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Wikipedia / Scapegoat tree

In computer science, a scapegoat tree is a self-balancing binary search tree, invented by Arne Andersson in 1989 and rediscovered by Igal Galperin and Ronald L. Rivest in 1993. lookup time (as expressed in big O notation amortized insertion and deletion time.

lookup time, scapegoat trees have no additional per-node memory overhead compared to a regular binary search tree: besides key and value, a node stores only two pointers to the child nodes. This makes scapegoat trees easier to implement and, due to data structure alignment, can reduce node overhead by up to one-third.

worst-case update performance.

Theory

A binary search tree is said to be weight-balanced if half the nodes are on the left of the root, and half on the right. An α-weight-balanced node is defined as meeting a relaxed weight balance criterion:

size(left) ≤ α*size(node)
size(right) ≤ α*size(node)

Where size can be defined recursively as:

function size(node) is
    if node = nil then
        return 0
    else
        return size(node->left) + size(node->right) + 1
    end if
end function

Even a degenerate tree (linked list) satisfies this condition if α=1, whereas an α=0.5 would only match almost complete binary trees.

A binary search tree that is α-weight-balanced must also be α-height-balanced, that is

height(tree) ≤ floor(log1/α(size(tree)))

By contraposition, a tree that is not α-height-balanced is not α-weight-balanced.

Scapegoat trees are not guaranteed to keep α-weight-balance at all times, but are always loosely α-height-balanced in that

height(scapegoat tree) ≤ floor(log1/α(size(tree))) + 1.

Violations of this height balance condition can be detected at insertion time, and imply that a violation of the weight balance condition must exist.

This makes scapegoat trees similar to red–black trees in that they both have restrictions on their height. They differ greatly though in their implementations of determining where the rotations (or in the case of scapegoat trees, rebalances) take place. Whereas red–black trees store additional 'color' information in each node to determine the location, scapegoat trees find a scapegoat which isn't α-weight-balanced to perform the rebalance operation on. This is loosely similar to AVL trees, in that the actual rotations depend on 'balances' of nodes, but the means of determining the balance differs greatly. Since AVL trees check the balance value on every insertion/deletion, it is typically stored in each node; scapegoat trees are able to calculate it only as needed, which is only when a scapegoat needs to be found.

Unlike most other self-balancing search trees, scapegoat trees are entirely flexible as to their balancing. They support any α such that 0.5 < α < 1. A high α value results in fewer balances, making insertion quicker but lookups and deletions slower, and vice versa for a low α. Therefore in practical applications, an α can be chosen depending on how frequently these actions should be performed.

Operations

Lookup

. This is in contrast to splay trees. The reduced node memory overhead compared to other self-balancing binary search trees can further improve locality of reference and caching.

Insertion

Insertion is implemented with the same basic ideas as an unbalanced binary search tree, however with a few significant changes.

When finding the insertion point, the depth of the new node must also be recorded. This is implemented via a simple counter that gets incremented during each iteration of the lookup, effectively counting the number of edges between the root and the inserted node. If this node violates the α-height-balance property (defined above), a rebalance is required.

To rebalance, an entire subtree rooted at a scapegoat undergoes a balancing operation. The scapegoat is defined as being an ancestor of the inserted node which isn't α-weight-balanced. There will always be at least one such ancestor. Rebalancing any of them will restore the α-height-balanced property.

One way of finding a scapegoat, is to climb from the new node back up to the root and select the first node that isn't α-weight-balanced.

storage space, usually allocated on the stack, or parent pointers. This can actually be avoided by pointing each child at its parent as you go down, and repairing on the walk back up.

To determine whether a potential node is a viable scapegoat, we need to check its α-weight-balanced property. To do this we can go back to the definition:

size(left) ≤ α*size(node)
size(right) ≤ α*size(node)

However a large optimization can be made by realizing that we already know two of the three sizes, leaving only the third to be calculated.

Consider the following example to demonstrate this. Assuming that we're climbing back up to the root:

size(parent) = size(node) + size(sibling) + 1

But as:

size(inserted node) = 1.

The case is trivialized down to:

size[x+1] = size[x] + size(sibling) + 1

Where x = this node, x + 1 = parent and size(sibling) is the only function call actually required.

Once the scapegoat is found, the subtree rooted at the scapegoat is completely rebuilt to be perfectly balanced. time by traversing the nodes of the subtree to find their values in sorted order and recursively choosing the median as the root of the subtree.

amortized time.

Sketch of proof for cost of insertion

per insertion.

Define the Imbalance of a node v to be the absolute value of the difference in size between its left node and right node minus 1, or 0, whichever is greater. In other words:

Additionally, we define the imbalance of a subtree to be the sum of the imbalance of the nodes in the subtree.

Lemma 1:, is Big Omega notation.)

Proof:

.

Lemma 2:.

Proof:

The rebuilt tree is perfectly balanced, such that the size of subtrees rooted at the same level differ by at most 1. By the above definition of imbalance, the imbalance of all nodes in the subtree is then 0.

Theorem 3:.

Proof:

by Lemmas 1 and 2, the cost of repairing each unit of imbalance is

.

to fix, resulting in an amortized cost of rebalancing per insertion insertion of

.

By adding the costs of the initial search phase and rebalancing, the amortized cost of insertion is thus

.

This bound holds for all 0.5 < α < 1.

Deletion

Scapegoat trees are unusual in that deletion is easier than insertion. To enable deletion, scapegoat trees need to store an additional value with the tree data structure. This property, which we will call MaxNodeCount simply represents the highest achieved NodeCount. It is set to NodeCount whenever the entire tree is rebalanced, and after insertion is set to max(MaxNodeCount, NodeCount).

To perform a deletion, we simply remove the node as you would in a simple binary search tree, but if

NodeCount ≤ α*MaxNodeCount

then we rebalance the entire tree about the root, remembering to set MaxNodeCount to NodeCount.

.

Sketch of proof for cost of deletion

:

Etymology

The name Scapegoat tree "[...] is based on the common wisdom that, when something goes wrong, the first thing people tend to do is find someone to blame (the scapegoat)." In the Bible, a scapegoat is an animal that is ritually burdened with the sins of others, and then driven away.

See also

References