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Chapter 2.10

Chapter 2.10. Recursive Algorithms and Backtracking. Step-up function : Fac(n) = (n)*Fac(n-1) Trivial solution : Fac(1) = 1 Recursive procedure :. Recursive Algorithms Factorial. PROCEDURE Fac(n:CARDINAL): CARDINAL; BEGIN IF n > 1 THEN RETURN n*Fac(n-1) ELSE RETURN 1

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Chapter 2.10

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  1. Chapter 2.10 Recursive Algorithms and Backtracking

  2. Step-up function :Fac(n) = (n)*Fac(n-1) Trivial solution :Fac(1) = 1 Recursive procedure : Recursive AlgorithmsFactorial • PROCEDURE Fac(n:CARDINAL): CARDINAL; • BEGIN • IF n > 1 • THENRETURN n*Fac(n-1) • ELSERETURN 1 • END (* IF *) • END Fac;

  3. Alternative Iterative Procedure : Recursive AlgorithmsFactorial • PROCEDURE Fac(n:CARDINAL): CARDINAL; • VAR f : CARDINAL; • BEGIN • f := 1; • WHILE n > 1DO • f := f * n ; n := n - 1 • END; (* WHILE *) • RETURN f • END Fac;

  4. Step-up function : Fib(n) = Fib(n-1)+Fib(n-2) Trivial solution : Fib(1) = 1; Fib(0) = 0 Recursive procedure : Recursive AlgorithmsFibonaci • PROCEDURE Fib(n:CARDINAL): CARDINAL; • BEGIN • IF n > 1 • THEN RETURN Fib(n-1)+Fib(n-2) • ELSIF n=1THEN RETURN1 • ELSERETURN0 • END (* IF *) • END Fib;

  5. Alternative Iterative Procedure : Recursive AlgorithmsFibonaci • PROCEDURE Fib(n:CARDINAL): CARDINAL; • VAR i,fn,fnm1,fnm2 : CARDINAL; • BEGINIF n = 0THEN RETURN0 • ELSIF n = 1THEN RETURN1 • ELSE fnm2 := 0; fnm1 := 1; • FOR i := 2TO n DO • fn := fnm1 + fnm2; • fnm2 := fnm1; fnm1 := fn • END; (* FOR *) • RETURN fn • END (* IF *) • END Fib;

  6. 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 Recursive AlgorithmsTowers of Hanoi

  7. Recursive procedure : Recursive AlgorithmsTowers of Hanoi • PROCEDURE MoveTower (Height: CARDINAL • From,Towards,Using:CHAR); • PROCEDURE MoveDisk..... • BEGIN • IF Height > 0 THEN • MoveTower(Height-1,From,Using,Towards) • MoveDisk(From,Towards); • MoveTower(Height-1,Using,Towards,From) • END (* IF *) • END MoveTower;

  8. Simultaneous activations Simple Algorithm Complex Algorithm Few ... YES Many Never ??? Recursive AlgorithmsConclusion

  9. Backtracking AlgorithmsA treelike maze

  10. Generic Search with Backtracking Select a successor node Allowed node ? No Yes Record selected node Final node ? Not yet Yes Call recursively the backtracking procedure for the next node Display the solution Erase previous node selection UNTIL all successor nodes have been explored

  11. The Eight Queens Problem

  12. PROCEDURE Try (Row : CARDINAL); FOR Col := 1 TO 8 DO Board[Row,Col] safe ? No Yes Put queen on Board[Row,Col] (Row = 8) ? Not yet Yes Try(Row+1) Display the Board Remove queen from Board[Row,Col] Eight Queens Procedure

  13. 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; Eight Queens - Try

  14. Eight Queens - Board Extended board for determination of safe positions

  15. 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 f END Safe;

  16. 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.

  17. 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 *)

  18. ./. n6 2075 Hz ./. n5 1925 Hz F = ? ./. n4 1625 Hz ./. n3 1475 Hz ./. n2 1025 Hz ./. n1 875 Hz Frequency Synthetizer SCM = 175 855 554 875

  19. PROCEDURE Try(Fr : CARDINAL); FOR Fact[Fr] := MinFact[Fr] TO MaxFact[Fr] DO Acceptable factor ? No Yes Update frequency range for oscillator (Fr = 6) ? Not yet Yes Try(Fr+1) Print Factors Restore previous frequency range Frequency synthetizer Procedure

  20. Traveling Salesman with Bactracking Select next town Total distance < Min ? No Yes Record next town Final town ? Not yet Yes Call recursively the backtracking procedure for the next town Min := Total distance Erase previous town UNTIL all possible next towns have been selected

  21. yh xl xr yl Recursive Fractals (1) Graphical libraries: FROM Graph IMPORT Init, Plot, Rectangle, _WHITE, _BLUE, _clrLIGHTRED, _clrWHITE; Basic building block: a rectangle PROCEDURE Rectangle(xl,yl,xr,yh,color)

  22. d x,y d Recursive Fractals(2) To draw a square of size 2d centered in x,y: PROCEDURE Box(x,y,d:CARDINAL,color); BEGIN Rectangle(x-d,y-d,x+d,y+d,color) END Box;

  23. Recursive Fractals(3) To draw an elementary fractal box : 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);

  24. Recursive Fractals(4) To draw a series of n fractal boxes : 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); END END FractalBox

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