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I have some problems with code for my classes. Even though it works correctly, I run out of time for half of the examples.
Here's the task (I really did my best trying to translate it):
You have a permutation of numbers 1,2,...,n for some n. All consecutive numbers of permutations together create sequence a1, a2, an. Your task is to count, how many arithmetic substrings of a sequence of length 3 exist.
Input:
In first line there is a number n (1 <= n <= 200 000). In the second line there is n numbers a1, a2...an representing our permutation.
Output:
The program needs to print out amount of arithmetic substrings of length 3 for permutations from entry. You can assume that the result won't be bigger than 1 000 000.
Is there any way to make it work faster? Thanks for help!
#include <iostream>
using namespace std;
int main()
{
int input_length;
cin >> input_length;
int correct_sequences = 0;
bool* whether_itroduced = new bool[input_length + 1]{0}; // true - if number was already introduced and false otherwise.
for (int i = 0; i < input_length; i++)
{
int a;
cin >> a;
whether_itroduced[a] = true;
int range = min(input_length - a, a - 1); // max or min number that may be in the subsequence e.g. if introduced number a = 3, and all numbers are six, max range is 2 (3 - 2 = 1 and 3 + 2 = 5, so the longest possible subsequence is 1, 3, 5)
for (int r = range * -1; r <= range; r++) // r - there is a formula used to count arithmetic sequences -> an-1 = a1-r, an = a1, an+1 = a1+r, I have no idea how to explain it
{
if (r == 0) continue; // r cannot be 0
if (whether_itroduced[a - r] && !whether_itroduced[a + r])
correct_sequences++;
}
}
cout << correct_sequences;
}
example
input:
5
1 5 4 2 3
output:
2
// 1,2,3 and 5,4,3
I need to write a program which is printing n pairs of prime numbers and the those pairs are :
p q
where p and q are prime numbers and q = p+2.
Input example :
n = 3
3 5 //
5 7 //
11 13 //
I'm pretty much nowhere still... So, someone?
#include <iostream>
#include <cmath>
int twins(int n)
{
for (int i = 0; i < n; i++)
{
???
}
}
int main()
{
std::cout<<twins(5);
return 0;
}
Here is the top-level simple pseudo-code for such a beast:
def printTwinPrimes(count):
currNum = 3
while count > 0:
if isPrime(currNum) and isPrime(currNum + 2):
print currnum, currnum + 2
count = count - 1
currNum = currNum + 2
It simply starts at 3 (since we know 2,4 is impossible as a twin-prime pair because 4 is composite). For each possibility, it checks whether it constitutes a twin-prime pair and prints it if so.
So all you need to do (other than translating that into real code) is to create isPrime(), for which there are countless examples on the net.
For completeness, here's a simple one, by no means the most efficient but adequate for beginners:
def isPrime(num):
if num < 2:
return false
root = 2
while root * root <= num:
if num % root == 0:
return false
root = root + 1
return true
Though you could make that more efficient by using the fact that all primes other than two or three are of the form 6n±1, n >= 1(a):
def isPrime(num):
if num < 2: return false
if num == 2 or num == 3: return true
if num % 2 == 0 or num % 3 == 0: return false
if num % 6 is neither 1 nor 5: return false
root = 5
adder = 2 # initial adder 2, 5 -> 7
while root * root <= num:
if num % root == 0:
return false
root = root + adder # checks 5, 7, 11, 13, 17, 19, ...
adder = 6 - adder # because alternate 2, 4 to give 6n±1
return true
In fact, you can use this divisibility trick to see if an arbitraily large number stored as a string is likely to be a prime. You just have to check if the number below it or above it is divisible by six. If not, the number is definitely not a prime. If so, more (slower) checks will be needed to fully ascertain primality.
A number is divisible by six only if it is divisible by both two and three. It's easy to tell the former, even numbers end with an even digit.
But it's also reasonably easy to tell if it's divisible by three since, in that case, the sum of the individual digits will also be divisible by three. For example, lets' use 31415926535902718281828459.
The sum of all those digits is 118. How do we tell if that's a multiple of three? Why, using exactly the same trick recursively:
118: 1 + 1 + 8 = 10
10: 1 + 0 = 1
Once you're down to a single digit, it'll be 0, 3, 6, or 9 if the original number was a multiple of three. Any other digit means it wasn't (such as in this case).
(a) If you divide any non-negative number by six and the remainder is 0, 2 or 4, then it's even and therefore non-prime (2 is the exception case here):
6n + 0 = 2(3n + 0), an even number.
6n + 2 = 2(3n + 1), an even number.
6n + 4 = 2(3n + 2), an even number.
If the remainder is 3, then it is divisible by 3 and therefore non-prime (3 is the exception case here):
6n + 3 = 3(2n + 1), a multiple of three.
That leaves just the remainders 1 and 5, and those numbers are all of that form 6n±1.
Might not be the most efficient but you can calculate all primes till n, store them in a vector then only print those which have a difference of 2
#include <iostream>
#include<vector>
using namespace std;
void pr(int n, vector<int>& v)
{
for (int i=2; i<n; i++)
{
bool prime=true;
for (int j=2; j*j<=i; j++)
{
if (i % j == 0)
{
prime=false;
break;
}
}
if(prime) v.push_back(i);
}
}
int main()
{
vector<int> v;
pr(50, v);
for(int i = 0;i < v.size()-1; i++) {
if(v[i+1]-v[i] == 2) {
cout << v[i+1] << " " << v[i] << endl;
}
}
return 0;
}
I think is the efficient algo for you and easy to understand. You can change the value of k as per your constraints.
#include <iostream>
#include <cstring>
using namespace std;
int n,p=2,savePrime=2,k=100000;
void printNPrime(int n)
{
bool prime[k];
memset(prime, true, sizeof(prime));
while(n>0)
{
if (prime[p] == true)
{
if(p-savePrime == 2)
{
cout<<savePrime<<" "<<p<<endl;
n--;
}
// Update all multiples of p
for (int i=p*2; i<=k; i += p)
prime[i] = false;
savePrime=p;
}
p++;
}
}
int main() {
cin>>n;
printNPrime(n);
return 0;
}
I was trying to solve this problem.
An integer M and a non-empty zero-indexed array A consisting of N
non-negative integers are given. All integers in array A are less than
or equal to M.
A pair of integers (P, Q), such that 0 ≤ P ≤ Q < N, is called a slice
of array A. The slice consists of the elements A[P], A[P + 1], ...,
A[Q]. A distinct slice is a slice consisting of only unique numbers.
That is, no individual number occurs more than once in the slice.
For example, consider integer M = 6 and array A such that:
A[0] = 3
A[1] = 4
A[2] = 5
A[3] = 5
A[4] = 2
There are exactly nine distinct slices: (0, 0), (0, 1), (0, 2), (1,
1), (1,2), (2, 2), (3, 3), (3, 4) and (4, 4).
The goal is to calculate the number of distinct slices.
Thanks in advance.
#include <algorithm>
#include <cstring>
#include <cmath>
#define MAX 100002
// you can write to stdout for debugging purposes, e.g.
// cout << "this is a debug message" << endl;
using namespace std;
bool check[MAX];
int solution(int M, vector<int> &A) {
memset(check, false, sizeof(check));
int base = 0;
int fibot = 0;
int sum = 0;
while(fibot < A.size()){
if(check[A[fibot]]){
base = fibot;
}
check[A[fibot]] = true;
sum += fibot - base + 1;
fibot += 1;
}
return min(sum, 1000000000);
}
The solution is not correct because your algorithm is wrong.
First of all, let me show you a counter example. Let A = {2, 1, 2}. The first iteration: base = 0, fibot = 0, sum += 1. That's right. The second one: base = 0, fibot = 1, sum += 2. That's correct, too. The last step: fibot = 2, check[A[fibot]] is true, thus, base = 2. But it should be 1. So your code returns1 + 2 + 1 = 4 while the right answer 1 + 2 + 2 = 5.
The right way to do it could be like this: start with L = 0. For each R from 0 to n - 1, keep moving the L to the right until the subarray contais only distinct values (you can maintain the number of occurrences of each value in an array and use the fact that A[R] is the only element that can occur more than once).
There is one more issue with your code: the sum variable may overflow if int is 32-bit type on the testing platform (for instance, if all elements of A are distinct).
As for the question WHY your algorithm is incorrect, I have no idea why it should be correct in the first place. Can you prove it? The base = fibot assignment looks quite arbitrary to me.
I would like to share the explanation of the algorithm that I have implemented in C++ followed by the actual implementation.
Notice that the minimum amount of distinct slices is N because each element is a distinct one-item slice.
Start the back index from the first element.
Start the front index from the first element.
Advance the front until we find a duplicate in the sequence.
In each iteration, increment the counter with the necessary amount, this is the difference between front and back.
If we reach the maximum counts at any iteration, just return immediately for slight optimisation.
In each iteration of the sequence, record the elements that have occurred.
Once we have found a duplicate, advance the back index one ahead of the duplicate.
While we advance the back index, clear all the occurred elements since we start a new slice beyond those elements.
The runtime complexity of this solution is O(N) since we go through each
element.
The space complexity of this solution is O(M) because we have a hash to store
the occurred elements in the sequences. The maximum element of this hash is M.
int solution(int M, vector<int> &A)
{
int N = A.size();
int distinct_slices = N;
vector<bool> seq_hash(M + 1, false);
for (int back = 0, front = 0; front < N; ++back) {
while (front < N and !seq_hash[A[front]]) { distinct_slices += front - back; if (distinct_slices > 1000000000) return 1000000000; seq_hash[A[front++]] = true; }
while (front < N and back < N and A[back] != A[front]) seq_hash[A[back++]] = false;
seq_hash[A[back]] = false;
}
return distinct_slices;
}
100% python solution that helped me, thanks to https://www.martinkysel.com/codility-countdistinctslices-solution/
def solution(M, A):
the_sum = 0
front = back = 0
seen = [False] * (M+1)
while (front < len(A) and back < len(A)):
while (front < len(A) and seen[A[front]] != True):
the_sum += (front-back+1)
seen[A[front]] = True
front += 1
else:
while front < len(A) and back < len(A) and A[back] != A[front]:
seen[A[back]] = False
back += 1
seen[A[back]] = False
back += 1
return min(the_sum, 1000000000)
Solution with 100% using Ruby
LIMIT = 1_000_000_000
def solution(_m, a)
a.each_with_index.inject([0, {}]) do |(result, slice), (back, i)|
return LIMIT if result >= LIMIT
slice[back] = true
a[(i + slice.size)..-1].each do |front|
break if slice[front]
slice[front] = true
end
slice.delete back
[result + slice.size, slice]
end.first + a.size
end
Using Caterpillar algorithm and the formula that S(n+1) = S(n) + n + 1 where S(n) is count of slices for n-element array java solution could be:
public int solution(int top, int[] numbers) {
int len = numbers.length;
long count = 0;
if (len == 1) return 1;
int front = 0;
int[] counter = new int[top + 1];
for (int i = 0; i < len; i++) {
while(front < len && counter[numbers[front]] == 0 ) {
count += front - i + 1;
counter[numbers[front++]] = 1;
}
while(front < len && numbers[i] != numbers[front] && i < front) {
counter[numbers[i++]] = 0;
}
counter[numbers[i]] = 0;
if (count > 1_000_000_000) {
return 1_000_000_000;
}
}
return count;
}
Can anybody find any potentially more efficient algorithms for accomplishing the following task?:
For any given permutation of the integers 0 thru 7, return the index which describes the permutation lexicographically (indexed from 0, not 1).
For example,
The array 0 1 2 3 4 5 6 7 should return an index of 0.
The array 0 1 2 3 4 5 7 6 should return an index of 1.
The array 0 1 2 3 4 6 5 7 should return an index of 2.
The array 1 0 2 3 4 5 6 7 should return an index of 5039 (that's 7!-1 or factorial(7)-1).
The array 7 6 5 4 3 2 1 0 should return an index of 40319 (that's 8!-1). This is the maximum possible return value.
My current code looks like this:
int lexic_ix(int* A){
int value = 0;
for(int i=0 ; i<7 ; i++){
int x = A[i];
for(int j=0 ; j<i ; j++)
if(A[j]<A[i]) x--;
value += x*factorial(7-i); // actual unrolled version doesn't have a function call
}
return value;
}
I'm wondering if there's any way I can reduce the number of operations by removing that inner loop, or if I can reduce conditional branching in any way (other than unrolling - my current code is actually an unrolled version of the above), or if there are any clever bitwise hacks or filthy C tricks to help.
I already tried replacing
if(A[j]<A[i]) x--;
with
x -= (A[j]<A[i]);
and I also tried
x = A[j]<A[i] ? x-1 : x;
Both replacements actually led to worse performance.
And before anyone says it - YES this is a huge performance bottleneck: currently about 61% of the program's runtime is spent in this function, and NO, I don't want to have a table of precomputed values.
Aside from those, any suggestions are welcome.
Don't know if this helps but here's an other solution :
int lexic_ix(int* A, int n){ //n = last index = number of digits - 1
int value = 0;
int x = 0;
for(int i=0 ; i<n ; i++){
int diff = (A[i] - x); //pb1
if(diff > 0)
{
for(int j=0 ; j<i ; j++)//pb2
{
if(A[j]<A[i] && A[j] > x)
{
if(A[j]==x+1)
{
x++;
}
diff--;
}
}
value += diff;
}
else
{
x++;
}
value *= n - i;
}
return value;
}
I couldn't get rid of the inner loop, so complexity is o(n log(n)) in worst case, but o(n) in best case, versus your solution which is o(n log(n)) in all cases.
Alternatively, you can replace the inner loop by the following to remove some worst cases at the expense of another verification in the inner loop :
int j=0;
while(diff>1 && j<i)
{
if(A[j]<A[i])
{
if(A[j]==x+1)
{
x++;
}
diff--;
}
j++;
}
Explanation :
(or rather "How I ended with that code", I think it is not that different from yours but it can make you have ideas, maybe)
(for less confusion I used characters instead and digit and only four characters)
abcd 0 = ((0 * 3 + 0) * 2 + 0) * 1 + 0
abdc 1 = ((0 * 3 + 0) * 2 + 1) * 1 + 0
acbd 2 = ((0 * 3 + 1) * 2 + 0) * 1 + 0
acdb 3 = ((0 * 3 + 1) * 2 + 1) * 1 + 0
adbc 4 = ((0 * 3 + 2) * 2 + 0) * 1 + 0
adcb 5 = ((0 * 3 + 2) * 2 + 1) * 1 + 0 //pb1
bacd 6 = ((1 * 3 + 0) * 2 + 0) * 1 + 0
badc 7 = ((1 * 3 + 0) * 2 + 1) * 1 + 0
bcad 8 = ((1 * 3 + 1) * 2 + 0) * 1 + 0 //First reflexion
bcda 9 = ((1 * 3 + 1) * 2 + 1) * 1 + 0
bdac 10 = ((1 * 3 + 2) * 2 + 0) * 1 + 0
bdca 11 = ((1 * 3 + 2) * 2 + 1) * 1 + 0
cabd 12 = ((2 * 3 + 0) * 2 + 0) * 1 + 0
cadb 13 = ((2 * 3 + 0) * 2 + 1) * 1 + 0
cbad 14 = ((2 * 3 + 1) * 2 + 0) * 1 + 0
cbda 15 = ((2 * 3 + 1) * 2 + 1) * 1 + 0 //pb2
cdab 16 = ((2 * 3 + 2) * 2 + 0) * 1 + 0
cdba 17 = ((2 * 3 + 2) * 2 + 1) * 1 + 0
[...]
dcba 23 = ((3 * 3 + 2) * 2 + 1) * 1 + 0
First "reflexion" :
An entropy point of view. abcd have the fewest "entropy". If a character is in a place it "shouldn't" be, it creates entropy, and the earlier the entropy is the greatest it becomes.
For bcad for example, lexicographic index is 8 = ((1 * 3 + 1) * 2 + 0) * 1 + 0 and can be calculated that way :
value = 0;
value += max(b - a, 0); // = 1; (a "should be" in the first place [to create the less possible entropy] but instead it is b)
value *= 3 - 0; //last index - current index
value += max(c - b, 0); // = 1; (b "should be" in the second place but instead it is c)
value *= 3 - 1;
value += max(a - c, 0); // = 0; (a "should have been" put earlier, so it does not create entropy to put it there)
value *= 3 - 2;
value += max(d - d, 0); // = 0;
Note that the last operation will always do nothing, that's why "i
First problem (pb1) :
For adcb, for example, the first logic doesn't work (it leads to an lexicographic index of ((0* 3+ 2) * 2+ 0) * 1 = 4) because c-d = 0 but it creates entropy to put c before b. I added x because of that, it represents the first digit/character that isn't placed yet. With x, diff cannot be negative.
For adcb, lexicographic index is 5 = ((0 * 3 + 2) * 2 + 1) * 1 + 0 and can be calculated that way :
value = 0; x=0;
diff = a - a; // = 0; (a is in the right place)
diff == 0 => x++; //x=b now and we don't modify value
value *= 3 - 0; //last index - current index
diff = d - b; // = 2; (b "should be" there (it's x) but instead it is d)
diff > 0 => value += diff; //we add diff to value and we don't modify x
diff = c - b; // = 1; (b "should be" there but instead it is c) This is where it differs from the first reflexion
diff > 0 => value += diff;
value *= 3 - 2;
Second problem (pb2) :
For cbda, for example, lexicographic index is 15 = ((2 * 3 + 1) * 2 + 1) * 1 + 0, but the first reflexion gives : ((2 * 3 + 0) * 2 + 1) * 1 + 0 = 13 and the solution to pb1 gives ((2 * 3 + 1) * 2 + 3) * 1 + 0 = 17. The solution to pb1 doesn't work because the two last characters to place are d and a, so d - a "means" 1 instead of 3. I had to count the characters placed before that comes before the character in place, but after x, so I had to add an inner loop.
Putting it all together :
I then realised that pb1 was just a particular case of pb2, and that if you remove x, and you simply take diff = A[i], we end up with the unnested version of your solution (with factorial calculated little by little, and my diff corresponding to your x).
So, basically, my "contribution" (I think) is to add a variable, x, which can avoid doing the inner loop when diff equals 0 or 1, at the expense of checking if you have to increment x and doing it if so.
I also checked if you have to increment x in the inner loop (if(A[j]==x+1)) because if you take for example badce, x will be b at the end because a comes after b, and you will enter the inner loop one more time, encountering c. If you check x in the inner loop, when you encounter d you have no choice but doing the inner loop, but x will update to c, and when you encounter c you will not enter the inner loop. You can remove this check without breaking the program
With the alternative version and the check in the inner loop it makes 4 different versions. The alternative one with the check is the one in which you enter the less the inner loop, so in terms of "theoretical complexity" it is the best, but in terms of performance/number of operations, I don't know.
Hope all of this helps (since the question is rather old, and I didn't read all the answers in details). If not, I still had fun doing it. Sorry for the long post. Also I'm new on Stack Overflow (as a member), and not a native speaker, so please be nice, and don't hesitate to let me know if I did something wrong.
Linear traversal of memory already in cache really doesn't take much times at all. Don't worry about it. You won't be traversing enough distance before factorial() overflows.
Move the 8 out as a parameter.
int factorial ( int input )
{
return input ? input * factorial (input - 1) : 1;
}
int lexic_ix ( int* arr, int N )
{
int output = 0;
int fact = factorial (N);
for ( int i = 0; i < N - 1; i++ )
{
int order = arr [ i ];
for ( int j = 0; j < i; j++ )
order -= arr [ j ] < arr [ i ];
output += order * (fact /= N - i);
}
return output;
}
int main()
{
int arr [ ] = { 11, 10, 9, 8, 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0 };
const int length = 12;
for ( int i = 0; i < length; ++i )
std::cout << lexic_ix ( arr + i, length - i ) << std::endl;
}
Say, for a M-digit sequence permutation, from your code, you can get the lexicographic SN formula which is something like: Am-1*(m-1)! + Am-2*(m-2)! + ... + A0*(0)! , where Aj range from 0 to j. You can calculate SN from A0*(0)!, then A1*(1)!, ..., then Am-1 * (m-1)!, and add these together(suppose your integer type does not overflow), so you do not need calculate factorials recursively and repeatedly. The SN number is a range from 0 to M!-1 (because Sum(n*n!, n in 0,1, ...n) = (n+1)!-1)
If you are not calculating factorials recursively, I cannot think of anything that could make any big improvement.
Sorry for posting the code a little bit late, I just did some research, and find this:
http://swortham.blogspot.com.au/2011/10/how-much-faster-is-multiplication-than.html
according to this author, integer multiplication can be 40 times faster than integer division. floating numbers are not so dramatic though, but here is pure integer.
int lexic_ix ( int arr[], int N )
{
// if this function will be called repeatedly, consider pass in this pointer as parameter
std::unique_ptr<int[]> coeff_arr = std::make_unique<int[]>(N);
for ( int i = 0; i < N - 1; i++ )
{
int order = arr [ i ];
for ( int j = 0; j < i; j++ )
order -= arr [ j ] < arr [ i ];
coeff_arr[i] = order; // save this into coeff_arr for later multiplication
}
//
// There are 2 points about the following code:
// 1). most modern processors have built-in multiplier, \
// and multiplication is much faster than division
// 2). In your code, you are only the maximum permutation serial number,
// if you put in a random sequence, say, when length is 10, you put in
// a random sequence, say, {3, 7, 2, 9, 0, 1, 5, 8, 4, 6}; if you look into
// the coeff_arr[] in debugger, you can see that coeff_arr[] is:
// {3, 6, 2, 6, 0, 0, 1, 2, 0, 0}, the last number will always be zero anyway.
// so, you will have good chance to reduce many multiplications.
// I did not do any performance profiling, you could have a go, and it will be
// much appreciated if you could give some feedback about the result.
//
long fac = 1;
long sn = 0;
for (int i = 1; i < N; ++i) // start from 1, because coeff_arr[N-1] is always 0
{
fac *= i;
if (coeff_arr[N - 1 - i])
sn += coeff_arr[N - 1 - i] * fac;
}
return sn;
}
int main()
{
int arr [ ] = { 3, 7, 2, 9, 0, 1, 5, 8, 4, 6 }; // try this and check coeff_arr
const int length = 10;
std::cout << lexic_ix(arr, length ) << std::endl;
return 0;
}
This is the whole profiling code, I only run the test in Linux, code was compiled using G++8.4, with '-std=c++11 -O3' compiler options. To be fair, I slightly rewrote your code, pre-calculate the N! and pass it into the function, but it seems this does not help much.
The performance profiling for N = 9 (362,880 permutations) is:
Time durations are: 34, 30, 25 milliseconds
Time durations are: 34, 30, 25 milliseconds
Time durations are: 33, 30, 25 milliseconds
The performance profiling for N=10 (3,628,800 permutations) is:
Time durations are: 345, 335, 275 milliseconds
Time durations are: 348, 334, 275 milliseconds
Time durations are: 345, 335, 275 milliseconds
The first number is your original function, the second is the function re-written that gets N! passed in, the last number is my result. The permutation generation function is very primitive and runs slowly, but as long as it generates all permutations as testing dataset, that is alright. By the way, these tests are run on a Quad-Core 3.1Ghz, 4GBytes desktop running Ubuntu 14.04.
EDIT: I forgot a factor that the first function may need to expand the lexi_numbers vector, so I put an empty call before timing. After this, the times are 333, 334, 275.
EDIT: Another factor that could influence the performance, I am using long integer in my code, if I change those 2 'long' to 2 'int', the running time will become: 334, 333, 264.
#include <iostream>
#include <vector>
#include <chrono>
using namespace std::chrono;
int factorial(int input)
{
return input ? input * factorial(input - 1) : 1;
}
int lexic_ix(int* arr, int N)
{
int output = 0;
int fact = factorial(N);
for (int i = 0; i < N - 1; i++)
{
int order = arr[i];
for (int j = 0; j < i; j++)
order -= arr[j] < arr[i];
output += order * (fact /= N - i);
}
return output;
}
int lexic_ix1(int* arr, int N, int N_fac)
{
int output = 0;
int fact = N_fac;
for (int i = 0; i < N - 1; i++)
{
int order = arr[i];
for (int j = 0; j < i; j++)
order -= arr[j] < arr[i];
output += order * (fact /= N - i);
}
return output;
}
int lexic_ix2( int arr[], int N , int coeff_arr[])
{
for ( int i = 0; i < N - 1; i++ )
{
int order = arr [ i ];
for ( int j = 0; j < i; j++ )
order -= arr [ j ] < arr [ i ];
coeff_arr[i] = order;
}
long fac = 1;
long sn = 0;
for (int i = 1; i < N; ++i)
{
fac *= i;
if (coeff_arr[N - 1 - i])
sn += coeff_arr[N - 1 - i] * fac;
}
return sn;
}
std::vector<std::vector<int>> gen_permutation(const std::vector<int>& permu_base)
{
if (permu_base.size() == 1)
return std::vector<std::vector<int>>(1, std::vector<int>(1, permu_base[0]));
std::vector<std::vector<int>> results;
for (int i = 0; i < permu_base.size(); ++i)
{
int cur_int = permu_base[i];
std::vector<int> cur_subseq = permu_base;
cur_subseq.erase(cur_subseq.begin() + i);
std::vector<std::vector<int>> temp = gen_permutation(cur_subseq);
for (auto x : temp)
{
x.insert(x.begin(), cur_int);
results.push_back(x);
}
}
return results;
}
int main()
{
#define N 10
std::vector<int> arr;
int buff_arr[N];
const int length = N;
int N_fac = factorial(N);
for(int i=0; i<N; ++i)
arr.push_back(N-i-1); // for N=10, arr is {9, 8, 7, 6, 5, 4, 3, 2, 1, 0}
std::vector<std::vector<int>> all_permus = gen_permutation(arr);
std::vector<int> lexi_numbers;
// This call is not timed, only to expand the lexi_numbers vector
for (auto x : all_permus)
lexi_numbers.push_back(lexic_ix2(&x[0], length, buff_arr));
lexi_numbers.clear();
auto t0 = high_resolution_clock::now();
for (auto x : all_permus)
lexi_numbers.push_back(lexic_ix(&x[0], length));
auto t1 = high_resolution_clock::now();
lexi_numbers.clear();
auto t2 = high_resolution_clock::now();
for (auto x : all_permus)
lexi_numbers.push_back(lexic_ix1(&x[0], length, N_fac));
auto t3 = high_resolution_clock::now();
lexi_numbers.clear();
auto t4 = high_resolution_clock::now();
for (auto x : all_permus)
lexi_numbers.push_back(lexic_ix2(&x[0], length, buff_arr));
auto t5 = high_resolution_clock::now();
std::cout << std::endl << "Time durations are: " << duration_cast<milliseconds> \
(t1 -t0).count() << ", " << duration_cast<milliseconds>(t3 - t2).count() << ", " \
<< duration_cast<milliseconds>(t5 - t4).count() <<" milliseconds" << std::endl;
return 0;
}
There are n people numbered from 1 to n. I have to write a code which produces and print all different combinations of k people from these n. Please explain the algorithm used for that.
I assume you're asking about combinations in combinatorial sense (that is, order of elements doesn't matter, so [1 2 3] is the same as [2 1 3]). The idea is pretty simple then, if you understand induction / recursion: to get all K-element combinations, you first pick initial element of a combination out of existing set of people, and then you "concatenate" this initial element with all possible combinations of K-1 people produced from elements that succeed the initial element.
As an example, let's say we want to take all combinations of 3 people from a set of 5 people. Then all possible combinations of 3 people can be expressed in terms of all possible combinations of 2 people:
comb({ 1 2 3 4 5 }, 3) =
{ 1, comb({ 2 3 4 5 }, 2) } and
{ 2, comb({ 3 4 5 }, 2) } and
{ 3, comb({ 4 5 }, 2) }
Here's C++ code that implements this idea:
#include <iostream>
#include <vector>
using namespace std;
vector<int> people;
vector<int> combination;
void pretty_print(const vector<int>& v) {
static int count = 0;
cout << "combination no " << (++count) << ": [ ";
for (int i = 0; i < v.size(); ++i) { cout << v[i] << " "; }
cout << "] " << endl;
}
void go(int offset, int k) {
if (k == 0) {
pretty_print(combination);
return;
}
for (int i = offset; i <= people.size() - k; ++i) {
combination.push_back(people[i]);
go(i+1, k-1);
combination.pop_back();
}
}
int main() {
int n = 5, k = 3;
for (int i = 0; i < n; ++i) { people.push_back(i+1); }
go(0, k);
return 0;
}
And here's output for N = 5, K = 3:
combination no 1: [ 1 2 3 ]
combination no 2: [ 1 2 4 ]
combination no 3: [ 1 2 5 ]
combination no 4: [ 1 3 4 ]
combination no 5: [ 1 3 5 ]
combination no 6: [ 1 4 5 ]
combination no 7: [ 2 3 4 ]
combination no 8: [ 2 3 5 ]
combination no 9: [ 2 4 5 ]
combination no 10: [ 3 4 5 ]
From Rosetta code
#include <algorithm>
#include <iostream>
#include <string>
void comb(int N, int K)
{
std::string bitmask(K, 1); // K leading 1's
bitmask.resize(N, 0); // N-K trailing 0's
// print integers and permute bitmask
do {
for (int i = 0; i < N; ++i) // [0..N-1] integers
{
if (bitmask[i]) std::cout << " " << i;
}
std::cout << std::endl;
} while (std::prev_permutation(bitmask.begin(), bitmask.end()));
}
int main()
{
comb(5, 3);
}
output
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4
Analysis and idea
The whole point is to play with the binary representation of numbers
for example the number 7 in binary is 0111
So this binary representation can also be seen as an assignment list as such:
For each bit i if the bit is set (i.e is 1) means the ith item is assigned else not.
Then by simply computing a list of consecutive binary numbers and exploiting the binary representation (which can be very fast) gives an algorithm to compute all combinations of N over k.
The sorting at the end (of some implementations) is not needed. It is just a way to deterministicaly normalize the result, i.e for same numbers (N, K) and same algorithm same order of combinations is returned
For further reading about number representations and their relation to combinations, permutations, power sets (and other interesting stuff), have a look at Combinatorial number system , Factorial number system
PS: You may want to check out my combinatorics framework Abacus which computes many types of combinatorial objects efficiently and its routines (originaly in JavaScript) can be adapted easily to many other languages.
If the number of the set would be within 32, 64 or a machine native primitive size, then you can do it with a simple bit manipulation.
template<typename T>
void combo(const T& c, int k)
{
int n = c.size();
int combo = (1 << k) - 1; // k bit sets
while (combo < 1<<n) {
pretty_print(c, combo);
int x = combo & -combo;
int y = combo + x;
int z = (combo & ~y);
combo = z / x;
combo >>= 1;
combo |= y;
}
}
this example calls pretty_print() function by the dictionary order.
For example. You want to have 6C3 and assuming the current 'combo' is 010110.
Obviously the next combo MUST be 011001.
011001 is :
010000 | 001000 | 000001
010000 : deleted continuously 1s of LSB side.
001000 : set 1 on the next of continuously 1s of LSB side.
000001 : shifted continuously 1s of LSB to the right and remove LSB bit.
int x = combo & -combo;
this obtains the lowest 1.
int y = combo + x;
this eliminates continuously 1s of LSB side and set 1 on the next of it (in the above case, 010000 | 001000)
int z = (combo & ~y)
this gives you the continuously 1s of LSB side (000110).
combo = z / x;
combo >> =1;
this is for 'shifted continuously 1s of LSB to the right and remove LSB bit'.
So the final job is to OR y to the above.
combo |= y;
Some simple concrete example :
#include <bits/stdc++.h>
using namespace std;
template<typename T>
void pretty_print(const T& c, int combo)
{
int n = c.size();
for (int i = 0; i < n; ++i) {
if ((combo >> i) & 1)
cout << c[i] << ' ';
}
cout << endl;
}
template<typename T>
void combo(const T& c, int k)
{
int n = c.size();
int combo = (1 << k) - 1; // k bit sets
while (combo < 1<<n) {
pretty_print(c, combo);
int x = combo & -combo;
int y = combo + x;
int z = (combo & ~y);
combo = z / x;
combo >>= 1;
combo |= y;
}
}
int main()
{
vector<char> c0 = {'1', '2', '3', '4', '5'};
combo(c0, 3);
vector<char> c1 = {'a', 'b', 'c', 'd', 'e', 'f', 'g'};
combo(c1, 4);
return 0;
}
result :
1 2 3
1 2 4
1 3 4
2 3 4
1 2 5
1 3 5
2 3 5
1 4 5
2 4 5
3 4 5
a b c d
a b c e
a b d e
a c d e
b c d e
a b c f
a b d f
a c d f
b c d f
a b e f
a c e f
b c e f
a d e f
b d e f
c d e f
a b c g
a b d g
a c d g
b c d g
a b e g
a c e g
b c e g
a d e g
b d e g
c d e g
a b f g
a c f g
b c f g
a d f g
b d f g
c d f g
a e f g
b e f g
c e f g
d e f g
In Python, this is implemented as itertools.combinations
https://docs.python.org/2/library/itertools.html#itertools.combinations
In C++, such combination function could be implemented based on permutation function.
The basic idea is to use a vector of size n, and set only k item to 1 inside, then all combinations of nchoosek could obtained by collecting the k items in each permutation.
Though it might not be the most efficient way require large space, as combination is usually a very large number. It's better to be implemented as a generator or put working codes into do_sth().
Code sample:
#include <vector>
#include <iostream>
#include <iterator>
#include <algorithm>
using namespace std;
int main(void) {
int n=5, k=3;
// vector<vector<int> > combinations;
vector<int> selected;
vector<int> selector(n);
fill(selector.begin(), selector.begin() + k, 1);
do {
for (int i = 0; i < n; i++) {
if (selector[i]) {
selected.push_back(i);
}
}
// combinations.push_back(selected);
do_sth(selected);
copy(selected.begin(), selected.end(), ostream_iterator<int>(cout, " "));
cout << endl;
selected.clear();
}
while (prev_permutation(selector.begin(), selector.end()));
return 0;
}
and the output is
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4
This solution is actually a duplicate with
Generating combinations in c++
Here is an algorithm i came up with for solving this problem. You should be able to modify it to work with your code.
void r_nCr(const unsigned int &startNum, const unsigned int &bitVal, const unsigned int &testNum) // Should be called with arguments (2^r)-1, 2^(r-1), 2^(n-1)
{
unsigned int n = (startNum - bitVal) << 1;
n += bitVal ? 1 : 0;
for (unsigned int i = log2(testNum) + 1; i > 0; i--) // Prints combination as a series of 1s and 0s
cout << (n >> (i - 1) & 1);
cout << endl;
if (!(n & testNum) && n != startNum)
r_nCr(n, bitVal, testNum);
if (bitVal && bitVal < testNum)
r_nCr(startNum, bitVal >> 1, testNum);
}
You can see an explanation of how it works here.
I have written a class in C# to handle common functions for working with the binomial coefficient, which is the type of problem that your problem falls under. It performs the following tasks:
Outputs all the K-indexes in a nice format for any N choose K to a file. The K-indexes can be substituted with more descriptive strings or letters. This method makes solving this type of problem quite trivial.
Converts the K-indexes to the proper index of an entry in the sorted binomial coefficient table. This technique is much faster than older published techniques that rely on iteration. It does this by using a mathematical property inherent in Pascal's Triangle. My paper talks about this. I believe I am the first to discover and publish this technique.
Converts the index in a sorted binomial coefficient table to the corresponding K-indexes. I believe it is also faster than the other solutions.
Uses Mark Dominus method to calculate the binomial coefficient, which is much less likely to overflow and works with larger numbers.
The class is written in .NET C# and provides a way to manage the objects related to the problem (if any) by using a generic list. The constructor of this class takes a bool value called InitTable that when true will create a generic list to hold the objects to be managed. If this value is false, then it will not create the table. The table does not need to be created in order to perform the 4 above methods. Accessor methods are provided to access the table.
There is an associated test class which shows how to use the class and its methods. It has been extensively tested with 2 cases and there are no known bugs.
To read about this class and download the code, see Tablizing The Binomial Coeffieicent.
It should be pretty straight forward to port the class over to C++.
The solution to your problem involves generating the K-indexes for each N choose K case. For example:
int NumPeople = 10;
int N = TotalColumns;
// Loop thru all the possible groups of combinations.
for (int K = N - 1; K < N; K++)
{
// Create the bin coeff object required to get all
// the combos for this N choose K combination.
BinCoeff<int> BC = new BinCoeff<int>(N, K, false);
int NumCombos = BinCoeff<int>.GetBinCoeff(N, K);
int[] KIndexes = new int[K];
BC.OutputKIndexes(FileName, DispChars, "", " ", 60, false);
// Loop thru all the combinations for this N choose K case.
for (int Combo = 0; Combo < NumCombos; Combo++)
{
// Get the k-indexes for this combination, which in this case
// are the indexes to each person in the problem set.
BC.GetKIndexes(Loop, KIndexes);
// Do whatever processing that needs to be done with the indicies in KIndexes.
...
}
}
The OutputKIndexes method can also be used to output the K-indexes to a file, but it will use a different file for each N choose K case.
This templated function works with the vector of any type as an input.
Combinations are returned as a vector of vectors.
/*
* Function return all possible combinations of k elements from N-size inputVector.
* The result is returned as a vector of k-long vectors containing all combinations.
*/
template<typename T> std::vector<std::vector<T>> getAllCombinations(const std::vector<T>& inputVector, int k)
{
std::vector<std::vector<T>> combinations;
std::vector<int> selector(inputVector.size());
std::fill(selector.begin(), selector.begin() + k, 1);
do {
std::vector<int> selectedIds;
std::vector<T> selectedVectorElements;
for (int i = 0; i < inputVector.size(); i++) {
if (selector[i]) {
selectedIds.push_back(i);
}
}
for (auto& id : selectedIds) {
selectedVectorElements.push_back(inputVector[id]);
}
combinations.push_back(selectedVectorElements);
} while (std::prev_permutation(selector.begin(), selector.end()));
return combinations;
}
You can use the "count_each_combination" and "for_each_combination" functions from the combinations library from Howard Hinnant to generate all the combinations for take k from n.
#include <vector>
#include "combinations.h"
std::vector<std::vector<u_int8_t> >
combinationsNoRepetitionAndOrderDoesNotMatter (long int subsetSize, std::vector<uint8_t> setOfNumbers)
{
std::vector<std::vector<u_int8_t> > subsets{};
subsets.reserve (count_each_combination (setOfNumbers.begin (), setOfNumbers.begin () + subsetSize, setOfNumbers.end ()));
for_each_combination (setOfNumbers.begin (), setOfNumbers.begin () + subsetSize, setOfNumbers.end (), [&subsets] (auto first, auto last) {
subsets.push_back (std::vector<uint8_t>{ first, last });
return false;
});
return subsets;
}
int main(){
combinationsNoRepetitionAndOrderDoesNotMatter (6, { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36 });
}
Benchmark on a Intel(R) Core(TM) i5-8600K CPU # 3.60GHz:
g++
benchmark name samples iterations estimated
mean low mean high mean
std dev low std dev high std dev
-------------------------------------------------------------------------------
combinations no repetition and
order does not matter 6 from 36 100 1 10.2829 s
92.5451 ms 92.3971 ms 92.9411 ms
1.15617 ms 532.604 us 2.48342 ms
clang++
benchmark name samples iterations estimated
mean low mean high mean
std dev low std dev high std dev
-------------------------------------------------------------------------------
combinations no repetition and
order does not matter 6 from 36 100 1 11.0786 s
88.1275 ms 87.8212 ms 89.3204 ms
2.82107 ms 400.665 us 6.67526 ms
Behind the link below is a generic C# answer to this problem: How to format all combinations out of a list of objects. You can limit the results only to the length of k pretty easily.
https://stackoverflow.com/a/40417765/2613458
It can also be done using backtracking by maintaining a visited array.
void foo(vector<vector<int> > &s,vector<int> &data,int go,int k,vector<int> &vis,int tot)
{
vis[go]=1;
data.push_back(go);
if(data.size()==k)
{
s.push_back(data);
vis[go]=0;
data.pop_back();
return;
}
for(int i=go+1;i<=tot;++i)
{
if(!vis[i])
{
foo(s,data,i,k,vis,tot);
}
}
vis[go]=0;
data.pop_back();
}
vector<vector<int> > Solution::combine(int n, int k) {
vector<int> data;
vector<int> vis(n+1,0);
vector<vector<int> > sol;
for(int i=1;i<=n;++i)
{
for(int i=1;i<=n;++i) vis[i]=0;
foo(sol,data,i,k,vis,n);
}
return sol;
}
I thought my simple "all possible combination generator" might help someone, i think its a really good example for building something bigger and better
you can change N (characters) to any you like by just removing/adding from string array (you can change it to int as well). Current amount of characters is 36
you can also change K (size of the generated combinations) by just adding more loops, for each element, there must be one extra loop. Current size is 4
#include<iostream>
using namespace std;
int main() {
string num[] = {"0","1","2","3","4","5","6","7","8","9","a","b","c","d","e","f","g","h","i","j","k","l","m","n","o","p","q","r","s","t","u","v","w","x","y","z" };
for (int i1 = 0; i1 < sizeof(num)/sizeof(string); i1++) {
for (int i2 = 0; i2 < sizeof(num)/sizeof(string); i2++) {
for (int i3 = 0; i3 < sizeof(num)/sizeof(string); i3++) {
for (int i4 = 0; i4 < sizeof(num)/sizeof(string); i4++) {
cout << num[i1] << num[i2] << num[i3] << num[i4] << endl;
}
}
}
}}
Result
0: A A A
1: B A A
2: C A A
3: A B A
4: B B A
5: C B A
6: A C A
7: B C A
8: C C A
9: A A B
...
just keep in mind that the amount of combinations can be ridicules.
--UPDATE--
a better way to generate all possible combinations would be with this code, which can be easily adjusted and configured in the "variables" section of the code.
#include<iostream>
#include<math.h>
int main() {
//VARIABLES
char chars[] = { 'A', 'B', 'C' };
int password[4]{0};
//SIZES OF VERIABLES
int chars_length = sizeof(chars) / sizeof(char);
int password_length = sizeof(password) / sizeof(int);
//CYCKLE TROUGH ALL OF THE COMBINATIONS
for (int i = 0; i < pow(chars_length, password_length); i++){
//CYCKLE TROUGH ALL OF THE VERIABLES IN ARRAY
for (int i2 = 0; i2 < password_length; i2++) {
//IF VERIABLE IN "PASSWORD" ARRAY IS THE LAST VERIABLE IN CHAR "CHARS" ARRRAY
if (password[i2] == chars_length) {
//THEN INCREMENT THE NEXT VERIABLE IN "PASSWORD" ARRAY
password[i2 + 1]++;
//AND RESET THE VERIABLE BACK TO ZERO
password[i2] = 0;
}}
//PRINT OUT FIRST COMBINATION
std::cout << i << ": ";
for (int i2 = 0; i2 < password_length; i2++) {
std::cout << chars[password[i2]] << " ";
}
std::cout << "\n";
//INCREMENT THE FIRST VERIABLE IN ARRAY
password[0]++;
}}
To make it more complete, the following answer covers the case that the data set contains duplicate values. The function is written close to the style of std::next_permutation() so that it is easy to follow up.
template< class RandomIt >
bool next_combination(RandomIt first, RandomIt n_first, RandomIt last)
{
if (first == last || n_first == first || n_first == last)
{
return false;
}
RandomIt it_left = n_first;
--it_left;
RandomIt it_right = n_first;
bool reset = false;
while (true)
{
auto it = std::upper_bound(it_right, last, *it_left);
if (it != last)
{
std::iter_swap(it_left, it);
if (reset)
{
++it_left;
it_right = it;
++it_right;
std::size_t left_len = std::distance(it_left, n_first);
std::size_t right_len = std::distance(it_right, last);
if (left_len < right_len)
{
std::swap_ranges(it_left, n_first, it_right);
std::rotate(it_right, it_right+left_len, last);
}
else
{
std::swap_ranges(it_right, last, it_left);
std::rotate(it_left, it_left+right_len, n_first);
}
}
return true;
}
else
{
reset = true;
if (it_left == first)
{
break;
}
--it_left;
it_right = n_first;
}
}
return false;
}
The full data set is represented in the range [first, last). The current combination is represented in the range [first, n_first) and the range [n_first, last) holds the complement set of the current combination.
As a combination is irrelevant to its order, [first, n_first) and [n_first, last) are kept in ascending order to avoid duplication.
The algorithm works by increasing the last value A on the left side by swapping with the first value B on the right side that is greater than A. After the swapping, both sides are still ordered. If no such value B exists on the right side, then we start to consider increasing the second last on the left side until all values on the left side are not less than the right side.
An example of drawing 2 elements from a set by the following code:
std::vector<int> seq = {1, 1, 2, 2, 3, 4, 5};
do
{
for (int x : seq)
{
std::cout << x << " ";
}
std::cout << "\n";
} while (next_combination(seq.begin(), seq.begin()+2, seq.end()));
gives:
1 1 2 2 3 4 5
1 2 1 2 3 4 5
1 3 1 2 2 4 5
1 4 1 2 2 3 5
1 5 1 2 2 3 4
2 2 1 1 3 4 5
2 3 1 1 2 4 5
2 4 1 1 2 3 5
2 5 1 1 2 3 4
3 4 1 1 2 2 5
3 5 1 1 2 2 4
4 5 1 1 2 2 3
It is trivial to retrieve the first two elements as the combination result if needed.
The basic idea of this solution is to mimic the way you enumerate all the combinations without repetitions by hand in high school. Let com be List[int] of length k and nums be List[int] the given n items, where n >= k.
The idea is as follows:
for x[0] in nums[0,...,n-1]
for x[1] in nums[idx_of_x[0] + 1,..,n-1]
for x[2] in nums [idx_of_x[1] + 1,...,n-1]
..........
for x[k-1] in nums [idx_of_x[k-2]+1, ..,n-1]
Obviously, k and n are variable arguments, which makes it impossible to write explicit multiple nested for-loops. This is where the recursion comes to rescue the issue.
Statement len(com) + len(nums[i:]) >= k checks whether the remaining unvisited forward list of items can provide k iitems. By forward, I mean you should not walk the nums backward for avoiding the repeated combination, which consists of same set of items but in different order. Put it in another way, in these different orders, we can choose the order these items appear in the list by scaning the list forward. More importantly, this test clause internally prunes the recursion tree such that it only contains n choose k recursive calls. Hence, the running time is O(n choose k).
from typing import List
class Solution:
def combine(self, n: int, k: int) -> List[List[int]]:
assert 1 <= n <= 20
assert 1 <= k <= n
com_sets = []
self._combine_recurse(k, list(range(1, n+1)), [], com_sets)
return com_sets
def _combine_recurse(self, k: int, nums: List[int], com: List[int], com_set: List[List[int]]):
"""
O(C_n^k)
"""
if len(com) < k:
for i in range(len(nums)):
# Once again, don't com.append() since com should not be global!
if len(com) + len(nums[i:]) >= k:
self._combine_recurse(k, nums[i+1:], com + [nums[i]], com_set)
else:
if len(com) == k:
com_set.append(com)
print(com)
sol = Solution()
sol.combine(5, 3)
[1, 2, 3]
[1, 2, 4]
[1, 2, 5]
[1, 3, 4]
[1, 3, 5]
[1, 4, 5]
[2, 3, 4]
[2, 3, 5]
[2, 4, 5]
[3, 4, 5]
Try this:
#include <iostream>
#include <string>
#include <vector>
using namespace std;
void combo(vector<char> &alphabet, int n, vector<string> &result, string curr) {
if (n == 0) {
result.push_back(curr);
return;
}
for (int i = 0; i < alphabet.size(); i++) {
combo(alphabet, n - 1, result, curr + alphabet[i]);
}
return;
}
int main() {
//N items
vector<char> alphabet = {'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', 'k', 'l', 'm', 'n','o','p','q','r','s','t','u','v','w','x','y','z'};
vector<string> result;
//K is 4
combo(alphabet, 4, result, "");
for (auto s : result) {
cout << s << endl;
}
return 0;
}