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</tr><tr><td></td>  <td align="center"><font face="Times New Roman, Times, Serif" size="3">1</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">10</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">100</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr><tr><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">100</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="center"><font face="Times New Roman, Times, Serif" size="3">1</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">100</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">10,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr><tr><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="center"><font face="Times New Roman, Times, Serif" size="3">1</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000,000,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr><tr><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">10,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="center"><font face="Times New Roman, Times, Serif" size="3">1</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">10,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">100,000,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td><td valign="top"><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td align="right"><font face="Times New Roman, Times, Serif" size="3">1,000,000,000,000</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr><tr><td colspan="5" valign="top"><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">As you can see, each time the exponent increases by 1, the number of steps is multiplied by an additional order of magnitude (factor of 10). That is, if <i>N</i> is made 10 times greater, the work involved in an <i>N</i></font><font face="Times New Roman, Times, Serif" size="2"><sup>2</sup></font><font face="Times New Roman, Times, Serif" size="3"> algorithm increases by a factor of 100, and the work involved in an <i>N</i></font><font face="Times New Roman, Times, Serif" size="2"><sup>3</sup></font><font face="Times New Roman, Times, Serif" size="3"> algorithm increases by a factor of 1000. To put this in more concrete terms, an algorithm with a doubly nested loop, in which each loop depends on the number of data values, takes 1000 steps for 10 input values and 1 trillion steps for 10,000 values. On a computer that executes 1 million instructions per second, the latter case would take more than 11 days to run.</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr><tr><td colspan="5" valign="top"><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3">The table also shows that the steps outside of the innermost loop account for an insignificant portion of the total number of steps as <i>N</i> gets bigger. Because the innermost</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table></td></tr></table><br /><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td>  <td colspan="3" height="12"></td>  <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td>  <td><font face="Times New Roman, Times, Serif" size="3"><i>(text box continues on next page)</i></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>  </td>			</tr>				<tr>				<td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_284.html">&lt;&nbsp;previous page</a></td>				<td align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_285</strong></td>				<td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_286.html">next page&nbsp;&gt;</a></td>			</tr>		</table>		</body>	</html>

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