Chapter 6 Hashing
Dictionary operations:
Dictionary implementations and complexity of operations
unsorted array—
sorted array
linked list
Hashing
1.
2.
3.
Potential problems we have to work around:
Open Hashing (Separate Chaining)
Notation:
A[0]…A[m-1]
h
Hash Functions
Characteristics of a good hash function
1.
2.
Hash Functions
A commonly used hash function is obtained by choosing a prime
number p as the table size with function h(key) = key (mod p).
Consider the hash function h(x) = x2 (mod 10).
Two other hashing strategies--truncation and folding:
Treating the key as a string
More hash functions
function h(s)
z0
for i  0 to k-1 do
z  (z + s[i]*W[i])mod q
z  z mod m
return z
Also read about the polynomial hash function on p. 205.
Complexity of the Find Operation
Assumptions:
1.
2.
Space required:
Worst case time complexity
The load factor of a hash table is the ratio of the number of items
in the table to the size of the table. notation:  = n/m.
A probe is an access into the data structure.
a. unsuccessful search—
b. successful search—
Closed Hashing—Open Addressing
Collision resolution strategies
linear probing
0
1
X
2
X
3
X
4
5
6
X
7
8
X
9
10
Suppose the next item to be placed in the table hashes to 3.
Claim (without proof) The expected number of probes is
½ [1 + 1/(1 - 2)] for insertions and unsuccessful searches and
½ [1 + 1/(1 - )] for successful searches.
Hash Table Deletions
Other Collision Resolution Strategies
Quadratic probing is a collision resolution strategy in which i2 is
the increment where i is the number of collisions encountered.
Double hashing uses a second hash function h2 to help
determine where the element will be placed.
Rehashing is used to rebuild the hash table when it gets too full.