Resolving collisions: 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.

Open addressing: Linear Probing hash function: (h(k) + i ) mod m for 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.

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

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...

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

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

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

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.

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

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 is h2(k) = p  (k mod p) where p is a prime less than m.

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