Join-based tree algorithms
In computer science, join-based tree algorithms are a class of algorithms for self-balancing binary search trees.
The algorithmic framework is based on a single operation join. Under this framework, the join operation captures all balancing criteria of different balancing schemes, and all other functions join have generic implementation across different balancing schemes. The join-based algorithms can be applied to at least four balancing schemes: AVL trees, red-black trees, weight-balanced trees and treaps.
The join operation takes as input two binary balanced trees and of the same balancing scheme, and a key, and outputs a new balanced binary tree whose in-order traversal is the in-order traversal of, then then the in-order traversal of. In particular, if the trees are search trees, which means that the in-order of the trees maintain a total ordering on keys, it must satisfy the condition that all keys in are smaller than and all keys in are greater than.
History
The join operation was first defined by Tarjan on red-black trees, which runs in worst-case logarithmic time. Later Sleator and Tarjan described a join algorithm for splay trees which runs in amortized logarithmic time. Later Adams extended join to weight-balanced trees and used it for fast set-set functions including union, intersection and set difference. In 1998, Blelloch and Reid-Miller extended join on treaps, and proved the bound of the set functions to be for two trees of size and, which is optimal in the comparison model. They also brought up parallelism in Adams' algorithm by using a divide-and-conquer scheme. In 2016, Blelloch et al. formally proposed the join-based algorithms, and formalized the join algorithm for four different balancing schemes: AVL trees, red-black trees, weight-balanced trees and treaps. In the same work they proved that Adams' algorithms on union, intersection and difference are work-optimal on all the four balancing schemes.Join algorithms
The function join considers rebalancing the tree, and thus depends on the input balancing scheme. If the two trees are balanced, join simply creates a new node with left subtree, root and right subtree. Suppose that is heavier than . Join follows the right spine of until a node which is balanced with. At this point a new node with left child, root and right child is created to replace c. The new node may invalidate the balancing invariant. This can be fixed with rotations.The following is the join algorithms on different balancing schemes.
The join algorithm for AVL trees:
function joinRightAVL
= expose
if <= h
T'=Node
if <= h then return Node
else return rotateLeft
else
T' = joinRightAVL
T = Node
if <= h return T
else return rotateLeft
function joinLeftAVL
/* symmetric to joinRightAVL */
function join
if > h return joinRightAVL
if > h return joinLeftAVL
return Node
Here of a node the height of. expose= means to extract a tree node 's left child, the key of the node, and the right child. Node means to create a node of left child, key, and right child.
The join algorithm for red-black trees:
function joinRightRB
if r = ⌊r/2⌋ × 2:
return Node
else
= expose
T' = Node
if and :
T'.right.right.color = black
return rotateLeft
else return T'
function joinLeftRB
/* symmetric to joinRightRB */
function join
if ⌊r/2⌋ > ⌊r/2⌋ × 2:
T' = joinRightRB
if and :
T'.color = black
return T'
else if ⌊r/2⌋ > ⌊r/2⌋ × 2
/* symmetric */
else if and
Node
else
Node
Here of a node means twice the black height of a black node, and the twice the black height of a red node. expose= means to extract a tree node 's left child, the key of the node, the color of the node and the right child. Node means to create a node of left child, key, color and right child.
The join algorithm for weight-balanced trees:
function joinRightWB
=expose
if balance return Node
else
T' = joinRightWB
= expose
if return Node
else if and balance)
return rotateLeft
else return rotateLeft
/* symmetric to joinRightWB */
function join
if return joinRightWB
if return joinLeftWB
Node
Here balance means two weights and are balanced. expose= means to extract a tree node 's left child, the key of the node and the right child. Node means to create a node of left child, key and right child.
Join-based algorithms
In the following, expose= means to extract a tree node 's left child, the key of the node and the right child. Node means to create a node of left child, key and right child. right and left extracts the right child and the left child of a tree node, respectively. extract the key of a node. "" means that two statements and can run in parallel.Split
To split a tree into two trees, those smaller than key x, and those larger than key x, we first draw a path from the root by inserting x into the tree. After this insertion, all values less than x will be found on the left of the path, and all values greater than x will be found on the right. By applying Join, all the subtrees on the left side are merged bottom-up using keys on the path as intermediate nodes from bottom to top to form the left tree, and the right part is asymmetric. For some applications, Split also returns a boolean value denoting if x appears in the tree. The cost of Split is, order of the height of the tree.The split algorithm is as follows:
function split
if return
=expose
if return
if
=split
return
if
=split
return )
Join2
This function is defined similarly as join but without the middle key. It first splits out the last key of the left tree, and then join the rest part of the left tree with the right tree with.The algorithm is as follows:
function splitLast
=expose
if return
=splitLast
return
function join2
if return R
=splitLast
return join
The cost is for a tree of size.
Insert and delete
The insertion and deletion algorithms, when making use of join can be independent of balancing schemes. For an insertion, the algorithm compares the key to be inserted with the key in the root, inserts it to the left/right subtree if the key is smaller/greater than the key in the root, and joins the two subtrees back with the root. A deletion compares the key to be deleted with the key in the root. If they are equal, return join2 on the two subtrees. Otherwise, delete the key from the corresponding subtree, and join the two subtrees back with the root.The algorithms are as follows:
function insert
if return Node
=expose
if return join
if return join
return T
function delete
if return nil
=expose
if return join
if return join
return join2
Both insertion and deletion requires time if.
Set–set functions
Several set operations have been defined on weight-balanced trees: union, intersection and set difference. The union of two weight-balanced trees and representing sets and, is a tree that represents. The following recursive function computes this union:function union:
if t1 = nil:
return t2
if t2 = nil:
return t1
= split t2 on t1.root
nl = union || nr = union
return join
Similarly, the algorithms of intersection and set-difference are as follows:
function intersection:
if return nil
= split t2 on t1.root
nl = intersection || nr = intersection
if return join
else return join2
function difference:
if return nil
if return t1
= split t2 on t1.root
nl = difference || nr = difference
return join2
The complexity of each of union, intersection and difference is for two weight-balanced trees of sizes and. This complexity is optimal in terms of the number of comparisons. More importantly, since the recursive calls to union, intersection or difference are independent of each other, they can be executed in parallel with a parallel depth. When, the join-based implementation applies the same computation as in a single-element insertion or deletion if the root of the larger tree is used to split the smaller tree.
Build
The algorithm for building a tree can make use of the union algorithm, and use the divide-and-conquer scheme:function build:
if return nil
if return Node
L = build || R =
return union
This algorithm costs work and has depth. A more-efficient algorithm makes use of a parallel sorting algorithm.
function buildSorted:
if return nil
if return Node
L = build || R =
return join
function build:
A'=sort
return buildSorted
This algorithm costs work and has depth assuming the sorting algorithm has work and depth.
Filter
This function selects all entries in a tree satisfying an indicator, and return a tree containing all selected entries. It recursively filters the two subtrees, and join them with the root if the root satisfies, otherwise join2 the two subtrees.function filter:
if return nil
L = filter || R =
if
This algorithm costs work and depth on a tree of size, assuming has constant cost.