j.tiberghien chapter 2.10 recursive algorithms and backtracking
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J.Tiberghien
Chapter 2.10
Recursive Algorithmsand
Backtracking
J.Tiberghien
Recursive Algorithms
Factorial• Step-up function : Fac(n) = (n)*Fac(n-1)• Trivial solution : Fac(1) = 1• Recursive procedure :
PROCEDURE Fac(n:CARDINAL): CARDINAL;BEGIN
IF n > 1THEN RETURN n*Fac(n-1)ELSE RETURN 1
END (* IF *)END Fac;
J.Tiberghien
Recursive Algorithms
Factorial• Alternative Iterative Procedure :
PROCEDURE Fac(n:CARDINAL): CARDINAL; VAR f : CARDINAL;BEGIN f := 1;
WHILE n > 1 DOf := f * n ; n := n - 1
END; (* WHILE *)RETURN f
END Fac;
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Recursive Algorithms
Fibonaci• Step-up function : Fib(n) = Fib(n-1)+Fib(n-2)• Trivial solution : Fib(1) = 1; Fib(0) = 0• Recursive procedure :
PROCEDURE Fib(n:CARDINAL): CARDINAL;BEGIN
IF n > 1THEN RETURN Fib(n-1)+Fib(n-2)ELSIF n=1 THEN RETURN 1ELSE RETURN 0
END (* IF *)END Fib;
J.Tiberghien
Recursive Algorithms
Fibonaci• Alternative Iterative Procedure :
PROCEDURE Fib(n:CARDINAL): CARDINAL; VAR i,fn,fnm1,fnm2 : CARDINAL;BEGIN IF n = 0 THEN RETURN 0
ELSIF n = 1 THEN RETURN 1 ELSE fnm2 := 0; fnm1 := 1; FOR i := 2 TO n DO fn := fnm1 + fnm2; fnm2 := fnm1; fnm1 := fn END; (* FOR *) RETURN fn END (* IF *)
END Fib;
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Recursive AlgorithmsTowers of Hanoi• The problem :
Moving one tower with n rings from A to B using C
• Step-up function :– Move the n-1 upper rings from A to C
– Move one ring from A to B
– Move the n-1 rings from C to B
• Trivial solution :– Moving a tower with no rings
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Recursive AlgorithmsTowers of Hanoi• Recursive procedure :
PROCEDURE MoveTower (Height: CARDINAL From,Towards,Using:CHAR); PROCEDURE MoveDisk.....BEGIN
IF Height > 0 THENMoveTower(Height-1,From,Using,Towards)MoveDisk(From,Towards);MoveTower(Height-1,Using,Towards,From)
END (* IF *)END MoveTower;
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Recursive Algorithms Conclusion
Simultaneous activations
Few
Many
ComplexAlgorithm
YES
???
SimpleAlgorithm
...
Never
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Backtracking Algorithms
A treelike maze
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Generic Search with Backtracking
UNTIL all successor nodes have been explored
Select a successor node
Record selected node
Erase previous node selection
Allowed node ?No Yes
Call recursivelythe backtracking
procedure forthe next node
Final node ?Not yet Yes
Display thesolution
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The Eight Queens Problem
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Eight Queens Procedure
FOR Col := 1 TO 8 DO
Put queen on Board[Row,Col]
Remove queen from Board[Row,Col]
Board[Row,Col] safe ?No Yes
Try(Row+1)
(Row = 8) ?Not yet Yes
Display the Board
PROCEDURE Try (Row : CARDINAL);
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Eight Queens - Try
PROCEDURE Try(Row : INTEGER); VAR Col : INTEGER; PROCEDURE Safe … … … ;BEGIN FOR Col := 1 TO 8 DO IF Safe(Col,Row) THEN Board[Col,Row] := FALSE; IF Row < 8 THEN Try(Row + 1) ELSE PrintSolution END; Board[Col,Row] := TRUE END (* IF *) END (* FOR *)END Try;
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Eight Queens - Board
Extended board for determination of safe positions
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Eight Queens - Safe
PROCEDURE Safe(Col,Row : INTEGER):BOOLEAN; VAR r : INTEGER; f : BOOLEAN;BEGIN f := TRUE; FOR r := 1 TO Row-1 DO f := f AND Board[Col,r] AND Board[Col+Row-r,r] AND Board[Col-Row+r,r] END; (* FOR *) RETURN fEND Safe;
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Eight Queens - Main program
BEGIN FOR Col := -6 TO 15 DO FOR Row := 1 TO 8 DO Board[Col,Row] := TRUE END; (* FOR Rows*) END; (* FOR Cols*) Try(1)END Queens.
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Eight Queens - Non recursive FOR Col1 := 1 TO 8 DO
IF Safe(Col1,1) THEN Board[Col1,1] := FALSE; FOR Col2 := 1 TO 8 DO IF Safe(Col2,2) THEN Board[Col2,2] := FALSE; ... FOR Col8 := 1 TO 8 DO IF Safe(Col8,8) THEN Board[Col8,8] := FALSE; PrintSolution Board[Col8,8] := TRUE; END; END; (* FOR 8 *) ... Board[Col2,2] := TRUE END; (* IF 2 *) END; (* FOR 2 *) Board[Col1,1] := TRUE
END; (* IF 1 *) END; (* FOR 1 *)
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Frequency Synthetizer
2075 Hz
1925 Hz
1625 Hz
1475 Hz
1025 Hz
875 Hz./. n1
./. n2
./. n3
./. n4
./. n5
./. n6
F = ?
SCM =175 855 554 875
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Frequency synthetizer Procedure
FOR Fact[Fr] := MinFact[Fr] TO MaxFact[Fr] DO
Update frequency range for oscillator
Restore previous frequency range
Acceptable factor ?No Yes
Try(Fr+1)
(Fr = 6) ?Not yet Yes
Print Factors
PROCEDURE Try(Fr : CARDINAL);
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Traveling Salesman with Bactracking
UNTIL all possible next towns have been selected
Select next town
Record next town
Erase previous town
Total distance < Min ?No Yes
Call recursivelythe backtracking
procedure forthe next town
Final town ?Not yet Yes
Min := Total distance
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Recursive Fractals (1)
Basic building block: a rectangle
PROCEDURE Rectangle(xl,yl,xr,yh,color)
Graphical libraries:
FROM Graph IMPORT Init, Plot, Rectangle, _WHITE, _BLUE, _clrLIGHTRED, _clrWHITE;
yl
yh
xrxl
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Recursive Fractals(2)
PROCEDURE Box(x,y,d:CARDINAL,color);BEGIN Rectangle(x-d,y-d,x+d,y+d,color)END Box;
To draw a square of size 2d centered in x,y:
d
d
x,y
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Recursive Fractals(3)
Rectangle(x-d,y-d,x+d,y+d);Box(x-d,y-d,d DIV 2);Box(x-d,y+d,d DIV 2);Box(x+d,y-d,d DIV 2);Box(x+d,y+d,d DIV 2);
To draw an elementary fractal box :
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Recursive Fractals(4)
PROCEDURE FractalBox(x,y,d,n:CARDINAL);BEGIN Rectangle(x-d,y-d,x+d,y+d); n := n-1 IF n > 0 THEN FractalBox(x-d,y-d,d DIV 2,n); FractalBox(x-d,y+d,d DIV 2,n); FractalBox(x+d,y-d,d DIV 2,n); FractalBox(x+d,y+d,d DIV 2,n); ENDEND FractalBox
To draw a series of n fractal boxes :
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