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# divide - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Part 1 Divide and conquer 2. Assorted subproblems 7 3. Alike subproblems 19 Divide-and-conquer reasoning – breaking large problems into small ones – is useful in many contexts. Each example of it has unique features, but two broad reasoning cate- gories stand out. In the first category, you break the large problem into unlike, or assorted subproblems. An example is estimating the number of piano tuners in New York or, since this problem was made famous by Fermi, in Chicago, where Fermi spent much of his ca- reer. You might break it into fragments such as the number of pianos, how often each one is tuned, and how long it takes to tune a piano. In the second category, you break the large problem into similar or identical subproblems. An example is merge sort, which breaks a list to be sorted into two halves, each sorted using merge sort – an example of recursion. The next two chapters contain extended examples in each category.
Chapter 2 Assorted subproblems For the first example of dividing into unlike subproblems, we estimate the spacing between pits on a CD ROM. Then we estimate the amount of oil that the United States imports annually. 2.1 Pits on a CDROM Q: What is the spacing between the pits on a CDROM? The pits (indentations) are the memory elements, each pit storing one bit of information. A quick estimate comes from turning over a CDROM and enjoying the brilliant colors. The colors arise because the arrangement of pits diffracts visible light by a significant angle, and the angle depends strongly on the wavelength (or color). So the pits are spaced comparably to the wavelength of light, say about 1 µ m. A second estimate might come from knowing a bit about the laser in a CD player or in a CDROM drive. It is a near-infrared laser, so its wavelength – which will be comparable to the pit size and spacing – is slightly longer than visible-light wavelengths. Since visible- light wavelengths range from 350 to 700 nm or from 0 . 4 to 0 . 8 µ m, a reasonable estimate for the pit spacing is again 1 µ m. These two estimates agree, which increases our confidence in each estimate. Here is why. Because the methods are so different, an error in one method is likely to be significantly different from an error in the second method. Therefore, if the estimates agree, they are probably both reasonable. The lesson is to use as many diverse methods as you can. The third method uses divide-and-conquer reasoning. The capacity and area together de- termine the pit spacing, if we make the useful approximation that the pits are regularly spaced. [This approximation is an example of discarding information, which is the ex- tended topic of Part 3 .] The area is A (10 cm ) 2 .

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divide - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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