4 Splay Trees

8.4

4 Splay Trees

Splay trees are an efficient data structure for mutable dictionaries with totally ordered keys. They were described in the paper “Self-Adjusting Binary Search Trees” by Daniel Sleator and Robert Tarjan in Journal of the ACM 32(3) pp652-686.

A splay-tree is a ordered dictionary (dict? and ordered-dict?).

Operations on splay-trees are not thread-safe. If a key in a splay-tree is mutated, the splay-tree’s internal invariants may be violated, causing its behavior to become unpredictable.

procedure
(make-splay-tree [ ord
#:key-contract key-contract
#:value-contract value-contract]) → splay-tree?
  ord : order? = datum-order
  key-contract : contract? = any/c
  value-contract : contract? = any/c

Makes a new empty splay-tree. The splay tree uses ord to order keys; in addition, the domain contract of ord is combined with key-contract to check keys.

Examples:

> (define splay-tree
    (make-splay-tree (order 'string-order string? string=? string<?)))
> (splay-tree-set! splay-tree "dot" 10)
> (splay-tree-set! splay-tree "cherry" 500)
> (dict-map splay-tree list)
'(("cherry" 500) ("dot" 10))
> (splay-tree-ref splay-tree "dot")
10
> (splay-tree-remove! splay-tree "cherry")
> (splay-tree-count splay-tree)
1
> (splay-tree-set! splay-tree 'pear 3)
splay-tree-set!: contract violation
  expected: string?
  given: 'pear
  in: the key argument of
      (->i
       ((s splay-tree?)
        (key (s) (key-c s))
        (v (s) (val-c s)))
       (_r void?))
  contract from:
      <pkgs>/data-lib/data/splay-tree.rkt
  blaming: top-level
   (assuming the contract is correct)
  at: <pkgs>/data-lib/data/splay-tree.rkt:609:2

procedure
(make-adjustable-splay-tree [ #:key-contract key-contract
#:value-contract value-contract])
→ splay-tree?
key-contract : contract? = any/c
value-contract : contract? = any/c

Makes a new empty splay-tree that permits only exact integers as keys (in addition to any constraints imposed by key-contract). The resulting splay tree answers true to adjustable-splay-tree? and supports efficient key adjustment.

Examples:

> (define splay-tree (make-adjustable-splay-tree))
> (splay-tree-set! splay-tree 3 'apple)
> (splay-tree-set! splay-tree 6 'cherry)
> (dict-map splay-tree list)
'((3 apple) (6 cherry))
> (splay-tree-ref splay-tree 3)
'apple
> (splay-tree-remove! splay-tree 6)
> (splay-tree-count splay-tree)
1

procedure
(splay-tree? x) → boolean?
x : any/c

Returns #t if x is a splay-tree, #f otherwise.

procedure
(adjustable-splay-tree? x) → boolean?
x : any/c

Returns #t if x is a splay-tree that supports key adjustment; see splay-tree-contract! and splay-tree-expand!.

procedure
(splay-tree-ref s key [default]) → any
  s : splay-tree?
  key : any/c
  default : any/c = (lambda () (error ....))
procedure
(splay-tree-set! s key value) → void?
  s : splay-tree?
  key : any/c
  value : any/c
procedure
(splay-tree-remove! s key) → void?
  s : splay-tree?
  key : any/c
procedure
(splay-tree-count s) → exact-nonnegative-integer?
  s : splay-tree?
procedure
(splay-tree-iterate-first s) → (or/c #f splay-tree-iter?)
  s : splay-tree?
procedure
(splay-tree-iterate-next s iter) → (or/c #f splay-tree-iter?)
  s : splay-tree?
  iter : splay-tree-iter?
procedure
(splay-tree-iterate-key s iter) → any/c
  s : splay-tree?
  iter : splay-tree-iter?
procedure
(splay-tree-iterate-value s iter) → any/c
  s : splay-tree?
  iter : splay-tree-iter?

Implementations of dict-ref, dict-set!, dict-remove!, dict-count, dict-iterate-first, dict-iterate-next, dict-iterate-key, and dict-iterate-value, respectively.

procedure
(splay-tree-remove-range! s from to) → void?
  s : splay-tree?
  from : any/c
  to : any/c

Removes all keys in [from, to); that is, all keys greater than or equal to from and less than to.

This operation takes O(N) time, or O(log N) time if (adjustable-splay-tree? s).

procedure
(splay-tree-contract! s from to) → void?
  s : adjustable-splay-tree?
  from : exact-integer?
  to : exact-integer?

Like splay-tree-remove-range!, but also decreases the value of all keys greater than or equal to to by (- to from).

This operation is only allowed on adjustable splay trees, and it takes O(log N) time.

procedure
(splay-tree-expand! s from to) → void?
  s : adjustable-splay-tree?
  from : exact-integer?
  to : exact-integer?

Increases the value of all keys greater than or equal to from by (- to from).

This operation is only allowed on adjustable splay trees, and it takes O(log N) time.

procedure
(splay-tree-iterate-least s) → (or/c #f splay-tree-iter?)
  s : splay-tree
procedure
(splay-tree-iterate-greatest s) → (or/c #f splay-tree-iter?)
  s : splay-tree
procedure
(splay-tree-iterate-least/>? s key) → (or/c #f splay-tree-iter?)
  s : splay-tree?
  key : any/c
procedure
(splay-tree-iterate-least/>=? s key)
→ (or/c #f splay-tree-iter?)
  s : splay-tree?
  key : any/c
procedure
(splay-tree-iterate-greatest/<? s key)
→ (or/c #f splay-tree-iter?)
  s : splay-tree?
  key : any/c
procedure
(splay-tree-iterate-greatest/<=? s key)
→ (or/c #f splay-tree-iter?)
  s : splay-tree?
  key : any/c

Implementations of dict-iterate-least, dict-iterate-greatest, dict-iterate-least/>?, dict-iterate-least/>=?, dict-iterate-greatest/<?, and dict-iterate-greatest/<=?, respectively.

procedure
(splay-tree-iter? x) → boolean?
x : any/c

Returns #t if x represents a position in a splay-tree, #f otherwise.

procedure
(splay-tree->list s) → (listof pair?)
s : splay-tree?

Returns an association list with the keys and values of s, in order.

1	Imperative Queues
2	Growable Vectors
3	Orders and Ordered Dictionaries
4	Splay Trees
5	Skip Lists
6	Interval Maps
7	Binary Heaps
8	Integer Sets
9	Bit Vectors
10	Union-Find: Sets with only Canonical Elements
11	Enumerations
	Bibliography