Naive Analysis of Linear Probing in Java

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2031 Naive Analysis of Linear Probing
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To estimate the performance of linear probing, we make two assumptions:
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The simplistic analysis of linear probing is based on the assumption that successive probes are independentThis assumption is not true and thus the analysis underestimates the costs of searching and insertion
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1 the hash table is large, and 2 each probe in the hash table is independent of the previous probe
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Assumption 1 is reasonable; otherwise, we would not be bothering with a hash table Assumption 2 says that, if the fraction of the table that is full is A, each time we examine a cell the probability that it is occupied is also 2, independent of any previous probes Independence is an important statistical property that greatly simplifies the analysis of random events Unfortunately, as discussed in Section 2032, the assumption of independence is not only unjustified, but it also is erroneous Thus the naive analysis that we perform is incorrect Even so, it is helpful because it tells us what we can hope to achieve if we are more careful about how collisions are resolved As mentioned earlier in the chapter, the performance of the hash table depends on how full the table is Its fullness is given by the load factor
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DEFINITION: The load factor, A, of a probing hash table is the fraction of the table that is full The load factor ranges from 0 (empty) to 1 (completely full)
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We can now give a simple but incorrect analysis of linear probing in Theorem 20 I
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The load factor of a probing hash table is the fraction of the table that is full It ranges from 0 (empty) to 1 (full)
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/findependence of probes is assumed, the average number of cells examined in an insertion using linear probing is 1 / ( 1 - A)
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Theorem 201
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For a table wit/ cr load factor of A, the probability of any cell's being empty is 1 - A Consequently, the expected number of independent trials required to,find an empty cell is 1/( 1 - A)
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Proof
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In the proof of Theorem 201 we use the fact that, if the probability of some event's occurring is p, then on average Ilp trials are required until the event occurs, provided that the trials are independent For example, the expected number of coin flips until a heads occurs is two, and the expected number of rolls of a single six-sided die until a 4 occurs is six, assuming independence
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Illustration of primary clustering in linear probing (b) versus no clustering (a) and the less significant secondary clustering in quadratic
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probing (c) Long lines represent occupied cells, and the load factor
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2032 What Really Happens: Primary Clustering
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The effect of primary clustering is the formation of large clusters of occu~ied cells, making insertions into the cluster expensive (and then the insertion makes the cluster even larger)
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Unfortunately, independence does not hold, as shown in Figure 205 Part (a) shows the result o<filling a hash table to 70 percent capacity, if all successive probes are independent Part (b) shows the result of linear probing Note the group of clusters: the phenomenon known as primary clustering In primary clustering, large blocks of occupied cells are formed Any
key that hashes into this cluster requires excessive attempts to resolve the collision, and then it adds to the size of the cluster Not only d~ items that collide because of identical hash functions cause degenerate performance, but also an item that collides with an alternative location for another item causes poor performance The mathematical analysis required to take this phenomenon into account is complex but has been solved, yielding Theorem 202
Theorem 202
The average number of cells examined in an insertion using linear probing is roughly ( 1 + 1/( 1 - i1)2)/2
Proof
The proof is beyond the scope of this text See reference [6]
Linear Probing
For a half-full table, we obtain 25 as the average number of cells examined during an insertion This outcome is almost the same as what the naive analysis indicated The main difference occurs as A gets close to 1 For instance, if the table is 90 percent full, A = 09 The naive analysis suggests that 10 cells would have to be examined-a lot but not completely out of the question However, by Theorem 202, the real answer is that some 50 cells need to be examined That is excessive (especially as this number is only an average and thus some insertions must be worse)
Primary clustering is a problem at high load factors For halfempty tables, the effect is not disastrous