Stanford編程方法學公開課作業 1 ----關於Karel 中點尋找問題的討論 （Midpoint finding）

原創 young2203 2020-06-02 04:32
問題描述：Karel需要尋找到第一行的中點位置，並在那裏放下一個beeper，具體描述請參考英文原文

Problem 3

As an exercise in solving algorithmic problems, program Karel to place a single beeper at the center of 1st Street. For example, if Karel starts in the world 

it should end with Karel standing on a beeper in the following position: 

Note that the final configuration  of the world should have only a single beeper at the midpoint of 1st Street.  Along the way, Karel is allowed to place additional beepers wherever it wants to, but must pick them all up again before it finishes. 

In solving this problem, you may count on the following facts about the world: 

  • Karel starts at 1st Avenue and 1st Street, facing east, with  an infinite number of beepers in its bag. 
  • The initial state of the world includes no interior walls or beepers. 
  • The world need not be square, but you may assume that it is at least as tall as it is wide. 

Your program, moreover, can assume the following simplifications: 

  • If the width of the world is odd, Karel must put the beeper in the center square. If the width is even, Karel may drop the beeper on either of the two center squares. 
  • It does not matter which direction Karel is facing at the end of the run. 

There are many different algorithms you can use to solve this problem. The interesting part of this assignment is to come up with a strategy that works.



解決思路：（可以提出以下四種解題思路）
1. 先將第一行填滿，然後左右各取出一個，並在第二行放下一個，當第一行beeper取完的時候，第二行的beeper數目即爲第一行長度的一半
代碼如下：
  1. /*
  2.  * File: MidpointFindingKarel.java
  3.  * -------------------------------
  4.  * THe MidpointFindingKarel class will leave a beeper on the corner 
  5.  * closest to the center of 1st Street
  6.  * (or either of the two central corners if 1st Street has an even
  7.  * number of corners).  The world may be of any size, but you are allowed to
  8.  * assume that it is at least as tall as it is wide.
  9.  */
  10. 

  11. import stanford.karel.*;
  12. 

  13. public class MidpointFindingKarel extends SuperKarel {
  14. 

  15. 	public void run() {
  16. 		fillFirstRow();		
  17. 		//midpoint finding part
  18. 		while (beepersPresent()) {
  19. 			collectLeftEnd();
  20. 			secondRowAddOne();
  21. 			collectRightEnd();	
  22. 			
  23. 			//step forward one more to make sure there is no more beeper
  24. 			if (frontIsClear()) { //check whether all beepers have been collected
  25. 				move(); //if width is odd number Karel will finally arrive (1,1)
  26. 			} //if width is even number, Karel will finally arrive mid-point
  27. 		}
  28. 		
  29. 		//go to the midpoint of the first row, put beeper then collect beepers
  30. 		//on the 2nd row
  31. 		goToMidpoint();
  32. 		putBeeper();
  33. 		collectSecondRow();	
  34. 		goToOrigin();
  35. 		moveToBeeper();
  36. 	}
  37. 	
  38. 	/*
  39. 	 * fillFirstRow method is defined to fill the 1st row with beepers
  40. 	 * Karel will put only one beeper at each corner
  41. 	 * precondition: facing east, at (1,1) point
  42. 	 * postcondition: facing east, (1,1) point
  43. 	 */
  44. 	private void fillFirstRow() {
  45. 		goToOrigin(); //go to (1,1) facing east
  46. 		while (frontIsClear()) {
  47. 			putBeeper();
  48. 			move();
  49. 		}
  50. 		putBeeper(); //Off-By-One-Bug
  51. 		//make sure go back to (1,1) point facing east
  52. 		turnAround();
  53. 		moveToWall();
  54. 		turnAround();
  55. 	}
  56. 

  57. 	
  58. 	/*
  59. 	 * Collect the most left beeper on the 1st row, if there is any
  60. 	 * Precondition: any case
  61. 	 * Postcondition: facing east, stop at the position where Karel 
  62. 	 * just collected one beeper, 1st row
  63. 	 */
  64. 	private void collectLeftEnd() {
  65. 		goToOrigin();
  66. 		moveToBeeper();
  67. 		if (beepersPresent()) {
  68. 			pickBeeper();
  69. 		}		
  70. 	}
  71. 	
  72. 	/*Similar as collectLeftEnd, while in this method, Karel will start
  73. 	  from the most right end, and try to collect the beeper at the 
  74. 	  most right end
  75. 	*/
  76. 	private void collectRightEnd() {
  77. 		goToOrigin();
  78. 		moveToWall(); //Reach the right end of the 1st row
  79. 		turnAround(); //To face west
  80. 		moveToBeeper();
  81. 		if (beepersPresent()) {
  82. 			pickBeeper();
  83. 		}
  84. 	}
  85. 	
  86. 	/*
  87. 	 * go to 2nd row, add one more beeper
  88. 	 * precondition: anywhere at 1st row facing east
  89. 	 * postcondition: on 2nd row, at the most end where no more beeper exist
  90. 	 */
  91. 	private void secondRowAddOne() {
  92. 		goToOrigin();
  93. 		bigTurnLeft();
  94. 		turnRight();
  95. 		while( beepersPresent()) { //arrive at the end of existing beepers
  96. 			move();
  97. 		}
  98. 		putBeeper();
  99. 	}
  100. 	
  101. 	
  102. 	/*
  103. 	 * Make sure Karel will go to the midpoint of the first row
  104. 	 * Precondition: any case
  105. 	 * Postcondition: arrive at the mid point of the 1st row, facing south
  106. 	 */
  107. 	private void goToMidpoint() {
  108. 		goToOrigin();
  109. 		moveToWall(); //reach right end 
  110. 		bigTurnLeft();
  111. 		turnLeft();
  112. 		moveToBeeper();
  113. 		bigTurnLeft();		
  114. 	}
  115. 	
  116. 	private void collectSecondRow() {
  117. 		goToOrigin();
  118. 		bigTurnLeft(); //facing north
  119. 		turnRight(); //facing east
  120. 		while (beepersPresent() || frontIsBlocked()) {
  121. 			pickBeeper();
  122. 			move();
  123. 		}
  124. 	}
  125. 	
  126. 	/*Move Karel back to origin of map (1,1), make it facing east
  127. 	 * Precondition: can be any case
  128. 	 * Postcondition: standing on (1,1) facing east
  129. 	 */
  130. 	private void goToOrigin() { 
  131. 		while (notFacingSouth()) {
  132. 			turnLeft();
  133. 		}
  134. 		moveToWall();
  135. 		while (notFacingWest()) {
  136. 			turnLeft();
  137. 		}
  138. 		moveToWall();
  139. 		while (notFacingEast()) {
  140. 			turnLeft(); //Karel will face east
  141. 		}
  142. 	}
  143. 	
  144. 	
  145. 	/*
  146. 	 * move Karel to first beeper on its way
  147. 	 * precondition: facing east/west
  148. 	 * postcondition: share same orientation as in precondition, standing on 
  149. 	 * the corner where first beeper appears on its way
  150. 	 */
  151. 	private void moveToBeeper() { 
  152. 		while ( noBeepersPresent() && frontIsClear() ) {
  153. 			move();
  154. 		}
  155. 	}
  156. 	
  157. 	/*
  158. 	 * go to upper/lower row above without changing column number, 
  159. 	 * make a big turn left
  160. 	 * precondition: any case
  161. 	 * postcondition: same column, arrive at upper/lower row 
  162. 	 * which is one above/under the row in precondition, 
  163. 	 * direction will turn left
  164. 	 */
  165. 	private void bigTurnLeft() {
  166. 		turnLeft();
  167. 		move(); 
  168. 	}
  169. 	
  170. 	// Similar as in bigTurnLeft, but in this case make a big
  171. 	//turn right
  172. 	private void bigTurnRight() {
  173. 		turnRight();
  174. 		move(); 
  175. 	}
  176. 	
  177. 	private void moveToWall() { //move Karel to wall
  178. 		while (frontIsClear()) {
  179. 			move();
  180. 		}
  181. 	}
  182. 	
  183. }// end of public class
2. 對於方法1中的思路進行進一步優化，其實左側取一個右側再取一個完全可以兼併成爲從左側出發一次取兩個，同時放一個到第二行，這將大量節省Karel的運動過程，優化後代碼如下：
  1. /*
  2.  * File: MidpointFindingKarel.java
  3.  * -------------------------------
  4.  * THe MidpointFindingKarel class will leave a beeper on the corner 
  5.  * closest to the center of 1st Street
  6.  * (or either of the two central corners if 1st Street has an even
  7.  * number of corners).  The world may be of any size, but you are allowed to
  8.  * assume that it is at least as tall as it is wide.
  9.  */
  10. 

  11. import stanford.karel.*;
  12. 

  13. public class MidpointFindingKarel extends SuperKarel {
  14. 

  15. 	public void run() {
  16. 		fillFirstRow();		
  17. 		//midpoint finding part
  18. 		while (beepersPresent()) {
  19. 			secondRowAddOne();
  20. 			collectLeftEnd();
  21. 			//step forward one more to make sure there is no more beeper
  22. 			if (frontIsClear()) { //check whether all beepers have been collected
  23. 				move(); //if width is odd number Karel will finally arrive (1,1)
  24. 			} //if width is even number, Karel will finally arrive mid-point
  25. 			
  26. 		}
  27. 		
  28. 		//go to the midpoint of the first row, put beeper then collect beepers
  29. 		//on the 2nd row
  30. 		goToMidpoint();
  31. 		putBeeper();
  32. 		collectSecondRow();	
  33. 		goToOrigin();
  34. 		moveToBeeper();
  35. 	}
  36. 	
  37. 	/*
  38. 	 * fillFirstRow method is defined to fill the 1st row with beepers
  39. 	 * Karel will put only one beeper at each corner
  40. 	 * precondition: facing east, at (1,1) point
  41. 	 * postcondition: facing east, at the most right end of 1st row
  42. 	 */
  43. 	private void fillFirstRow() {
  44. 		goToOrigin(); //go to (1,1) facing east
  45. 		while (frontIsClear()) {
  46. 			putBeeper();
  47. 			move();
  48. 		}
  49. 		putBeeper(); //Off-By-One-Bug
  50. 	}
  51. 

  52. 	
  53. 	/*
  54. 	 * Collect the most left beeper on the 1st row, if there is any
  55. 	 * Precondition: any case
  56. 	 * Postcondition: facing east, stop at the position where Karel 
  57. 	 * just collected one beeper, 1st row
  58. 	 */
  59. 	private void collectLeftEnd() {
  60. 		goToOrigin();
  61. 		moveToBeeper();
  62. 		if ( beepersPresent() ) {
  63. 			pickBeeper();
  64. 		}
  65. 		if ( frontIsClear() ) {
  66. 			move();
  67. 		}
  68. 		if ( beepersPresent() ) {
  69. 			pickBeeper();
  70. 		}
  71. 	}
  72. 	
  73. 

  74. 	/*
  75. 	 * go to 2nd row, add one more beeper
  76. 	 * precondition: anywhere at 1st row facing east
  77. 	 * postcondition: on 2nd row, at the most end where no more beeper exist
  78. 	 */
  79. 	private void secondRowAddOne() {
  80. 		goToOrigin();
  81. 		bigTurnLeft();
  82. 		turnRight();
  83. 		while( beepersPresent()) { //arrive at the end of existing beepers
  84. 			move();
  85. 		}
  86. 		putBeeper();
  87. 	}
  88. 		
  89. 	/*
  90. 	 * Make sure Karel will go to the midpoint of the first row
  91. 	 * Precondition: any case
  92. 	 * Postcondition: arrive at the mid point of the 1st row, facing south
  93. 	 */
  94. 	private void goToMidpoint() {
  95. 		goToOrigin();
  96. 		moveToWall(); //reach right end 
  97. 		bigTurnLeft();
  98. 		turnLeft();
  99. 		moveToBeeper();
  100. 		bigTurnLeft();		
  101. 	}
  102. 	
  103. 	private void collectSecondRow() {
  104. 		goToOrigin();
  105. 		bigTurnLeft(); //facing north
  106. 		turnRight(); //facing east
  107. 		while (beepersPresent() || frontIsBlocked()) {
  108. 			pickBeeper();
  109. 			move();
  110. 		}
  111. 	}
  112. 	
  113. 	/*Move Karel back to origin of map (1,1), make it facing east
  114. 	 * Precondition: can be any case
  115. 	 * Postcondition: standing on (1,1) facing east
  116. 	 */
  117. 	private void goToOrigin() { 
  118. 		while (notFacingSouth()) {
  119. 			turnLeft();
  120. 		}
  121. 		moveToWall();
  122. 		while (notFacingWest()) {
  123. 			turnLeft();
  124. 		}
  125. 		moveToWall();
  126. 		while (notFacingEast()) {
  127. 			turnLeft(); //Karel will face east
  128. 		}
  129. 	}
  130. 	
  131. 	
  132. 	/*
  133. 	 * move Karel to first beeper on its way
  134. 	 * precondition: facing east/west
  135. 	 * postcondition: share same orientation as in precondition, standing on 
  136. 	 * the corner where first beeper appears on its way
  137. 	 */
  138. 	private void moveToBeeper() { 
  139. 		while ( noBeepersPresent() && frontIsClear() ) {
  140. 			move();
  141. 		}
  142. 	}
  143. 	
  144. 	/*
  145. 	 * go to upper/lower row above without changing column number, 
  146. 	 * make a big turn left
  147. 	 * precondition: any case
  148. 	 * postcondition: same column, arrive at upper/lower row 
  149. 	 * which is one above/under the row in precondition, 
  150. 	 * direction will turn left
  151. 	 */
  152. 	private void bigTurnLeft() {
  153. 		turnLeft();
  154. 		move(); 
  155. 	}
  156. 		
  157. 	private void moveToWall() { //move Karel to wall
  158. 		while (frontIsClear()) {
  159. 			move();
  160. 		}
  161. 	}
  162. 	
  163. }// end of public class
3. 對於方法2中的思路進行進一步優化，其實karel完全可以將用於中點位置計數的beeper直接放在第一行，這樣Karel的運動過程將會得到進一步的簡化，換言之，Karel將從（1,1）出發，一次取兩個beeper，然後折回（1,1），放下一個beeper，之後再去取兩個beeper，折回在（1,2）放下一個beeper，以此類推，直至開始時放置的beeper全部取完。具體代碼如下：
  1. /*
  2.  * File: MidpointFindingKarel.java
  3.  * -------------------------------
  4.  * THe MidpointFindingKarel class will leave a beeper on the corner 
  5.  * closest to the center of 1st Street
  6.  * (or either of the two central corners if 1st Street has an even
  7.  * number of corners).  The world may be of any size, but you are allowed to
  8.  * assume that it is at least as tall as it is wide.
  9.  */
  10. 

  11. import stanford.karel.*;
  12. 

  13. public class MidpointFindingKarel extends SuperKarel {
  14. 

  15. 	public void run() {
  16. 		fillFirstRow();	
  17. 		goToOrigin();
  18. 		//midpoint finding part
  19. 		while (beepersPresent()) {
  20. 			collectLeftEnd();
  21. 			firstRowAddOne();
  22. 			move(); //go to blank place without any beeper
  23. 			moveToBeeper();
  24. 		}
  25. 		
  26. 		//go to the midpoint of the first row, put beeper then collect beepers
  27. 		//on the 2nd row
  28. 		goToMidpoint();
  29. 	}
  30. 	
  31. 	/*
  32. 	 * fillFirstRow method is defined to fill the 1st row with beepers
  33. 	 * Karel will put only one beeper at each corner
  34. 	 * precondition: facing east, at (1,1) point
  35. 	 * postcondition: facing east, at the most right end of 1st row
  36. 	 */
  37. 	private void fillFirstRow() {
  38. 		goToOrigin(); //go to (1,1) facing east
  39. 		while (frontIsClear()) {
  40. 			putBeeper();
  41. 			move();
  42. 		}
  43. 		putBeeper(); //Off-By-One-Bug
  44. 	}
  45. 

  46. 	
  47. 	/*
  48. 	 * Collect the most left beeper on the 1st row, if there is any
  49. 	 * Precondition: any case
  50. 	 * Postcondition: facing east, stop at the position where Karel 
  51. 	 * just collected one beeper, 1st row
  52. 	 */
  53. 	private void collectLeftEnd() {
  54. 		if ( beepersPresent() ) {
  55. 			pickBeeper();
  56. 		}
  57. 		if ( frontIsClear() ) {
  58. 			move();
  59. 		}
  60. 		if ( beepersPresent() ) {
  61. 			pickBeeper();
  62. 		}
  63. 	}
  64. 	
  65. 

  66. 	/*
  67. 	 * go to 2nd row, add one more beeper
  68. 	 * precondition: anywhere at 1st row facing east
  69. 	 * postcondition: on 2nd row, at the most end where no more beeper exist
  70. 	 */
  71. 	private void firstRowAddOne() {
  72. 		goToOrigin();
  73. 		while( beepersPresent()) { //arrive at the end of existing beepers
  74. 			move();
  75. 		}
  76. 		putBeeper();
  77. 	}
  78. 		
  79. 	/*
  80. 	 * Make sure Karel will go to the midpoint of the first row
  81. 	 * Precondition: any case
  82. 	 * Postcondition: arrive at the mid point of the 1st row, facing west
  83. 	 */
  84. 	private void goToMidpoint() {
  85. 		goToOrigin();
  86. 		while ( beepersPresent() ) {			
  87. 			pickBeeper();
  88. 			move();
  89. 		} 
  90. 		turnAround(); //OBOB, Karel will move one more step 
  91. 		move(); //at the end of this part
  92. 		putBeeper();
  93. 	}
  94. 	
  95. 	/*Move Karel back to origin of map (1,1), make it facing east
  96. 	 * Precondition: can be any case
  97. 	 * Postcondition: standing on (1,1) facing east
  98. 	 */
  99. 	private void goToOrigin() { 
  100. 		while (notFacingSouth()) {
  101. 			turnLeft();
  102. 		}
  103. 		moveToWall();
  104. 		while (notFacingWest()) {
  105. 			turnLeft();
  106. 		}
  107. 		moveToWall();
  108. 		while (notFacingEast()) {
  109. 			turnLeft(); //Karel will face east
  110. 		}
  111. 	}
  112. 	
  113. 	
  114. 	/*
  115. 	 * move Karel to first beeper on its way
  116. 	 * precondition: facing east/west
  117. 	 * postcondition: share same orientation as in precondition, standing on 
  118. 	 * the corner where first beeper appears on its way
  119. 	 */
  120. 	private void moveToBeeper() { 
  121. 		while ( noBeepersPresent() && frontIsClear() ) {
  122. 			move();
  123. 		}
  124. 	}
  125. 	
  126. 	/*
  127. 	 * go to upper/lower row above without changing column number, 
  128. 	 * make a big turn left
  129. 	 * precondition: any case
  130. 	 * postcondition: same column, arrive at upper/lower row 
  131. 	 * which is one above/under the row in precondition, 
  132. 	 * direction will turn left
  133. 	 */
  134. 	private void bigTurnLeft() {
  135. 		turnLeft();
  136. 		move(); 
  137. 	}
  138. 		
  139. 	private void moveToWall() { //move Karel to wall
  140. 		while (frontIsClear()) {
  141. 			move();
  142. 		}
  143. 	}
  144. 	
  145. }// end of public class

4. 對於方案1還有另一種優化的思路，那就是擺脫計數器的限制，左右各取出一個，一直這樣取下去，直至取完，那麼Karel此時必然自然位於中點位置從而無需計數，其運動過程將更爲簡單快捷。編程的要點在於要保證Karel始終執行往復運動直至beeper被全部取盡，代碼如下：
  1. /*
  2.  * File: MidpointFindingKarel.java
  3.  * -------------------------------
  4.  * THe MidpointFindingKarel class will leave a beeper on the corner 
  5.  * closest to the center of 1st Street
  6.  * (or either of the two central corners if 1st Street has an even
  7.  * number of corners).  The world may be of any size, but you are allowed to
  8.  * assume that it is at least as tall as it is wide.
  9.  */
  10. import stanford.karel.*;
  11. 

  12. public class MidpointFindingKarel extends SuperKarel {
  13. 	public void run() {
  14. 		fillFirstRow();	
  15. 		goToOrigin();
  16. 		//midpoint finding part
  17. 		while (beepersPresent()) {		
  18. 			pickBeeper();						
  19. 			move();
  20. 			moveToEnd();					
  21. 		}		
  22. 		putBeeper();
  23. 	}
  24. 	
  25. 	/*
  26. 	 * fillFirstRow method is defined to fill the 1st row with beepers
  27. 	 * Karel will put only one beeper at each corner
  28. 	 * precondition: facing east, at (1,1) point
  29. 	 * postcondition: facing east, at the end of 1st row
  30. 	 */
  31. 	private void fillFirstRow() {
  32. 		goToOrigin(); //go to (1,1) facing east
  33. 		while (frontIsClear()) {
  34. 			putBeeper();
  35. 			move();
  36. 		}
  37. 		putBeeper(); //Off-By-One-Bug
  38. 	}
  39. 	
  40. 	/*
  41. 	 * Karel will move to the last beeper of the beeper chain
  42. 	 * Precondition: standing at one end facing that chain
  43. 	 * Postcondition: standing at another end facing that chain 
  44. 	 * which means the orientation is changed oppositely 
  45. 	 */
  46. 	private void moveToEnd() {		
  47. 		while ( beepersPresent() && frontIsClear() ) {
  48. 			move();
  49. 		}
  50. 		if ( frontIsBlocked() && beepersPresent() ) { //for the case arrive at the wall
  51. 			turnAround();
  52. 		} else {
  53. 			turnAround();
  54. 			move();
  55. 		}
  56. 		
  57. 	}
  58. 	
  59. 	/*Move Karel back to origin of map (1,1), make it facing east
  60. 	 * Precondition: can be any case
  61. 	 * Postcondition: standing on (1,1) facing east
  62. 	 */
  63. 	private void goToOrigin() { 
  64. 		while (notFacingSouth()) {
  65. 			turnLeft();
  66. 		}
  67. 		moveToWall();
  68. 		while (notFacingWest()) {
  69. 			turnLeft();
  70. 		}
  71. 		moveToWall();
  72. 		while (notFacingEast()) {
  73. 			turnLeft(); //Karel will face east
  74. 		}
  75. 	}
  76. 	
  77. 	private void moveToWall() { //move Karel to wall
  78. 		while (frontIsClear()) {
  79. 			move();
  80. 		}
  81. 	}
  82. 	
  83. }// end of public class

討論：
從思想性上，方案4遠優於其他方案，而且實現過程簡潔漂亮；但就擴展簡易性而言不如方案2、3，方案2，3稍作修改就可以實現在1/3，1/4... 處放置一個beeper，而方案4的擴展性相對而言不是特別明顯（以1/3爲例，需要左側取一個，右側取兩個）



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