4.21. Implementing an Unordered List: Linked Lists¶
In order to implement an unordered list, we will construct what is
commonly known as a linked list. Recall that we need to be sure that
we can maintain the relative positioning of the items. However, there is
no requirement that we maintain that positioning in contiguous memory.
For example, consider the collection of items shown in
Figure 1. It appears that these values have been placed
randomly. If we can maintain some explicit information in each item,
namely the location of the next item (see Figure 2), then the
relative position of each item can be expressed by simply following the
link from one item to the next.
It is important to note that the location of the first item of the list
must be explicitly specified. Once we know where the first item is, the
first item can tell us where the second is, and so on. The external
reference is often referred to as the head of the list. Similarly,
the last item needs to know that there is no next item.
The basic building block for the linked list implementation is the
node. Each node object must hold at least two pieces of information.
First, the node must contain the list item itself. We will call this the
data field of the node. In addition, each node must hold a reference
to the next node. Listing 1 shows the Python
implementation. To construct a node, you need to supply the initial data
value for the node. Evaluating the assignment statement below will yield
a node object containing the value 93 (see Figure 3). You
should note that we will typically represent a node object as shown in
Figure 4. The Node class also includes the usual methods
to access and modify the data and the next reference.
The special Python reference value None will play an important role
in the Node class and later in the linked list itself. A reference
to None will denote the fact that there is no next node. Note in the
constructor that a node is initially created with next set to
None. Since this is sometimes referred to as “grounding the node,”
we will use the standard ground symbol to denote a reference that is
referring to None. It is always a good idea to explicitly assign
None to your initial next reference values.
As we suggested above, the unordered list will be built from a
collection of nodes, each linked to the next by explicit references. As
long as we know where to find the first node (containing the first
item), each item after that can be found by successively following the
next links. With this in mind, the UnorderedList class must maintain
a reference to the first node. Listing 2 shows the
constructor. Note that each list object will maintain a single reference
to the head of the list.
Initially when we construct a list, there are no items. The assignment
creates the linked list representation shown in
Figure 5. As we discussed in the Node class, the
special reference None will again be used to state that the head of
the list does not refer to anything. Eventually, the example list given
earlier will be represented by a linked list as shown in
Figure 6. The head of the list refers to the first node
which contains the first item of the list. In turn, that node holds a
reference to the next node (the next item) and so on. It is very
important to note that the list class itself does not contain any node
objects. Instead it contains a single reference to only the first node
in the linked structure.
The isEmpty method, shown in Listing 3, simply checks to
see if the head of the list is a reference to None. The result of
the boolean expression self.head==None will only be true if there
are no nodes in the linked list. Since a new list is empty, the
constructor and the check for empty must be consistent with one another.
This shows the advantage to using the reference None to denote the
“end” of the linked structure. In Python, None can be compared to
any reference. Two references are equal if they both refer to the same
object. We will use this often in our remaining methods.
So, how do we get items into our list? We need to implement the add
method. However, before we can do that, we need to address the important
question of where in the linked list to place the new item. Since this
list is unordered, the specific location of the new item with respect to
the other items already in the list is not important. The new item can
go anywhere. With that in mind, it makes sense to place the new item in
the easiest location possible.
Recall that the linked list structure provides us with only one entry
point, the head of the list. All of the other nodes can only be reached
by accessing the first node and then following next links. This
means that the easiest place to add the new node is right at the head,
or beginning, of the list. In other words, we will make the new item the
first item of the list and the existing items will need to be linked to
this new first item so that they follow.
The linked list shown in Figure 6 was built by calling
the add method a number of times.
Note that since 31 is the first item added to the list, it will
eventually be the last node on the linked list as every other item is
added ahead of it. Also, since 54 is the last item added, it will become
the data value in the first node of the linked list.
The add method is shown in Listing 4. Each item of the list
must reside in a node object. Line 2 creates a new node and places the
item as its data. Now we must complete the process by linking the new
node into the existing structure. This requires two steps as shown in
Figure 7. Step 1 (line 3) changes the next reference
of the new node to refer to the old first node of the list. Now that the
rest of the list has been properly attached to the new node, we can
modify the head of the list to refer to the new node. The assignment
statement in line 4 sets the head of the list.
The order of the two steps described above is very important. What
happens if the order of line 3 and line 4 is reversed? If the
modification of the head of the list happens first, the result can be
seen in Figure 8. Since the head was the only external
reference to the list nodes, all of the original nodes are lost and can
no longer be accessed.
The next methods that we will implement–size, search, and
remove–are all based on a technique known as linked list
traversal. Traversal refers to the process of systematically visiting
each node. To do this we use an external reference that starts at the
first node in the list. As we visit each node, we move the reference to
the next node by “traversing” the next reference.
To implement the size method, we need to traverse the linked list
and keep a count of the number of nodes that occurred.
Listing 5 shows the Python code for counting the number of
nodes in the list. The external reference is called current and is
initialized to the head of the list in line 2. At the start of the
process we have not seen any nodes so the count is set to \(0\).
Lines 4–6 actually implement the traversal. As long as the current
reference has not seen the end of the list (None), we move current
along to the next node via the assignment statement in line 6. Again,
the ability to compare a reference to None is very useful. Every
time current moves to a new node, we add \(1\) to count.
Finally, count gets returned after the iteration stops.
Figure 9 shows this process as it proceeds down the list.
Searching for a value in a linked list implementation of an unordered
list also uses the traversal technique. As we visit each node in the
linked list we will ask whether the data stored there matches the item
we are looking for. In this case, however, we may not have to traverse
all the way to the end of the list. In fact, if we do get to the end of
the list, that means that the item we are looking for must not be
present. Also, if we do find the item, there is no need to continue.
Listing 6 shows the implementation for the search method.
As in the size method, the traversal is initialized to start at
the head of the list (line 2). We also use a boolean variable called
found to remember whether we have located the item we are searching
for. Since we have not found the item at the start of the traversal,
found can be set to False (line 3). The iteration in line 4
takes into account both conditions discussed above. As long as there are
more nodes to visit and we have not found the item we are looking for,
we continue to check the next node. The question in line 5 asks whether
the data item is present in the current node. If so, found can be
set to True.
As an example, consider invoking the search method looking for the
Since 17 is in the list, the traversal process needs to move only to the
node containing 17. At that point, the variable found is set to
True and the while condition will fail, leading to the return
value seen above. This process can be seen in Figure 10.
The remove method requires two logical steps. First, we need to
traverse the list looking for the item we want to remove. Once we find
the item (recall that we assume it is present), we must remove it. The
first step is very similar to search. Starting with an external
reference set to the head of the list, we traverse the links until we
discover the item we are looking for. Since we assume that item is
present, we know that the iteration will stop before current gets to
None. This means that we can simply use the boolean found in the
When found becomes True, current will be a reference to the
node containing the item to be removed. But how do we remove it? One
possibility would be to replace the value of the item with some marker
that suggests that the item is no longer present. The problem with this
approach is the number of nodes will no longer match the number of
items. It would be much better to remove the item by removing the entire
In order to remove the node containing the item, we need to modify the
link in the previous node so that it refers to the node that comes after
current. Unfortunately, there is no way to go backward in the linked
list. Since current refers to the node ahead of the node where we
would like to make the change, it is too late to make the necessary
The solution to this dilemma is to use two external references as we
traverse down the linked list. current will behave just as it did
before, marking the current location of the traverse. The new reference,
which we will call previous, will always travel one node behind
current. That way, when current stops at the node to be removed,
previous will be referring to the proper place in the linked list
for the modification.
Listing 7 shows the complete remove method. Lines 2–3
assign initial values to the two references. Note that current
starts out at the list head as in the other traversal examples.
previous, however, is assumed to always travel one node behind
current. For this reason, previous starts out with a value of
None since there is no node before the head (see
Figure 11). The boolean variable found will again be
used to control the iteration.
In lines 6–7 we ask whether the item stored in the current node is the
item we wish to remove. If so, found can be set to True. If we
do not find the item, previous and current must both be moved
one node ahead. Again, the order of these two statements is crucial.
previous must first be moved one node ahead to the location of
current. At that point, current can be moved. This process is
often referred to as “inch-worming” as previous must catch up to
current before current moves ahead. Figure 12 shows
the movement of previous and current as they progress down the
list looking for the node containing the value 17.
Once the searching step of the remove has been completed, we need to
remove the node from the linked list. Figure 13 shows the
link that must be modified. However, there is a special case that needs
to be addressed. If the item to be removed happens to be the first item
in the list, then current will reference the first node in the
linked list. This also means that previous will be None. We said
earlier that previous would be referring to the node whose next
reference needs to be modified in order to complete the remove. In this
case, it is not previous but rather the head of the list that needs
to be changed (see Figure 14).
Line 12 allows us to check whether we are dealing with the special case
described above. If previous did not move, it will still have the
value None when the boolean found becomes True. In that case
(line 13) the head of the list is modified to refer to the node after
the current node, in effect removing the first node from the linked
list. However, if previous is not None, the node to be removed is
somewhere down the linked list structure. In this case the previous
reference is providing us with the node whose next reference must be
changed. Line 15 uses the setNext method from previous to
accomplish the removal. Note that in both cases the destination of the
reference change is current.getNext(). One question that often
arises is whether the two cases shown here will also handle the
situation where the item to be removed is in the last node of the linked
list. We leave that for you to consider.
You can try out the UnorderedList class in ActiveCode 1.
The remaining methods append, insert, index, and pop are
left as exercises. Remember that each of these must take into account
whether the change is taking place at the head of the list or someplace
else. Also, insert, index, and pop require that we name the
positions of the list. We will assume that position names are integers
starting with 0.
Part I: Implement the append method for UnorderedList. What is the time complexity of the method you created?
Part II: In the previous problem, you most likely created an append method that was \(O(n)\) If you add an instance variable to the UnorderedList class you can create an append method that is \(O(1)\). Modify your append method to be \(O(1)\) Be Careful! To really do this correctly you will need to consider a couple of special cases that may require you to make a modification to the add method as well.