Treap - Treap

Treap

The treap was first described by Cecilia R. Aragon and Raimund Seidel in 1989; its name is a portmanteau of tree and heap. It is a Cartesian tree in which each key is given a (randomly chosen) numeric priority. As with any binary search tree, the inorder traversal order of the nodes is the same as the sorted order of the keys. The structure of the tree is determined by the requirement that it be heap-ordered: that is, the priority number for any non-leaf node must be greater than or equal to the priority of its children. Thus, as with Cartesian trees more generally, the root node is the maximum-priority node, and its left and right subtrees are formed in the same manner from the subsequences of the sorted order to the left and right of that node.

An equivalent way of describing the treap is that it could be formed by inserting the nodes highest-priority-first into a binary search tree without doing any rebalancing. Therefore, if the priorities are independent random numbers (from a distribution over a large enough space of possible priorities to ensure that two nodes are very unlikely to have the same priority) then the shape of a treap has the same probability distribution as the shape of a random binary search tree, a search tree formed by inserting the nodes without rebalancing in a randomly chosen insertion order. Because random binary search trees are known to have logarithmic height with high probability, the same is true for treaps.

Specifically, the treap supports the following operations:

• To search for a given key value, apply a standard binary search algorithm in a binary search tree, ignoring the priorities.
• To insert a new key x into the treap, generate a random priority y for x. Binary search for x in the tree, and create a new node at the leaf position where the binary search determines a node for x should exist. Then, as long as x is not the root of the tree and has a larger priority number than its parent z, perform a tree rotation that reverses the parent-child relation between x and z.
• To delete a node x from the treap, if x is a leaf of the tree, simply remove it. If x has a single child z, remove x from the tree and make z be the child of the parent of x (or make z the root of the tree if x had no parent). Finally, if x has two children, swap its position in the tree with the position of its immediate successor z in the sorted order, resulting in one of the previous cases. In this final case, the swap may violate the heap-ordering property for z, so additional rotations may need to be performed to restore this property.
• To split a treap into two smaller treaps, those smaller than key x, and those larger than key x, insert x into the treap with maximum priority—larger than the priority of any node in the treap. After this insertion, x will be the root node of the treap, all values less than x will be found in the left subtreap, and all values greater than x will be found in the right subtreap. This costs as much as a single insertion into the treap.
• Merging two treaps (assumed to be the product of a former split), one can safely assume that the greatest value in the first treap is less than the smallest value in the second treap. Insert a value x, such that x is larger than this max-value in the first treap, and smaller than the min-value in the second treap, and assign it the minimum priority. After insertion it will be a leaf node, and can easily be deleted. The result is one treap merged from the two original treaps. This is effectively "undoing" a split, and costs the same.

Aragon and Seidel also suggest assigning higher priorities to frequently accessed nodes, for instance by a process that, on each access, chooses a random number and replaces the priority of the node with that number if it is higher than the previous priority. This modification would cause the tree to lose its random shape; instead, frequently accessed nodes would be more likely to be near the root of the tree, causing searches for them to be faster.

Blelloch and Reid-Miller describe an application of treaps to a problem of maintaining sets of items and performing set union, set intersection, and set difference operations, using a treap to represent each set. Naor and Nissim describe another application, for maintaining authorization certificates in public-key cryptosystems.