Average case successful find cost

Assume a table with load factor
α
= N/M. Consider random hashing, so clustering is not a problem; each probe location is generated randomly, and independently

For a key in the table, the number of probes required to successfully find it is equal to the number of probes taken when it was inserted in the table. The insertion of each new key increases the load factor, starting from 0 and going to
α
.

Therefore, the average number of probes for successful find with random hashing is

With linear probing, clusters form, which leads to longer probe sequences. It can be shown that the average number of probes for successful find with linear probing is

These average case time costs are bad, bounded only by M, when
α
is close to 1; but are good (1.8 and 2.5 respectively) when
α
is .75 or less, independent of M
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