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Unformatted text preview: Outline • Deﬁnition • Insertion • Deletion • Search • Analysis Lecture 15, Double Hashing p. 1 Double Hashing: Deﬁnitions • Motivation: What if we have a lot of collisions in an openaddressing hashtable? Other than linear probing, how can we scatter keys better? • Two hash functions: ﬁrst hash function as before; the second hash function decides the increment step. • Double hashing: use a second hash function to get a ﬁxed increment to use for the probe sequence. • What is a good second hashing function? To reach every possible slot in the table. Be relative prime to the table size.
Lecture 15, Double Hashing p. 2 Operations: • Insertion
– First slot is determined by the ﬁrst hash function – If there is a collision, the ﬁxed increment is determined by the second hash function – Until an empty slot is reached, or the table is full. • Deletion
– Mark it, instead of deleting it • Search
– First slot to search: determined by the ﬁrst hash function – If it is not, the ﬁxed increment is determined by the second hash function – Until an empty slot is reached, or the key is found. Lecture 15, Double Hashing p. 3 Analysis: on Search/Insertion • Assumption. Each probe equally likely hits each table position
∞ ( n )i i=0 M 1 = 1−(n/M ) (a geometric • A miss: series) • A insertion: the same as above • A hit: the same as insert the key as the j th key.
– Expected time to insert the ﬁrst key: 1 – Expected time to insert the second key, i.e. a 1 table with one key: 1−1/M – Expected time to insert the third key, a table 1 with two keys: 1−2/M Lecture 15, Double Hashing p. 4 – Expected time to insert the forth key, a table 1 with three keys: 1−3/M – The total expected time to insert n keys: 1 + 1 1 1 + 1−(2/M ) + ... + 1−((n−1)/M ) 1−(1/M )
M M M – = 1 + M −1 + M −2 + ... + M −n+1 (diﬀerence of two harmonic series) 1 – total time M ln 1−n/M 1 – average time M/n ln 1−n/M Lecture 15, Double Hashing p. 5 ...
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 Spring '09
 ORTIZ

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