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CS 361, Lecture 15
Jared Saia
University of New Mexico
Outline
Dictionary ADT
Hash Tables
Dictionary ADT
A dictionary ADT implements the following operations
Insert(x)
: puts the item x into the dictionary
Delete(x)
: deletes the item x from the dictionary
IsIn(x)
: returns true iff the item x is in the dictionary
Dictionary ADT
nary asFrequently, we think of the items being stored in the dictio-
keys
The keys typically have
records
associated with them which
implementationare carried around with the key but not used by the ADT
Thus we can implement functions like:
Insert(k,r)
puts the item (k,r) into the dictionary if the
key k is not already there, otherwise returns an error
Delete(k)
: deletes the item with key k from the dictionary
Lookup(k)
: returns the item (k,r) if k is in the dictionary,
otherwise returns null
Implementing Dictionaries
linked listThe simplest way to implement a dictionary ADT is with a
Let
l be a linked list data structure,
assume we have the
following operations defined for
l
head(l): returns a pointer to the head of the list
next(p): given a pointer
p
into the list, returns a pointer
to the next element in the list if such exists, null otherwise
previous(p):
given
a
pointer
p
into
the
list,
returns
a
null otherwisepointer to the previous element in the list if such exists,
of that itemkey(p): given a pointer into the list, returns the key value
value of that itemrecord(p): given a pointer into the list, returns the record
In-Class Exercise
Implement a dictionary with a linked list
to the item with key k if it is in the dictionary or null otherwiseQ1: Write the operation Lookup(k) which returns a pointer
Q2: Write the operation Insert(k,r)
Q3: Write the operation Delete(k)
Q4:
For a dictionary with
n
elements, what is the runtime
of all of these operations for the linked list data structure?
In-Class Exercise
Q5:
Describe how you would use this dictionary ADT to
book.count the number of occurences of each word in an online
Q6:
If
m
is the total number of words in the online book,
and
n
is the number of unique words, what is the runtime of
the algorithm for the previous question?
Dictionaries
This linked list implementation of dictionaries is very slow
Q: Can we do better?
A: Yes, with hash tables, AVL trees, etc
Hash Tables
“Key” Idea:
An element with key
k
is stored in slot
h ( k ),
where
h is a
hash function
mapping
U
into the set
,... , m
Main problem: Two keys can now hash to the same slot
Q: How do we resolve this problem?
making the table large enoughA1: Try to prevent it by hashing keys to “random” slots and
A2: Chaining
A3: Open Addressing
Chained Hash
CH-Delete(T,x){delete x from the list T[h(key(x))];}CH-Search(T,k){search for elem with key k in list T[h(k)];} CH-Insert(T,x){Insert x at the head of list T[h(key(x))];}linked list. In chaining, all elements that hash to the same slot are put in a
Analysis
CH-Insert and CH-Delete take
O
(1) time if the list is doubly
linked and there are no duplicate keys
Q: How long does CH-Search take?
A: It depends.
In particular,
depends on the
load factor
α
=
n/m
(i.e. average number of elems in a list)
CH-Search Analysis
Worst case analysis: everyone hashes to one slot so Θ(
n )
For average case, make the
simple uniform hashing
assump-
mtion: any given elem is equally likely to hash into any of the
slots, indep. of the other elems
Let
n i be a random variable giving the length of the list at
the
i -th slot
Then time to do a search for key
k
is 1 +
(^) n h ( k )
CH-Search Analysis
Q: What is
E
n h ( k ) )?
A: We know that
h ( k ) is uniformly distributed among
, .., m
Thus,
E
n h ( k ) ) =
∑ m − 1
i
/m
n i =
n/m
α
Hash Functions
Want each key to be equally likely to hash to any of the
m
slots, independently of the other keys
terns that might exist in the dataKey idea is to use the hash function to “break up” any pat-
easily convert strings to naturaly numbers)We will always assume a key is a natural number (can e.g.
Division Method
h ( k ) =
k
mod
m
Want
m
to be a
prime number
, which is not too close to a
power of 2
Why?
Multiplication Method
h ( k ) =
b m (^) ∗ (^) ( kA
mod 1)
c
kA
mod 1 means the fractional part of
kA
Advantage: value of
m
is not critical, need not be a prime
A
2 works well in practice
