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: Linear Probing
The
data structure uses an array of lists, where the
th
list stores all elements
such that
. An alternative,
called open addressing
is to store the elements directly in an
array,
, with each array location in
storing at most one value.
This approach is taken by the
described in this
section. In some places, this data structure is described as open
addressing with linear probing.
The main idea behind a
is that we would, ideally, like
to store the element
with hash value
in the table location
![$ \mathtt{t[i]}$](img785.png){kind=small}
. If we cannot do this (because some element is already stored
there) then we try to store it at location
![$ \ensuremath{\mathtt{t}}[(\ensuremath{\mathtt{i}}+1)\bmod\ensuremath{\mathtt{t.length}}]$](img875.png){kind=small}
;
if that's not possible, then we try
![$ \ensuremath{\mathtt{t}}[(\ensuremath{\mathtt{i}}+2)\bmod\ensuremath{\mathtt{t.length}}]$](img876.png){kind=small}
,
and so on, until we find a place for
.
There are three types of entries stored in
:
that we are representing;
values: at array locations where no data has ever been
stored; and
values: at array locations where data was once stored but that
has since been deleted.
, that keeps track of the number of elements
in the
, a counter,
, keeps track of the number of
elements of Types 1 and 3. That is,
is equal to
plus the number
of
values in
. To make this work efficiently, we need
to be considerably larger than
, so that there are lots of
values in
. The operations on a
therefore maintain
the invariant that
.
To summarize, a
contains an array,
, that stores
data elements, and integers
and
that keep track of the number of
data elements and non-
values of
, respectively. Because many
hash functions only work for table sizes that are a power of 2, we also
keep an integer
and maintain the invariant that
.
array<T> t; int n; // number of values in T int q; // number of non-null entries in T int d; // t.length = 2^d
The
operation in a
is simple. We start
at array entry
where
and search entries
,
,
, and so on,
until we find an index
such that, either,
![$ \mathtt{t[i']=x}$](img882.png){kind=small}
, or
![$ \mathtt{t[i']=null}$](img883.png){kind=small}
.
In the former case we return
![$ \mathtt{t[i']}$](img884.png){kind=small}
. In the latter case, we conclude
that
is not contained in the hash table and return
.
T find(T x) {
int i = hash(x);
while (t[i] != null) {
if (t[i] != del && t[i] == x) return t[i];
i = (i == t.length-1) ? 0 : i + 1; // increment i
}
return null;
}
The
operation is also fairly easy to implement. After checking
that
is not already stored in the table (using
), we search
,
,
,
and so on, until we find a
or
and store
at that location,
increment
, and
, if appropriate.
bool add(T x) {
if (find(x) != null) return false;
if (2*(q+1) > t.length) resize(); // max 50% occupancy
int i = hash(x);
while (t[i] != null && t[i] != del)
i = (i == t.length-1) ? 0 : i + 1; // increment i
if (t[i] == null) q++;
n++;
t[i] = x;
return true;
}
By now, the implementation of the
operation should be obvious.
We search
,
,
, and so on until we find an index
such that
or
. In the former case, we set
![$ \mathtt{t[i']=del}$](img885.png){kind=small}
and return
. In the latter case we conclude that
was not stored in the
table (and therefore cannot be deleted) and return
.
T remove(T x) {
int i = hash(x);
while (t[i] != null) {
T y = t[i];
if (y != del && x == y) {
t[i] = del;
n--;
if (8*n < t.length) resize(); // min 12.5% occupancy
return y;
}
i = (i == t.length-1) ? 0 : i + 1; // increment i
}
return null;
}
The correctness of the
,
, and
methods is
easy to verify, though it relies on the use of
values. Notice
that none of these operations ever sets a non-
entry to
.
Therefore, when we reach an index
such that
, this is
a proof that the element,
, that we are searching for is not stored
in the table;
has always been
, so there is no reason that
a previous
operation would have proceeded beyond index
.
The
method is called by
when the number of non-
entries exceeds
or by
when the number of
data entries is less than
. The
method works
like the
methods in other array-based data structures.
We find the smallest non-negative integer
such that
. We reallocate the array
so that it has size
,
and then we insert all the elements in the old version of
into the
newly-resized copy of
. While doing this, we reset
equal to
since the newly-allocated
contains no
values.
void resize() {
d = 1;
while ((1<<d) < 3*n) d++;
array<T> tnew(1<<d, null);
q = n;
// insert everything into tnew
for (int k = 0; k < t.length; k++) {
if (t[k] != null && t[k] != del) {
int i = hash(t[k]);
while (tnew[i] != null)
i = (i == tnew.length-1) ? 0 : i + 1;
tnew[i] = t[k];
}
}
t = tnew;
}
Notice that each operation,
,
, or
, finishes
as soon as (or before) it discovers the first
entry in
.
The intuition behind the analysis of linear probing is that, since at
least half the elements in
are equal to
, an operation should
not take long to complete because it will very quickly come across a
entry. We shouldn't rely too heavily on this intuition, though,
because it would lead us to (the incorrect) conclusion that the expected
number of locations in
examined by an operation is at most 2.
For the rest of this section, we will assume that all hash values are
independently and uniformly distributed in
.
This is not a realistic assumption, but it will make it possible for
us to analyze linear probing. Later in this section we will describe a
method, called tabulation hashing, that produces a hash function that is
``good enough'' for linear probing. We will also assume that all indices
into the positions of
are taken modulo
, so that
is really a shorthand for
![$ \ensuremath{\mathtt{t}}[\ensuremath{\mathtt{i}}\bmod\ensuremath{\mathtt{t.length}}]$](img889.png){kind=small}
.
We say that a run of length 
that starts at
occurs when all
the table entries
![$ \ensuremath{\mathtt{t[i]}}, \ensuremath{\mathtt{t[i+1]}},\ldots,\ensuremath{\mathtt{t}}[\ensuremath{\mathtt{i}}+k-1]$](img890.png){kind=small}
are non-
and
![$ \ensuremath{\mathtt{t}}[\ensuremath{\mathtt{i}}-1]=\ensuremath{\mathtt{t}}[\ensuremath{\mathtt{i}}+k]=\ensuremath{\mathtt{null}}$](img891.png){kind=small}
. The number of non-
elements of
is exactly
and the
method ensures that, at all times,
. There are
elements
that have been inserted into
since the last
operation.
By our assumption, each of these has a hash value,
,
that is uniform and independent of the rest. With this setup, we can
prove the main lemma required to analyze linear probing.
starts at
, then there are exactly
elements
such that
.
The probability that this occurs is exactly
elements, these elements must hash to
one of the locations and the remaining
elements must hash to
the other
table locations.5.1
In the following derivation we will cheat a little and replace 
with
. Stirling's Approximation (Section 1.3.2) shows that this
is only a factor of
from the truth. This is just done to
make the derivation simpler; Exercise 5.4 asks the reader to
redo the calculation more rigorously using Stirling's Approximation in
its entirety.
The value of 
is maximized when
is minimum, and the data
structure maintains the invariant that
, so
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[Stirling's approximation] | |||
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, which holds
for all 
.) Since
, this completes
the proof.
Using Lemma 5.4 to prove upper-bounds on the expected
running time of
,
, and
is now fairly
straightforward. Consider the simplest case, where we execute
for some value
that has never been stored in the
. In this case,
is a random value in
independent of the contents of
. If
is part of a run of length , then the time it takes to execute the
operation is at most 
. Thus, the expected running time
can be upper-bounded by
contributes to the inner sum times for a total contribution of 
, so the above sum can be rewritten as
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is an exponentially decreasing
series.5.2Therefore, we conclude that the expected running time of the
operation
for a value
that is not contained in a
is 
.
If we ignore the cost of the
operation, then the above
analysis gives us all we need to analyze the cost of operations on a
.
First of all, the analysis of
given above applies to the
operation when
is not contained in the table. To analyze
the
operation when
is contained in the table, we need only
note that this is the same as the cost of the
operation that
previously added
to the table. Finally, the cost of a
operation is the same as the cost of a
operation.
In summary, if we ignore the cost of calls to
, all operations
on a
run in expected time. Accounting for the
cost of resize can be done using the same type of amortized analysis
performed for the
data structure in Section 2.1.
The following theorem summarizes the performance of the
data structure:
implements the
interface. Ignoring the
cost of calls to
, a
supports the operations
,
, and
in expected time per
operation.
Furthermore, beginning with an empty
, any
sequence of 
and
operations results in a total
of 
time spent during all calls to
.
While analyzing the
structure, we made a very strong
assumption: That for any set of elements,
,
the hash values
are independently and
uniformly distributed over the set
. One way to
achieve this is to store a giant array,
, of length
,
where each entry is a random
-bit integer, independent of all the
other entries. In this way, we could implement
by extracting
a
-bit integer from
![$ \mathtt{tab[x.hashCode()]}$](img934.png){kind=small}
:
int idealHash(T x) {
return tab[hashCode(x) >> w-d];
}
Unfortunately, storing an array of size
is prohibitive in terms
of memory usage. The approach used by tabulation hashing is to,
instead, treat
-bit integers as being comprised of
integers, each having only
bits. In this way, tabulation hashing
only needs
arrays each of length
. All the entries
in these arrays are independent
-bit integers. To obtain the value
of
we split
up into
-bit integers
and use these as indices into these arrays. We then combine all these
values with the bitwise exclusive-or operator to obtain
.
The following code shows how this works when
and
:
int hash(T x) {
unsigned h = hashCode(x);
return (tab[0][h&0xff]
^ tab[1][(h>>8)&0xff]
^ tab[2][(h>>16)&0xff]
^ tab[3][(h>>24)&0xff])
>> (w-d);
}
In this case,
is a two-dimensional array with four columns and
rows.
One can easily verify that, for any
,
is uniformly
distributed over
. With a little work, one
can even verify that any pair of values have independent hash values.
This implies tabulation hashing could be used in place of multiplicative
hashing for the
implementation.
However, it is not true that any set of
distinct values gives a set
of
independent hash values. Nevertheless, when tabulation hashing is
used, the bound of Theorem 5.2 still holds. References for
this are provided at the end of this chapter.
is greater
than the probability that a run of length starts at
, since the definition of does not include the requirement
.
such
that, for all 
,
.