Resolving collisions: Open addressing

1. Resolving collisions: Open addressing • Open addressing • Store all elements within the table • The space we save from the chain pointers is used instead to make the array larger. • If there is a collision, probe the table in a systematic way to find an empty slot. • If the table fills up, we need to enlarge it and rehash all the keys.

2. Open addressing: Linear Probing • hash function:(h(k) + i ) mod mfor i=0, 1,...,m-1 • Insert : Start with the location where the key hashed and do a sequential search for an empty slot. • Search : Start with the location where the key hashed and do a sequential search until you either find the key (success) or find an empty slot (failure). • Delete : (lazy deletion) follow same route but mark slot as DELETED rather than EMPTY, otherwise subsequent searches will fail.

3. Open addressing: Linear Probing • Advantage: very easy to implement • Disadvantage:primary clustering • Long sequences of used slots build up with gaps between them. Every insertion requires several probes and adds to the cluster. • The average length of a probe sequence when inserting is

4. Open addressing: Quadratic Probing • Probe the table at slots ( h(k) + i2 ) mod m for i =0, 1,2, 3, ..., m-1 • Ease of computation: • Not as easy as linear probing. • Do we really have to compute a power? • Clustering • Primary clustering is avoided, since the probes are not sequential. But...

5. Open addressing: Quadratic Probing • Probe sequence for hash value 3 in a table of size 16: 3 + 02 = 3 3 + 12 = 4 3 + 22 = 7 3 + 32 = 12 3 + 42 = 3 3 + 52 = 12 3 + 62 = 7 3 + 72 = 4 3 + 82 = 3 3 + 92 = 4 3 + 102 = 7 3 + 112 = 12 3 + 122 = 3 3 + 132 = 12 3 + 142 = 7 3 + 152 = 4

6. Open addressing: Quadratic Probing • Probe sequence for hash value 3 in a table of size 19: Why did this happen? 3 + 02 = 3 3 + 12 = 4 3 + 22 = 7 3 + 32 = 12 3 + 42 = 0 3 + 52 = 9 3 + 62 = 1 3 + 72 = 14 3 + 82 = 10 3 + 92 = 8 3 + 102 = 8 3 + 112 = 10 3 + 122 = 14 3 + 132 = 1 3 + 142 = 9 3 + 152 = 0 3 + 162 = 12 3 + 172 = 7 3 + 182 = 4 i2 is not a good idea after all. Use a small variant instead: h(k)+(-1)i-1 ((i+1)/1)2

7. Open addressing: Quadratic Probing An element can always be inserted, All probes will be at different indices Prime table size  <= 0.5

8. Open addressing: Quadratic Probing • Disadvantage: secondary clustering: • if h(k1)==h(k2) the probing sequences for k1 and k2 are exactly the same. • Is this really bad? • In practice, not so much • It becomes an issue when the load factor is high.

9. Open addressing: Double hashing • The hash function is (h(k)+i h2(k)) mod m • In English: use a second hash function to obtain the next slot. • The probing sequence is: h(k), h(k)+h2(k), h(k)+2h2(k), h(k)+3h3(k), ... • Performance : • Much better than linear or quadratic probing. • Does not suffer from clustering • BUT requires computation of a second function

10. Open addressing: Double hashing • The choice of h2(k) is important • It must never evaluate to zero • consider h2(k)=k mod 9 for k=81 • The choice of m is important • If it is not prime, we may run out of alternate locations very fast. • If m and h2(k) are relatively prime, we’ll end up probing the entire table. • A good choice for h2 ish2(k) = p  (k mod p) where p is a prime less than m.

11. Open addressing: Random hashing • Use a pseudorandom number generator to obtain the sequence of slots that are probed.