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. It provides worst-case O ( log n ) {\displaystyle O(\log n)} lookup time (as expressed in big O notation, with n {\displaystyle n} as the number of entries) and O ( log n ) {\displaystyle O(\log n)} amortized insertion and deletion time. Unlike most other self-balancing binary search trees which also provide worst case O ( log n ) {\displaystyle O(\log n)} 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. Instead of the small incremental rebalancing operations used by most balanced tree algorithms, scapegoat trees rarely but expensively choose a "scapegoat" and completely rebuilds the subtree rooted at the scapegoat into a complete binary tree. Thus, scapegoat trees have O ( n ) {\displaystyle O(n)} 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 Lookup is not modified from a standard binary search tree, and has a worst-case time of O ( log n ) {\displaystyle O(\log n)} . This is in contrast to splay trees which have a worst-case time of O ( n ) {\displaystyle O(n)} . 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. Climbing back up to the root requires O ( log n ) {\displaystyle O(\log n)} 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
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