Balanced Binary Trees

Sets and maps are important and useful abstractions. We've seen various ways to implement an abstract data type for sets and maps, since data structures that implement sets can be used to implement maps as well. It's time to look at an implementation of sets that is asymptotically efficient and useful in practice: balanced binary trees.

Binary trees have two advantages above the asymptotically more efficient hash table: first, they support nondestructive update with the same asymptotic efficiency. Second, they store their values (or keys, in the case of a map) in order, which makes range queries and in-order iteration possible.

For simplicity, we will implement sets of some type we will call
`value`

. We assume we have a comparison function ```
compare:
value*value -> order
```

.

The signature that we will work with is a little different from that in Lecture 8:

signature SET = sig (* a "set" is an immutable set of values. *) type set type value (* empty is the empty set, ∅ *) val empty : set (* add(x,s) is {x} union s. *) val add: value*set -> set (* union(x,y) is x ∪ y. *) val union: set*set -> set (* contains(x,s) is whether x is a member of s (i.e., x∈s)*) val contains: value*set -> bool (* size(s) is the number of elements in s *) val size: set->int (* fold over the elements of the set *) val fold: ((value * 'b)->'b) -> 'b -> set -> 'b end

An important property of a search tree is that it can be used to implement an
**ordered set** or ordered map easily:
a set (map) that abstractly keeps its elements in sorted order.
Although the signature above doesn't show them, ordered sets
generally provide operations for finding the minimum and maximum elements of the
set, for iterating over all the elements between two elements, and for
extracting (or iterating over) ordered subsets of the elements between a range.

datatype btree = Empty | Node of {value: value, left:btree, right:btree} type set = btree

A binary search tree is a binary tree with the following representation invariant:
For
any node `n`

, every node in `#left n`

has a value less than
that of `n`

, and every node in `#right n`

has a value more
than that of `n`

. And the entire left and right subtrees satisfy the
same invariant.

Given such a tree, how do you perform a lookup operation? Start from the root, and at every node, if the value of the node is what you are looking for, you are done; otherwise, recursively look up in the left or right subtree depending on the value stored at the node. In code:

fun contains (n:int, t:btree): bool = (case t of Empty => false | Node {value,left,right} => (case compare (value,n) of EQUAL => true | GREATER => contains (n,left) | LESSER => contains (n,right)))

Addition is similar: you perform a lookup until you find the empty node that should contain the value. In code:

fun add (n:int, t:btree):btree = (case t of Empty => Node {value=n, left=Empty, right=Empty} | Node {value,left,right} => (case compare (value,n) of EQUAL => t | GREATER => Node {value=value, left=add (n,left), right=right} | LESSER => Node {value=value, left=left, right=add (n,right)}))

What is the running time of those operations? Since `add`

is just a lookup
with an extra node creation, we focus on the lookup operation. Clearly, an
analysis of the code shows that `add`

is *O*(height
of the tree). What's the worst-case height of a tree? Clearly, a tree of *n*
nodes all in a single long branch (imagine adding the numbers 1,2,3,4,5,6,7
in order into a binary search tree). So the worst-case running time of lookup is
still *O*(*n*) (for *n*
the number of nodes in the tree).

Some useful code resources:

- An implementation of binary search trees that includes both add and remove operations.
- Support for printing out trees

What is a good shape for a tree that would allow for fast lookup? A
**perfect binary tree** has the largest number of nodes n for
a given height h: n = 2^{h+1}-1. Therefore h = lg(n+1)-1 = O(lg n).

^ 50 | / \ | 25 75 height=3 | / \ / \ n=15 | 10 30 60 90 | / \ / \ / \ / \ V 4 12 27 40 55 65 80 99

If a tree with *n* nodes is kept balanced,
its height is *O*(lg* n*), which leads
to a lookup operation running in time *O*(lg* n*).

How can we keep a tree balanced? It can become unbalanced during element addition or deletion. Most approaches involve adding or deleting an element just like in a normal binary search tree, followed by some kind of tree surgery to rebalance the tree. Similarly, element deletion proceeds as in a binary search tree, followed by some corrective rebalancing action. Examples of balanced binary search tree data structures include

- AVL (or height-balanced) trees (1962)
- 2-3 trees (1970's)
- Red-black trees

In each of these, we ensure asymptotic complexity of O(lg n) by enforcing a stronger invariant on the data structure than just the binary search tree invariant.

Red-black trees are a fairly simple, efficient data structure for maintaining
a balanced binary tree. The idea is to strengthen the rep invariant
a trees has height logarithmic in n. To help enforce the invariant, we
color each node of the tree either *red* or *black*. Where it
matters, we consider the color of an empty tree to be black.

datatype color = Red | Black datatype rbtree = Empty | Node of {color: color, value: int, left:rbtree, right:rbtree} type set = rbtree

Here are the new conditions we add to the binary search tree rep invariant:

- No red node has a red parent.
- Every path from the root to an empty node has the same number of black nodes.

If a tree satisfies these two conditions, it must also be the case that every subtree of the tree also satisfies the conditions. If a subtree violated either of the conditions, the whole tree would also.

With this invariant, the longest possible path from the root to an empty
node would alternately contain red and black nodes; therefore it is at most
twice as long as the shortest possible path, which only contains black nodes. If
*n* is the number of nodes in the tree, the longest
path cannot have a length greater than twice the length of the paths in a
perfect binary tree: 2 lg *n*,
which is O(lg* n*). Therefore,
the tree has height O(lg* n*) and
the operations are all as asymptotically efficient as we could expect.

How do we check for membership in red-black trees? Exactly the same way as for general binary trees:

fun contains (n: int, t:rbtree): bool = (case t of Empty => false | Node {value,left,right,...} => (case compare (value, n) of EQUAL => true | GREATER => contains (n,left) | LESSER => contains (n,right)))

More interesting is the `add`

operation. We add by replacing
the empty node that a standard add into a binary
search tree would. We also color the new node red to ensure that
invariant #2 is preserved. However, we may destroy invariant #1 in doing so, by
producing two red nodes, one the parent of the other. In order to restore this
invariant we will need to consider not only the two red nodes, but their parent.
Otherwise, the red-red conflict cannot be fixed while preserving black depth. The next figure shows all
the possible cases that may arise:

1234B_{z}B_{z}B_{x}B_{x}/ \ / \ / \ / \ R_{y}d R_{x}d a R_{z}a R_{y}/ \ / \ / \ / \ R_{x}c a R_{y}R_{y}d b R_{z}/ \ / \ / \ / \ a b b c b c c d

Notice that in each of these trees, the values of the nodes in a,b,c,d must
have the same relative ordering with respect to x, y, and z:
a<x<b<y<c<z<d. Therefore, we can perform a local **tree
rotation** to restore the invariant locally, while possibly breaking
invariant 1 one level up in the tree:

R_{y}/ \ B_{x}B_{z}/ \ / \ a b c d

By performing a rebalance of the tree at that level, and all the levels above, we can locally (and incrementally) enforce invariant #1. In the end, we may end up with two red nodes, one of them the root and the other the child of the root; this we can easily correct by coloring the root black. The SML code (which really shows the power of pattern matching!) is as follows:

fun add (n:int, t:rbtree): rbtree = let (* Definition: a tree t satisfies the "reconstruction invariant" if it is * black and satisfies the rep invariant, or if it is red and its children * satisfy the rep invariant and have the same black height. *) (* makeBlack(t) is a tree that satisfies the rep invariant. Requires: t satisfies the reconstruction invariant Algorithm: Make a tree identical to t but with a black root. *) fun makeBlack (t:rbtree): rbtree = case t of Empty => Empty | Node {color,value,left,right} => Node {color=Black, value=value, left=left, right=right} (* Construct the result of a red-black tree rotation. *) fun rotate(x: value, y: value, z: value, a: rbtree, b: rbtree, c:rbtree, d: rbtree): rbtree = Node {color=Red, value=y, left= Node {color=Black, value=x, left=a, right=b}, right=Node {color=Black, value=z, left=c, right=d}} (* balance(t) is a tree that satisfies the reconstruction invariant and * contains all the same values as t. * Requires: one of the children of t satisfies the rep invariant and * the other satisfies the reconstruction invariant. Both children * have the same black height. *) fun balance (t:rbtree): rbtree = case t of (*1*) Node {color=Black, value=z, left= Node {color=Red, value=y, left=Node {color=Red, value=x, left=a, right=b}, right=c}, right=d} => rotate(x,y,z,a,b,c,d) | (*2*) Node {color=Black, value=z, left=Node {color=Red, value=x, left=a, right=Node {color=Red, value=y, left=b, right=c}}, right=d} => rotate(x,y,z,a,b,c,d) | (*3*) Node {color=Black, value=x, left=a, right=Node {color=Red, value=z, left=Node {color=Red, value=y, left=b, right=c}, right=d}} => rotate(x,y,z,a,b,c,d) | (*4*) Node {color=Black, value=x, left=a, right=Node {color=Red, value=y, left=b, right=Node {color=Red, value=z, left=c, right=d}}} => rotate(x,y,z,a,b,c,d) | _ => t (* Add x into t, returning a tree that satisfies the reconstruction invariant. *) fun walk (t:rbtree):rbtree = case t of Empty => Node {color=Red, value=n, left=Empty, right=Empty} | Node {color,value,left,right} => (case compare (value,n) of EQUAL => t | GREATER => balance (Node {color=color, value=value, left=walk left, right=right}) | LESSER => balance (Node {color=color, value=value, left=left, right=walk right})) in makeBlack (walk (t)) end

This code walks back up the tree from the point of insertion fixing the
invariants at every level. At red nodes we don't try to fix the invariant; we
let the recursive walk go back until a black node is found. When the walk
reaches the top the color of the root node is restored to black, which is needed
if `balance`

rotates the root.

Deletion of elements from a red-black tree is also possible, but requires the consideration of more cases. Deleting a black element from the tree creates the possibility that some path in the tree has too few black nodes; the solution is to consider that path to contain a "doubly-black" node. A series of tree rotations can then eliminate the doubly-black node, by propagating the blackness up until a red node can be converted to a black node.