A DLList (doubly-linked list) is very similar to an SLList except
that each node
in a DLList has references to both the node
that follows it and the node
that precedes it.
class Node { T x; Node prev, next; }
When implementing an SLList, we saw that there were always several
special cases to worry about. For example, removing the last element
from an SLList or adding an element to an empty SLList requires care
to ensure that
and
are correctly updated. In a DLList,
the number of these special cases increases considerably. Perhaps the
cleanest way to take care of all these special cases in a DLList is to
introduce a
node.
This is a node that does not contain any data,
but acts as a placeholder so that there are no special nodes; every node
has both a
and a
, with
acting as the node that
follows the last node in the list and that precedes the first node in
the list. In this way, the nodes of the list are (doubly-)linked into
a cycle, as illustrated in Figure 3.2.
int n; Node dummy; DLList() { dummy = new Node(); dummy.next = dummy; dummy.prev = dummy; n = 0; }
Finding the node with a particular index in a DLList is easy; we can
either start at the head of the list (
) and work forward,
or start at the tail of the list (
) and work backward.
This allows us to reach the
th node in
time:
Node getNode(int i) { Node p = null; if (i < n / 2) { p = dummy.next; for (int j = 0; j < i; j++) p = p.next; } else { p = dummy; for (int j = n; j > i; j--) p = p.prev; } return p; }
The
and
operations are now also easy. We first find the
th node and then get or set its
value:
T get(int i) { if (i < 0 || i > n - 1) throw new IndexOutOfBoundsException(); return getNode(i).x; } T set(int i, T x) { if (i < 0 || i > n - 1) throw new IndexOutOfBoundsException(); Node u = getNode(i); T y = u.x; u.x = x; return y; }
The running time of these operations is dominated by the time it takes
to find the
th node, and is therefore
.
If we have a reference to a node
in a DLList and we want to insert a
node
before
, then this is just a matter of setting
,
, and then adjusting
and
. (See Figure 3.3.)
Thanks to the dummy node, there is no need to worry about
or
not existing.
Node addBefore(Node w, T x) { Node u = new Node(); u.x = x; u.prev = w.prev; u.next = w; u.next.prev = u; u.prev.next = u; n++; return u; }
Now, the list operation
is trivial to implement. We find the
th node in the DLList and insert a new node
that contains
just before it.
void add(int i, T x) { if (i < 0 || i > n) throw new IndexOutOfBoundsException(); addBefore(getNode(i), x); }
The only non-constant part of the running time of
is the time
it takes to find the
th node (using
). Thus,
runs in
time.
Removing a node
from a DLList is easy. We only need to adjust
pointers at
and
so that they skip over
. Again, the
use of the dummy node eliminates the need to consider any special cases:
void remove(Node w) { w.prev.next = w.next; w.next.prev = w.prev; n--; }
Now the
operation is trivial. We find the node with index
and remove it:
T remove(int i) { if (i < 0 || i > n - 1) throw new IndexOutOfBoundsException(); Node w = getNode(i); remove(w); return w.x; }
Again, the only expensive part of this operation is finding the
th node
using
, so
runs in
time.
The following theorem summarizes the performance of a DLList:
It is worth noting that, if we ignore the cost of the
operation, then all operations on a DLList take constant time.
Thus, the only expensive part of operations on a DLList is finding
the relevant node. Once we have the relevant node, adding, removing,
or accessing the data at that node takes only constant time.
This is in sharp contrast to the array-based List implementations of Chapter 2; in those implementations, the relevant array item can be found in constant time. However, addition or removal requires shifting elements in the array and, in general, takes non-constant time.
For this reason, linked list structures are well-suited to applications where references to list nodes can be obtained through external means. An example of this is the LinkedHashSet data structure found in the Java Collections Framework, in which a set of items is stored in a doubly-linked list and the nodes of the doubly-linked list are stored in a hash table (discussed in Chapter 5). When elements are removed from a LinkedHashSet, the hash table is used to find the relevant list node in constant time and then the list node is deleted (also in constant time).
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