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17.1 Embedded references
We have seen examples of attributes that refer to other objects, which we called embedded references (see Section 12.8). A common data structure, the linked list, takes advantage of this feature.
Linked lists are made up of nodes, where each node contains a reference to the next node in the list. In addition, each node contains a unit of data called the cargo.
A linked list is considered a recursive data structure because it has a recursive definition.
A linked list is either:
17.2 The Node class
As usual when writing a new class, we'll start with the initialization and __str__ methods so that we can test the basic mechanism of creating and displaying the new type:
As usual, the parameters for the initialization method are optional. By default, both the cargo and the link, next, are set to None.
The string representation of a node is just the string representation of the cargo. Since any value can be passed to the str function, we can store any value in a list.
To test the implementation so far, we can create a Node and print it:
>>> node = Node("test")
To make it interesting, we need a list with more than one node:
>>> node1 = Node(1)
This code creates three nodes, but we don't have a list yet because the nodes are not linked. The state diagram looks like this:
To link the nodes, we have to make the first node refer to the second and the second node refer to the third:
>>> node1.next = node2
The reference of the third node is None, which indicates that it is the end of the list. Now the state diagram looks like this:
17.3 Lists as collections
Lists are useful because they provide a way to assemble multiple objects into a single entity, sometimes called a collection. In the example, the first node of the list serves as a reference to the entire list.
To pass the list as an argument, we only have to pass a reference to the first node. For example, the function printList takes a single node as an argument. Starting with the head of the list, it prints each node until it gets to the end:
To invoke this function, we pass a reference to the first node:
Inside printList we have a reference to the first node of the list, but there is no variable that refers to the other nodes. We have to use the next value from each node to get to the next node.
To traverse a linked list, it is common to use a loop variable like node to refer to each of the nodes in succession.
This diagram shows the nodes in the list and the values that node takes on:
By convention, lists are often printed in brackets with commas between the elements, as in [1, 2, 3]. As an exercise, modify printList so that it generates output in this format.
17.4 Lists and recursion
It is natural to express many list operations using recursive methods. For example, the following is a recursive algorithm for printing a list backwards:
Of course, Step 2, the recursive call, assumes that we have a way of
printing a list backward. But if we assume that the recursive
All we need are a base case and a way of proving that for any list, we will eventually get to the base case. Given the recursive definition of a list, a natural base case is the empty list, represented by None:
The first line handles the base case by doing nothing. The next two lines split the list into head and tail. The last two lines print the list. The comma at the end of the last line keeps Python from printing a newline after each node.
We invoke this function as we invoked printList:
The result is a backward list.
You might wonder why printList and printBackward are functions and not methods in the Node class. The reason is that we want to use None to represent the empty list and it is not legal to invoke a method on None. This limitation makes it awkward to write list-manipulating code in a clean object-oriented style.
17.5 Infinite lists
There is nothing to prevent a node from referring back to an earlier node in the list, including itself. For example, this figure shows a list with two nodes, one of which refers to itself:
If we invoke printList on this list, it will loop forever. If we invoke printBackward, it will recurse infinitely. This sort of behavior makes infinite lists difficult to work with.
Nevertheless, they are occasionally useful. For example, we might represent a number as a list of digits and use an infinite list to represent a repeating fraction.
Regardless, it is problematic that we cannot prove that printList and printBackward terminate. The best we can do is the hypothetical statement, "If the list contains no loops, then these functions will terminate." This sort of claim is called a precondition. It imposes a constraint on one of the arguments and describes the behavior of the function if the constraint is satisfied. You will see more examples soon.
17.6 The fundamental ambiguity theorem
One part of printBackward might have raised an eyebrow:
head = list
After the first assignment, head and list have the same type and the same value. So why did we create a new variable?
The reason is that the two variables play different roles. We think of head as a reference to a single node, and we think of list as a reference to the first node of a list. These "roles" are not part of the program; they are in the mind of the programmer.
In general we can't tell by looking at a program what role a variable plays. This ambiguity can be useful, but it can also make programs difficult to read. We often use variable names like node and list to document how we intend to use a variable and sometimes create additional variables to disambiguate.
We could have written printBackward without head and tail, which makes it more concise but possibly less clear:
def printBackward(list) :
Looking at the two function calls, we have to remember that printBackward treats its argument as a collection and print treats its argument as a single object.
The fundamental ambiguity theorem describes the ambiguity that is inherent in a reference to a node:
A variable that refers to a node might treat the node as a single object or as the first in a list of nodes.
17.7 Modifying lists
There are two ways to modify a linked list. Obviously, we can change the cargo of one of the nodes, but the more interesting operations are the ones that add, remove, or reorder the nodes.
As an example, let's write a function that removes the second node in the list and returns a reference to the removed node:
Again, we are using temporary variables to make the code more readable. Here is how to use this function:
This state diagram shows the effect of the operation:
What happens if you invoke this function and pass a list with only one element (a singleton)? What happens if you pass the empty list as an argument? Is there a precondition for this function? If so, fix the function to handle a violation of the precondition in a reasonable way.
17.8 Wrappers and helpers
It is often useful to divide a list operation into two functions. For example, to print a list backward in the format [3 2 1] we can use the printBackward function to print 3 2 1 but we need a separate function to print the brackets. Let's call it printBackwardNicely:
def printBackwardNicely(list) :
Again, it is a good idea to check functions like this to see if they work with special cases like an empty list or a singleton.
When we use this function elsewhere in the program, we invoke printBackwardNicely directly, and it invokes printBackward on our behalf. In that sense, printBackwardNicely acts as a wrapper, and it uses printBackward as a helper.
17.9 The LinkedList class
There are some subtle problems with the way we have been implementing lists. In a reversal of cause and effect, we'll propose an alternative implementation first and then explain what problems it solves.
First, we'll create a new class called LinkedList. Its attributes are an integer that contains the length of the list and a reference to the first node. LinkedList objects serve as handles for manipulating lists of Node objects:
class LinkedList :
One nice thing about the LinkedList class is that it provides a natural place to put wrapper functions like printBackwardNicely, which we can make a method of the LinkedList class:
Just to make things confusing, we renamed printBackwardNicely. Now there are two methods named printBackward: one in the Node class (the helper); and one in the LinkedList class (the wrapper). When the wrapper invokes self.head.printBackward, it is invoking the helper, because self.head is a Node object.
Another benefit of the LinkedList class is that it makes it easier to add or remove the first element of a list. For example, addFirst is a method for LinkedLists; it takes an item of cargo as an argument and puts it at the beginning of the list:
Some lists are "well formed"; others are not. For example, if a list contains a loop, it will cause many of our methods to crash, so we might want to require that lists contain no loops. Another requirement is that the length value in the LinkedList object should be equal to the actual number of nodes in the list.
Requirements like these are called invariants because, ideally, they should be true of every object all the time. Specifying invariants for objects is a useful programming practice because it makes it easier to prove the correctness of code, check the integrity of data structures, and detect errors.
One thing that is sometimes confusing about invariants is that there are times when they are violated. For example, in the middle of addFirst, after we have added the node but before we have incremented length, the invariant is violated. This kind of violation is acceptable; in fact, it is often impossible to modify an object without violating an invariant for at least a little while. Normally, we require that every method that violates an invariant must restore the invariant.
Warning: the HTML version of this document is generated from Latex and may contain translation errors. In particular, some mathematical expressions are not translated correctly.