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I'm trying to write a Haskell program that calculates multiples. Basically, when given two integers a and b, I want to find how many integers 1 ≤ bi ≤ b are multiple of any integer 2 ≤ ai ≤ a. For example, if a = 3 and b = 30, I want to know how many integers in the range of 1-30 are a multiple of 2 or 3; there are 20 such integers: 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30.
I have a C program that does this. I'm trying to get this translated into Haskell, but part of the difficulty is getting around the loops that I've used since Haskell doesn't use loops. I appreciate any and all help in translating this!
My C program for reference (sorry if formatting is off):
#define PRIME_RANGE 130
#define PRIME_CNT 32
#define UPPER_LIMIT (1000000000000000ull) //10^15
#define MAX_BASE_MULTIPLES_COUNT 25000000
typedef struct
{
char primeFactorFlag;
long long multiple;
}multipleInfo;
unsigned char primeFlag[PRIME_RANGE + 1];
int primes[PRIME_CNT];
int primeCnt = 0;
int maxPrimeStart[PRIME_CNT];
multipleInfo baseMultiples[MAX_BASE_MULTIPLES_COUNT];
multipleInfo mergedMultiples[MAX_BASE_MULTIPLES_COUNT];
int baseMultiplesCount, mergedMultiplesCount;
void findOddMultiples(int a, long long b, long long *count);
void generateBaseMultiples(void);
void mergeLists(multipleInfo listSource[], int countS, multipleInfo
listDest[], int *countD);
void sieve(void);
int main(void)
{
int i, j, a, n, startInd, endInd;
long long b, multiples;
//Generate primes
sieve();
primes[primeCnt] = PRIME_RANGE + 1;
generateBaseMultiples();
baseMultiples[baseMultiplesCount].multiple = UPPER_LIMIT + 1;
//Input and Output
scanf("%d", &n);
for(i = 1; i <= n; i++)
{
scanf("%d%lld", &a, &b);
//If b <= a, all are multiple except 1
if(b <= a)
printf("%lld\n",b-1);
else
{
//Add all even multiples
multiples = b / 2;
//Add all odd multiples
findOddMultiples(a, b, &multiples);-
printf("%lld\n", multiples);
}
}
return 0;
}
void findOddMultiples(int a, long long b, long long *count)
{
int i, k;
long long currentNum;
for(k = 1; k < primeCnt && primes[k] <= a; k++)
{
for(i = maxPrimeStart[k]; i < maxPrimeStart[k + 1] &&
baseMultiples[i].multiple <= b; i++)
{
currentNum = b/baseMultiples[i].multiple;
currentNum = (currentNum + 1) >> 1; // remove even multiples
if(baseMultiples[i].primeFactorFlag) //odd number of factors
(*count) += currentNum;
else
(*count) -= currentNum;
}
}
}
void addTheMultiple(long long value, int primeFactorFlag)
{
baseMultiples[baseMultiplesCount].multiple = value;
baseMultiples[baseMultiplesCount].primeFactorFlag = primeFactorFlag;
baseMultiplesCount++;
}
void generateBaseMultiples(void)
{
int i, j, t, prevCount;
long long curValue;
addTheMultiple(3, 1);
mergedMultiples[0] = baseMultiples[0];
mergedMultiplesCount = 1;
maxPrimeStart[1] = 0;
prevCount = mergedMultiplesCount;
for(i = 2; i < primeCnt; i++)
{
maxPrimeStart[i] = baseMultiplesCount;
addTheMultiple(primes[i], 1);
for(j = 0; j < prevCount; j++)
{
curValue = mergedMultiples[j].multiple * primes[i];
if(curValue > UPPER_LIMIT)
break;
addTheMultiple(curValue, 1 - mergedMultiples[j].primeFactorFlag);
}
if(i < primeCnt - 1)
mergeLists(&baseMultiples[prevCount], baseMultiplesCount - prevCount, mergedMultiples, &mergedMultiplesCount);
prevCount = mergedMultiplesCount;
}
maxPrimeStart[primeCnt] = baseMultiplesCount;
}
void mergeLists(multipleInfo listSource[], int countS, multipleInfo listDest[], int *countD)
{
int limit = countS + *countD;
int i1, i2, j, k;
//Copy one list in unused safe memory
for(j = limit - 1, k = *countD - 1; k >= 0; j--, k--)
listDest[j] = listDest[k];
//Merge the lists
for(i1 = 0, i2 = countS, k = 0; i1 < countS && i2 < limit; k++)
{
if(listSource[i1].multiple <= listDest[i2].multiple)
listDest[k] = listSource[i1++];
else
listDest[k] = listDest[i2++];
}
while(i1 < countS)
listDest[k++] = listSource[i1++];
while(i2 < limit)
listDest[k++] = listDest[i2++];
*countD = k;
}
void sieve(void)
{
int i, j, root = sqrt(PRIME_RANGE);
primes[primeCnt++] = 2;
for(i = 3; i <= PRIME_RANGE; i+= 2)
{
if(!primeFlag[i])
{
primes[primeCnt++] = i;
if(root >= i)
{
for(j = i * i; j <= PRIME_RANGE; j += i << 1)
primeFlag[j] = 1;
}
}
}
}
First, unless I'm grossly misunderstanding, the number of multiples you have there is wrong. The number of multiples of 2 between 1 and 30 is 15, and the number of multiples of 3 between 1 and 30 is 10, so there should be 25 numbers there.
EDIT: I did misunderstand; you want unique multiples.
To get unique multiples, you can use Data.Set, which has the invariant that the elements of the Set are unique and ordered ascendingly.
If you know you aren't going to exceed x = maxBound :: Int, you can get even better speedups using Data.IntSet. I've also included some test cases and annotated with comments what they run at on my machine.
{-# LANGUAGE BangPatterns #-}
{-# OPTIONS_GHC -O2 #-}
module Main (main) where
import System.CPUTime (getCPUTime)
import Data.IntSet (IntSet)
import qualified Data.IntSet as IntSet
main :: IO ()
main = do
test 3 30 -- 0.12 ms
test 131 132 -- 0.14 ms
test 500 300000 -- 117.63 ms
test :: Int -> Int -> IO ()
test !a !b = do
start <- getCPUTime
print (numMultiples a b)
end <- getCPUTime
print $ "Needed " ++ show ((fromIntegral (end - start)) / 10^9) ++ " ms.\n"
numMultiples :: Int -> Int -> Int
numMultiples !a !b = IntSet.size (foldMap go [2..a])
where
go :: Int -> IntSet
go !x = IntSet.fromAscList [x, x+x .. b]
I'm not really into understanding your C, so I implemented a solution afresh using the algorithm discussed here. The N in the linked algorithm is the product of the primes up to a in your problem description.
So first we'll need a list of primes. There's a standardish trick for getting a list of primes that is at once very idiomatic and relatively efficient:
primes :: [Integer]
primes = 2:filter isPrime [3..]
-- Doesn't work right for n<2, but we never call it there, so who cares?
isPrime :: Integer -> Bool
isPrime n = go primes n where
go (p:ps) n | p*p>n = True
| otherwise = n `rem` p /= 0 && go ps n
Next up: we want a way to iterate over the positive square-free divisors of N. This can be achieved by iterating over the subsets of the primes less than a. There's a standard idiomatic way to get a powerset, namely:
-- import Control.Monad
-- powerSet :: [a] -> [[a]]
-- powerSet = filterM (const [False, True])
That would be a fine component to use, but since at the end of the day we only care about the product of each powerset element and the value of the Mobius function of that product, we would end up duplicating a lot of multiplications and counting problems. It's cheaper to compute those two things directly while producing the powerset. So:
-- Given the prime factorization of a square-free number, produce a list of
-- its divisors d together with mu(d).
divisorsWithMu :: Num a => [a] -> [(a, a)]
divisorsWithMu [] = [(1, 1)]
divisorsWithMu (p:ps) = rec ++ [(p*d, -mu) | (d, mu) <- rec] where
rec = divisorsWithMu ps
With that in hand, we can just iterate and do a little arithmetic.
f :: Integer -> Integer -> Integer
f a b = b - sum
[ mu * (b `div` d)
| (d, mu) <- divisorsWithMu (takeWhile (<=a) primes)
]
And that's all the code. Crunched 137 lines of C down to 15 lines of Haskell -- not bad! Try it out in ghci:
> f 3 30
20
As an additional optimization, one could consider modifying divisorsWithMu to short-circuit when its divisor is bigger than b, as we know such terms will not contribute to the final sum. This makes a noticeable difference for large a, as without it there are exponentially many elements in the powerset. Here's how that modification looks:
-- Given an upper bound and the prime factorization of a square-free number,
-- produce a list of its divisors d that are no larger than the upper bound
-- together with mu(d).
divisorsWithMuUnder :: (Ord a, Num a) => a -> [a] -> [(a, a)]
divisorsWithMuUnder n [] = [(1, 1)]
divisorsWithMuUnder n (p:ps) = rec ++ [(p*d, -mu) | (d, mu) <- rec, p*d<=n]
where rec = divisorsWithMuUnder n ps
f' :: Integer -> Integer -> Integer
f' a b = b - sum
[ mu * (b `div` d)
| (d, mu) <- divisorsWithMuUnder b (takeWhile (<=a) primes)
]
Not much more complicated; the only really interesting difference is that there's now a condition in the list comprehension. Here's an example of f' finishing quickly for inputs that would take infeasibly long with f:
> f' 100 100000
88169
With data-ordlist package mentioned by Daniel Wagner in the comments, it is just
f a b = length $ unionAll [ [p,p+p..b] | p <- takeWhile (<= a) primes]
That is all. Some timings, for non-compiled code run inside GHCi:
~> f 100 (10^5)
88169
(0.05 secs, 48855072 bytes)
~> f 131 (3*10^6)
2659571
(0.55 secs, 1493586480 bytes)
~> f 131 132
131
(0.00 secs, 0 bytes)
~> f 500 300000
274055
(0.11 secs, 192704760 bytes)
Compiling will surely make the memory consumption a non-issue, by converting the length to a counting loop.
You'll have to use recursion in place of loops.
In (most) procedural or object-orientated languages, you should hardly ever (never?) be using recursion. It is horribly inefficient, as a new stack frame must be created each time the recursive function is called.
However, in a functional language, like Haskell, the compiler is often able to optimize the recursion away into a loop, which makes it much faster then its procedural counterparts.
I've converted your sieve function into a set of recursive functions in C. I'll leave it to you to convert it into Haskell:
int main(void) {
//...
int root = sqrt(PRIME_RANGE);
primes[primeCnt++] = 2;
sieve(3, PRIME_RANGE, root);
//...
}
void sieve(int i, int end, int root) {
if(i > end) {
return;
}
if(!primeFlag[i]) {
primes[primeCnt++] = i;
if(root >= i) {
markMultiples(i * i, PRIME_RANGE, i);
}
}
i += 2;
sieve(i, end, root);
}
void markMultiples(int j, int end, int prime) {
if(j > end) {
return;
}
primeFlag[j] = 1;
j += i << 1;
markMultiples(j, end, prime);
}
The point of recursion is that the same function is called repeatedly, until a condition is met. The results of one recursive call are passed onto the next call, until the condition is met.
Also, why are you bit-fiddling instead of just multiplying or dividing by 2? Any half-decent compiler these days can convert most multiplications and divisions by 2 into a bit-shift.
I want to write a function that takes an array of letters as an argument and a number of those letters to select.
Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:
8! / ((8 - 3)! * 3!) = 56
Arrays (or words) in return consisting of 3 letters each.
Art of Computer Programming Volume 4: Fascicle 3 has a ton of these that might fit your particular situation better than how I describe.
Gray Codes
An issue that you will come across is of course memory and pretty quickly, you'll have problems by 20 elements in your set -- 20C3 = 1140. And if you want to iterate over the set it's best to use a modified gray code algorithm so you aren't holding all of them in memory. These generate the next combination from the previous and avoid repetitions. There are many of these for different uses. Do we want to maximize the differences between successive combinations? minimize? et cetera.
Some of the original papers describing gray codes:
Some Hamilton Paths and a Minimal Change Algorithm
Adjacent Interchange Combination Generation Algorithm
Here are some other papers covering the topic:
An Efficient Implementation of the Eades, Hickey, Read Adjacent Interchange Combination Generation Algorithm (PDF, with code in Pascal)
Combination Generators
Survey of Combinatorial Gray Codes (PostScript)
An Algorithm for Gray Codes
Chase's Twiddle (algorithm)
Phillip J Chase, `Algorithm 382: Combinations of M out of N Objects' (1970)
The algorithm in C...
Index of Combinations in Lexicographical Order (Buckles Algorithm 515)
You can also reference a combination by its index (in lexicographical order). Realizing that the index should be some amount of change from right to left based on the index we can construct something that should recover a combination.
So, we have a set {1,2,3,4,5,6}... and we want three elements. Let's say {1,2,3} we can say that the difference between the elements is one and in order and minimal. {1,2,4} has one change and is lexicographically number 2. So the number of 'changes' in the last place accounts for one change in the lexicographical ordering. The second place, with one change {1,3,4} has one change but accounts for more change since it's in the second place (proportional to the number of elements in the original set).
The method I've described is a deconstruction, as it seems, from set to the index, we need to do the reverse – which is much trickier. This is how Buckles solves the problem. I wrote some C to compute them, with minor changes – I used the index of the sets rather than a number range to represent the set, so we are always working from 0...n.
Note:
Since combinations are unordered, {1,3,2} = {1,2,3} --we order them to be lexicographical.
This method has an implicit 0 to start the set for the first difference.
Index of Combinations in Lexicographical Order (McCaffrey)
There is another way:, its concept is easier to grasp and program but it's without the optimizations of Buckles. Fortunately, it also does not produce duplicate combinations:
The set that maximizes , where .
For an example: 27 = C(6,4) + C(5,3) + C(2,2) + C(1,1). So, the 27th lexicographical combination of four things is: {1,2,5,6}, those are the indexes of whatever set you want to look at. Example below (OCaml), requires choose function, left to reader:
(* this will find the [x] combination of a [set] list when taking [k] elements *)
let combination_maccaffery set k x =
(* maximize function -- maximize a that is aCb *)
(* return largest c where c < i and choose(c,i) <= z *)
let rec maximize a b x =
if (choose a b ) <= x then a else maximize (a-1) b x
in
let rec iterate n x i = match i with
| 0 -> []
| i ->
let max = maximize n i x in
max :: iterate n (x - (choose max i)) (i-1)
in
if x < 0 then failwith "errors" else
let idxs = iterate (List.length set) x k in
List.map (List.nth set) (List.sort (-) idxs)
A small and simple combinations iterator
The following two algorithms are provided for didactic purposes. They implement an iterator and (a more general) folder overall combinations.
They are as fast as possible, having the complexity O(nCk). The memory consumption is bound by k.
We will start with the iterator, which will call a user provided function for each combination
let iter_combs n k f =
let rec iter v s j =
if j = k then f v
else for i = s to n - 1 do iter (i::v) (i+1) (j+1) done in
iter [] 0 0
A more general version will call the user provided function along with the state variable, starting from the initial state. Since we need to pass the state between different states we won't use the for-loop, but instead, use recursion,
let fold_combs n k f x =
let rec loop i s c x =
if i < n then
loop (i+1) s c ##
let c = i::c and s = s + 1 and i = i + 1 in
if s < k then loop i s c x else f c x
else x in
loop 0 0 [] x
In C#:
public static IEnumerable<IEnumerable<T>> Combinations<T>(this IEnumerable<T> elements, int k)
{
return k == 0 ? new[] { new T[0] } :
elements.SelectMany((e, i) =>
elements.Skip(i + 1).Combinations(k - 1).Select(c => (new[] {e}).Concat(c)));
}
Usage:
var result = Combinations(new[] { 1, 2, 3, 4, 5 }, 3);
Result:
123
124
125
134
135
145
234
235
245
345
Short java solution:
import java.util.Arrays;
public class Combination {
public static void main(String[] args){
String[] arr = {"A","B","C","D","E","F"};
combinations2(arr, 3, 0, new String[3]);
}
static void combinations2(String[] arr, int len, int startPosition, String[] result){
if (len == 0){
System.out.println(Arrays.toString(result));
return;
}
for (int i = startPosition; i <= arr.length-len; i++){
result[result.length - len] = arr[i];
combinations2(arr, len-1, i+1, result);
}
}
}
Result will be
[A, B, C]
[A, B, D]
[A, B, E]
[A, B, F]
[A, C, D]
[A, C, E]
[A, C, F]
[A, D, E]
[A, D, F]
[A, E, F]
[B, C, D]
[B, C, E]
[B, C, F]
[B, D, E]
[B, D, F]
[B, E, F]
[C, D, E]
[C, D, F]
[C, E, F]
[D, E, F]
May I present my recursive Python solution to this problem?
def choose_iter(elements, length):
for i in xrange(len(elements)):
if length == 1:
yield (elements[i],)
else:
for next in choose_iter(elements[i+1:], length-1):
yield (elements[i],) + next
def choose(l, k):
return list(choose_iter(l, k))
Example usage:
>>> len(list(choose_iter("abcdefgh",3)))
56
I like it for its simplicity.
Lets say your array of letters looks like this: "ABCDEFGH". You have three indices (i, j, k) indicating which letters you are going to use for the current word, You start with:
A B C D E F G H
^ ^ ^
i j k
First you vary k, so the next step looks like that:
A B C D E F G H
^ ^ ^
i j k
If you reached the end you go on and vary j and then k again.
A B C D E F G H
^ ^ ^
i j k
A B C D E F G H
^ ^ ^
i j k
Once you j reached G you start also to vary i.
A B C D E F G H
^ ^ ^
i j k
A B C D E F G H
^ ^ ^
i j k
...
Written in code this look something like that
void print_combinations(const char *string)
{
int i, j, k;
int len = strlen(string);
for (i = 0; i < len - 2; i++)
{
for (j = i + 1; j < len - 1; j++)
{
for (k = j + 1; k < len; k++)
printf("%c%c%c\n", string[i], string[j], string[k]);
}
}
}
The following recursive algorithm picks all of the k-element combinations from an ordered set:
choose the first element i of your combination
combine i with each of the combinations of k-1 elements chosen recursively from the set of elements larger than i.
Iterate the above for each i in the set.
It is essential that you pick the rest of the elements as larger than i, to avoid repetition. This way [3,5] will be picked only once, as [3] combined with [5], instead of twice (the condition eliminates [5] + [3]). Without this condition you get variations instead of combinations.
Short example in Python:
def comb(sofar, rest, n):
if n == 0:
print sofar
else:
for i in range(len(rest)):
comb(sofar + rest[i], rest[i+1:], n-1)
>>> comb("", "abcde", 3)
abc
abd
abe
acd
ace
ade
bcd
bce
bde
cde
For explanation, the recursive method is described with the following example:
Example: A B C D E
All combinations of 3 would be:
A with all combinations of 2 from the rest (B C D E)
B with all combinations of 2 from the rest (C D E)
C with all combinations of 2 from the rest (D E)
I found this thread useful and thought I would add a Javascript solution that you can pop into Firebug. Depending on your JS engine, it could take a little time if the starting string is large.
function string_recurse(active, rest) {
if (rest.length == 0) {
console.log(active);
} else {
string_recurse(active + rest.charAt(0), rest.substring(1, rest.length));
string_recurse(active, rest.substring(1, rest.length));
}
}
string_recurse("", "abc");
The output should be as follows:
abc
ab
ac
a
bc
b
c
In C++ the following routine will produce all combinations of length distance(first,k) between the range [first,last):
#include <algorithm>
template <typename Iterator>
bool next_combination(const Iterator first, Iterator k, const Iterator last)
{
/* Credits: Mark Nelson http://marknelson.us */
if ((first == last) || (first == k) || (last == k))
return false;
Iterator i1 = first;
Iterator i2 = last;
++i1;
if (last == i1)
return false;
i1 = last;
--i1;
i1 = k;
--i2;
while (first != i1)
{
if (*--i1 < *i2)
{
Iterator j = k;
while (!(*i1 < *j)) ++j;
std::iter_swap(i1,j);
++i1;
++j;
i2 = k;
std::rotate(i1,j,last);
while (last != j)
{
++j;
++i2;
}
std::rotate(k,i2,last);
return true;
}
}
std::rotate(first,k,last);
return false;
}
It can be used like this:
#include <string>
#include <iostream>
int main()
{
std::string s = "12345";
std::size_t comb_size = 3;
do
{
std::cout << std::string(s.begin(), s.begin() + comb_size) << std::endl;
} while (next_combination(s.begin(), s.begin() + comb_size, s.end()));
return 0;
}
This will print the following:
123
124
125
134
135
145
234
235
245
345
static IEnumerable<string> Combinations(List<string> characters, int length)
{
for (int i = 0; i < characters.Count; i++)
{
// only want 1 character, just return this one
if (length == 1)
yield return characters[i];
// want more than one character, return this one plus all combinations one shorter
// only use characters after the current one for the rest of the combinations
else
foreach (string next in Combinations(characters.GetRange(i + 1, characters.Count - (i + 1)), length - 1))
yield return characters[i] + next;
}
}
Simple recursive algorithm in Haskell
import Data.List
combinations 0 lst = [[]]
combinations n lst = do
(x:xs) <- tails lst
rest <- combinations (n-1) xs
return $ x : rest
We first define the special case, i.e. selecting zero elements. It produces a single result, which is an empty list (i.e. a list that contains an empty list).
For n > 0, x goes through every element of the list and xs is every element after x.
rest picks n - 1 elements from xs using a recursive call to combinations. The final result of the function is a list where each element is x : rest (i.e. a list which has x as head and rest as tail) for every different value of x and rest.
> combinations 3 "abcde"
["abc","abd","abe","acd","ace","ade","bcd","bce","bde","cde"]
And of course, since Haskell is lazy, the list is gradually generated as needed, so you can partially evaluate exponentially large combinations.
> let c = combinations 8 "abcdefghijklmnopqrstuvwxyz"
> take 10 c
["abcdefgh","abcdefgi","abcdefgj","abcdefgk","abcdefgl","abcdefgm","abcdefgn",
"abcdefgo","abcdefgp","abcdefgq"]
And here comes granddaddy COBOL, the much maligned language.
Let's assume an array of 34 elements of 8 bytes each (purely arbitrary selection.) The idea is to enumerate all possible 4-element combinations and load them into an array.
We use 4 indices, one each for each position in the group of 4
The array is processed like this:
idx1 = 1
idx2 = 2
idx3 = 3
idx4 = 4
We vary idx4 from 4 to the end. For each idx4 we get a unique combination
of groups of four. When idx4 comes to the end of the array, we increment idx3 by 1 and set idx4 to idx3+1. Then we run idx4 to the end again. We proceed in this manner, augmenting idx3,idx2, and idx1 respectively until the position of idx1 is less than 4 from the end of the array. That finishes the algorithm.
1 --- pos.1
2 --- pos 2
3 --- pos 3
4 --- pos 4
5
6
7
etc.
First iterations:
1234
1235
1236
1237
1245
1246
1247
1256
1257
1267
etc.
A COBOL example:
01 DATA_ARAY.
05 FILLER PIC X(8) VALUE "VALUE_01".
05 FILLER PIC X(8) VALUE "VALUE_02".
etc.
01 ARAY_DATA OCCURS 34.
05 ARAY_ITEM PIC X(8).
01 OUTPUT_ARAY OCCURS 50000 PIC X(32).
01 MAX_NUM PIC 99 COMP VALUE 34.
01 INDEXXES COMP.
05 IDX1 PIC 99.
05 IDX2 PIC 99.
05 IDX3 PIC 99.
05 IDX4 PIC 99.
05 OUT_IDX PIC 9(9).
01 WHERE_TO_STOP_SEARCH PIC 99 COMP.
* Stop the search when IDX1 is on the third last array element:
COMPUTE WHERE_TO_STOP_SEARCH = MAX_VALUE - 3
MOVE 1 TO IDX1
PERFORM UNTIL IDX1 > WHERE_TO_STOP_SEARCH
COMPUTE IDX2 = IDX1 + 1
PERFORM UNTIL IDX2 > MAX_NUM
COMPUTE IDX3 = IDX2 + 1
PERFORM UNTIL IDX3 > MAX_NUM
COMPUTE IDX4 = IDX3 + 1
PERFORM UNTIL IDX4 > MAX_NUM
ADD 1 TO OUT_IDX
STRING ARAY_ITEM(IDX1)
ARAY_ITEM(IDX2)
ARAY_ITEM(IDX3)
ARAY_ITEM(IDX4)
INTO OUTPUT_ARAY(OUT_IDX)
ADD 1 TO IDX4
END-PERFORM
ADD 1 TO IDX3
END-PERFORM
ADD 1 TO IDX2
END_PERFORM
ADD 1 TO IDX1
END-PERFORM.
Another C# version with lazy generation of the combination indices. This version maintains a single array of indices to define a mapping between the list of all values and the values for the current combination, i.e. constantly uses O(k) additional space during the entire runtime. The code generates individual combinations, including the first one, in O(k) time.
public static IEnumerable<T[]> Combinations<T>(this T[] values, int k)
{
if (k < 0 || values.Length < k)
yield break; // invalid parameters, no combinations possible
// generate the initial combination indices
var combIndices = new int[k];
for (var i = 0; i < k; i++)
{
combIndices[i] = i;
}
while (true)
{
// return next combination
var combination = new T[k];
for (var i = 0; i < k; i++)
{
combination[i] = values[combIndices[i]];
}
yield return combination;
// find first index to update
var indexToUpdate = k - 1;
while (indexToUpdate >= 0 && combIndices[indexToUpdate] >= values.Length - k + indexToUpdate)
{
indexToUpdate--;
}
if (indexToUpdate < 0)
yield break; // done
// update combination indices
for (var combIndex = combIndices[indexToUpdate] + 1; indexToUpdate < k; indexToUpdate++, combIndex++)
{
combIndices[indexToUpdate] = combIndex;
}
}
}
Test code:
foreach (var combination in new[] {'a', 'b', 'c', 'd', 'e'}.Combinations(3))
{
System.Console.WriteLine(String.Join(" ", combination));
}
Output:
a b c
a b d
a b e
a c d
a c e
a d e
b c d
b c e
b d e
c d e
Here is an elegant, generic implementation in Scala, as described on 99 Scala Problems.
object P26 {
def flatMapSublists[A,B](ls: List[A])(f: (List[A]) => List[B]): List[B] =
ls match {
case Nil => Nil
case sublist#(_ :: tail) => f(sublist) ::: flatMapSublists(tail)(f)
}
def combinations[A](n: Int, ls: List[A]): List[List[A]] =
if (n == 0) List(Nil)
else flatMapSublists(ls) { sl =>
combinations(n - 1, sl.tail) map {sl.head :: _}
}
}
If you can use SQL syntax - say, if you're using LINQ to access fields of an structure or array, or directly accessing a database that has a table called "Alphabet" with just one char field "Letter", you can adapt following code:
SELECT A.Letter, B.Letter, C.Letter
FROM Alphabet AS A, Alphabet AS B, Alphabet AS C
WHERE A.Letter<>B.Letter AND A.Letter<>C.Letter AND B.Letter<>C.Letter
AND A.Letter<B.Letter AND B.Letter<C.Letter
This will return all combinations of 3 letters, notwithstanding how many letters you have in table "Alphabet" (it can be 3, 8, 10, 27, etc.).
If what you want is all permutations, rather than combinations (i.e. you want "ACB" and "ABC" to count as different, rather than appear just once) just delete the last line (the AND one) and it's done.
Post-Edit: After re-reading the question, I realise what's needed is the general algorithm, not just a specific one for the case of selecting 3 items. Adam Hughes' answer is the complete one, unfortunately I cannot vote it up (yet). This answer's simple but works only for when you want exactly 3 items.
I had a permutation algorithm I used for project euler, in python:
def missing(miss,src):
"Returns the list of items in src not present in miss"
return [i for i in src if i not in miss]
def permutation_gen(n,l):
"Generates all the permutations of n items of the l list"
for i in l:
if n<=1: yield [i]
r = [i]
for j in permutation_gen(n-1,missing([i],l)): yield r+j
If
n<len(l)
you should have all combination you need without repetition, do you need it?
It is a generator, so you use it in something like this:
for comb in permutation_gen(3,list("ABCDEFGH")):
print comb
https://gist.github.com/3118596
There is an implementation for JavaScript. It has functions to get k-combinations and all combinations of an array of any objects. Examples:
k_combinations([1,2,3], 2)
-> [[1,2], [1,3], [2,3]]
combinations([1,2,3])
-> [[1],[2],[3],[1,2],[1,3],[2,3],[1,2,3]]
Lets say your array of letters looks like this: "ABCDEFGH". You have three indices (i, j, k) indicating which letters you are going to use for the current word, You start with:
A B C D E F G H
^ ^ ^
i j k
First you vary k, so the next step looks like that:
A B C D E F G H
^ ^ ^
i j k
If you reached the end you go on and vary j and then k again.
A B C D E F G H
^ ^ ^
i j k
A B C D E F G H
^ ^ ^
i j k
Once you j reached G you start also to vary i.
A B C D E F G H
^ ^ ^
i j k
A B C D E F G H
^ ^ ^
i j k
...
function initializePointers($cnt) {
$pointers = [];
for($i=0; $i<$cnt; $i++) {
$pointers[] = $i;
}
return $pointers;
}
function incrementPointers(&$pointers, &$arrLength) {
for($i=0; $i<count($pointers); $i++) {
$currentPointerIndex = count($pointers) - $i - 1;
$currentPointer = $pointers[$currentPointerIndex];
if($currentPointer < $arrLength - $i - 1) {
++$pointers[$currentPointerIndex];
for($j=1; ($currentPointerIndex+$j)<count($pointers); $j++) {
$pointers[$currentPointerIndex+$j] = $pointers[$currentPointerIndex]+$j;
}
return true;
}
}
return false;
}
function getDataByPointers(&$arr, &$pointers) {
$data = [];
for($i=0; $i<count($pointers); $i++) {
$data[] = $arr[$pointers[$i]];
}
return $data;
}
function getCombinations($arr, $cnt)
{
$len = count($arr);
$result = [];
$pointers = initializePointers($cnt);
do {
$result[] = getDataByPointers($arr, $pointers);
} while(incrementPointers($pointers, count($arr)));
return $result;
}
$result = getCombinations([0, 1, 2, 3, 4, 5], 3);
print_r($result);
Based on https://stackoverflow.com/a/127898/2628125, but more abstract, for any size of pointers.
Here you have a lazy evaluated version of that algorithm coded in C#:
static bool nextCombination(int[] num, int n, int k)
{
bool finished, changed;
changed = finished = false;
if (k > 0)
{
for (int i = k - 1; !finished && !changed; i--)
{
if (num[i] < (n - 1) - (k - 1) + i)
{
num[i]++;
if (i < k - 1)
{
for (int j = i + 1; j < k; j++)
{
num[j] = num[j - 1] + 1;
}
}
changed = true;
}
finished = (i == 0);
}
}
return changed;
}
static IEnumerable Combinations<T>(IEnumerable<T> elements, int k)
{
T[] elem = elements.ToArray();
int size = elem.Length;
if (k <= size)
{
int[] numbers = new int[k];
for (int i = 0; i < k; i++)
{
numbers[i] = i;
}
do
{
yield return numbers.Select(n => elem[n]);
}
while (nextCombination(numbers, size, k));
}
}
And test part:
static void Main(string[] args)
{
int k = 3;
var t = new[] { "dog", "cat", "mouse", "zebra"};
foreach (IEnumerable<string> i in Combinations(t, k))
{
Console.WriteLine(string.Join(",", i));
}
}
Hope this help you!
Another version, that forces all the first k to appear firstly, then all the first k+1 combinations, then all the first k+2 etc.. It means that if you have sorted array, the most important on the top, it would take them and expand gradually to the next ones - only when it is must do so.
private static bool NextCombinationFirstsAlwaysFirst(int[] num, int n, int k)
{
if (k > 1 && NextCombinationFirstsAlwaysFirst(num, num[k - 1], k - 1))
return true;
if (num[k - 1] + 1 == n)
return false;
++num[k - 1];
for (int i = 0; i < k - 1; ++i)
num[i] = i;
return true;
}
For instance, if you run the first method ("nextCombination") on k=3, n=5 you'll get:
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
But if you'll run
int[] nums = new int[k];
for (int i = 0; i < k; ++i)
nums[i] = i;
do
{
Console.WriteLine(string.Join(" ", nums));
}
while (NextCombinationFirstsAlwaysFirst(nums, n, k));
You'll get this (I added empty lines for clarity):
0 1 2
0 1 3
0 2 3
1 2 3
0 1 4
0 2 4
1 2 4
0 3 4
1 3 4
2 3 4
It's adding "4" only when must to, and also after "4" was added it adds "3" again only when it must to (after doing 01, 02, 12).
Array.prototype.combs = function(num) {
var str = this,
length = str.length,
of = Math.pow(2, length) - 1,
out, combinations = [];
while(of) {
out = [];
for(var i = 0, y; i < length; i++) {
y = (1 << i);
if(y & of && (y !== of))
out.push(str[i]);
}
if (out.length >= num) {
combinations.push(out);
}
of--;
}
return combinations;
}
Clojure version:
(defn comb [k l]
(if (= 1 k) (map vector l)
(apply concat
(map-indexed
#(map (fn [x] (conj x %2))
(comb (dec k) (drop (inc %1) l)))
l))))
Algorithm:
Count from 1 to 2^n.
Convert each digit to its binary representation.
Translate each 'on' bit to elements of your set, based on position.
In C#:
void Main()
{
var set = new [] {"A", "B", "C", "D" }; //, "E", "F", "G", "H", "I", "J" };
var kElement = 2;
for(var i = 1; i < Math.Pow(2, set.Length); i++) {
var result = Convert.ToString(i, 2).PadLeft(set.Length, '0');
var cnt = Regex.Matches(Regex.Escape(result), "1").Count;
if (cnt == kElement) {
for(int j = 0; j < set.Length; j++)
if ( Char.GetNumericValue(result[j]) == 1)
Console.Write(set[j]);
Console.WriteLine();
}
}
}
Why does it work?
There is a bijection between the subsets of an n-element set and n-bit sequences.
That means we can figure out how many subsets there are by counting sequences.
e.g., the four element set below can be represented by {0,1} X {0, 1} X {0, 1} X {0, 1} (or 2^4) different sequences.
So - all we have to do is count from 1 to 2^n to find all the combinations. (We ignore the empty set.) Next, translate the digits to their binary representation. Then substitute elements of your set for 'on' bits.
If you want only k element results, only print when k bits are 'on'.
(If you want all subsets instead of k length subsets, remove the cnt/kElement part.)
(For proof, see MIT free courseware Mathematics for Computer Science, Lehman et al, section 11.2.2. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/ )
short python code, yielding index positions
def yield_combos(n,k):
# n is set size, k is combo size
i = 0
a = [0]*k
while i > -1:
for j in range(i+1, k):
a[j] = a[j-1]+1
i=j
yield a
while a[i] == i + n - k:
i -= 1
a[i] += 1
All said and and done here comes the O'caml code for that.
Algorithm is evident from the code..
let combi n lst =
let rec comb l c =
if( List.length c = n) then [c] else
match l with
[] -> []
| (h::t) -> (combi t (h::c))#(combi t c)
in
combi lst []
;;
Here is a method which gives you all combinations of specified size from a random length string. Similar to quinmars' solution, but works for varied input and k.
The code can be changed to wrap around, ie 'dab' from input 'abcd' w k=3.
public void run(String data, int howMany){
choose(data, howMany, new StringBuffer(), 0);
}
//n choose k
private void choose(String data, int k, StringBuffer result, int startIndex){
if (result.length()==k){
System.out.println(result.toString());
return;
}
for (int i=startIndex; i<data.length(); i++){
result.append(data.charAt(i));
choose(data,k,result, i+1);
result.setLength(result.length()-1);
}
}
Output for "abcde":
abc abd abe acd ace ade bcd bce bde cde
Short javascript version (ES 5)
let combine = (list, n) =>
n == 0 ?
[[]] :
list.flatMap((e, i) =>
combine(
list.slice(i + 1),
n - 1
).map(c => [e].concat(c))
);
let res = combine([1,2,3,4], 3);
res.forEach(e => console.log(e.join()));
Another python recusive solution.
def combination_indicies(n, k, j = 0, stack = []):
if len(stack) == k:
yield list(stack)
return
for i in range(j, n):
stack.append(i)
for x in combination_indicies(n, k, i + 1, stack):
yield x
stack.pop()
list(combination_indicies(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]]
I created a solution in SQL Server 2005 for this, and posted it on my website: http://www.jessemclain.com/downloads/code/sql/fn_GetMChooseNCombos.sql.htm
Here is an example to show usage:
SELECT * FROM dbo.fn_GetMChooseNCombos('ABCD', 2, '')
results:
Word
----
AB
AC
AD
BC
BD
CD
(6 row(s) affected)
Here is my proposition in C++
I tried to impose as little restriction on the iterator type as i could so this solution assumes just forward iterator, and it can be a const_iterator. This should work with any standard container. In cases where arguments don't make sense it throws std::invalid_argumnent
#include <vector>
#include <stdexcept>
template <typename Fci> // Fci - forward const iterator
std::vector<std::vector<Fci> >
enumerate_combinations(Fci begin, Fci end, unsigned int combination_size)
{
if(begin == end && combination_size > 0u)
throw std::invalid_argument("empty set and positive combination size!");
std::vector<std::vector<Fci> > result; // empty set of combinations
if(combination_size == 0u) return result; // there is exactly one combination of
// size 0 - emty set
std::vector<Fci> current_combination;
current_combination.reserve(combination_size + 1u); // I reserve one aditional slot
// in my vector to store
// the end sentinel there.
// The code is cleaner thanks to that
for(unsigned int i = 0u; i < combination_size && begin != end; ++i, ++begin)
{
current_combination.push_back(begin); // Construction of the first combination
}
// Since I assume the itarators support only incrementing, I have to iterate over
// the set to get its size, which is expensive. Here I had to itrate anyway to
// produce the first cobination, so I use the loop to also check the size.
if(current_combination.size() < combination_size)
throw std::invalid_argument("combination size > set size!");
result.push_back(current_combination); // Store the first combination in the results set
current_combination.push_back(end); // Here I add mentioned earlier sentinel to
// simplyfy rest of the code. If I did it
// earlier, previous statement would get ugly.
while(true)
{
unsigned int i = combination_size;
Fci tmp; // Thanks to the sentinel I can find first
do // iterator to change, simply by scaning
{ // from right to left and looking for the
tmp = current_combination[--i]; // first "bubble". The fact, that it's
++tmp; // a forward iterator makes it ugly but I
} // can't help it.
while(i > 0u && tmp == current_combination[i + 1u]);
// Here is probably my most obfuscated expression.
// Loop above looks for a "bubble". If there is no "bubble", that means, that
// current_combination is the last combination, Expression in the if statement
// below evaluates to true and the function exits returning result.
// If the "bubble" is found however, the ststement below has a sideeffect of
// incrementing the first iterator to the left of the "bubble".
if(++current_combination[i] == current_combination[i + 1u])
return result;
// Rest of the code sets posiotons of the rest of the iterstors
// (if there are any), that are to the right of the incremented one,
// to form next combination
while(++i < combination_size)
{
current_combination[i] = current_combination[i - 1u];
++current_combination[i];
}
// Below is the ugly side of using the sentinel. Well it had to haave some
// disadvantage. Try without it.
result.push_back(std::vector<Fci>(current_combination.begin(),
current_combination.end() - 1));
}
}
Here is a code I recently wrote in Java, which calculates and returns all the combination of "num" elements from "outOf" elements.
// author: Sourabh Bhat (heySourabh#gmail.com)
public class Testing
{
public static void main(String[] args)
{
// Test case num = 5, outOf = 8.
int num = 5;
int outOf = 8;
int[][] combinations = getCombinations(num, outOf);
for (int i = 0; i < combinations.length; i++)
{
for (int j = 0; j < combinations[i].length; j++)
{
System.out.print(combinations[i][j] + " ");
}
System.out.println();
}
}
private static int[][] getCombinations(int num, int outOf)
{
int possibilities = get_nCr(outOf, num);
int[][] combinations = new int[possibilities][num];
int arrayPointer = 0;
int[] counter = new int[num];
for (int i = 0; i < num; i++)
{
counter[i] = i;
}
breakLoop: while (true)
{
// Initializing part
for (int i = 1; i < num; i++)
{
if (counter[i] >= outOf - (num - 1 - i))
counter[i] = counter[i - 1] + 1;
}
// Testing part
for (int i = 0; i < num; i++)
{
if (counter[i] < outOf)
{
continue;
} else
{
break breakLoop;
}
}
// Innermost part
combinations[arrayPointer] = counter.clone();
arrayPointer++;
// Incrementing part
counter[num - 1]++;
for (int i = num - 1; i >= 1; i--)
{
if (counter[i] >= outOf - (num - 1 - i))
counter[i - 1]++;
}
}
return combinations;
}
private static int get_nCr(int n, int r)
{
if(r > n)
{
throw new ArithmeticException("r is greater then n");
}
long numerator = 1;
long denominator = 1;
for (int i = n; i >= r + 1; i--)
{
numerator *= i;
}
for (int i = 2; i <= n - r; i++)
{
denominator *= i;
}
return (int) (numerator / denominator);
}
}
Suppose I have an array of M elements, all numbers, negative or positive or zero.
Can anyone suggest an algorithm to select N elements from the array, such that the sum of these N elements is the smallest possible positive number?
Take this array for example:
-1000,-700,-400,-200,-100,-50,10,100,300,600,800,1200
Now I have to select any 5 elements such that their sum is the smallest possible positive number.
Formulation
For i = 1, ..., M:
Let a_i be the ith number in your list of candidates
Let x_i denote whether the ith number is included in your set of N chosen numbers
Then you want to solve the following integer programming problem.
minimize: sum(a_i * x_i)
with respect to: x_i
subject to:
(1) sum(a_i * x_i) >= 0
(2) sum(x_i) = N
(3) x_i in {0, 1}
You can apply an integer program solver "out of the box" to this problem to find the optimal solution or a suboptimal solution with controllable precision.
Resources
Integer programming
Explanation of branch-and-bound integer program solver
If you want to find the best possible solution, you can simply use brute force ie. try all posible combinations of fiwe numbers.
Something like this very quick and dirty algorithm:
public List<Integer> findLeastPositivSum(List<Integer> numbers) {
List<Integer> result;
Integer resultSum;
List<Integer> subresult, subresult2, subresult3, subresult4, subresult5;
for (int i = 0; i < numbers.size() - 4; i++) {
subresult = new ArrayList<Integer>();
subresult.add(numbers.get(i));
for (int j = i + 1; j < numbers.size() - 3; j++) {
subresult2 = new ArrayList<Integer>(subresult);
subresult2.add(j);
for (int k = j + 1; k < numbers.size() - 2; k++) {
subresult3 = new ArrayList<Integer>(subresult2);
subresult3.add(k);
for (int l = k + 1; l < numbers.size() - 1; l++) {
subresult4 = new ArrayList<Integer>(subresult3);
subresult4.add(k);
for (int m = l + 1; m < numbers.size(); m++) {
subresult5 = new ArrayList<Integer>(subresult4);
subresult5.add(k);
Integer subresultSum = sum(subresult5);
if (subresultSum > 0) {
if (result == null || resultSum > subresultSum) {
result = subresult;
}
}
}
}
}
}
}
return result;
}
public Integer sum(List<Integer> list) {
Integer result = 0;
for (Integer integer : list) {
result += integer;
}
return result;
}
This is really quick and dirty algorithm, it can be done more elegantly. I can provide cleaner algorithm e.g. using recursion.
It can be also further optimized. E.g. you can remove similar numbers from input list as first step.
Let initial array be shorted already, or i guess this will work even when it isnt shorted..
N -> Length of array
M -> Element req.
R[] -> Answer
TEMP[] -> For calculations
minSum -> minSum
A[] -> Initial input
All above variables are globally defined
int find(int A[],int start,int left)
{
if(left=0)
{
//sum elements in TEMP[] and save it as curSum
if(curSum<minSum)
{
minSum=curSum;
//assign elements from TEMP[] to R[] (i.e. our answer)
}
}
for(i=start;i<=(N-left);i++)
{
if(left==M)
curSum=0;
TEMP[left-1]=A[i];
find(A[],i+1,left-1);
}
}
// Made it in hurry so maybe some error would be existing..
Working solution on ideone :
http://ideone.com/YN8PeW
I suppose Kadane’s Algorithm would do the trick, although it is for the maximum sum but I have also implemented it to find the minimum sum, though can't find the code right now.
Here's something sub optimal in Haskell, which (as with many of my ideas) could probably be further and better optimized. It goes something like this:
Sort the array (I got interesting results by trying both ascending and descending)
B N = first N elements of the array
B (i), for i > N = best candidate; where (assuming integers) if they are both less than 1, the candidates are compared by the absolute value of their sums; if they are both 1 or greater, by their sums; and if only one candidate is greater than 0 then that candidate is chosen. If a candidate's sum is 1, return that candidate as the answer. The candidates are:
B (i-1), B (i-1)[2,3,4..N] ++ array [i], B (i-1)[1,3,4..N] ++ array [i]...B (i-1)[1,2..N-1] ++ array [i]
B (i-2)[2,3,4..N] ++ array [i], B (i-2)[1,3,4..N] ++ array [i]...B (i-2)[1,2..N-1] ++ array [i]
...
B (N)[2,3,4..N] ++ array [i], B (N)[1,3,4..N] ++ array [i]...B (N)[1,2..N-1] ++ array [i]
Note that for the part of the array where the numbers are negative (in the case of ascending sort) or positive (in the case of descending sort), step 3 can be done immediately without calculations.
Output:
*Main> least 5 "desc" [-1000,-700,-400,-200,-100,-50,10,100,300,600,800,1200]
(10,[-1000,600,300,100,10])
(0.02 secs, 1106836 bytes)
*Main> least 5 "asc" [-1000,-700,-400,-200,-100,-50,10,100,300,600,800,1200]
(50,[300,100,-200,-100,-50])
(0.02 secs, 1097492 bytes)
*Main> main -- 10000 random numbers ranging from -100000 to 100000
(1,[-106,4,-40,74,69])
(1.77 secs, 108964888 bytes)
Code:
import Data.Map (fromList, insert, (!))
import Data.List (minimumBy,tails,sort)
import Control.Monad.Random hiding (fromList)
array = [-1000,-700,-400,-200,-100,-50,10,100,300,600,800,1200]
least n rev arr = comb (fromList listStart) [fst (last listStart) + 1..m]
where
m = length arr
r = if rev == "asc" then False else True
sorted = (if r then reverse else id) (sort arr)
listStart = if null lStart
then [(n,(sum $ take n sorted,take n sorted))]
else lStart
lStart = zip [n..]
. takeWhile (all (if r then (>0) else (<0)) . snd)
. foldr (\a b -> let c = take n (drop a sorted) in (sum c,c) : b) []
$ [0..]
s = fromList (zip [1..] sorted)
comb list [] = list ! m
comb list (i:is)
| fst (list ! (i-1)) == 1 = list ! (i-1)
| otherwise = comb updatedMap is
where updatedMap = insert i bestCandidate list
bestCandidate = comb' (list!(i - 1)) [i - 1,i - 2..n] where
comb' best [] = best
comb' best (j:js)
| fst best == 1 = best
| otherwise =
let s' = map (\x -> (sum x,x))
. (take n . map (take (n - 1)) . tails . cycle)
$ snd (list!j)
t = s!i
candidate = minimumBy compare' (map (add t) s')
in comb' (minimumBy compare' [candidate,best]) js
add x y#(a,b) = (x + a,x:b)
compare' a#(a',_) b#(b',_)
| a' < 1 = if b' < 1 then compare (abs a') (abs b') else GT
| otherwise = if b' < 1 then LT else compare a' b'
rnd :: (RandomGen g) => Rand g Int
rnd = getRandomR (-100000,100000)
main = do
values <- evalRandIO (sequence (replicate (10000) rnd))
putStrLn (show $ least 5 "desc" values)
Assumption: M is the original array
Pesudocode
S = sort(M);
R = [];
sum = 0;
for(i=0, i < length(S); i++){
sum = sum + S[i];
if(sum < 1){
R.push(S[i]);
}else{
return R;
}
}