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Figure 81 1
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A binary search routine, using recursion
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A recursively drawn ruler
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Figure 812 shows the result of running a Java program that draws ruler markings Here, we consider the problem of marking 1 inch In the middle is the longest mark In Figure 812, to the left of the middle is a miniaturized version of the ruler and to the right of the middle is a second miniaturized version This result suggests a recursive algorithm that first draws the middle line and then draws the left and right halves
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1 / / Java code to draw Figure 812 2 void drawRuler( Graphics g, int left, int right, int level 3 i 4 if( level < 1 ) 5 return;
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int mid
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left + right
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9 10 11 12 13 1 gdrawLine( mid, 80, mid, 80 - level * 5 drawRuler( g, left, mid - 1 , level - 1 ) ; drawRuler ( g, mid + 1, right, level - 1 )
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You do not have to understand the details of drawing lines and shapes in Java to understand this program You simply need to know that a Graphics object is something that gets drawn to The drawRuler method in Figure 813 is our recursive routine It uses the drawLine method, which is part of the Graphics class The method drawLine draws a line from one (x, y) coordinate to another (x, y) coordinate, where coordinates are offset from the topleft corner Our routine draws markings at leve1 different heights; each recursive call is one level deeper (in Figure 812 there are eight levels) It first disposes of the base case at lines 4 and 5 Then the midpoint mark is drawn at line 9 Finally, the two miniatures are drawn recursively at lines 1 1 and 12 In the online code, we include extra code to slow down the drawing In that way, we can see the order in which the lines are drawn by the recursive algorithm
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Shown in Figure 814(a) is a seemingly complex pattern, called a fractal star, which we can easily draw by using recursion The entire canvas is initially gray (not shown), the pattern is formed by drawing white squares onto the gray background The last square drawn is over the center Figure 814(b) shows the drawing immediately before the last square is added Thus prior to the last square being drawn, four miniature versions have been drawn, one in each of the four quadrants This pattern provides the information needed to derive the recursive algorithm As with the previous example, the method drawFractal uses a Java library routine In this case, fil lRect draws a rectangle; its upper left-hand comer and dimensions must be specified The code is shown in Figure 815
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(a) A fractal star outline drawn by the code shown in Figure 815 (b) The same star immediately before the last square is added
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1 / / Draw picture in Figure 814 2 void drawFractal( Graphics g , int xCenter, 3 int yCenter, int boundingDim ) 4 I 5 int side = boundingDim / 2; 6 7 if( side < 1 ) 8 return; 9 10 / / Compute corners 11 int left = xcenter - side / 2; 12 int top = yCenter - side / 2; 13 int right = xCenter + side / 2; 14 int bottom = yCenter + side / 2; 15 / / Recursively draw four quadrants 16 17 drawFractal( g, left, top, boundingDim / 2 ) ; 18 drawFractal( g, left, bottom, boundingDim / 2 1 ; 19 drawFractal( g, right, top, boundingDim / 2 ) ; 20 drawFractal( g, right, bottom, boundingDim / 2 ) ; 21 / / Draw central square, overlapping quadrants 22 23 gfillRect( left, top, right - left, bottom - top 24 } Figure 815
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