Recursive Algorithms

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Chapter 2.10

• Recursive Algorithms
• and
• Backtracking

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Recursive AlgorithmsFactorial

• Step-up function Fac(n) (n)Fac(n-1)
• Trivial solution Fac(1) 1
• Recursive procedure

PROCEDURE Fac(nCARDINAL) CARDINAL BEGIN IF n gt
1 THEN RETURN nFac(n-1) ELSE RETURN 1 END ( IF
) END Fac
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Recursive AlgorithmsFactorial

• Alternative Iterative Procedure

PROCEDURE Fac(nCARDINAL) CARDINAL VAR f
CARDINAL BEGIN f 1 WHILE n gt 1 DO f f
n n n - 1 END ( WHILE ) RETURN f END Fac
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Recursive AlgorithmsFibonaci

• Step-up function Fib(n) Fib(n-1)Fib(n-2)
• Trivial solution Fib(1) 1 Fib(0) 0
• Recursive procedure

PROCEDURE Fib(nCARDINAL) CARDINAL BEGIN IF n gt
1 THEN RETURN Fib(n-1)Fib(n-2) ELSIF n1 THEN
RETURN 1 ELSE RETURN 0 END ( IF ) END
Fib
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Recursive AlgorithmsFibonaci

• Alternative Iterative Procedure

PROCEDURE Fib(nCARDINAL) 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,UsingCHAR) PROCEDURE
MoveDisk..... BEGIN IF Height gt 0
THEN MoveTower(Height-1,From,Using,Towards) MoveDi
sk(From,Towards) MoveTower(Height-1,Using,Towards
,From) END ( IF ) END MoveTower
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Recursive Algorithms Conclusion
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Backtracking AlgorithmsA treelike maze
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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
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The Eight Queens Problem
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Eight Queens Procedure
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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
• BoardCol,Row FALSE
• IF Row lt 8 THEN Try(Row 1)
• ELSE PrintSolution
• END
• BoardCol,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
BoardCol,r AND BoardColRow-r,r
AND BoardCol-Rowr,r END ( FOR
) RETURN f END Safe
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Eight Queens - Main program
BEGIN FOR Col -6 TO 15 DO FOR Row 1
TO 8 DO BoardCol,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
BoardCol1,1 FALSE FOR Col2 1 TO 8
DO IF Safe(Col2,2) THEN BoardCol2,2
FALSE ... FOR Col8 1 TO
8 DO IF Safe(Col8,8) THEN
BoardCol8,8 FALSE
PrintSolution BoardCol8,8
TRUE END END ( FOR 8
) ... BoardCol2,2 TRUE
END ( IF 2 ) END ( FOR 2 )
BoardCol1,1 TRUE END ( IF 1 ) END (
FOR 1 )
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Frequency Synthetizer
SCM 175 855 554 875
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Frequency synthetizer Procedure
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Traveling Salesman with Bactracking
Select next town
Total distance lt 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
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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)
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Recursive Fractals(2)
To draw a square of size 2d centered in x,y
PROCEDURE Box(x,y,dCARDINAL,color) BEGIN
Rectangle(x-d,y-d,xd,yd,color) END Box
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Recursive Fractals(3)
To draw an elementary fractal box
Rectangle(x-d,y-d,xd,yd) Box(x-d,y-d,d DIV
2) Box(x-d,yd,d DIV 2) Box(xd,y-d,d DIV
2) Box(xd,yd,d DIV 2)
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Recursive Fractals(4)
To draw a series of n fractal boxes
PROCEDURE FractalBox(x,y,d,nCARDINAL) BEGIN
Rectangle(x-d,y-d,xd,yd) n n-1 IF n gt 0
THEN FractalBox(x-d,y-d,d DIV 2,n)
FractalBox(x-d,yd,d DIV 2,n)
FractalBox(xd,y-d,d DIV 2,n)
FractalBox(xd,yd,d DIV 2,n) END END
FractalBox
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