Quadratic Probing in Java

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1 / / Generic hash function - - used if no other matches 2 template <class Object> 3 unsigned int hash( const Object & key ) 4 { 5 unsigned int hashVal = 0; 6 7 const char *keyp = reinterpret-cast<const char * > ( &key 8 for( size-t i = 0; i < sizeof( Object ) ; i++ ) 9 hashVal = 37 * hashVal + keyp[ i I ; 10 11 return hashVal; 12 1
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Figure 2016 A generic hash function
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2042 Analysis of Quadratic Probing
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Quadratic probing has not yet been mathematically analyzed, although we know that it eliminates primary clustering In quadratic probing, elements that hash to the same position probe the same alternative cells, which is known as secondary clustering Again, the independence of successive probes cannot be assumed Secondary clustering is a slight theoretical blemish Simulation results suggest that it generally causes less than an extra onehalf probe per search and that this increase is true only for high load factors Figure 205 illustrates the difference between linear probing and quadratic probing and shows that quadratic probing does not suffer from as much clustering as does linear probing Techniques that eliminate secondary clustering are available The most popular is double hashing, in which a second hash function is used to drive the collision resolution Specifically, we probe at a distance Hash,(X), 2Hash2(X),and so on The second hash function must be carefully chosen (eg, it should never evaluate to O), and all cells must be capable of being probed A function such as Hash,(X) = R - (X mod R ) , with R a prime smaller than M, generally works well Double hashing is theoretically interesting because it can be shown to use essentially the same number of probes as the purely random analysis of linear probing would imply However, it is somewhat more complicated than quadratic probing to implement and requires careful attention to some details There seems to be no good reason not to use a quadratic probing strategy, unless the overhead of maintaining a half-empty table is burdensome That would be the case in other programming languages if the items being stored were very large
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In secondary clustering, elements that hash to the same position probe the same alternative cells Secondary clustering is a minor theoretical blemish
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Double hashing is a hashing technique that does not suffer from secondary clustering A second hash function is used to drive the collision resolution
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205 Separate Chaining Hashing
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Separate chaining hashing is a spaceefficient alternative to quadratic probing in which an array of linked lists is maintained It is less sensitive to high load factors
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For separate chaining hashing, a reasonable load factor is 1O A lower load factor does not significantly improve performance; a moderately higher load factor is acceptable and can save space
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A popular and space-efficient alternative to quadratic probing is separate chaining hashing in which an array of linked lists is maintained For an array of linked lists, Lo, L,, , L,_ ,, the hash function tells us in which list to insert an item X and then, during a f i n d , which list contains X The idea is that, although searching a linked list is a linear operation, if the lists are sufficiently short, the search time will be very fast In particular, suppose that the load factor, N/M, is A , which is not bounded by 1 O Thus the average list has length A, making the expected number of probes for an insertion or unsuccessful search A and the expected number of probes for a successful search 1 + A / 2 The reason is that a successful search must occur in a nonempty list, and in such a list we expect to have to traverse halfway down the list The relative cost of a successful search versus an unsuccessful search is unusual in that, if A < 2, the successful search is more expensive than the unsuccessful search This condition makes sense, however, because many unsuccessful searches encounter an empty linked list A typical load factor is 1 O; a lower load factor does not significantly enhance performance, but it costs extra space The appeal of separate chaining hashing is that performance is not affected by a moderately increasing load factor; thus rehashing can be avoided For languages that do not allow dynamic array expansion, this consideration is significant Furthermore, the expected number of probes for a search is less than in quadratic probing, particularly for unsuccessful searches We can implement separate chaining hashing by using our existing linked list classes However, because the header node adds space overhead and is not really needed, if space were at a premium we could elect not to reuse components and instead implement a simple stacklike list The coding effort turns out to be remarkably light Also, the space overhead is essentially one pointer per node, plus an additional pointer per list; for example, when the load factor is 1 O, it is two pointers per item This feature could be important if the size of an item is large In that case, we have the same tradeoffs as with the array and linked list implementations of stacks
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