Quadratic Probing

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probability of getting a somewhat lengthier probe sequence, linear probing is not a terrible strategy Because it is so easy to implement, any method we use to remove primary clustering must be of comparable complexity Otherwise, we expend too much time in saving only a fraction of a probe One such method is quadratic probing

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Quadratic Probing

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Quadratic probing examines cells ' 9 4 9 99 and so on, away from the original probe point

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Quadratic probing is a collision resolution method that eliminates the primary clustering problem of linear probing by examining certain cells away from the original probe point Its name is derived from the use of the formula F ( i ) = i 2 to resolve collisions Specifically, if the hash function evaluates to H and a search in cell H is inconclusive, we try cells H + 1 *, H + 22 H + 3*, , H + i2 (employing wraparound) in sequence This strategy differs from the linear probing strategy of searching H + I , H + 2, H + 3, , H + i Figure 206 shows the table that results when quadratic probing is used instead of linear probing for the insertion sequence shown in Figure 204 When 49 collides with 89 the first alternative attempted is one cell away This cell is empty, so 49 is placed there Next, 58 collides at position 8 The cell at position 9 (which is one away) is tried, but another collision occurs A vacant

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Remember that subsequent probe points are a quadratic number of positions from the original probe point

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After insel 89 After insert 18 After insert 49 After insert 58 After insert 9

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A quadratic probing hash table after each insertion (note that the table size was poorly chosen because it is not a prime number)

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cell is found at the next cell tried, which is 22 = 4 positions away from the original hash position Thus 58 is placed in cell 2 The same thing happens for 9 Note that the alternative locations for items that hash to position 8 and the alternative locations for the items that hash to position 9 are not the same The long probe sequence to insert 58 did not affect the subsequent insertion of 9, which contrasts with what happened with linear probing We need to consider a few details before we write code In linear probing, each probe tries a different cell Does quadratic probing guarantee that, when a cell is tried, we have not already tried it during the course of the current access Does quadratic probing guarantee that when we are inserting X and the table is not full, X will be inserted Linear probing is easily implemented Quadratic probing appears to require multiplication and mod operations Does this apparent added complexity make quadratic probing impractical What happens (in both linear probing and quadratic probing) if the load factor gets too high Can we dynamically expand the table, as is typically done with other array-based data structures

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If the table size is prime and the load factor is no larger than 05, all probes will be to different locations and an item can always be inserted

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Fortunately, the news is relatively good on all cases If the table size is prime and the load factor never exceeds 05, we can always place a new item X and no cell is probed twice during an access However, for these guarantees to hold, we need to ensure that the table size is a prime number We prove this case in Theorem 204 For completeness, Figure 207 shows a routine that generates prime numbers, using the algorithm shown in Figure 108 (a more complex algorithm is not warranted)

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Theorem 204

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lf quadratic probing

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is used and the table size is prime, then a new element can always be inserted ifthe fable is at least h a l f e m p ~ Furthermore, in the course of the insertion, no cell is probed twice

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