[This file and the accompanying source code were derived from
Oleg's code for Gambit available from
http://okmij.org/ftp/Scheme/lib/treap.scm
]
An implementation of an ordered dictionary data structure, based
on randomized search trees (treaps) by Seidel and Aragon:
R. Seidel and C. R. Aragon. Randomized Search Trees.
Algorithmica, 16(4/5):464-497, 1996.
This code defines a treap object that implements an ordered dictionary
mapping of keys to values. The object responds to a variety of query and
update messages, including efficient methods for finding the minimum and
maximum keys and their associated values as well as traversing of a
treap in an ascending or descending order of keys. Looking up an arbitrary
or the min/max keys, and deleting the min/max keys require no more
key comparisons than the depth of the treap, which is O(log n) where
n is the total number of keys in the treap. Arbitrary key deletion and
insertions run in O(log n) _amortized_ time.
This code is inspired by a Stefan Nilsson's article "Treaps in Java"
(Dr.Dobb's Journal, July 1997, p.40-44) and by the Java implementation
of treaps described in the article. Yet this Scheme code has been
developed completely from scratch, using the description of the algorithm
given in the article, and insight gained from examining the Java source
code. As a matter of fact, treap insertion and deletion algorithms
implemented in this code differ from the ones described in the article
and implemented in the Java code; this Scheme implementation uses fewer
assignments and comparisons (see below for details). Some insight as
to a generic tree interface gleaned from wttree.scm, "Weight balanced
trees" by Stephen Adams (a part of The Scheme Library, slib2b1).
A treap is a regular binary search tree, with one extension. The extension
is that every node in a tree, beside a key, a value, and references to
its left and right children, has an additional constant field, a priority.
The value of this field is set (to a random integer number) when the node
is constructed, and is not changed afterwards. At any given moment,
the priority of every non-leaf node never exceeds the priorities of its
children. When a new node is inserted, we check that this invariant holds;
otherwise, we perform a right or left rotation that swaps a parent and
its kid, keeping the ordering of keys intact; the changes may need to be
propagated recursively up. The priority property, and the fact they are
chosen at random, makes a treap look like a binary search tree built by
a random sequence of insertions. As the article shows, this makes a treap
a balanced tree: the expected length of an average search path is roughly
1.4log2(n)-1, and the expected length of the longest search path is about
4.3log2(n). See the Stefan Nilsson's article for more details.
(make-treap key-compare) -> treap
Creates a treap object. Here KEY-COMPARE-PROC is a user-supplied
function
KEY-COMPARE-PROC key1 key2
that takes two keys and returns a negative, positive, or zero number
depending on how the first key compares to the second. The treap uses
SRFI-27's DEFAULT-RANDOM-SOURCE so you may want to initialize this
once with (random-source-randomize! default-random-source).
(treap? thing) -> boolean
Type predicate for treaps.
(treap-get treap key [default-clause]) -> key-value pair or value of default-clause
Searches the treap for an association with a given KEY, and returns a
(key . value) pair of the found association. If an association with
the KEY cannot be located in the treap, the PROC returns the result of
evaluating the DEFAULT-CLAUSE. If the default clause is omitted, an
error is signalled. The KEY must be comparable to the keys in the
treap by a key-compare predicate (which has been specified when the
treap was created)
(treap-get-min treap) -> key-value pair
(treap-get-max treap) -> key-value pair
return a (key . value) pair for an association in the treap with the
smallest/largest key. If the treap is empty, an error is signalled.
(treap-delete-min! treap) -> key-value pair
(treap-delete-max! treap) -> key-value pair
remove the min/max key and the corresponding association from the
treap. Return a (key . value) pair of the removed association. If
the treap is empty, an error is signalled.
(treap-empty? treap) -> boolean
returns #t if the treap is empty.
(treap-size treap) -> integer
returns the size (the number of associations) in the treap.
(treap-depth treap) -> integer
returns the depth of the tree. It requires the complete traversal of
the tree, so use sparingly.
(treap-clear! treap) -> unspecific
removes all associations from the treap (thus making it empty).
(treap-put! treap key value) -> key-value pair or #f
adds the corresponding association to the treap. If an association
with the same KEY already exists, its value is replaced with the VALUE
(and the old (key . value) association is returned). Otherwise, the
return value is #f.
(treap-delete! treap key [default-clause]) -> key-value pair
searches the treap for an association with a given KEY, deletes it,
and returns a (key . value) pair of the found and deleted association.
If an association with the KEY cannot be located in the treap, the
PROC returns the result of evaluating the DEFAULT-CLAUSE. If the
default clause is omitted, an error is signalled.
(treap-for-each-ascending treap proc) -> unspecific
applies the given procedure PROC to each (key . value) association of
the treap, from the one with the smallest key all the way to the one
with the max key, in an ascending order of keys. The treap must not
be empty.
(treap-for-each-descending treap proc) -> unspecific
applies the given procedure PROC to each (key . value) association of
the treap, in the descending order of keys. The treap must not be
empty.
(treap-debugprint treap) -> unspecific
prints out all the nodes in the treap, for debug purposes.
Notes on the algorithm
As the DDJ paper shows, insertion of a node into a treap is a simple
recursive algorithm, Example 1 of the paper. This algorithm is implemented
in the accompanying source [Java] code as
<BLOCKQUOTE>
private Tree insert(Tree node, Tree tree) {
if (tree == null) return node;
int comp = node.key.compareTo(tree.key);
if (comp < 0) {
tree.left = insert(node, tree.left);
if (tree.prio > tree.left.prio)
tree = tree.rotateRight();
} else if (comp > 0) {
tree.right = insert(node, tree.right);
if (tree.prio > tree.right.prio)
tree = tree.rotateLeft();
} else {
keyFound = true;
prevValue = tree.value;
tree.value = node.value;
}
return tree;
}
</BLOCKQUOTE>
This algorithm, however, is not as efficient as it could be. Suppose we
try to insert a node which turns out to already exist in the tree,
at a depth D. The algorithm above would descend into this node in the
winding phase of the algorithm, replace the node's value, and, in the
unwinding phase of the recursion, would perform D assignments of the kind
tree.left = insert(node, tree.left);
and D comparisons of nodes' priorities. None of these priority checks and
assignments are actually necessary: as we haven't added any new node,
the tree structure hasn't changed.
Therefore, the present Scheme code implements a different insertion
algorithm, which avoids doing unnecessary operations. The idea is simple:
if we insert a new node into some particular branch of the treap and verify
that this branch conforms to the treap priority invariant, we are certain
the invariant holds throughout the entire treap, and no further checks
(up the tree to the root) are necessary. In more detail:
- Starting from the root, we recursively descend until we find
a node with a given key, or a place a new node can be inserted.
- We insert the node and check to see if its priority is less than
that of its parent. If this is the case, we left- or right-rotate
the tree to make the old parent a child of the new node, and the
new node a new root of this particular branch. We return this new
parent as an indication that further checks up the tree are
necessary. If the new node conforms to the parent's priority, we
insert it and return #f
- On the way up, we check the priorities again and rotate the tree
to restore the priority invariant at the current level if needed.
- If no changes are made at the current level, we return a flag #f
meaning that no further changes or checks are necessary at the
higher levels.
Thus, if a new node was originally inserted at a level D in the tree (level
0 being the root) but then migrated up by L levels (because of its priority),
the original insertion algorithm would perform (D-1) assignments,
(D-1) priority checks, plus L rotations (at a cost of 2 assignments in the
treap each). Our algorithm does only (L+1) node priority checks and
max(2(L-1)+2,1) assignments.
Note if priorities are really (uniformly) random, L is uniformly distributed
over [0,D], so the average cost of our algorithm is
D/2 +1 checks and D assignments
compared to
D-1 checks and 2D-1 assignments
for the original algorithm described in the DDJ paper.
The similar gripe applies to the Java implementation of a node deletion
algorithm:
<BLOCKQUOTE>
private Tree delete(Ordered key, Tree t) {
if (t == null) return null;
int comp = key.compareTo(t.key);
if (comp < 0)
t.left = delete(key, t.left);
else if (comp > 0)
t.right = delete(key, t.right);
else {
keyFound = true;
prevValue = t.value;
t = t.deleteRoot();
}
return t;
}
</BLOCKQUOTE>
The algorithm as implemented looks fully-recursive. Furthermore, deletion
of a node at a level D in the treap involves at least D assignments, most
of them being unnecessary. Indeed, if a node being deleted is a leaf, only
one change to the tree is needed to detach the node. Deleting a node
obviously requires a left- or a right-kid field of the node's parent be
modified (cleared). This change, however does NOT need to be propagated up:
deleting of a node does not violate neither ordering nor priority invariants
of the treap; all changes are confined to a branch rooted at the
parent of the deleted node.
This Scheme code implements node deletion algorithm in the optimal way,
performing only those assignments which are absolutely necessary, and
replacing full recursions with tail recursions (which are simply iterations).
Our implementation is also simpler and clearer, making use of a helper
procedure join! to join two treap branches (while keeping both treap
invariants intact). The deletion algorithm can then be expressed as
replacing a node with a join of its two kids; compare this explanation
to the one given in the DDJ paper!
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