ppt - Texas A&M University

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025-612-0001
981-101-0002
451-229-0004
Chapter 9:
Maps, Dictionaries, Hashing
Nancy Amato
Parasol Lab, Dept. CSE, Texas A&M University
Acknowledgement: These slides are adapted from slides provided with Data Structures and
Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004)
http://parasol.tamu.edu
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Ordered Maps (§9.3)
Hash Tables (§9.2)
Dictionaries & Hashing
Dictionaries
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• The map ADT models a
searchable collection of keyelement items
• The main operations of a map
are searching, inserting, and
deleting items
• Multiple items with the same key
are not allowed
• Applications:
• credit card authorization
• mapping host names (e.g.,
(e.g., 128.148.34.101)
• find(k): if M has an entry with
key k, return an iterator p
referring to this element, else,
return special end iterator.
• put(k, v): if M has no entry
with key k, then add entry (k,
v) to M, otherwise replace the
value of the entry with v;
return iterator to the
inserted/modified entry
• erase(k) or erase(p): remove
from M entry with key k or
iterator p; An error occurs if
there is no such element.
• size(), empty()
Dictionaries & Hashing
Dictionaries
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• A direct address table is a map in which
• The keys are in the range {0,1,2,…,N}
• Stored in an array of size N - T[0,N]
• Item with key k stored in T[k]
• Performance:
• insertItem, find, and removeElement all take O(1) time
• Space - requires space O(N), independent of n, the number of items
stored in the map
• The direct address table is not space efficient unless the range of
the keys is close to the number of elements to be stored in the
map, I.e., unless n is close to N.
Dictionaries & Hashing
Dictionaries
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• The dictionary ADT models a
searchable collection of key• find(k): if the dictionary has an
element items
entry with key k, returns an iterator
p to an arbitrary elt.
• The main difference from a map
• findAll(k): Return iterators (b,e) s.t.
is that multiple items with the
that all entries with key k are
same key are allowed
between them.
• Any data structure that supports
• insert(k, v): insert entry (k, v) into
a dictionary also supports a map
D, return iterator to it.
• Applications:
• Dictionary which has multiple
definitions for the same word
• erase(k), erase(p): remove
arbitrary entry with key k or entry
referenced by p. Error occurs if
there is no such entry
• Begin(), end(): return iterator to
first or just beyond last entry of D
• size(), isEmpty()
Dictionaries & Hashing
Dictionaries
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Dictionary - Log File
(unordered sequence implementation)
• A log file is a dictionary implemented by means of an unsorted
sequence
• We store the items of the dictionary in a sequence (based on a doubly-linked
lists or a circular array), in arbitrary order
• Performance:
• insert takes O(1) time since we can insert the new item at the beginning or at
the end of the sequence
• find and erase take O(n) time since in the worst case (the item is not found)
we traverse the entire sequence to look for an item with the given key
• Space - can be O(n), where n is the number of elements in the dictionary
• The log file is effective only for dictionaries of small size or for
dictionaries on which insertions are the most common operations, while
searches and removals are rarely performed (e.g., historical record of
Dictionaries & Hashing
Dictionaries
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Map/Dictionary implementations
• n - #elements in map/Dictionary
Vectors
Insert
Find
Space
Log File
O(1)
O(n)
O(n)
(map only)
O(1)
O(1)
O(N)
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• An Ordered Map/Dictionary
supports the usual
map/dictionary operations, but
also maintains an order relation
for the keys.
• Naturally supports
• Look-Up Tables - store dictionary
in a vector by non-decreasing
order of the keys
• Binary Search
• In addition to the generic
the following functions:
• closestBefore(k): return the
position of an item with the
largest key less than or equal
to k
• closestAfter(k): return the
position of an item with the
smallest key greater than or
equal to k
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Lookup Table
• A lookup table is a dictionary implemented by means of a sorted
sequence
• We store the items of the dictionary in an array-based sequence,
sorted by key
• We use an external comparator for the keys
• Performance:
• find takes O(log n) time, using binary search
• insertItem takes O(n) time since in the worst case we have to shift
n/2 items to make room for the new item
• removeElement take O(n) time since in the worst case we have to
shift n/2 items to compact the items after the removal
• The lookup table is effective only for dictionaries of small size or
for dictionaries on which searches are the most common
operations, while insertions and removals are rarely performed
(e.g., credit card authorizations)
Dictionaries
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AM
Example of Ordered Map: Binary Search
• Binary search performs operation find(k) on a dictionary
implemented by means of an array-based sequence, sorted by key
• similar to the high-low game
• at each step, the number of candidate items is halved
• terminates after a logarithmic number of steps
• Example: find(7)
0
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m
l
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m
l
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h
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h
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l
m
h
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l=m =h
Dictionaries
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AM
Map/Dictionary implementations
• n - #elements in map/Dictionary
Insert
Find
Space
O(1)
O(n)
O(n)
only)
O(1)
O(N)
Lookup Table
(ordered map/dictionary)
O(logn)
O(n)
Log File
Vectors
O(n)
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Hash Tables
• Hashing
• Hash table (an array) of size N, H[0,N]
• Hash function h that maps keys to indices in H
• Issues
• Hash functions - need method to transform key to an index in H that
will have nice properties.
• Collisions - some keys will map to the same index of H (otherwise
we have a Direct Address Table). Several methods to resolve the
collisions
• Chaining - put elts that hash to same location in a linked list
• Open addressing - if a collision occurs, have a method to select another
location in the table.
• Probe sequences
Dictionaries & Hashing
Dictionaries
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Hash Functions and Hash
Tables
• A hash function h maps keys of a given type to integers
in a fixed interval [0, N - 1]
• Example:
h(x) = x mod N
is a hash function for integer keys
• The integer h(x) is called the hash value of key x
• A hash table for a given key type consists of
• Hash function h
• Array (called table) of size N
• When implementing a dictionary with a hash table, the goal is to
store item (k, o) at index i = h(k)
Dictionaries & Hashing
Dictionaries
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Example
0 
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025-612-0001
981-101-0002
451-229-0004
…
• We design a hash table for a
dictionary storing items
(SSN, Name), where SSN
(social security number) is a
nine-digit positive integer
• Our hash table uses an
array of size N = 10,000 and
the hash function
h(x) = last four digits of x
9997 
9998
9999 
Dictionaries & Hashing
Dictionaries
200-751-9998
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Collisions
• Collisions occur when • Example with Division
different elements are Method
mapped to the same
cell
• h(k) = k mod N
• collisions must be
resolved
• Chaining (store in list
outside the table)
another cell in the table)
• If N=10, then
• h(k)=0 for k=0,10,20, …
• h(k)= 1 for k=1, 11, 21, etc
• …
Dictionaries & Hashing
Dictionaries
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Collision Resolution with Chaining
• Collisions occur when 0 
1
different elements are 2  025-612-0001
mapped to the same
3 
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451-229-0004
981-101-0004
cell
• Chaining: let each cell
in the table point to a • Chaining is simple,
but requires
that map there
outside the table
Dictionaries & Hashing
Dictionaries
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Exercise: chaining
• Assume you have a hash table H with N=9 slots
(H[0,8]) and let the hash function be h(k)=k mod N.
• Demonstrate (by picture) the insertion of the following
keys into a hash table with collisions resolved by
chaining.
• 5, 28, 19, 15, 20, 33, 12, 17, 10
Dictionaries & Hashing
Dictionaries
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- Linear Probing
colliding item is placed in a
different cell of the table
• Linear probing handles
collisions by placing the colliding
item in the next (circularly)
available table cell. So the i-th
cell checked is:
• Example:
• h(x) = x mod 13
• Insert keys 18, 41, 22, 44,
59, 32, 31, 73, in this
order
• H(k,i) = (h(k)+i)mod N
• Each table cell inspected is
referred to as a “probe”
• Colliding items lump together,
causing future collisions to
cause a longer sequence of
probes
0 1 2 3 4 5 6 7 8 9 10 11 12
18 44 59 32 22 31 73
0 1 2 3 4 5 6 7 8 9 10 11 12
41
Dictionaries & Hashing
Dictionaries
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Search with Linear Probing
• Consider a hash table A
that uses linear probing
• find(k)
• We start at cell h(k)
• We probe consecutive
locations until one of the
following occurs
• An item with key k is
found, or
• An empty cell is found, or
• N cells have been
unsuccessfully probed
Algorithm find(k)
i  h(k)
p0
repeat
c  A[i]
if c = 
return Position(null)
else if c.key () = k
return Position(c)
else
i  (i + 1) mod N
pp+1
until p = N
return Position(null)
Dictionaries & Hashing
Dictionaries
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• To handle insertions and
deletions, we introduce a
special object, called
AVAILABLE, which replaces
deleted elements
• removeElement(k)
• We search for an item with
key k
• If such an item (k, o) is found,
we replace it with the special
item AVAILABLE and we
return the position of this item
• Else, we return a null position
• insertItem(k, o)
• We throw an exception if
the table is full
• We start at cell h(k)
• We probe consecutive
cells until one of the
following occurs
• A cell i is found that is
either empty or stores
AVAILABLE, or
• N cells have been
unsuccessfully probed
• We store item (k, o) in cell
i
Dictionaries & Hashing
Dictionaries
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Exercise: Linear Probing
• Assume you have a hash table H with N=11 slots
(H[0,10]) and let the hash function be h(k)=k mod N.
• Demonstrate (by picture) the insertion of the following
keys into a hash table with collisions resolved by linear
probing.
• 10, 22, 31, 4, 15, 28, 17, 88, 59
Dictionaries & Hashing
Dictionaries
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Double Hashing
• Double hashing uses a
secondary hash function h2(k)
and handles collisions by
placing an item in the first
available cell of the series
h(k,i) =(h1(k) + ih2(k)) mod N
for i = 0, 1, … , N - 1
• The secondary hash function
h2(k) cannot have zero values
• The table size N must be a
prime to allow probing of all the
cells
• Common choice of
compression map for the
secondary hash function:
h2(k) = q - k mod q
where
• q<N
• q is a prime
• The possible values for
h2(k) are
1, 2, … , q
Dictionaries & Hashing
Dictionaries
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Example of Double Hashing
• Consider a hash table
storing integer keys
that handles collision
with double hashing
• N = 13
• h1(k) = k mod 13
• h2(k) = 7 - k mod 7
• Insert keys 18, 41,
22, 44, 59, 32, 31, 73,
in this order
0 1 2 3 4 5 6 7 8 9 10 11 12
31
41
18 32 59 73 22 44
0 1 2 3 4 5 6 7 8 9 10 11 12
Dictionaries & Hashing
Dictionaries
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Exercise: Double Hashing
• Assume you have a hash table H with N=11
slots (H[0,10]) and let the hash functions for
double hashing be
• h(k,i)=(h1(k) + ih2(k))mod N
• h1(k)=k mod N
• h2(k)=1 + (k mod (N-1))
• Demonstrate (by picture) the insertion of the
following keys into H
• 10, 22, 31, 4, 15, 28, 17, 88, 59
Dictionaries & Hashing
Dictionaries
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Performance of Hashing
• In the worst case, searches,
insertions and removals on a
hash table take O(n) time
• The worst case occurs when all
the keys inserted into the
dictionary collide
• The load factor a = n/N affects the
performance of a hash table
• Assuming that the hash values
are like random numbers, it can
be shown that the expected
number of probes for an insertion
1 / (1 - a)
• The expected running time of
operations in a hash table is
O(1)
• In practice, hashing is very
is not close to 100%
• Applications of hash tables:
• small databases
• compilers
• browser caches
Dictionaries & Hashing
Dictionaries
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Uniform Hashing Assumption
• The probe sequence of a key k is the sequence of slots
that will be probed when looking for k
• In open addressing, the probe sequence is h(k,0), h(k,1), h(k,2),
h(k,3), …
• Uniform Hashing Assumption: Each key is equally
likely to have any one of the N! permutations of {0,1, 2,
…, N-1} as is probe sequence
• Note: Linear probing and double hashing are far from
achieving Uniform Hashing
• Linear probing: N distinct probe sequences
• Double Hashing: N2 distinct probe sequences
Dictionaries & Hashing
Dictionaries
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Performance of Uniform Hashing
• Theorem: Assuming uniform hashing and an openaddress hash table with load factor a = n/N < 1, the
expected number of probes in an unsuccessful search
is at most 1/(1-a).
• Exercise: compute the expected number of probes in
an unsuccessful search in an open address hash table
with a = ½ , a=3/4, and a = 99/100.
Dictionaries
Dictionaries & Hashing
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Universal Hashing
• A family of hash functions is
universal if, for any
0<i,j<M-1,
• Theorem: The set of
all
functions,
h,
as
• Choose p as a prime
defined here, is
between M and 2M.
universal.
• Randomly select 0<a<p and
• Pr(h(j)=h(i)) < 1/N.
0<b<p, and define
h(k)=(ak+b mod p) mod N
Dictionaries & Hashing
Dictionaries
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Maps/Dictionaries
• n = #elements in map/dictionary,
• N=#possible keys (it could be N>>n) or size of hash table
Vectors
Insert
Find
Space
Log File
O(1)
O(n)
O(n)
(map only)
O(1)
O(1)
O(N)
Lookup Table
(ordered map/dictionary)
O(n)
O(logn)
O(n)
Hashing
(chaining)
O(1)
O(n/N)
O(n+N)
Hashing
O(1/(1-n/N))
O(1/(1-n/N)) O(N)
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