144936 664579
20M: 2.169 seconds 721544444 1270607 */
2B: projected time: at least 16 minutes but likely much more (*)
/* check_prime__nums:
time taken No of divisions No of primes
10M: 4.650 seconds 1746210131 664579
20M: 12.585 seconds 4677014576 1270607 */
2B: projected time: at least 3 hours but likely much more (*)
I also changed the type of the number of divisions counter to long int because it was wrapping around the data type limit. So they could have been misinterpreting that.
But the run time wasn't being affected by that. A wall clock is a wall clock.
Most likely explanation seems to be a sloppy testing by the OP, with different values used in each test case, by mistake.
(*) The time projection was made by the empirical orders of growth analysis:
100**1.32 * 2.169 / 60 = 15.8
100**1.45 * 12.585 / 3600 = 2.8
Empirical orders of growth, as measured on the given range of sizes, were noticeably better for the list algorithm, n1.32 vs. the n1.45 for the testing by all numbers. This is entirely expected from theoretical complexity, since there are fewer primes than all numbers up to n, by a factor of log n, for a total complexity of O(n1.5/log n) vs. O(n1.5). It is also highly unlikely for any implementational discrepancy to beat an actual algorithmic advantage.
I have tested three C++ program codes which load the data from the memory to CPU, do the simple + or x operations (count the time here) and then report the result. These three codes have the same structure but with different data types (int,double,float).
The testing result is: The time is 2x when the data size is 2x for both three codes.
However, I have the following observations.
Observation 1: the time is 2x slower when the optimization is not used. However, it is strange since the loading time (bottleneck) should not be affected by the compiler.
Observation 2: the time for double type program code is roughly 2x faster than both the int and float type program codes when no compiler optimization is added into it and the data size is fixed (256MB,512MB,1024MB,2048MB,4096MB). It is also strange, since double should be the slowest one.
Remark for Observation 2: The time for both three codes are similar when I add the compiler optimization (O,O2,O3).
The attched Code is here:
int main()
{
float value;
double totalTimeDifference;
const int numberOFElements=178956970; //4GB for 6 arrays in total
float*FLOAT_Array_one=new float[numberOFElements];
float*FLOAT_Array_two=new float[numberOFElements];
float*FLOAT_Array_three=new float[numberOFElements];
float*FLOAT_Array_four=new float[numberOFElements];
float*FLOAT_Array_five=new float[numberOFElements];
float*FLOAT_Array=new float[numberOFElements];
srand(time(NULL));
for(int i=0;i<numberOFElements;i++)
{
FLOAT_Array_one[i]=rand()% 400;
FLOAT_Array_two[i]=rand()% 400;
FLOAT_Array_four[i]=rand()% 400;
FLOAT_Array_five[i]=rand()% 400;
}
timeval tim1;
timeval tim2;
gettimeofday(&tim1,NULL);
//****************************//
for(int i=0;i<numberOFElements;i++)
{
FLOAT_Array[i]=FLOAT_Array_one[i]+FLOAT_Array_two[i];
}
//****************************//
//****************************//
for(int i=0;i<numberOFElements;i++)
{
FLOAT_Array_three[i]=FLOAT_Array_four[i]*FLOAT_Array_five[i];
}
//****************************//
gettimeofday(&tim2,NULL);
double t1=tim1.tv_sec+(tim1.tv_usec/1000000.0);
double t2=tim2.tv_sec+(tim2.tv_usec/1000000.0);
for(int i=0;i<numberOFElements;i++)
{
if(i%2==0)
value=value+FLOAT_Array[i]+FLOAT_Array_three[i];
else
value=value-FLOAT_Array[i]-FLOAT_Array_three[i];
}
totalTimeDifference=t2-t1;
cout<<value<<endl;
cout<<totalTimeDifference<<endl;
}
A few educated guesses:
Since you are doing arithmetic operations in your "loading" between the two time checks, it may be employing SSE streaming floating-point math instructions only when optimized. This should result in a significant speedup.
If you are on a 64-bit OS, memory access for two half words takes more time than one whole one. On a 64-bit program, a float takes only 32-bits, and subsequent accesses to half words take more time than a single access to a whole word. The part I don't understand about that, however, is how optimizations are able to get around this.
I am C++ student and I am working on creating a random number generator.
Infact I should say my algorithm selects a number within a defined range.
I am writing this just because of my curiosity.
I am not challenging existing library functions.
I always use library functions when writing applications based on randomness but I am again stating that I just want to make it because of my curiosity.
I would also like to know if there is something wrong with my algorithm or with my approach.
Because i googled how PRNGs work and on some sites they said that a mathematical algorithm is there and a predefined series of numbers and a seed just sets the pointer in a different point in the series and after some intervals the sequence repeats itself.
My algorithm just starts moving to and fro in the array of possible values and the seed breaks the loop with different values each time. I don't i this approach is wrong. I got answers suggesting a different algorithm but they didn't explain What's wrong with my current algorithm?
Yes,there was a problem with my seed as it was not precise and made results little predictable as here:-
cout<
<
rn(50,100);
The results in running four times are 74,93,56,79.
See the pattern of "increasing order".
And for large ranges patterns could be seen easily.I got an answer on getting good seeds but that too recommended a new algorithm(but didn't say why?).
An alternative way could be to shuffle my array randomly generating a new sequence every time.And the pattern of increasing order will go off.Any help with that rearranging too will also be good.Here is the code below.And if my function is not possible please notify me.
Thanking you in anticipation.
int rn(int lowerlt, int upperlt)
{
/* Over short ranges, results are satisfactory.
* I want to make it effective for big ranges.
*/
const int size = upperlt - lowerlt; // Constant size of the integer array.
int ar[size]; // Array to store all possible values within defined range.
int i, x, ret; // Variables to control loops and return value.
long pointer = 0; //pointer variable. The one which breaks the main loop.
// Loop to initialize the array with possible values..
for (i=0, x=lowerlt; x <= upperlt; i++, x++)
ar[i]=x;
long seed = time(0);
//Main loop . To find the random number.
for (i=0; pointer <= seed; i++, pointer++)
{
ret = ar[i];
if (i == size-1)
{
// Reverse loop.
for (; i >= 0; i--)
{
ret=ar[i];
}
}
}
return ret;
}
Caveat: From your post, aside from your random generator algorithm, one of your problems is getting a good seed value, so I'll address that part of it.
You could use /dev/random to get a seed value. That would be a great place to start [and would be sufficient on its own], but might be considered "cheating" from some perspective.
So, here are some other sources of "entropy":
Use a higher resolution time of day clock source: gettimeofday or clock_gettime(CLOCK_REALTIME,...) call it "cur_time". Use only the microsecond or nanosecond portion respectively, call it "cur_nano". Note that cur_nano is usually pretty random all by itself.
Do a getpid(2). This has a few unpredictable bits because between invocations other programs are starting and we don't know how many.
Create a new temp file and get the file's inode number [then delete it]. This varies slowly over time. It may be the same on each invocation [or not]
Get the high resolution value for the system's time of day clock when the system was booted, call it "sysboot".
Get the high resolution value for the start time of your "session": When your program's parent shell was started, call it "shell_start".
If you were using Linux, you could compute a checksum of /proc/interrupts as that's always changing. For other systems, get some hash of the number of interrupts of various types [should be available from some type of syscall].
Now, create some hash of all of the above (e.g.):
dev_random * cur_nano * (cur_time - sysboot) * (cur_time - shell_start) *
getpid * inode_number * interrupt_count
That's a simple equation. You could enhance it with some XOR and/or sum operations. Experiment until you get one that works for you.
Note: This only gives you the seed value for your PRNG. You'll have to create your PRNG from something else (e.g. earl's linear algorithm)
unsigned int Random::next() {
s = (1664525 * s + 1013904223);
return s;
}
's' is growing with every call of that function.
Correct is
unsigned int Random::next() {
s = (1664525 * s + 1013904223) % xxxxxx;
return s;
}
Maybe use this function
long long Factor = 279470273LL, Divisor = 4294967291LL;
long long seed;
next()
{
seed = (seed * Factor) % Divisor;
}
OK, here's my situation :
I have a function - let's say U64 calc (U64 x) - which takes a 64-bit integer parameter, performs some CPU-intensive operation, and returns a 64-bit value
Now, given that I know ALL possible inputs (the xs) of that function beforehand (there are some 16000 though), I thought it might be better to pre-calculate them and then fetch them on demand (from an array-like structure).
The ideal situation would be to store them all in some array U64 CALC[] and retrieve them by index (the x again)
And here's the issue : I may know what the possible inputs for my calc function are, but they are most definitely NOT consecutive (e.g. not from 1 to 16000, but values that may go as low as 0 and as high as some trillions - always withing a 64-bit range)
E.G.
X CALC[X]
-----------------------
123123 123123123
12312 12312312
897523 986123
etc.
And here comes my question :
How would you store them?
What workaround would you prefer?
Now, given that these values (from CALC) will have to be accessed some thousands-to-millions of times, per sec, which would be the best solution performance-wise?
Note : I'm no mentioning anything I've thought of or tried so as not to turn the answers into some debate of A vs B type-of-thing, and mostly not influence anyone...
I would use some sort of hash function that creates an index to a u64 pair where one is the value the key was created from and the other the replacement value. Technically the index could be three bytes long (assuming 16 million -"16000 thousand" - pairs) if you need to conserve space but I'd use u32s. If the stored value does not match the value computed on (hash collision) you'd enter an overflow handler.
You need to determine a custom hashing algorithm to fit your data
Since you know the size of the data you don't need algorithms that allow the data to grow.
I'd be wary of using some standard algorithm because they seldom fit specific data
I'd be wary of using a C++ method unless you are sure the code is WYSIWYG (doesn't generate a lot of invisible calls)
Your index should be 25% larger than the number of pairs
Run through all possible inputs and determine min, max, average and standard deviation for the number of collisions and use these to determine the acceptable performance level. Then profile to achieve the best possible code.
The required memory space (using a u32 index) comes out to (4*1.25)+8+8 = 21 bytes per pair or 336 MeB, no problem on a typical PC.
________ EDIT________
I have been challenged by "RocketRoy" to put my money where my mouth is. Here goes:
The problem has to do with collision handling in a (fixed size) hash index. To set the stage:
I have a list of n entries where a field in the entry contains the value v that the hash is computed from
I have a vector of n*1.25 (approximately) indeces such that the number of indeces x is a prime number
A prime number y is computed which is a fraction of x
The vector is initialized to say -1 to denote unoccupied
Pretty standard stuff I think you'll agree.
The entries in the list are processed and the hash value h computed and modulo'd and used as an index into the vector and the index to the entry is placed there.
Eventually I encounter the situation where the vector entry pointed to by the index is occupied (doesn't contain -1) - voilà, a collision.
So what do I do? I keep the original h as ho, add y to h and take modulo x and get a new index into the vector. If the entry is unoccupied I use it, otherwise I continue with add y modulo x until I reach ho. In theory, this will happen because both x and y are prime numbers. In practice x is larger than n so it won't.
So the "re-hash" that RocketRoy claims is very costly is no such thing.
The tricky part with this method - as with all hashing methods - is to:
Determine a suitable value for x (becomes less tricky the larger x finally used)
Determine the algorithm a for h=a(v)%x such that a the h's index reasonably evenly ("randomly") into the index vector with as few collisions as possible
Determine a suitable value for y such that collisions index reasonably evenly ("randomly") into the index vector
________ EDIT________
I'm sorry I've taken so long to produce this code ... other things have had higher priorities.
Anyway, here is the code which proves that hashing has better prospects for quick lookups than a binary tree. It runs through a bunch of hashing index sizes and algorithms to aid in finding the most suitable combo for the specific data. For every algorithm the code will print the first index size such that no lookup takes longer than fourteen searches (worst case for binary searching) and an average lookup takes less than 1.5 searches.
I have a fondness for prime numbers in these types of applications, in case you haven't noticed.
There are many ways of creating a hashing algorithm with its mandatory overflow handling. I opted for simplicity assuming it will translate into speed ... and it does. On my laptop with an i5 M 480 # 2.67 GHz an average lookup requires between 55 and 60 clock cycles (comes out to around 45 million lookups per second). I implemented a special get operation with a constant number of indeces and ditto rehash value and the cycle count dropped to 40 (65 million lookups per second). If you look at the line calling getOpSpec the index i is xor'ed with 0x444 to exercise the caches to achieve more "real world"-like results.
I must again point out that the program suggests suitable combinations for the specific data. Other data may require a different combo.
The source code contains both the code for generating the 16000 unsigned long long pairs and for testing different constants (index sizes and rehash values):
#include <windows.h>
#define i8 signed char
#define i16 short
#define i32 long
#define i64 long long
#define id i64
#define u8 char
#define u16 unsigned short
#define u32 unsigned long
#define u64 unsigned long long
#define ud u64
#include <string.h>
#include <stdio.h>
u64 prime_find_next (const u64 value);
u64 prime_find_previous (const u64 value);
static inline volatile unsigned long long rdtsc_to_rax (void)
{
unsigned long long lower,upper;
asm volatile( "rdtsc\n"
: "=a"(lower), "=d"(upper));
return lower|(upper<<32);
}
static u16 index[65536];
static u64 nindeces,rehshFactor;
static struct PAIRS {u64 oval,rval;} pairs[16000] = {
#include "pairs.h"
};
struct HASH_STATS
{
u64 ninvocs,nrhshs,nworst;
} getOpStats,putOpStats;
i8 putOp (u16 index[], const struct PAIRS data[], const u64 oval, const u64 ci)
{
u64 nworst=1,ho,h,i;
i8 success=1;
++putOpStats.ninvocs;
ho=oval%nindeces;
h=ho;
do
{
i=index[h];
if (i==0xffff) /* unused position */
{
index[h]=(u16)ci;
goto added;
}
if (oval==data[i].oval) goto duplicate;
++putOpStats.nrhshs;
++nworst;
h+=rehshFactor;
if (h>=nindeces) h-=nindeces;
} while (h!=ho);
exhausted: /* should not happen */
duplicate:
success=0;
added:
if (nworst>putOpStats.nworst) putOpStats.nworst=nworst;
return success;
}
i8 getOp (u16 index[], const struct PAIRS data[], const u64 oval, u64 *rval)
{
u64 ho,h,i;
i8 success=1;
ho=oval%nindeces;
h=ho;
do
{
i=index[h];
if (i==0xffffu) goto not_found; /* unused position */
if (oval==data[i].oval)
{
*rval=data[i].rval; /* fetch the replacement value */
goto found;
}
h+=rehshFactor;
if (h>=nindeces) h-=nindeces;
} while (h!=ho);
exhausted:
not_found: /* should not happen */
success=0;
found:
return success;
}
volatile i8 stop = 0;
int main (int argc, char *argv[])
{
u64 i,rval,mulup,divdown,start;
double ave;
SetThreadAffinityMask (GetCurrentThread(), 0x00000004ull);
divdown=5; //5
while (divdown<=100)
{
mulup=3; // 3
while (mulup<divdown)
{
nindeces=16000;
while (nindeces<65500)
{
nindeces= prime_find_next (nindeces);
rehshFactor=nindeces*mulup/divdown;
rehshFactor=prime_find_previous (rehshFactor);
memset (index, 0xff, sizeof(index));
memset (&putOpStats, 0, sizeof(struct HASH_STATS));
i=0;
while (i<16000)
{
if (!putOp (index, pairs, pairs[i].oval, (u16) i)) stop=1;
++i;
}
ave=(double)(putOpStats.ninvocs+putOpStats.nrhshs)/(double)putOpStats.ninvocs;
if (ave<1.5 && putOpStats.nworst<15)
{
start=rdtsc_to_rax ();
i=0;
while (i<16000)
{
if (!getOp (index, pairs, pairs[i^0x0444]. oval, &rval)) stop=1;
++i;
}
start=rdtsc_to_rax ()-start+8000; /* 8000 is half of 16000 (pairs), for rounding */
printf ("%u;%u;%u;%u;%1.3f;%u;%u\n", (u32)mulup, (u32)divdown, (u32)nindeces, (u32)rehshFactor, ave, (u32) putOpStats.nworst, (u32) (start/16000ull));
goto found;
}
nindeces+=2;
}
printf ("%u;%u\n", (u32)mulup, (u32)divdown);
found:
mulup=prime_find_next (mulup);
}
divdown=prime_find_next (divdown);
}
SetThreadAffinityMask (GetCurrentThread(), 0x0000000fu);
return 0;
}
It was not possible to include the generated pairs file (an answer is apparently limited to 30000 characters). But send a message to my inbox and I'll mail it.
And these are the results:
3;5;35569;21323;1.390;14;73
3;7;33577;14389;1.435;14;60
5;7;32069;22901;1.474;14;61
3;11;35107;9551;1.412;14;59
5;11;33967;15427;1.446;14;61
7;11;34583;22003;1.422;14;59
3;13;34253;7901;1.439;14;61
5;13;34039;13063;1.443;14;60
7;13;32801;17659;1.456;14;60
11;13;33791;28591;1.436;14;59
3;17;34337;6053;1.413;14;59
5;17;32341;9511;1.470;14;61
7;17;32507;13381;1.474;14;62
11;17;33301;21529;1.454;14;60
13;17;34981;26737;1.403;13;59
3;19;33791;5333;1.437;14;60
5;19;35149;9241;1.403;14;59
7;19;33377;12289;1.439;14;97
11;19;34337;19867;1.417;14;59
13;19;34403;23537;1.430;14;61
17;19;33923;30347;1.467;14;61
3;23;33857;4409;1.425;14;60
5;23;34729;7547;1.429;14;60
7;23;32801;9973;1.456;14;61
11;23;33911;16127;1.445;14;60
13;23;33637;19009;1.435;13;60
17;23;34439;25453;1.426;13;60
19;23;33329;27529;1.468;14;62
3;29;32939;3391;1.474;14;62
5;29;34543;5953;1.437;13;60
7;29;34259;8263;1.414;13;59
11;29;34367;13033;1.409;14;60
13;29;33049;14813;1.444;14;60
17;29;34511;20219;1.422;14;60
19;29;33893;22193;1.445;13;61
23;29;34693;27509;1.412;13;92
3;31;34019;3271;1.441;14;60
5;31;33923;5449;1.460;14;61
7;31;33049;7459;1.442;14;60
11;31;35897;12721;1.389;14;59
13;31;35393;14831;1.397;14;59
17;31;33773;18517;1.425;14;60
19;31;33997;20809;1.442;14;60
23;31;34841;25847;1.417;14;59
29;31;33857;31667;1.426;14;60
3;37;32569;2633;1.476;14;61
5;37;34729;4691;1.419;14;59
7;37;34141;6451;1.439;14;60
11;37;34549;10267;1.410;13;60
13;37;35117;12329;1.423;14;60
17;37;34631;15907;1.429;14;63
19;37;34253;17581;1.435;14;60
23;37;32909;20443;1.453;14;61
29;37;33403;26177;1.445;14;60
31;37;34361;28771;1.413;14;59
3;41;34297;2503;1.424;14;60
5;41;33587;4093;1.430;14;60
7;41;34583;5903;1.404;13;59
11;41;32687;8761;1.440;14;60
13;41;34457;10909;1.439;14;60
17;41;34337;14221;1.425;14;59
19;41;32843;15217;1.476;14;62
23;41;35339;19819;1.423;14;59
29;41;34273;24239;1.436;14;60
31;41;34703;26237;1.414;14;60
37;41;33343;30089;1.456;14;61
3;43;34807;2423;1.417;14;59
5;43;35527;4129;1.413;14;60
7;43;33287;5417;1.467;14;61
11;43;33863;8647;1.436;14;60
13;43;34499;10427;1.418;14;78
17;43;34549;13649;1.431;14;60
19;43;33749;14897;1.429;13;60
23;43;34361;18371;1.409;14;59
29;43;33149;22349;1.452;14;61
31;43;34457;24821;1.428;14;60
37;43;32377;27851;1.482;14;81
41;43;33623;32057;1.424;13;59
3;47;33757;2153;1.459;14;61
5;47;33353;3547;1.445;14;61
7;47;34687;5153;1.414;13;59
11;47;34519;8069;1.417;14;60
13;47;34549;9551;1.412;13;59
17;47;33613;12149;1.461;14;61
19;47;33863;13687;1.443;14;60
23;47;35393;17317;1.402;14;59
29;47;34747;21433;1.432;13;60
31;47;34871;22993;1.409;14;59
37;47;34729;27337;1.425;14;59
41;47;33773;29453;1.438;14;60
43;47;31253;28591;1.487;14;62
3;53;33623;1901;1.430;14;59
5;53;34469;3229;1.430;13;60
7;53;34883;4603;1.408;14;59
11;53;34511;7159;1.412;13;59
13;53;32587;7963;1.453;14;60
17;53;34297;10993;1.432;13;80
19;53;33599;12043;1.443;14;64
23;53;34337;14897;1.415;14;59
29;53;34877;19081;1.424;14;61
31;53;34913;20411;1.406;13;59
37;53;34429;24029;1.417;13;60
41;53;34499;26683;1.418;14;59
43;53;32261;26171;1.488;14;62
47;53;34253;30367;1.437;14;79
3;59;33503;1699;1.432;14;61
5;59;34781;2939;1.424;14;60
7;59;35531;4211;1.403;14;59
11;59;34487;6427;1.420;14;59
13;59;33563;7393;1.453;14;61
17;59;34019;9791;1.440;14;60
19;59;33967;10937;1.447;14;60
23;59;33637;13109;1.438;14;60
29;59;34487;16943;1.424;14;59
31;59;32687;17167;1.480;14;61
37;59;35353;22159;1.404;14;59
41;59;34499;23971;1.431;14;60
43;59;34039;24799;1.445;14;60
47;59;32027;25471;1.499;14;62
53;59;34019;30557;1.449;14;61
3;61;35059;1723;1.418;14;60
5;61;34351;2803;1.416;13;60
7;61;35099;4021;1.412;14;59
11;61;34019;6133;1.442;14;60
13;61;35023;7459;1.406;14;88
17;61;35201;9803;1.414;14;61
19;61;34679;10799;1.425;14;101
23;61;34039;12829;1.441;13;60
29;61;33871;16097;1.446;14;60
31;61;34147;17351;1.427;14;61
37;61;34583;20963;1.412;14;59
41;61;32999;22171;1.452;14;62
43;61;33857;23857;1.431;14;98
47;61;34897;26881;1.431;14;60
53;61;33647;29231;1.434;14;60
59;61;32999;31907;1.454;14;60
3;67;32999;1471;1.455;14;61
5;67;35171;2621;1.403;14;59
7;67;33851;3533;1.463;14;61
11;67;34607;5669;1.437;14;60
13;67;35081;6803;1.416;14;61
17;67;33941;8609;1.417;14;60
19;67;34673;9829;1.427;14;60
23;67;35099;12043;1.415;14;60
29;67;33679;14563;1.452;14;61
31;67;34283;15859;1.437;14;60
37;67;32917;18169;1.460;13;61
41;67;33461;20443;1.441;14;61
43;67;34313;22013;1.426;14;60
47;67;33347;23371;1.452;14;61
53;67;33773;26713;1.434;14;60
59;67;35911;31607;1.395;14;58
61;67;34157;31091;1.431;14;63
3;71;34483;1453;1.423;14;59
5;71;34537;2423;1.428;14;59
7;71;33637;3313;1.428;13;60
11;71;32507;5023;1.465;14;79
13;71;35753;6529;1.403;14;59
17;71;33347;7963;1.444;14;61
19;71;35141;9397;1.410;14;59
23;71;32621;10559;1.475;14;61
29;71;33637;13729;1.429;14;60
31;71;33599;14657;1.443;14;60
37;71;34361;17903;1.396;14;59
41;71;33757;19489;1.435;14;61
43;71;34583;20939;1.413;14;59
47;71;34589;22877;1.441;14;60
53;71;35353;26387;1.418;14;59
59;71;35323;29347;1.406;14;59
61;71;35597;30577;1.401;14;59
67;71;34537;32587;1.425;14;59
3;73;34613;1409;1.418;14;59
5;73;32969;2251;1.453;14;62
7;73;33049;3167;1.448;14;61
11;73;33863;5101;1.435;14;60
13;73;34439;6131;1.456;14;60
17;73;33629;7829;1.455;14;61
19;73;34739;9029;1.421;14;60
23;73;33071;10399;1.469;14;61
29;73;33359;13249;1.460;14;61
31;73;33767;14327;1.422;14;59
37;73;32939;16693;1.490;14;62
41;73;33739;18947;1.438;14;60
43;73;33937;19979;1.432;14;61
47;73;33767;21739;1.422;14;59
53;73;33359;24203;1.435;14;60
59;73;34361;27767;1.401;13;59
61;73;33827;28229;1.443;14;60
67;73;34421;31583;1.423;14;71
71;73;33053;32143;1.447;14;60
3;79;35027;1327;1.410;14;60
5;79;34283;2161;1.432;14;60
7;79;34439;3049;1.432;14;60
11;79;34679;4817;1.416;14;59
13;79;34667;5701;1.405;14;59
17;79;33637;7237;1.428;14;60
19;79;34469;8287;1.417;14;60
23;79;34439;10009;1.433;14;60
29;79;33427;12269;1.448;13;61
31;79;33893;13297;1.445;14;61
37;79;33863;15823;1.439;14;60
41;79;32983;17107;1.450;14;60
43;79;34613;18803;1.431;14;60
47;79;33457;19891;1.457;14;61
53;79;33961;22777;1.435;14;61
59;79;32983;24631;1.465;14;60
61;79;34337;26501;1.428;14;60
67;79;33547;28447;1.458;14;61
71;79;32653;29339;1.473;14;61
73;79;34679;32029;1.429;14;64
3;83;35407;1277;1.405;14;59
5;83;32797;1973;1.451;14;60
7;83;33049;2777;1.443;14;61
11;83;33889;4483;1.431;14;60
13;83;35159;5503;1.409;14;59
17;83;34949;7151;1.412;14;59
19;83;32957;7541;1.467;14;61
23;83;32569;9013;1.470;14;61
29;83;33287;11621;1.474;14;61
31;83;33911;12659;1.448;13;60
37;83;33487;14923;1.456;14;62
41;83;33587;16573;1.438;13;60
43;83;34019;17623;1.435;14;60
47;83;31769;17987;1.483;14;62
53;83;33049;21101;1.451;14;61
59;83;32369;23003;1.465;14;61
61;83;32653;23993;1.469;14;61
67;83;33599;27109;1.437;14;61
71;83;33713;28837;1.452;14;61
73;83;33703;29641;1.454;14;61
79;83;34583;32911;1.417;14;59
3;89;34147;1129;1.415;13;60
5;89;32797;1831;1.461;14;61
7;89;33679;2647;1.443;14;73
11;89;34543;4261;1.427;13;60
13;89;34603;5051;1.419;14;60
17;89;34061;6491;1.444;14;60
19;89;34457;7351;1.422;14;79
23;89;33529;8663;1.450;14;61
29;89;34283;11161;1.431;14;60
31;89;35027;12197;1.411;13;59
37;89;34259;14221;1.403;14;59
41;89;33997;15649;1.434;14;60
43;89;33911;16127;1.445;14;60
47;89;34949;18451;1.419;14;59
53;89;34367;20443;1.434;14;60
59;89;33791;22397;1.430;14;59
61;89;34961;23957;1.404;14;59
67;89;33863;25471;1.433;13;60
71;89;35149;28031;1.414;14;79
73;89;33113;27143;1.447;14;60
79;89;32909;29209;1.458;14;61
83;89;33617;31337;1.400;14;59
3;97;34211;1051;1.448;14;60
5;97;34807;1789;1.430;14;60
7;97;33547;2417;1.446;14;60
11;97;35171;3967;1.407;14;89
13;97;32479;4349;1.474;14;61
17;97;34319;6011;1.444;14;60
19;97;32381;6337;1.491;14;64
23;97;33617;7963;1.421;14;59
29;97;33767;10093;1.423;14;59
31;97;33641;10739;1.447;14;60
37;97;34589;13187;1.425;13;60
41;97;34171;14437;1.451;14;60
43;97;31973;14159;1.484;14;62
47;97;33911;16127;1.445;14;61
53;97;34031;18593;1.448;14;80
59;97;32579;19813;1.457;14;61
61;97;34421;21617;1.417;13;60
67;97;33739;23297;1.448;14;60
71;97;33739;24691;1.435;14;60
73;97;33863;25471;1.433;13;60
79;97;34381;27997;1.419;14;59
83;97;33967;29063;1.446;14;60
89;97;33521;30727;1.441;14;60
Cols 1 and 2 are used to calculate a rough relationship between the rehash value and the index size. The next two are the first index size/rehash factor combination which averages less than 1.5 searches for a lookup with a worst case of 14 searches. Then average and worst case. Finally, the last column is the average number of clock cycles per lookup. It does not take into account the time required to read the time stamp register.
The actual memory space for the best constants (# of indeces = 31253 and rehash factor = 28591) comes out to more than I initially indicated (16000*2*8 + 1,25*16000*2 => 296000 bytes). The actual size is 16000*2*8+31253*2 => 318506.
The fastest combination is an approximate ratio of 11/31 with an index size of 35897 and rehash value of 12721. This will average 1.389 (1 initial hash + 0.389 rehashes) with a maximum of 14 (1+13).
________ EDIT________
I removed the "goto found;" in main () to show all combinations and it shows that much better performance is possible, of course at the expense of a larger index size. For example the combination 57667 and 33797 yields and average of 1.192 and a maximum rehash of 6. The combination 44543 and 23399 yields a 1.249 average and 10 maximum rehashes (it saves (57667-44543)*2=26468 bytes of index table compared to 57667/33797).
Specialized functions with hard-coded hash index size and rehash factor will execute in 60-70% of the time compared to variables. This is probably due to the compiler (gcc 64-bit) substituting the modulo with multiplications and not having to fetch the values from memory locations as they will be coded as immediate values.
________ EDIT________
On the subject of caches I see two issues.
The first is data cacheing which I don't think will be possible because the lookup will just be a small step in some larger process and you run the risk of the table data's cache lines begin invalidated to a lesser or (probably) greater degree - if not entirely - by other data accesses in other steps of the larger process. I e the more code executed and data accessed in the process as a whole the less likely it will be that any pertinent lookup data will remain in the caches (this may or may not be pertinent to the OP's situation). To find an entry using (my) hashing you will encounter two cache misses (one to load the correct part of the index, and the other to load the area containg the entry itself) for every comparison that needs to be performed. Finding an entry on the first try will have cost two misses, the second try four etc. In my example the 60 clock cycle average cost per lookup implies that the table probably resided entirely in the L2 cache and with L1 not having to go there in a majority of the cases. My x86-64 CPU has L1-3, RAM wait states of approximately 4, 10, 40 and 100 which to me shows that RAM was completely kept out and L3 mostly.
The second is code cacheing which will have a more significant impact if it is small, tight, in-lined and with few control transfers (jumps and calls). My hash routine probably resides entirely in the L1 code cache. For more normal cases, the fewer the number of code cache line loads the faster it will be.
Make an array of structures of key val pairs.
Sort the array by key, put this in your program as static array, would only be 128kbyte.
Then in your program a simple binary look up by key will need on average only 14 key comparisons to find the right value. Should be able to approach speeds of 300 million look ups per second on modern pc.
You can sort with qsort and search with bsearch, both std lib functions.
Perform memonization, or in simple terms, cache the values you've computed already and calculate the new ones. You should hash the input and check the cache for that result. You can even start off with a set of cache values that you think the function would get called more often for. Besides that, I don't think you need to go to any extreme as the other answer suggest. Do things simple and when you are done with your application you can use a profiling tool to find bottle necks.
EDIT: Some code
#include <iostream>
#include <ctime>
using namespace std;
const int MAX_SIZE = 16000;
int preCalcData[MAX_SIZE] = {};
int getPrecalculatedResult(int x){
return preCalcData[x];
}
void setupPreCalcDataCache(){
for(int i = 0; i < MAX_SIZE; ++i){
preCalcData[i] = i*i; //or whatever calculation
}
}
int main(){
setupPreCalcDataCache();
cout << getPrecalculatedResult(0) << endl;
cout << getPrecalculatedResult(15999) << endl;
return 0;
}
I wouldn't worry about performance too much. This simple example, using an array and binary search lower_bound
#include <stdint.h>
#include <algorithm>
#include <cstdlib>
#include <iostream>
#include <memory>
const int N = 16000;
typedef std::pair<uint64_t, uint64_t> CALC;
CALC calc[N];
static inline bool cmp_calcs(const CALC &c1, const CALC &c2)
{
return c1.first < c2.first;
}
int main(int argc, char **argv)
{
std::iostream::sync_with_stdio(false);
for (int i = 0; i < N; ++i)
calc[i] = std::make_pair(i, i);
std::sort(&calc[0], &calc[N], cmp_calcs);
for (long i = 0; i < 10000000; ++i) {
int r = rand() % 16000;
CALC *p = std::lower_bound(&calc[0], &calc[N], std::make_pair(r, 0), cmp_calcs);
if (p->first == r)
std::cout << "found\n";
}
return 0;
}
and compiled with
g++ -O2 example.cpp
does, including setup, 10,000,000 searches in about 2 seconds on my 5 year old PC.
You need to store 16 thousand values efficiently, preferably in memory. We are assuming that the computation of these values is more time consuming than accessing them from storage.
You have at your disposal many different data structures to get the job done, including databases. If you access these values in queriable chunks, then the DB overhead may very well be absorbed and spread accross your processing.
You mentioned map and hashmap (or hashtable) already in your question tags, but these are probably not the best possible answers for your problem, although they could do a fair job, provided that the hashing function isn't more expensive than the direct computation of the target UINT64 value, which has to be your reference benchmark.
Van Emde Boas Trees
Many variants of B-Trees (used extensively in database engines, high performance filesystems),
Tries
Are probably much better suited. Having some experience with it, I would probably go for a B-tree: they support fairly well serialization. That should let you prepare your dataset in advance in a different program. VEB trees have a very good access time (O(log log(n)), but I don't know how easily they may be serialized.
Later on, if you need even more performance, it would also be interesting to know usage patterns of your "database" to figure out what caching techniques you could implement on top of the store.
Using std::pair is better than any of map for speed.
but if I were you, I firstly use a std::list to store the data, after I got them all, I move them into a simple vector, then retrieving goes very fast if you implement a simple binary tree search by yourself.