典型递归算法——常见hanoi算法之扩展

学习《算法分析》时的拙作，不要见笑！

递归算法的经典例子，是求解hanoi塔问题（请参照常见的算法课本）。在这里介绍一种更为通用的算法去解决在hanoi塔游戏过程中的自动移动问题。也就是说，常见的hanoi塔算法仅仅是本算法的特殊情况。

假设现在有x，y，z三条柱子，上面共有n个盘子。但是盘子在柱子上是随机分布的，当然，下面的盘子一定比上面的大，只能在柱子上取出最上面的盘子放到x,y,z之一柱子上且比柱子最上面的盘子小，规定盘子的标号分别从1到n，且盘子越大，盘子号越大。现在我们的问题是如何将所有的盘子依次排放到z柱子上去。

首先证明满足上面条件的所有情况均能移动到一个柱子上去。

不妨设z为目标柱子（x,y,z均为并列关系），分类讨论：

（一）   当n=1时，直接将第n个移到z柱子上。

（二）   当n大于1时

（1）  找出n号盘子的位置。

（2）  当n号盘子在z柱子上时，该问题变成总盘子为n-1的情况。

（3）  在x柱子上时，只许将n-1个盘子收集到y柱子上

然后将n号盘子移到z上即可变成n-1问题。

（4）  盘子在y柱子上时，只许将n-1个盘子收集到x柱子上

然后将n号盘子移到z上即可变成n-1问题。

（5）       执行（1）到（4）直到n=1。

     所以，当盘子是任意分布时（每一柱子上盘子均为上小下大），均可以将盘子收集到z柱子上。又因为x,y,z均为并列关系，可将所有盘子移动到任意柱子上。

以上证明过程也是算法描述过程。

（1）       为了方便更多数人理解，特用qbasic语言编写了一款hanoi塔游戏。（在qbasic v1.1上调试通过）。

（2）       为了减少代码长度，不支持鼠标，游戏在文本方式下运行。

DECLARE SUB initiate21 (n!)

DECLARE SUB printmode (n!)

DECLARE FUNCTION directionkey! ()

DECLARE SUB initiate22 (n!)

DECLARE SUB insert (a%(), num!)

DECLARE FUNCTION gamemode! ()

DECLARE SUB finish (auto1!)

DECLARE SUB check (chn!)

DECLARE SUB initiate1 ()

DECLARE SUB zhinput (n!)

DECLARE SUB hanoi (n!, x1%(), x2%(), x3%())

DECLARE FUNCTION search% (n!, x1%(), x2%(), x3%())

DECLARE SUB move (x1%(), n!, x3%(), delay!)

DECLARE SUB main (n!)

DECLARE FUNCTION nextkey$ ()

DECLARE SUB windows ()

DIM SHARED movenum(0)

DIM SHARED x%(9), y%(9), z%(9)

   CLS

   CALL zhinput(n)

   CALL initiate1

   choice = gamemode

   DO

    IF choice = 1 THEN CALL initiate21(n) ELSE initiate22 (n)

    CALL windows

    CALL main(n)

    n = n + 1

  LOOP

END

CONST wherex = 11

CONST wherey = 31

CONST wherez = 51

DATA  " Fature  game "," Normal  game "

 

 

SUB check (chn)

ch = 0

chn = 1

FOR i = 1 TO x%(0) + y%(0) + z%(0)

 IF z%(i) <> i THEN

  ch = 1

  EXIT FOR

 END IF

NEXT i

IF ch = 0 THEN chn = 0

END SUB

 

FUNCTION directionkey

  DIM c AS STRING * 2

  DO

    c$ = INKEY$

     IF ASC(c$) = 13 THEN

      directionkey = 0

      EXIT FUNCTION

     END IF

    IF c$ <> "" AND ASC(MID$(c$, 1, 1)) = 0 THEN

      SELECT CASE ASC(MID$(c$, 2, 1))

      CASE 72

        directionkey = 1

      CASE 75

        directionkey = 2

      CASE 77

        directionkey = 3

      CASE 80

        directionkey = 4

      END SELECT

       EXIT FUNCTION

    END IF

  LOOP

 

 

END FUNCTION

 

SUB finish (auto1)

SELECT CASE z%(0)

  CASE 1 TO 2

   c$ = "oh,finish!"

  CASE 3 TO 4

   c$ = "OK! hurry on!"

  CASE 5 TO 7

   c$ = "pass the level! very good! continue!"

  CASE 8

   c$ = "perfect! only one level left!"

  CASE 9

   c$ = "congratulations! you have finish it!"

END SELECT

COLOR 1, 2

IF auto1 = 0 THEN

  LOCATE 11, 40 - LEN(c$) / 2

  PRINT c$

END IF

c$ = "PRESS ANY KEY TO CONTINUE"

LOCATE 12, 40 - LEN(c$) / 2

PRINT c$

  SLEEP

IF z%(0) = 9 THEN END

END SUB

 

FUNCTION gamemode

 

  gamemodenum = 1

  CALL printmode(1)

  DO

  SELECT CASE directionkey

   CASE 0

     EXIT DO

   CASE 1

    gamemodenum = gamemodenum - 1

    IF gamemodenum < 1 THEN gamemodenum = 2

    CALL printmode(gamemodenum)

   CASE 4

    gamemodenum = gamemodenum + 1

    IF gamemodenum > 2 THEN gamemodenum = 1

    CALL printmode(gamemodenum)

   END SELECT

  LOOP

   gamemode = gamemodenum

 

END FUNCTION

 

SUB hanoi (n, x1%(), x2%(), x3%())

subx = search%(n, x1%(), x2%(), x3%())

IF n = 1 THEN

  SELECT CASE subx

  CASE 1

    movenum(0) = movenum(0) + 1

    CALL move(x1%(), n, x3%(), 1)

  CASE 2

    movenum(0) = movenum(0) + 1

    CALL move(x2%(), n, x3%(), 1)

  END SELECT

ELSE

  SELECT CASE subx

  CASE 1

    CALL hanoi(n - 1, x1%(), x3%(), x2%())

    movenum(0) = movenum(0) + 1

    CALL move(x1%(), n, x3%(), 1)

    CALL hanoi(n - 1, x1%(), x2%(), x3%())

  CASE 2

    CALL hanoi(n - 1, x2%(), x3%(), x1%())

     movenum(0) = movenum(0) + 1

    CALL move(x2%(), n, x3%(), 1)

    CALL hanoi(n - 1, x2%(), x1%(), x3%())

  CASE 3

    CALL hanoi(n - 1, x1%(), x2%(), x3%())

  END SELECT

END IF

END SUB

 

SUB initiate1

COLOR 1, 2 ' the game color

CLS

LOCATE 1, 37: PRINT "Hanoi"

LOCATE 3, 3: PRINT "press x,y,z to move"

LOCATE 4, 3: PRINT "press Esc to exit"

LOCATE 5, 3: PRINT "press a to auto run"

LOCATE 21, wherex: PRINT "X"

LOCATE 21, wherey: PRINT "Y"

LOCATE 21, wherez: PRINT "Z"

END SUB

 

SUB initiate21 (n)

 x%(0) = n: y%(0) = 0: z%(0) = 0

 FOR i = 1 TO 9

  x%(i) = 99: y%(i) = 99: z%(i) = 99

 NEXT i

 FOR i = 1 TO n

  x%(i) = i

 NEXT i

END SUB

 

SUB initiate22 (n)

  c$ = TIME$

  seed = VAL(MID$(c$, 7, 8))

  RANDOMIZE (seed)

  x%(0) = 0

  y%(0) = 0

  z%(0) = 0

  FOR i = 1 TO 9

    x%(i) = 99: y%(i) = 99: z%(i) = 99

  NEXT i

 

  FOR i = n TO 1 STEP -1

   insertnum = INT(RND(1) * 3 + 1)

   SELECT CASE insertnum

    CASE 1

    CALL insert(x%(), i)

    CASE 2

      CALL insert(y%(), i)

    CASE 3

      CALL insert(z%(), i)

    END SELECT

   NEXT i

 

END SUB

 

SUB insert (a%(), num)

 FOR i = 1 TO a%(0)

   a%(a%(0) + 2 - i) = a%(a%(0) - i + 1)

 NEXT i

 a%(1) = num

 a%(0) = a%(0) + 1

 

END SUB

 

SUB main (n)

movenum(0) = 0

DO

 subc$ = nextkey$

 subc1$ = subc$

 LOCATE 12, 62

 PRINT subc$; "--> ";

 auto = 0

 SELECT CASE subc$

 CASE "a"

   CALL hanoi(n, x%(), y%(), z%())

   auto = 1

 CASE "x"

   subc$ = nextkey$

   SELECT CASE subc$

   CASE "y"

    movenum(0) = movenum(0) + 1

    CALL move(x%(), 1, y%(), 0)

   CASE "z"

    movenum(0) = movenum(0) + 1

    CALL move(x%(), 1, z%(), 0)

   END SELECT

 CASE "y"

   subc$ = nextkey$

   SELECT CASE subc$

   CASE "x"

     movenum(0) = movenum(0) + 1

     CALL move(y%(), 1, x%(), 0)

   CASE "z"

     movenum(0) = movenum(0) + 1

     CALL move(y%(), 1, z%(), 0)

   END SELECT

 CASE "z"

   subc$ = nextkey$

   SELECT CASE subc$

   CASE "x"

     movenum(0) = movenum(0) + 1

     CALL move(z%(), 1, x%(), 0)

   CASE "y"

     movenum(0) = movenum(0) + 1

     CALL move(z%(), 1, y%(), 0)

   END SELECT

 CASE CHR$(27)

   END

 END SELECT

   LOCATE 12, 62: PRINT subc1$; "-->"; subc$

 chn = 1

 CALL check(chn)

 IF chn = 0 THEN CALL finish(auto): EXIT DO

 CALL windows

   LOCATE 11, 62

   PRINT "move:"; movenum(0)

LOOP

END SUB

 

SUB move (x1%(), n, x3%(), delay)

IF x1%(1) < x3%(1) THEN

 IF delay <> 0 THEN

  FOR i = 1 TO 5 'ctr the speed

  SOUND 0, 1

  NEXT i

 END IF

  movex = x1%(1)

  FOR i = 2 TO x1%(0)

   x1%(i - 1) = x1%(i)

  NEXT i

  x1%(i - 1) = 99

  x1%(0) = x1%(0) - 1

  FOR i = x3%(0) TO 1 STEP -1

   x3%(i + 1) = x3%(i)

  NEXT i

  x3%(1) = movex

  x3%(0) = x3%(0) + 1

  CALL windows

  LOCATE 11, 62

  PRINT "move:"; movenum(0)

ELSE

LOCATE 11, 1: PRINT "          ";

LOCATE 11, 1: PRINT "wrong";

END IF

END SUB

 

FUNCTION nextkey$

DO

nextc$ = INKEY$

IF nextc$ = "a" OR nextc$ = CHR$(27) OR nextc$ = "x" OR nextc$ = "y" OR nextc$ = "z" THEN

nextkey$ = nextc$

EXIT DO

END IF

LOOP

END FUNCTION

 

SUB printmode (n)

  COLOR 15, 1

  RESTORE

  FOR i = 1 TO 2

    READ c$

    LOCATE 12 + i, 40 - LEN(c$) / 2

    PRINT c$

  NEXT i

  RESTORE

  FOR i = 1 TO n

    READ c$

  NEXT i

  LOCATE 12 + n, 40 - LEN(c$) / 2

  COLOR 15, 2

  PRINT c$

END SUB

 

FUNCTION search% (n, x1%(), x2%(), x3%())

FOR i = 1 TO x1%(0)

 IF n = x1%(i) THEN search = 1

NEXT i

FOR i = 1 TO x2%(0)

IF n = x2%(i) THEN search = 2

NEXT i

FOR i = 1 TO x3%(0)

IF n = x3%(i) THEN search = 3

NEXT i

END FUNCTION

 

SUB windows

 VIEW PRINT 11 TO 20

 COLOR 4, 2

 CLS

 FOR i = 1 TO x%(0)

   FOR j = 0 TO 2 * x%(i)

    LOCATE 20 - x%(0) + i, wherex - x%(i) + j

    PRINT CHR$(219);

   NEXT j

 NEXT i

 FOR i = 1 TO y%(0)

   FOR j = 0 TO 2 * y%(i)

    LOCATE 20 - y%(0) + i, wherey - y%(i) + j

    PRINT CHR$(219);

   NEXT j

 NEXT i

 FOR i = 1 TO z%(0)

   FOR j = 0 TO 2 * z%(i)

    LOCATE 20 - z%(0) + i, wherez - z%(i) + j

    PRINT CHR$(219);

   NEXT j

 NEXT i

END SUB

 

SUB zhinput (n)

COLOR 1, 2

CLS

LOCATE 2, 2

INPUT "input a number (3--9) to start the game:", n

IF n < 3 OR n > 9 THEN CALL zhinput(n)

END SUB

说明：以上是完整的hanoi塔游戏源程序，其中的随机数种子取自系统时间。

最麻烦的是三根柱子，多于三根的就可以转换成三根的问题了不仿设有n根（n>2) 目标是移到第n根上，那么，首先将前三根的盘子移到第三根此时的柱子就变成n-2了重复以上过程即可转化成三根的问题了