2.5.1. Space Comparison¶

Now that you have seen two substantially different implementations for lists, it is natural to ask which is better. In particular, if you must implement a list for some task, which implementation should you choose?

Given a collection of elements to store, they take up some amount of space whether they are simple integers or large objects with many fields. Any container data structure like a list then requires some additional space to organize the elements being stored. This additional space is called overhead.

Array-based lists have the disadvantage that their size must be predetermined before the array can be allocated. Array-based lists cannot grow beyond their predetermined size. Whenever the list contains only a few elements, a substantial amount of space might be tied up in a largely empty array. This empty space is the overhead required by the array-based list. Linked lists have the advantage that they only need space for the objects actually on the list. There is no limit to the number of elements on a linked list, as long as there is free store memory available. The amount of space required by a linked list is \(\Theta(n)\), while the space required by the array-based list implementation is \(\Omega(n)\), but can be greater.

Array-based lists have the advantage that there is no wasted space for an individual element. Linked lists require that an extra pointer for the next field be added to every list node. So the linked list has these next pointers as overhead. If the element size is small, then the overhead for links can be a significant fraction of the total storage. When the array for the array-based list is completely filled, there is no wasted space, and so no overhead. The array-based list will then be more space efficient, by a constant factor, than the linked implementation.

A simple formula can be used to determine whether the array-based list or the linked list implementation will be more space efficient in a particular situation. Call \(n\) the number of elements currently in the list, \(P\) the size of a pointer in storage units (typically four bytes), \(E\) the size of a data element in storage units (this could be anything, from one bit for a Boolean variable on up to thousands of bytes or more for complex records), and \(D\) the maximum number of list elements that can be stored in the array. The amount of space required for the array-based list is \(DE\), regardless of the number of elements actually stored in the list at any given time. The amount of space required for the linked list is \(n(P + E)\). The smaller of these expressions for a given value \(n\) determines the more space-efficient implementation for \(n\) elements. In general, the linked implementation requires less space than the array-based implementation when relatively few elements are in the list. Conversely, the array-based implementation becomes more space efficient when the array is close to full. Using the equation, we can solve for \(n\) to determine the break-even point beyond which the array-based implementation is more space efficient in any particular situation. This occurs when

If \(P = E\), then the break-even point is at \(D/2\). This would happen if the element field is either a four-byte int value or a pointer, and the next field is a typical four-byte pointer. That is, the array-based implementation would be more efficient (if the link field and the element field are the same size) whenever the array is more than half full.

As a rule of thumb, linked lists are more space efficient when implementing lists whose number of elements varies widely or is unknown. Array-based lists are generally more space efficient when the user knows in advance approximately how large the list will become, and can be confident that the list will never grow beyond a certain limit.

2.5.2. Time Comparison¶

Array-based lists are faster for access by position. Positions can easily be adjusted forwards or backwards by the next and prev methods. These operations always take \(\Theta(1)\) time. In contrast, singly linked lists have no explicit access to the previous element, and access by position requires that we march down the list from the front (or the current position) to the specified position. Both of these operations require \(\Theta(n)\) time in the average and worst cases, if we assume that each position on the list is equally likely to be accessed on any call to prev or moveToPos.

Given a pointer to a suitable location in the list, the insert and remove methods for linked lists require only \(\Theta(1)\) time. Array-based lists must shift the remainder of the list up or down within the array. This requires \(\Theta(n)\) time in the average and worst cases. For many applications, the time to insert and delete elements dominates all other operations. For this reason, linked lists are often preferred to array-based lists.

When implementing the array-based list, an implementor could allow the size of the array to grow and shrink depending on the number of elements that are actually stored. This data structure is known as a dynamic array. For example, both the Java and C++/STL Vector classes implement a dynamic array, and JavaScript arrays are always dynamic. Dynamic arrays allow the programmer to get around the limitation on the traditional array that its size cannot be changed once the array has been created. This also means that space need not be allocated to the dynamic array until it is to be used. The disadvantage of this approach is that it takes time to deal with space adjustments on the array. Each time the array grows in size, its contents must be copied. A good implementation of the dynamic array will grow and shrink the array in such a way as to keep the overall cost for a series of insert/delete operations relatively inexpensive, even though an occasional insert/delete operation might be expensive. A simple rule of thumb is to double the size of the array when it becomes full, and to cut the array size in half when it becomes one quarter full. To analyze the overall cost of dynamic array operations over time, we need to use a technique known as amortized analysis.