document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209467>Question 1209467</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given an array 𝑎[1..𝑛] containing half zeros and half ones, we need to find a position in 𝑎 that contains the value 1. Consider the following two algorithms:<BR>\n" );
document.write( "Algorithm 1:<BR>\n" );
document.write( "i = 1<BR>\n" );
document.write( "while a[i] != 1:<BR>\n" );
document.write( "    i += 1<BR>\n" );
document.write( "return i\r<BR>\n" );
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document.write( "Algorithm 2:<BR>\n" );
document.write( "while True:<BR>\n" );
document.write( "    i = random(1, n)<BR>\n" );
document.write( "    if a[i] == 1:<BR>\n" );
document.write( "        return i<BR>\n" );
document.write( "The function random(1, n) returns a random integer between 1 and 𝑛.<BR>\n" );
document.write( "a) Compare the two algorithms.<BR>\n" );
document.write( "b) Which algorithm is better? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848894 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848894\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's analyze the two algorithms and compare them.\r<BR>\n" );
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document.write( "**a) Comparison of Algorithms**\r<BR>\n" );
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document.write( "* **Algorithm 1 (Linear Search):**\r<BR>\n" );
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document.write( "    * **Pros:** <BR>\n" );
document.write( "        * Guaranteed to find a 1 if it exists.<BR>\n" );
document.write( "        * Simple to implement.\r<BR>\n" );
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document.write( "    * **Cons:**<BR>\n" );
document.write( "        * Worst-case time complexity is O(n), where n is the size of the array. <BR>\n" );
document.write( "        * Can be slow if the first half of the array contains only zeros.\r<BR>\n" );
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document.write( "* **Algorithm 2 (Random Search):**\r<BR>\n" );
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document.write( "    * **Pros:** <BR>\n" );
document.write( "        * Can be very fast if a 1 is found early on.\r<BR>\n" );
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document.write( "    * **Cons:** <BR>\n" );
document.write( "        * There's no guarantee of finding a 1 in a finite number of steps. <BR>\n" );
document.write( "        * The worst-case scenario is theoretically infinite.<BR>\n" );
document.write( "        * The average-case time complexity is harder to analyze but generally less efficient than Algorithm 1.\r<BR>\n" );
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document.write( "**b) Which Algorithm is Better?**\r<BR>\n" );
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document.write( "* **In general, Algorithm 1 (Linear Search) is generally considered better.** \r<BR>\n" );
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document.write( "    * **Guaranteed Success:** It provides a guaranteed solution within a finite number of steps.<BR>\n" );
document.write( "    * **Predictable Performance:** Its worst-case time complexity is known and bounded.\r<BR>\n" );
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document.write( "* **When Algorithm 2 might be preferable (in very specific scenarios):**\r<BR>\n" );
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document.write( "    * If the array is extremely large and the probability of finding a 1 early on is very high (e.g., if the array is almost entirely filled with 1s), Algorithm 2 could be faster. However, this is a highly specific and unlikely scenario.\r<BR>\n" );
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document.write( "**In most practical situations, Algorithm 1 (Linear Search) is the more reliable and generally more efficient choice.**\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the goal is to find *any* position containing a 1. If the goal is to find the *first* position containing a 1, Algorithm 1 is the only suitable option.\r<BR>\n" );
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document.write( "I hope this explanation is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );