public int GetIndex(bool sorted, int[] kIndexes)
{
// This function returns the proper index to an entry in the sorted binomial coefficient table from
// the underlying values in KIndexes. For example, for the 13 chooose 5 example which
// corresponds to 5 card poker hand ranks, then AKQJT (which is the greatest hand in the table) would
// be passed as value 12, 11, 10, 9, and 8, and the return value would be 1286, which is the highest
// element. Note that if the Sorted flag is false, then the values in KIndexes will be put into sorted
// order and returned that way. The sorted flag must be set to false if KIndexes is not in descending order.
//
// Handle the N choose 1 case.
if (GroupSize == 1)
{
return(kIndexes[0]);
}
int LoopIndex, n, Index = 0;
int[] IndexArray;
if (!sorted)
{
ArraySorter <int> .SortDescending(kIndexes);
}
for (LoopIndex = 0; LoopIndex < GroupSize - 1; LoopIndex++)
{
IndexArray = Indexes[LoopIndex];
n = kIndexes[LoopIndex];
Index += IndexArray[n];
}
Index += kIndexes[GroupSize - 1];
return(Index);
}
public BigInteger GetRank(bool sorted, int[] kIndexes, int groupSize = -1)
{
// This method returns the rank of the combination in kIndexes. For example, with the 13 chooose 5 case which
// corresponds to 5 card poker hand ranks, then AKQJT (which is the greatest hand in the table) would
// be passed as value 12, 11, 10, 9, and 8, and the return value would be 1286, which is the highest
// element. Note that if the Sorted flag is false, then the values in KIndexes will be put into sorted
// descending order and returned that way. The sorted flag must be set to false if KIndexes is not in descending order.
//
// If the optional argument groupSize is specified, then it must be <= to the GroupSize used to create the instance.
//
groupSize = (groupSize == -1) ? GroupSize : groupSize;
// Handle the n choose 1 case.
if (groupSize == 1)
{
return(kIndexes[0]);
}
// Handle the n choose n case.
if (groupSize == NumItems)
{
return(0);
}
int loopIndex, n;
// The times that Pascal's triangle may not have been legitimately created are handled above.
// So, if it has not been created, then throw an exception.
if (PasTri == null)
{
string s = "BinCoeffBigInt:GetNumCombos: Error - Pascal's Triangle has not been created.";
ApplicationException ae = new ApplicationException(s);
throw ae;
}
BigInteger rank = 0;
BigInteger[] indexArray;
if (!sorted)
{
ArraySorter <int> .SortDescending(kIndexes);
}
int startIndex = GroupSize - groupSize;
int kIndex = 0;
for (loopIndex = startIndex; loopIndex < GroupSizeM1; loopIndex++)
{
indexArray = PasTri[loopIndex];
n = kIndexes[kIndex++];
rank += indexArray[n];
}
rank += kIndexes[groupSize - 1];
return(rank);
}
public uint GetRank(bool sorted, int[] kIndexes, out bool overflow, int groupSize = -1)
{
// This method returns the rank of the combination in kIndexes. For example, with the 13 chooose 5 case which
// corresponds to 5 card poker hand ranks, then AKQJT (which is the greatest hand in the table) would
// be passed as value 12, 11, 10, 9, and 8, and the return value would be 1286, which is the highest
// element. Note that if the Sorted flag is false, then the values in KIndexes will be put into sorted
// order and returned that way. The sorted flag must be set to false if KIndexes is not in descending order.
//
// If the optional argument groupSize is specified, then it must be <= to the GroupSize used to create the instance.
//
string s;
overflow = false;
groupSize = (groupSize == -1) ? GroupSize : groupSize;
if (groupSize == 0)
{
s = "BinCoeffL:GetNumCombos: groupSize equals zero. This is not allowed.";
ApplicationException ae = new ApplicationException(s);
throw ae;
}
// Handle the n choose 1 case.
if (groupSize == 1)
{
return((uint)kIndexes[0]);
}
// Handle the n choose n case.
if (groupSize == NumItems)
{
return(0);
}
int loopIndex, n;
// The times that Pascal's triangle may not have been legitimately created are handled above.
// So, if it has not been created, then throw an exception.
if (PasTri == null)
{
s = "BinCoeffL:GetNumCombos: Error - Pascal's Triangle has not been created. ";
ApplicationException ae = new ApplicationException(s);
throw ae;
}
uint rank = 0;
uint[] indexArray;
if (!sorted)
{
ArraySorter <int> .SortDescending(kIndexes);
}
int startIndex = GroupSize - groupSize;
int kIndex = 0;
// for (LoopIndex = 0; LoopIndex < GroupSize - 1; LoopIndex++)
for (loopIndex = startIndex; loopIndex < GroupSizeM1; loopIndex++)
{
indexArray = PasTri[loopIndex];
n = kIndexes[kIndex++];
// Check for overflow first.
if ((indexArray.Length > n) && (rank <= uint.MaxValue - indexArray[n]))
{
rank += indexArray[n];
}
else
{
overflow = true;
return(0);
}
}
if (rank <= uint.MaxValue - (ulong)kIndexes[groupSize - 1])
{
rank += (uint)kIndexes[groupSize - 1];
}
else
{
overflow = true;
return(0);
}
return(rank);
}
public ulong GetRank(int numItems, int groupSize, bool sorted, int[] kIndexes)
{
// This function returns the proper index to an entry in the sorted binomial coefficient table from
// the underlying values in KIndexes. For example, for the 13 chooose 5 example which
// corresponds to 5 card poker hand ranks, then AKQJT (which is the greatest hand in the table) would
// be passed as value 12, 11, 10, 9, and 8, and the return value would be 1286, which is the highest
// element. Note that if the Sorted flag is false, then the values in KIndexes will be put longo sorted
// order and returned that way. The sorted flag must be set to false if KIndexes is not in descending order.
//
// Notes: 13 choose 4 = 715. Start at 2nd index array. Examples:
// 13 choose 4 = 495 + 165 + 45 + 9 = 714 -> add one since these numbers start from zero -> 715.
// 13 choose 3 = 220 + 55 + 10 = 285 (+ 1) = 286.
// Both of the above cases are correct.
// 12 choose 5 = 792. Start at 1st index array, 2nd entry. Examples:
// 12 choose 5 = 462 + 210 + 84 + 28 + 7 = 791 (+ 1) = 792.
// 11 choose 5 = 256 + 126 + 56 + 21 + 6 = 461 (+ 1) = 462.
// Both of the above cases are correct.
// 12 choose 4 = 495. Start at 2nd index and start off at 2nd entry in array.
// 12 choose 4 = 494 (+ 1) = 495.
// 9 choose 3 = 83 (+ 1) = 84. Start at 3rd index,
// 9 choose 3 = 56 + 21 + 6 = 83 ( + 1) = 84.
// 8 choose 2 = 27 ( + 1)= 28. Start at
//
// Proof that C(n, k) = C(n, n-k):
// N! / K! (N - K)! = N! / (N - K) ! (N - (N - K))!
// K! (N - K)! = (N - K)! (N - (N - K))!
// K! = (N - (N - K))!
// K! = K!
// Benchmark of dynamic -> 18 times slower than static types:
// https://dev.to/mokenyon/benchmarking-and-exploring-c-s-dynamic-keyword-1l5i#:~:text=You%20can%20see%20that%20the,18x%20slower%20and%20allocated%20memory.
// http://ghcimdm4u.weebly.com/uploads/1/3/5/8/13589538/5.4.pdf - Proof of each binomial coefficient can be obtained from N choose K.
// https://www.mathsisfun.com/algebra/binomial-theorem.html#coefficients - shows how Pascal's triangle is composed of binomial coefficients.
// https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Book_of_Proof_(Hammack)/03%3A_Counting/3.06%3A_Pascal%E2%80%99s_Triangle_and_the_Binomial_Theorem
// The above link has a good explanation on Pascal's Triangle and the Binomial Theorem.
// https://math.stackexchange.com/questions/453843/why-does-pascals-triangle-give-the-binomial-coefficients - Why does Pascal's Triangle give the Binomial Coefficients?
// https://en.wikipedia.org/wiki/Combinatorial_number_system - Combinatorial number system (Ordering Combinations).
// https://www.developertyrone.com/blog/generating-the-mth-lexicographical-element-of-a-mathematical-combination/ - Generating the mth Lexicographical Element of a Mathematical Combination
// https://stackoverflow.com/questions/29010699/can-i-add-two-generic-values-in-java - indicates that Java has same issue as C# when adding generic variables.
// https://arxiv.org/pdf/1601.05794.pdf proof of n choose k = rank = c1 choose k + c2 choose (k - 1) + ... + cn choose 1,
// where c1, c2, ... ck are one of the kIndexes and k is groupSize. PROOF OF BIJECTION FOR COMBINATORIAL NUMBER SYSTEM.
//
// 2 ^ 32 = 4,294,967,296
// 34 C 17 = 2,333,606,220
// 35 C 16 = 4,059,928,950
// 35 C 17 = 4,537,567,650
// 50 C 25 = 126,410,606,437,752
// 2 ^ 64 = 18,446,744,073,709,551,616
// 67 C 33 = 14,226,520,737,620,288,370
// Calling BinCoeffBase.GetNumCombos(67, 33, out bool overflow) overflows.
// 68 C 33 = 27,640,097,433,090,845,976
//
// 100 C 50 = 100,891,344,545,564,193,334,812,497,256
// 200 C 100 = 90,548,514,656,103,281,165,404,177,077,484,163,874,504,589,675,413,336,841,320
// 1000 C 500 = 27028824094543656951561469362597527549615200844654828700739287510662542870552219389861248392450237016536260608502154610480220975
// 00506799175498942196995184754236654842637517333561624640797378873443645741611194976045710449857562878805146009942194267523669158
// 56603136862602484428109296905863799821216320
if ((numItems > NumItems) || (groupSize > GroupSize))
{
ApplicationException ae = new ApplicationException("BinCoeffL:GetRank: Input parameter(s) greater than expected.");
throw ae;
}
int loopIndex;
long n;
ulong rank = 0;
ulong[] indexArray;
if (!sorted)
{
ArraySorter <int> .SortDescending(kIndexes);
}
int groupOffset = GroupSize - groupSize;
int startIndex = NumItems - numItems;
for (loopIndex = startIndex; loopIndex < GroupSize - 1; loopIndex++)
{
indexArray = PasTri[loopIndex];
n = kIndexes[loopIndex + groupOffset];
rank += indexArray[n];
}
rank += (ulong)kIndexes[GroupSize - 1];
return(rank);
}
public ulong GetRank(bool sorted, int[] kIndexes, out bool overflow, int groupSize = -1)
{
// This function returns the proper index to an entry in the sorted binomial coefficient table from
// the underlying values in KIndexes. For example, for the 13 chooose 5 example which
// corresponds to 5 card poker hand ranks, then AKQJT (which is the greatest hand in the table) would
// be passed as value 12, 11, 10, 9, and 8, and the return value would be 1286, which is the highest
// element. Note that if the Sorted flag is false, then the values in KIndexes will be put into descending
// order and returned that way. The sorted flag must be set to false if KIndexes needs to be sorted.
// overflow is set to true if the operation overflows.
//
overflow = false;
groupSize = (groupSize == -1) ? GroupSize : groupSize;
// Handle the n choose n case.
if (groupSize == NumItems)
{
return(0);
}
// Handle the n choose 1 case.
if (groupSize == 1)
{
return((ulong)kIndexes[0]);
}
// The times that Pascal's triangle may not have been legitimately created are handled above.
// So, if it has not been created, then throw an exception.
if (PasTri == null)
{
string s = "BinCoeffL:GetNumCombos: Pascal's Triangle has not been created." +
"This could occur if the instance is created with 5 choose 5 and then 5 choose 3 is tried.";
ApplicationException ae = new ApplicationException(s);
throw ae;
}
ulong rank = 0;
int loopIndex;
long n;
ulong[] indexArray;
if (!sorted)
{
ArraySorter <int> .SortDescending(kIndexes);
}
int startIndex = GroupSize - groupSize;
int kIndex = 0;
for (loopIndex = startIndex; loopIndex < GroupSizeM1; loopIndex++)
{
indexArray = PasTri[loopIndex];
n = kIndexes[kIndex++];
// Check for overflow first.
if ((indexArray.Length > n) && (rank <= ulong.MaxValue - indexArray[n]))
{
rank += indexArray[n];
}
else
{
overflow = true;
return(0);
}
}
if (rank <= ulong.MaxValue - (ulong)kIndexes[groupSize - 1])
{
rank += (ulong)kIndexes[groupSize - 1];
}
else
{
overflow = true;
return(0);
}
return(rank);
}
