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 m for i0,
  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 ((i1)/1)2
7
Open addressing Quadratic Probing
An element can always be inserted, All probes
will be at different indices
Prime table size
? lt 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 k81
• 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, well 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.

11
Open addressing Random hashing

• Use a pseudorandom number generator to obtain the
  sequence of slots that are probed.