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CSE332: Data AbstractionsLecture 11: Hash Tables
Dan GrossmanSpring 2010
Hash Tables: Review •^
Aim for constant-time (i.e.,
O (1))
find
,^ insert
, and
delete
- “On average” under some reasonable assumptions -^
A hash table is an array of some fixed size– But growable as we’ll see Spring 2010
CSE332: Data Abstractions
E^
int^
table-index
collision?
collisionresolution
client
hash table library
TableSize –
hash table
Hash Tables: A Different ADT? •^
In terms of a Dictionary ADT for just
insert
,^ find
,^ delete
hash tables and balanced trees are just different data structures– Hash tables
O (1) on average (
assuming
few collisions)
O (
log
n ) worst-case
•^
Constant-time is better, right?– Yes, but you need “hashing to behave” (collisions)– Yes, but
findMin
,^ findMax
,^ predecessor
, and
successor
go from
O (
log
n ) to
O (
n )
- Why your textbook considers this to be a different ADT• Not so important to argue over the definitions Spring 2010
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CSE332: Data Abstractions
Collision resolution Collision:
When two keys map to the same location in the hash table We try to avoid it, but number-of-keys exceeds table sizeSo hash tables should support collision resolution
CSE332: Data Abstractions
Separate Chaining
Chaining: All keys that map to the same
table location are kept in a list(a.k.a. a “chain” or “bucket”) As easy as it soundsExample: insert 10, 22, 107, 12, 42 with
mod hashing and
TableSize
Spring 2010
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CSE332: Data Abstractions
Separate Chaining Spring 2010
CSE332: Data Abstractions
10
/^
Chaining: All keys that map to the same
table location are kept in a list(a.k.a. a “chain” or “bucket”) As easy as it soundsExample: insert 10, 22, 107, 12, 42 with
mod hashing and
TableSize
Separate Chaining Spring 2010
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CSE332: Data Abstractions
10
/ 22 /
Chaining: All keys that map to the same
table location are kept in a list(a.k.a. a “chain” or “bucket”) As easy as it soundsExample: insert 10, 22, 107, 12, 42 with
mod hashing and
TableSize
Separate Chaining Spring 2010
CSE332: Data Abstractions
10
/ 22 / 107 /
Chaining: All keys that map to the same
table location are kept in a list(a.k.a. a “chain” or “bucket”) As easy as it soundsExample: insert 10, 22, 107, 12, 42 with
mod hashing and
TableSize
More rigorous chaining analysis Definition: The load factor,
λ ,^ of a hash table is
Spring 2010
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N TableSize CSE332: Data Abstractions
←^
number of elements
Under chaining, the average number of elements per bucket is
λ
So if some inserts are followed by
random
finds, then on average:
•^
Each unsuccessful
find
compares against ____ items
•^
Each successful
find
compares against _____ items
More rigorous chaining analysis Definition: The load factor,
λ ,^ of a hash table is
Spring 2010
N TableSize CSE332: Data Abstractions
←^
number of elements
Under chaining, the average number of elements per bucket is
λ
So if some inserts are followed by
random
finds, then on average:
•^
Each unsuccessful
find
compares against
λ^ items
•^
Each successful
find
compares against
λ^ / 2
items
Alternative: Use empty space in the table •^
Another simple idea: If
h(key)
is already full,
(h(key)
+^
1)^
%^ TableSize
. If full, - try
(h(key)
+^
2)^
%^ TableSize
. If full, - try
(h(key)
+^
3)^
%^ TableSize
. If full…
•^
Example: insert 38, 19, 8, 109, 10 Spring 2010
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CSE332: Data Abstractions
Alternative: Use empty space in the table Spring 2010
CSE332: Data Abstractions
•^
Another simple idea: If
h(key)
is already full,
(h(key)
+^
1)^
%^ TableSize
. If full, - try
(h(key)
+^
2)^
%^ TableSize
. If full, - try
(h(key)
+^
3)^
%^ TableSize
. If full…
•^
Example: insert 38, 19, 8, 109, 10
Alternative: Use empty space in the table Spring 2010
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CSE332: Data Abstractions
•^
Another simple idea: If
h(key)
is already full,
(h(key)
+^
1)^
%^ TableSize
. If full, - try
(h(key)
+^
2)^
%^ TableSize
. If full, - try
(h(key)
+^
3)^
%^ TableSize
. If full…
•^
Example: insert 38, 19, 8, 109, 10
Alternative: Use empty space in the table Spring 2010
CSE332: Data Abstractions
•^
Another simple idea: If
h(key)
is already full,
(h(key)
+^
1)^
%^ TableSize
. If full, - try
(h(key)
+^
2)^
%^ TableSize
. If full, - try
(h(key)
+^
3)^
%^ TableSize
. If full…
•^
Example: insert 38, 19, 8, 109, 10
Alternative: Use empty space in the table Spring 2010
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CSE332: Data Abstractions
•^
Another simple idea: If
h(key)
is already full,
(h(key)
+^
1)^
%^ TableSize
. If full, - try
(h(key)
+^
2)^
%^ TableSize
. If full, - try
(h(key)
+^
3)^
%^ TableSize
. If full…
•^
Example: insert 38, 19, 8, 109, 10
Open addressing This is
one example
of open addressing
In general, open addressing means resolving collisions by trying a
sequence of other positions in the table. Trying the next spot is called probing
th i probe was
(h(key)
+^
i)^
%^ TableSize
- This is called linear probing
- In general have some probe function
f^ and use
h(key)
+^
f(i)
%^
TableSize
Open addressing does poorly with high load factor
λ
- So want larger tables– Too many probes means no more
O (1)
Spring 2010
CSE332: Data Abstractions
In a chart •^
Linear-probing performance degrades rapidly as table gets full– (Formula assumes “large table” but point remains)
-^
By comparison, chaining performance is linear in
λ^ and has no
trouble with
λ >
Spring 2010
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CSE332: Data Abstractions
Quadratic probing •^
We can avoid primary clustering by changing the probe function
-^
A common technique is quadratic probing:^ –^
f(i)
=^
(^2) i
th^ probe:
h(key)
%^
TableSize
st^ probe:
(h(key)
+^
1)^
%^ TableSize
nd^ probe:
(h(key)
+^
4)^
%^ TableSize
rd^ probe:
(h(key)
+^
9)^
%^ TableSize
probe:
(h(key)
+^
(^2) i ) %
TableSize
•^
Intuition: Probes quickly “leave the neighborhood” Spring 2010
CSE332: Data Abstractions
Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Quadratic Probing Example Spring 2010
CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Quadratic Probing Example Spring 2010
CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Quadratic Probing Example Spring 2010
CSE332: Data Abstractions
TableSize=10Insert: 8918495879
Another Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize = 7Insert: 76
Another Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize = 7Insert: 76
Another Quadratic Probing Example Spring 2010
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CSE332: Data Abstractions
TableSize = 7Insert: 76
Uh-oh: For all
n ,^
((nn)*
%^
7 is
or
- Excel shows takes “at least” 50 probes and a pattern• Proof uses induction and
(n
%^
((n-7)
c^ and
k ,^
(^2) (n +c)
%^
k^ =
((n-k)
2 +c)
%^
k
From bad news to good news • The bad news is: After
TableSize
quadratic probes, we will just
cycle through the same indices
-^
The good news:– Assertion #1: If
T^
=^ TableSize
is^ prime
and
λ^ < ½, then
quadratic probing will find an empty slot in at most
T/
probes
T^ and
^0
≤^ i,j
≤^
T/
where
i^
≠^ j
(h(key)
+^
(^2) i ) % T
≠^
(h(key)
+^
(^2) j ) % T
- Assertion #3: Assertion #2 is the “key fact” for provingAssertion # -^
So: If you keep
λ^ < ½, no need to detect cycles
Spring 2010
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CSE332: Data Abstractions
Clustering reconsidered •^
Quadratic probing does not suffer from primary clustering: noproblem with keys initially hashing to the same neighborhood
-^
But it’s no help if keys initially hash to the same index– Called secondary clustering
-^
Can avoid secondary clustering with a probe function thatdepends on the key: double hashing… Spring 2010
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CSE332: Data Abstractions
Double hashing Idea:
- Given two good hash functions
h^ and
g , it is very unlikely
that for some
key
,^ h(key)
g(key)
- So make the probe function
f(i)
=^
ig(key)*
Probe sequence:
th^ probe:
h(key)
%^
TableSize
st^ probe:
(h(key)
+^
g(key))
%^
TableSize
nd^ probe:
(h(key)
+^
2g(key))*
%^
TableSize
rd^ probe:
(h(key)
+^
3g(key))*
%^
TableSize
probe:
(h(key)
+^
ig(key)) %*
TableSize
Detail: Make sure
g(key)
can’t be
^0
Spring 2010
CSE332: Data Abstractions
Double-hashing analysis •^
Intuition: Since each probe is “jumping” by
g(key)
each time,
we “leave the neighborhood”
and
“go different places from other
initial collisions”
-^
But we could still have a problem like in quadratic probing wherewe are not “safe” (infinite loop despite room in table)– It is known that this cannot happen in at least one case:
-^ h(key)
=^
key
%^
p
-^ g(key)
=^
q^ –
(key
%^
q)
•^^2
<^
q^ <
p
-^ p
and
q^ are prime
Spring 2010
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CSE332: Data Abstractions
More double-hashing facts •^
Assume “uniform hashing”– Means probability of
g(key1)
%^
p^ ==
g(key2)
%^
p^ is
1/p
-^
Non-trivial facts we won’t prove:Average # of probes given
λ^ (in the limit as
TableSize
→^ ∞
- Unsuccessful search (intuitive):– Successful search (less intuitive): -^
Bottom line: unsuccessful bad (but not as bad as linear probing),but successful is not nearly as bad Spring 2010
CSE332: Data Abstractions
log
e^1
⎛^
⎜^
Hashing and comparing •^
Haven’t emphasized enough for a find or a delete of an item oftype
E , we
hash
E , but then as we go through the chain or keep
probing, we have to
compare
each item we see to
E.
•^
So a hash table needs a hash function and a comparator– In Project 2, you’ll use two function objects– The Java standard library uses a more OO approach where
each object has an
equals
method and a
hashCode
method: Spring 2010
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CSE332: Data Abstractions
class
Object
boolean
equals(Object
o)
int
hashCode()
Equal objects must hash the same •^
The Java library (and your project hash table) make a veryimportant assumption that clients must satisfy…
-^
OO way of saying it:
If^ a.equals(b)
, then we must require
a.hashCode()==b.hashCode()
-^
Function object way of saying i:
If^ c.compare(a,b)
0 , then we must require
h.hash(a)
h.hash(b)
•^
Why is this essential? Spring 2010
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CSE332: Data Abstractions
Java bottom line •^
Lots of Java libraries use hash tables, perhaps without yourknowledge
-^
So: If you ever override
equals
, you need to override
hashCode
also in a consistent way
- See CoreJava book, Chapter 5 for other “gotchas” with^ equals Spring 2010
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CSE332: Data Abstractions
Bad Example Spring 2010
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CSE332: Data Abstractions
class
PolarPoint
double
r^
=^ 0.0;
double
theta
=^
void
addToAngle(double
theta2)
{^
theta+=theta2;
…boolean
equals(Object
otherObject)
if(this==otherObject)
return
true;
if(otherObject==null)
return
false;
if(getClass()!=other.getClass())
return
false;
PolarPoint
other
=^
(PolarPoint)otherObject;
double
angleDiff
(theta
–^
other.theta)
%^
(2Math.PI);*
double
rDiff
=^
r^ –
other.r;
return
Math.abs(angleDiff)
<^
&&^
Math.abs(rDiff)
<^
wrong:
must
override
hashCode!
-^ Think about using a hash table holding points }
By the way: comparison has rules too We didn’t emphasize some important “rules” about comparison
functions for:– all our dictionaries– sorting (next major topic) In short, comparison must impose a consistent, total ordering:For all
a ,^
b , and
c ,
compare(a,b)
<^
0 , then
compare(b,a)
>^
compare(a,b)
0 , then
compare(b,a)
compare(a,b)
<^
0 and
compare(b,c)
<^
then
compare(a,c)
<^
Spring 2010
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CSE332: Data Abstractions
Final word on hashing •^
The hash table is one of the most important data structures– Supports only
find
,^ insert
, and
delete
efficiently
•^
Important to use a good hash function
-^
Important to keep hash table at a good size
-^
Side-comment: hash functions have uses beyond hash tables– Examples: Cryptography, check-sums Spring 2010
CSE332: Data Abstractions
