Purely functional
Examples of purely functional data structures
Linked lists
This example is taken from Okasaki. See the bibliography.
Singly linked lists are a bread and butter data structure in functional languages. In ML-derived languages and Haskell , they are purely functional because once a node in the list has been allocated, it cannot be modified, only copied or destroyed.
Consider the two lists:
xs = [0, 1, 2]
ys = [3, 4, 5]
These would be represented in memory by:
where a circle indicates a node in the list (the arrow out showing the second element of the node which is a pointer to another node).
Now concatenating the two lists:
zs = xs ++ ys
results in the following memory structure:
Notice that the nodes in list xs have been copied, but the nodes in ys are shared. As a result, the original lists (xs and ys) persist and have not been modified.
The reason for the copy is that the last node in xs (the node containing the original value 2) cannot be modified to point to the start of ys, because that would change the value of xs.
Trees
This example is taken from Okasaki. See the bibliography.
Consider a binary tree used for fast searching, where every node has the recursive invariant that subnodes on the left are less than the node, and subnodes on the right are greater than the node.
For instance, the set of data
xs = [a, b, c, d, f, g, h]
might be represented by the following binary search tree:
A function which inserts data into the binary tree and maintains the invariant is:
fun insert (x, E) = T (E, x, E)
| insert (x, s as T (a, y, b)) =
if x < y then T (insert (x, a), y, b)
else if x > y then T (a, y, insert (x, b))
else s
After executing
ys = insert ("e", xs)
we end up with the following:
Notice two points: Firstly the original tree (xs) persists. Secondly many common nodes are shared between the old tree and the new tree. Such persistence and sharing is difficult to manage without some form of garbage collection (GC) to automatically free up nodes which have no live references, and this is why GC is a feature commonly found in functional programming languages.
Benefits and applications
The persistence property of purely functional data structures can be advantageous in the development of many applications which deal with multiple versions of an object.
For example, say you have a thesaurus service on a website which uses a red-black tree to store its list of which words are synonyms for which other words. Because your thesaurus is comprehensive, this tree has about a million nodes. Now say you wish to add a feature that allows each user of your site to add their own custom words to their personal thesaurus. One way to do this is to make a copy of the tree for each user, and then add their custom words to it, but this is wasteful, because each user using the service simultaneously would require memory for at least a million nodes. Moreover, it would cause a significant processing delay to make the complete copy of the tree.
A better approach is to store the words in a purely functional red-black tree. Then, you simply take the original version and produce a new tree based on it for each set of custom words. Because these new trees share large amounts of structure with the main tree, the space overhead for each additional user is at most 2 klog2 n or about 40k nodes, where k is the number of custom nodes.
Besides their efficiency benefits, the inherent referential transparency of functional data structures tends to make purely functional computation more amenable to analysis and optimization, both formal and informal.
Reference cycles
Since every value in a purely functional computation is built up out of existing values, it would seem that it is impossible to create a cycle of references, resulting in a reference graph (a graph which has edges from objects to the objects they reference) that is a directed acyclic graph. However, in most functional languages, functions can be defined recursively; this capability allows recursive structures using functional suspensions. In lazy languages, such as Haskell, all data structures are explicitly suspended; in these languages any data structure can be recursive. Some other languages, such as Objective Caml, allow the explicit definition of recursive values.
See Also
External Links
- Purely Functional Data Structures thesis by Chris Okasaki (PDF format)
- Making Data-Structures Persistent by James R. Driscoll, Neil Sarnak, Daniel D. Sleator, Robert E. Tarjan (PDF)
- Fully Persistent Lists with Catenation by James R. Driscoll, Daniel D. Sleator, Robert E. Tarjan (PDF)
- Persistent Data Structures from MIT open course Advanced Algorithms
Bibliography
Purely functional data structures by Chris Okasaki, Cambridge University Press, 1998, ISBN 0-521-66350-4 .
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