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<b><font face="Arial, Helvetica, sans-serif" size="4" color="#000000">(3)<font color="#FF0000"> Theoretical evaluation of Overflow techniques</font>
<i><font color="#0033CC">Sn</font></i> -- the expected number of identifier comparisons needed
to locate a randomly chosen <i>x <font size="2">i</font></i><font size="2"> </font>, 1 <= i <= n.
<i><font color="#0033CC">Un</font></i> -- the expected number of identifier comparisons when a
search is made for an identifier not in the hash table
〖<font color="#FF0000">theorem 8.1</font>〗
Let <img src="image/a.gif" width="15" height="12" align="middle"> = n / b be the loading density of a hash table using a uniform
hash function f, Then
(1)<font color="#FF0000"> for linear open addressing</font>:
<img src="image/text8-a0.gif" width="222" height="69"> <img src="image/text8-a1.gif" width="185" height="66">
(2)<font color="#FF0000">for rehashing, random probing, and quadratic probing:</font>
<i><font color="#0033CC">Un</font></i>= 1/(1- <img src="image/a.gif" width="15" height="12" align="middle">) <i><font color="#0033CC">Sn</font></i> =(1 )(1- <img src="image/a.gif" width="15" height="12" align="middle"> )
(3) <font color="#FF0000"> for chaining:</font>
<i><font color="#0033CC">Un</font></i>= <img src="image/a.gif" width="15" height="12" align="middle"> <i><font color="#0033CC">Sn</font></i> = 1 + <img src="image/a.gif" width="15" height="12" align="middle"> /2
proof:
<img src="image/text8-a.gif" width="327" height="38"></font></b></pre>
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