Treap

Treaps support the following basic 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 build a treap we can simply insert n values in the treap where each takes ${\displaystyle O(\log n)}$ time. Therefore a treap can be built in ${\displaystyle O(n\log n)}$ time from a list values.

In addition to the single-element insert, delete and lookup operations, several fast "bulk" operations have been defined on treaps: union, intersection and set difference. These rely on two helper operations, split and join.

• 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.
• Joining two treaps that are 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. Create a new node with 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, assign it the minimum priority, then set its left child to the first heap and its right child to the second heap. Rotate as necessary to fix the heap order. After that, 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. More generally, the join operation can work on two treaps and a key with arbitrary priority (i.e., not necessary to be the highest).

Join performed on treaps ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$. Right child of ${\displaystyle T_{1}}$ after the join is defined as a join of its former right child and ${\displaystyle T_{2}}$.

The join algorithm is as follows:

function join(L, k, R)
    if prior(k, k(L)) and prior(k, k(R)) return Node(L, k, R)
    if prior(k(L), k(R)) return Node(left(L), k(L), join(right(L), k, R))
    return Node(join(L, k, left(R)), k(R), right(R))

To split ${\displaystyle T}$ by ${\displaystyle x}$, recursive split call is done to either left or right child of ${\displaystyle T}$.

The split algorithm is as follows:

function split(T, k)
    if (T = nil) return (nil, false, nil)
    (L, (m, c), R) = expose(T)
    if (k = m) return (L, true, R)
    if (k < m) 
        (L', b, R') = split(L, k)
        return (L', b, join(R', m, R))
    if (k > m) 
        (L', b, R') = split(R, k)
        return (join(L, m, L'), b, R'))

The union of two treaps t₁ and t₂, representing sets A and B is a treap t that represents A ∪ B. The following recursive algorithm computes the union:

function union(t1, t2):
    if t1 = nil:
        return t2
    if t2 = nil:
        return t1
    if priority(t1) < priority(t2):
        swap t1 and t2
    t<, t> ← split t2 on key(t1)
    return join(union(left(t1), t<), key(t1),
                    union(right(t1), t>))

Here, split is presumed to return two trees: one holding the keys less than its input key, one holding the greater keys. (The algorithm is non-destructive, but an in-place destructive version exists as well.)

The algorithm for intersection is similar, but requires the join helper routine. The complexity of each of union, intersection and difference is O(m log n/m) for treaps of sizes m and n, with m ≤ n. Moreover, since the recursive calls to union are independent of each other, they can be executed in parallel.[4]

Split and Union call Join but do not deal with the balancing criteria of treaps directly, such an implementation is usually called the "join-based" implementation.

Note that if hash values of keys are used as priorities and structurally equal nodes are merged already at construction, then each merged node will be a unique representation of a set of keys. Provided that there can only be one simultaneous root node representing a given set of keys, two sets can be tested for equality by pointer comparison, which is constant in time.

This technique can be used to enhance the merge algorithms to perform fast also when the difference between two sets is small. If input sets are equal, the union and intersection functions could break immediately returning one of the input sets as result, while the difference function should return the empty set.

Let d be the size of the symmetric difference. The modified merge algorithms will then also be bounded by O(d log n/d).[5][6]

The information stored per node in the randomized binary tree is simpler than in a treap (a small integer rather than a high-precision random number), but it makes a greater number of calls to the random number generator (O(log n) calls per insertion or deletion rather than one call per insertion) and the insertion procedure is slightly more complicated due to the need to update the numbers of descendants per node. A minor technical difference is that, in a treap, there is a small probability of a collision (two keys getting the same priority), and in both cases, there will be statistical differences between a true random number generator and the pseudo-random number generator typically used on digital computers. However, in any case, the differences between the theoretical model of perfect random choices used to design the algorithm and the capabilities of actual random number generators are vanishingly small.

Although the treap and the randomized binary search tree both have the same random distribution of tree shapes after each update, the history of modifications to the trees performed by these two data structures over a sequence of insertion and deletion operations may be different. For instance, in a treap, if the three numbers 1, 2, and 3 are inserted in the order 1, 3, 2, and then the number 2 is deleted, the remaining two nodes will have the same parent-child relationship that they did prior to the insertion of the middle number. In a randomized binary search tree, the tree after the deletion is equally likely to be either of the two possible trees on its two nodes, independently of what the tree looked like prior to the insertion of the middle number.

To insert an element at position pos we divide the array into two subsections [0...pos-1] and [pos..sz] by calling the split function and we get two trees ${\displaystyle T1}$ and ${\displaystyle T2}$ . Then we merge ${\displaystyle T1}$ with the new node by calling the join function. Finally we call the join function to merge ${\displaystyle T1}$ and ${\displaystyle T2}$.

We find the element to be deleted and perform a join on its children L and R. We then replace the element to be deleted with the tree that resulted from the join operation.

To perform this calculation we will proceed as follows:

• First we will create an additional field F to store the value of the target function for the range represented by that node. we will create a function that calculates the value F based on the values of the L and R children of the node. We will call this target function at the end of all functions that modify the tree, i.e., split and join.
• Second we need to process a query for a given range [A..B]: We will call the split function twice and split the treap into ${\displaystyle T1}$ which contains ${\displaystyle \{1..A-1\}}$, ${\displaystyle T2}$ which contains ${\displaystyle \{A..B\}}$, and ${\displaystyle T3}$ which contains ${\displaystyle \{B+1..n\}}$. After the query is answered we will call the join function twice to restore the original treap.

To perform this operation we will proceed as follows:

• We will create an extra field D which will contain the added value for the subtree. We will create a function push which will be used to propagate this change from a node to its children. We will call this function at the beginning of all functions which modify the tree, i.e., split and join so that after any changes made to the tree the information will not be lost.

To show that the subtree of a given node needs to be reversed for each node we will create an extra boolean field R and set its value to true. To propagate this change we will swap the children of the node and set R to true for all of them.