Mega Code Archive
How to compute binomial coefficient (fast)
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Title: How to compute binomial coefficient (fast)
interface
uses Sysutils, Funct, MyBigInt;
function CalculateBinomKoeff(n, k: Integer): string;
implementation
function CalculateBinomKoeff(n, k: Integer): string;
const
zaehler = 0;
nenner = 1;
Pos = 0;
var
m, i, j, iGgT: Integer;
arBruch: array [0..1] of array of Integer;
mbResult: TMyBigInt;
begin
if (n - k) k then k := n - k;
m := n - k;
for i := Low(arBruch) to High(arBruch) do SetLength(arBruch[i], m + 1);
for i := 1 to m do
begin
arBruch[zaehler][i] := i + k;
arBruch[nenner][i] := i;
end;
arBruch[zaehler][Pos] := 1; // max{m+1: arBruch[zaehler][i]=1 f¨¹r alle i=1...m}
arBruch[nenner][Pos] := 2; // max{m+1: arBruch[nenner][i]=1 f¨¹r alle i=1...m}
while arBruch[nenner][Pos] = m do
begin
for i := m downto arBruch[nenner][Pos] do
begin
for j := m downto arBruch[zaehler][Pos] do
begin
// Bestimmung des ggTs
iGgT := ggT(arBruch[nenner][i], arBruch[zaehler][j]);
if iGgT 1 then
begin
// K¨¹rzen
arBruch[zaehler][j] := Round(arBruch[zaehler][j] / iGgT);
arBruch[nenner][i] := Round(arBruch[nenner][i] / iGgT);
if arBruch[nenner][i] = 1 then break;
end;
end;
// Verschieben der Position im Zaehler
for j := arBruch[zaehler][Pos] to m do
if arBruch[zaehler][j] = 1 then arBruch[zaehler][Pos] := j + 1
else
break;
end;
for i := arBruch[nenner][Pos] to m do
if arBruch[nenner][i] = 1 then arBruch[nenner][Pos] := i + 1
else
break;
end;
mbResult := TMyBigInt.Create(1);
try
for i := arBruch[zaehler][Pos] to m do mbResult.Multiply(mbResult, arBruch[zaehler][i]);
Result := mbResult.ToString;
finally
FreeAndNil(mbResult);
end;
end;
end.
//*** Code: Funct ****************************************************
unit Funct;
interface
function ggT(x, y: Integer): Integer;
implementation
function ggT(x, y: Integer): Integer;
begin
if y x then Result := ggT(y, x)
else if y = 0 then Result := x
else
Result := ggT(y, x mod y);
end;
end.
//*** Code: MyBigInt *************************************************
unit MyBigInt;
interface
uses Sysutils, Math;
const
Base = 10;
type
TMyBigInt = class
private
Len: Integer;
Value: AnsiString;
procedure Trim;
procedure Shift(k: Integer);
procedure MultiplyAtom(Multiplier1: TMyBigInt; Multiplier2: Integer);
public
constructor Create(iValue: Integer = 0);
procedure Add(Addend1, Addend2: TMyBigInt);
procedure Multiply(Multiplier1, Multiplier2: TMyBigInt); overload;
procedure Multiply(Multiplier1: TMyBigInt; Multiplier2: Integer); overload;
function ToString: string;
procedure CopyFrom(mbCopy: TMyBigInt);
end;
implementation
constructor TMyBigInt.Create(iValue: Integer = 0);
var
sTmp: ShortString;
i: Integer;
begin
inherited Create;
sTmp := IntToStr(abs(iValue));
Len := Length(sTmp);
SetLength(Value, Len);
for i := 1 to Len do Value[i] := Chr(StrToInt(sTmp[Len - i + 1]));
end;
procedure TMyBigInt.Add(Addend1, Addend2: TMyBigInt);
{ zwei TMyBigInt miteinander addieren }
var
i, iCarry, iTemp: Integer;
begin
// L?nge der Wert-Strings angleichen
iTemp := max(Addend1.Len, Addend2.Len);
SetLength(Value, iTemp);
for i := Len + 1 to iTemp do Value[i] := #0; // F¨¹r den Fall Addend1/Addend2=Self
Len := iTemp;
// Berechnung von ¨¹bertrag und Summe
iCarry := 0;
for i := 1 to Len do
begin
iTemp := iCarry;
if i = Addend1.Len then iTemp := iTemp + Ord(Addend1.Value[i]);
if i = Addend2.Len then iTemp := iTemp + Ord(Addend2.Value[i]);
Value[i] := Char(iTemp mod Base);
iCarry := iTemp div Base;
end;
if iCarry 0 then
begin
Len := Len + 1;
SetLength(Value, Len);
Value[Len] := Char(iCarry);
end;
end;
procedure TMyBigInt.Multiply(Multiplier1, Multiplier2: TMyBigInt);
{ zwei TMyBigInt miteinander multipliziren }
var
mbResult, mbTemp: TMyBigInt;
i: Integer;
begin
mbResult := TMyBigInt.Create;
try
mbTemp := TMyBigInt.Create;
try
for i := 1 to Multiplier2.Len do
begin
// Multiplizieren nach der "Schulmethode"
mbTemp.MultiplyAtom(Multiplier1, Ord(Multiplier2.Value[i]));
mbTemp.Shift(i - 1);
mbResult.Add(mbResult, mbTemp);
end;
finally
FreeAndNil(mbTemp);
end;
CopyFrom(mbResult);
finally
FreeAndNil(mbResult);
end;
end;
procedure TMyBigInt.Multiply(Multiplier1: TMyBigInt; Multiplier2: Integer);
{ TMyBigInt und einen Integer multiplizieren }
var
mbTemp: TMyBigInt;
begin
mbTemp := TMyBigInt.Create(Multiplier2);
try
Multiply(Multiplier1, mbTemp);
finally
end;
end;
function TMyBigInt.ToString: string;
{ Zahl in einen String umwandeln }
var
i: Integer;
begin
Trim;
Result := '';
for i := Len downto 1 do Result := Result + IntToStr(Ord(Value[i]));
end;
procedure TMyBigInt.CopyFrom(mbCopy: TMyBigInt);
{ von mbCopy kopieren }
begin
Value := mbCopy.Value;
Len := mbCopy.Len;
end;
procedure TMyBigInt.Trim;
{ f¨¹hrende Nullen entfernen }
var
i, p: Integer;
begin
p := Len;
for i := Len downto 1 do
begin
if not (Value[i] in ['0']) then break;
p := i - 1;
end;
if p then
begin
SetLength(Value, p);
Len := p;
end;
end;
procedure TMyBigInt.Shift(k: Integer);
{ von hinten mit k Nullen auff¨¹llen, also mit Base^k multiplizieren }
var
i: Integer;
begin
if k = 0 then Exit;
SetLength(Value, Len + k);
for i := Len downto 1 do Value[i + k] := Value[i];
for i := 1 to k do Value[i] := #0;
Len := Len + k;
end;
procedure TMyBigInt.MultiplyAtom(Multiplier1: TMyBigInt; Multiplier2: Integer);
{ Multiplikation mit einer Ziffer }
var
i, iCarry, iTemp: Integer;
begin
// Multiplikation mit 1
if Multiplier2 = 1 then
begin
CopyFrom(Multiplier1);
Exit;
end;
SetLength(Value, Multiplier1.Len);
Len := Multiplier1.Len;
iCarry := 0;
for i := 1 to Len do
begin
iTemp := Ord(Multiplier1.Value[i]) * Multiplier2 + iCarry;
Value[i] := Char(iTemp mod Base);
iCarry := iTemp div Base;
end;
if iCarry 0 then
begin
Len := Len + 1;
SetLength(Value, Len);
Value[Len] := Char(iCarry);
end;
end;
end.
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