Hashing data structure in genaral ss.ppt

2
Preview
 A hash function is a function that:
 When applied to an Object, returns a number
 When applied to equal Objects, returns the same number
for each
 When applied to unequal Objects, is very unlikely to return
the same number for each
 Hash functions turn out to be very important for
searching, that is, looking things up fast
 This is their story....

3
Searching
 Consider the problem of searching an array for a given
value
 If the array is not sorted, the search requires O(n) time
 If the value isn’t there, we need to search all n elements
 If the value is there, we search n/2 elements on average
 If the array is sorted, we can do a binary search
 A binary search requires O(log n) time
 About equally fast whether the element is found or not
 It doesn’t seem like we could do much better
 How about an O(1), that is, constant time search?
 We can do it if the array is organized in a particular way

4
Hashing
 Suppose we were to come up with a “magic function”
that, given a value to search for, would tell us exactly
where in the array to look
 If it’s in that location, it’s in the array
 If it’s not in that location, it’s not in the array
 This function would have no other purpose
 If we look at the function’s inputs and outputs, they
probably won’t “make sense”
 This function is called a hash function because it
“makes hash” of its inputs

5
Example (ideal) hash function
 Suppose our hash function
gave us the following values:
hashCode("apple") = 5
hashCode("watermelon") = 3
hashCode("grapes") = 8
hashCode("cantaloupe") = 7
hashCode("kiwi") = 0
hashCode("strawberry") = 9
hashCode("mango") = 6
hashCode("banana") = 2
kiwi
banana
watermelon
apple
mango
cantaloupe
grapes
strawberry
0
1
2
3
4
5
6
7
8
9

6
Sets and tables
 Sometimes we just want a set
of things—objects are either
in it, or they are not in it
 Sometimes we want a map—
a way of looking up one thing
based on the value of another
 We use a key to find a place in
the map
 The associated value is the
information we are trying to
look up
 Hashing works the same for
both sets and maps
 Most of our examples will be
sets
robin
sparrow
hawk
seagull
bluejay
owl
. . .
141
142
143
144
145
146
147
148
robin info
sparrow info
hawk info
seagull info
bluejay info
owl info
key value

7
Finding the hash function
 How can we come up with this magic function?
 In general, we cannot--there is no such magic
function 
 In a few specific cases, where all the possible values are
known in advance, it has been possible to compute a
perfect hash function
 What is the next best thing?
 A perfect hash function would tell us exactly where to look
 In general, the best we can do is a function that tells us
where to start looking!

8
Example imperfect hash function
 Suppose our hash function gave
us the following values:
 hash("apple") = 5
hash("watermelon") = 3
hash("grapes") = 8
hash("cantaloupe") = 7
hash("kiwi") = 0
hash("strawberry") = 9
hash("mango") = 6
hash("banana") = 2
hash("honeydew") = 6
kiwi
banana
watermelon
apple
mango
cantaloupe
grapes
strawberry
0
1
2
3
4
5
6
7
8
9
• Now what?

9
Collisions
 When two values hash to the same array location,
this is called a collision
 Collisions are normally treated as “first come, first
served”—the first value that hashes to the location
gets it
 We have to find something to do with the second and
subsequent values that hash to this same location

10
Handling collisions
 What can we do when two different values attempt
to occupy the same place in an array?
 Solution #1: Search from there for an empty location
 Can stop searching when we find the value or an empty location
 Search must be end-around
 Solution #2: Use a second hash function
 ...and a third, and a fourth, and a fifth, ...
 Solution #3: Use the array location as the header of a
linked list of values that hash to this location
 All these solutions work, provided:
 We use the same technique to add things to the array as
we use to search for things in the array

11
Insertion, I
 Suppose you want to add
seagull to this hash table
 Also suppose:
 hashCode(seagull) = 143
 table[143] is not empty
 table[143] != seagull
 table[145] is empty
 Therefore, put seagull at
location 145
robin
sparrow
hawk
bluejay
owl
. . .
141
142
143
144
145
146
147
148
. . .
seagull

12
Searching, I
 Suppose you want to look up
seagull in this hash table
 Also suppose:
 hashCode(seagull) = 143
 table[145] == seagull !
 We found seagull at location
145
robin
sparrow
hawk
bluejay
owl
. . .
141
142
143
144
145
146
147
148
. . .
seagull

13
Searching, II
 Suppose you want to look up
cow in this hash table
 Also suppose:
 hashCode(cow) = 144
 table[144] != cow
 table[145] != cow
 table[146] is empty
 If cow were in the table, we
should have found it by now
 Therefore, it isn’t here
robin
sparrow
hawk
bluejay
owl
. . .
141
142
143
144
145
146
147
148
. . .
seagull

14
Insertion, II
 Suppose you want to add
hawk to this hash table
 Also suppose
 hashCode(hawk) = 143
 table[143] != hawk
 table[144] == hawk
 hawk is already in the table,
so do nothing
robin
sparrow
hawk
seagull
bluejay
owl
. . .
141
142
143
144
145
146
147
148
. . .

15
Insertion, III
 Suppose:
 You want to add cardinal to
this hash table
 hashCode(cardinal) = 147
 The last location is 148
 147 and 148 are occupied
 Solution:
 Treat the table as circular; after
148 comes 0
 Hence, cardinal goes in
location 0 (or 1, or 2, or ...)
robin
sparrow
hawk
seagull
bluejay
owl
. . .
141
142
143
144
145
146
147
148

16
Clustering
 One problem with the above technique is the tendency to
form “clusters”
 A cluster is a group of items not containing any open slots
 The bigger a cluster gets, the more likely it is that new
values will hash into the cluster, and make it ever bigger
 Clusters cause efficiency to degrade
 Here is a non-solution: instead of stepping one ahead, step n
locations ahead
 The clusters are still there, they’re just harder to see
 Unless n and the table size are mutually prime, some table locations
are never checked

17
Efficiency
 Hash tables are actually surprisingly efficient
 Until the table is about 70% full, the number of
probes (places looked at in the table) is typically
only 2 or 3
 Sophisticated mathematical analysis is required to
prove that the expected cost of inserting into a hash
table, or looking something up in the hash table, is
O(1)
 Even if the table is nearly full (leading to occasional
long searches), efficiency is usually still quite high

18
Solution #2: Rehashing
 In the event of a collision, another approach is to rehash: compute
another hash function
 Since we may need to rehash many times, we need an easily computable
sequence of functions
 Simple example: in the case of hashing Strings, we might take the
previous hash code and add the length of the String to it
 Probably better if the length of the string was not a component in
computing the original hash function
 Possibly better yet: add the length of the String plus the number
of probes made so far
 Problem: are we sure we will look at every location in the array?
 Rehashing is a fairly uncommon approach, and we won’t pursue
it any further here

19
Solution #3: Bucket hashing
 The previous solutions
used open hashing: all
entries went into a “flat”
(unstructured) array
 Another solution is to
make each array location
the header of a linked
list of values that hash to
that location
robin
sparrow
hawk
bluejay
owl
. . .
141
142
143
144
145
146
147
148
. . .
seagull

20
The hashCode function
 public int hashCode() is defined in Object
 Like equals, the default implementation of
hashCode just uses the address of the object—
probably not what you want for your own objects
 You can override hashCode for your own objects
 As you might expect, String overrides hashCode
with a version appropriate for strings
 Note that the supplied hashCode method can
return any possible int value (including negative
numbers)
 You have to adjust the returned int value to the size of
your hash table

21
Why do you care?
 Java provides HashSet, Hashtable, and HashMap for your use
 These classes are very fast and very easy to use
 They work great, without any additional effort, for Strings
 But...
 They will not work for your own objects unless either:
 You are satisfied with the inherited equals method (no object is equal to
any other, separately created object)
 Or:
 You have defined equals for your objects and
 You have also defined a hashCode method that is consistent with your
equals method (that is, equal objects have equal hash codes)

22
Writing your own hashCode()
 A hashCode() method must:
 Return a value that is (or can be converted to) a legal array index
 Always return the same value for the same input
 It can’t use random numbers, or the time of day
 Return the same value for equal inputs
 Must be consistent with your equals method
 It does not need to guarantee different values for different
inputs
 A good hashCode() method should:
 Make it unlikely that different objects have the same hash code
 Be efficient to compute
 Give a uniform distribution of values
 Not assign similar numbers to similar input values

23
Other considerations
 The hash table might fill up; we need to be
prepared for that
 Not a problem for a bucket hash, of course
 You cannot easily delete items from an open hash
table
 This would create empty slots that might prevent you
from finding items that hash before the slot but end up
after it
 Again, not a problem for a bucket hash
 Generally speaking, hash tables work best when
the table size is a prime number

24
Hash tables in Java
 Java provides classes Hashtable, HashMap, and HashSet (and
many other, more specialized ones)
 Hashtable and HashMap are maps: they associate keys with
values
 Hashtable is synchronized; that is, it can be accessed safely from
multiple threads
 Hashtable uses an open hash, and has a rehash method, to increase the
size of the table
 HashMap is newer, faster, and usually better, but it is not
synchronized
 HashMap uses a bucket hash, and has a remove method
 HashSet is just a set, not a collection, and is not synchronized

25
Hash table operations
 HashSet, Hashtable and HashMap are in java.util
 All have no-argument constructors, as well as
constructors that take an integer table size
 The maps have methods:
 public Object put(Object key, Object value)
 (Returns the previous value for this key, or null)
 public Object get(Object key)
 public void clear()
 public Set keySet()
 Dynamically reflects changes in the hash table
 ...and many others

26
Bottom line
 You do not have to write a hashCode() method if:
 You never use a built-in class that depends on it, or
 You put only Strings in hash sets, and use only Strings as
keys in hash maps (values don’t matter), or
 You are happy with equals meaning ==, and don’t override it
 You do have to write a hashCode() method if:
 You use a built-in hashing class for your own objects, and you
override equals for those objects
 Finally, if you ever override hashCode, you must also
override equals

Hashing data structure in genaral ss.ppt

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