(TODO)

We specify required time efficiency upper bounds using big-O notation. We note when, in some cases, there is "slack" between the required bound and the theoretically optimal bound for an operation. Implementations may use data structures with amortized time bounds, but should document which bounds hold in only an amortized sense. The use of randomized data structures with expected time bounds is discouraged.

A deque (conventionally pronounced "deck") is a queue data structure that supports efficient manipulation of either of its ends. It may be used in place of a (LIFO) stack or (FIFO) queue.

The *unlabeled finger tree* data structure can meet all these requirements rather conveniently.

Create a new deque containing element.... The leftmost element (if any) will be at the front of the deque and the rightmost element (if any) will be at the back. Takes O(n) time, where n is the number of elements.

ideque?}}} and {{{ideque-empty?}}} take O(1) time; {{{ideque-size may take O(n) time, though O(1) is optimal.

Return the front/back element. It is an error for deque to be empty. Each takes O(1) time.

(ideque-pop-front deque) (ideque-pop-back deque)Return a new deque with the front/back element removed. It is an error for deque to be empty. Each takes O(1) time.

(ideque-push-front deque x) (ideque-push-back deque x)Return a new deque with x}}} pushed to the front/back of {{{deque. Each takes O(1) time.

Return a new deque with the contents of the first argument deque followed by the others. Takes O(kn) time, where k is the number of deques and n is the number of elements involved, though O(k log n) is possible.

Conversion between deque and list structures. FIFO order is preserved, so the front of a list corresponds to the front of a deque. Each operation takes O(n) time.

A sorted set data structure stores a finite collection of unique orderable (a.k.a. comparable or sortable) elements.

These requirements can be satisfied by many flavors of *self-balancing binary trees.* Red-black trees, 1-2 brother trees, and labeled 2-3 finger trees are particularly convenient in an immutable context.

An *order predicate* for a set of orderable elements is a procedure precedes?}}} such that, for set elements {{{x}}} and {{{y,

returns non-#false}}} iff {{{x}}} precedes {{{y}}} in the set's ordering. It returns {{{#false}}} when {{{x}}} succeeds (comes after) {{{y or when the elements are tied for the same position.

It is essential that order predicates return #false in the case of equal elements, as implementations may test for element equality with expressions of the form

(and (not (precedes? x y)) (not (precedes? y x)))Note that <}}}, {{{char<?}}}, and {{{string<? are valid order predicates for sets of numbers, characters, and strings, respectively.

The time bounds stated below are all based on the assumption that evaluating an order predicate takes O(1) time.

Return a new set using the order predicate precedes?}}} and containing elements {{{element...}}}. If any subset of element arguments are equal according to {{{precedes? then only the leftmost of them is a member of the returned set. Takes O(n log n) time.

iset-size may take O(n) time, though O(1) is optimal. The other procedures take O(1) time.

Return a new set containing the union/intersection/difference/symmetric difference of the arguments. All the arguments must be sets sharing an equivalent order predicate. Runs in O(kn log n) time, where k is the number of sets and n is the number of elements involved; O(kn) time is optimal.

Returns non-#false}}} iff {{{x}}} is an element of {{{set}}}, or {{{#false otherwise. Takes O(log n) time.

(iset-min set) (iset-max set)Returns the least/greatest element of set}}}. It is an error for {{{set to be empty. Takes O(log n) time; O(1) is optimal.

A continuation-based universal update procedure. Attempts to find an element match}}} equal to {{{x}}} in {{{set}}}. When such a {{{match is found, calls

(present-proc match update remove)which must either tail-call

(update new-element ret)to replace match}}} with {{{new-element, or tail-call

(remove ret)to remove match}}} from {{{set}}}. In either case {{{ret}}} will be returned from the enclosing {{{iset-update call.

When no such match}}} is found, {{{iset-update calls

(absent-proc insert ignore)which should either tail-call

(insert ret)to insert x}}} into {{{set, or else tail-call

(ignore ret)to leave set}}} unchanged. Again, {{{ret}}} will be returned from the enclosing {{{iset-update call.

In any case, iset-update returns

(values new-set ret)where new-set}}} is a new set reflecting the indicated modification, if any; and {{{ret is the return value produced by one of the continuations. It runs in O(log n) time.

(This procedure is based on an analogous procedure for hash tables suggested by Alexey Radul and attributed to Taylor Campbell.)

(iset-find iset x [absent-thunk])Returns the element y}}} equal to {{{x}}} in {{{set}}}, or the result of evaluating {{{(absent-thunk)}}} if no such element exists. If {{{absent-thunk}}} is unspecified then a default procedure that always returns {{{#false is used. Takes O(log n) time.

(iset-include set x) (iset-exclude set x)Return a new set that certainly does/doesn't include x. Each operation takes O(log n) time.

(iset-predecessor set x [absent-thunk]) (iset-successor set x [absent-thunk])Returns the element that immediately precedes/succeeds x}}} in {{{set}}}'s ordering. If no such element exists because {{{x}}} is the minimum/maximum element, or because {{{set}}} is empty, returns the result of evaluating {{{(absent-thunk)}}}. If {{{absent-thunk}}} is unspecified then a default procedure that always returns {{{#false is used. Takes O(log n) time.

Returns a new set containing the elements of set}}} in the interval between {{{least}}} and {{{most}}}. If {{{include-least}}}/{{{include-most}}} is non-{{{#false}}} then the result includes elements equal to {{{least}}}/{{{most respectively; otherwise those elements are not included. Takes O(k log k + log n) time, where n is the number of elements in the set and k is the number of elements returned; O(k + log n) is optimal.

(iset-range< set x) (iset-range<= set x) (iset-range= set x) (iset-range>= set x) (iset-range> set x)Returns a new set containing only the elements of set}}} that are less/less-or-equal/equal/greater-or-equal/greater than {{{x. Takes O(k log k + log n) time, where n is the number of elements in the set and k is the number of elements returned; O(k + log n) is optimal.

Note that since set elements are unique, iset-range= returns a set of at most one element.

Returns a new set containing only those elements x}}} for which {{{(predicate? x)}}} returns non-{{{#false. Takes O(n log n) time; O(n) is optimal.

(iset-fold proc base set)The fundamental set iterator. Equivalent to, but may be more efficient than,

(fold proc base (iset->ordered-list set))Takes O(n) time.

(iset-map/monotone proc set [precedes?])Returns a new set containing the elements (proc x)}}} for every {{{x}}} in {{{set}}}. {{{proc}}} must be a ''monotone'' unary procedure that preserves the order and uniqueness of set elements. If {{{precedes?}}} is given, it is the order predicate of the new set; otherwise the new set uses the same order predicate as {{{set. Takes O(n) time.

(iset-map proc set [precedes?])As iset-map/monotone}}}, except the {{{proc is not required to be monotone. When multiple elements are mapped to the same value, only the value originating from the least pre-mapped element is retained. Takes O(n log n) time.

Returns a list containing the elements of set in increasing order. Takes O(n) time.

(ordered-list->iset precedes? list)Returns a set containing the elements of list}}} and using order predicate {{{precedes?}}}. It is an error for {{{list to be anything other than a proper list of elements in increasing order. Takes O(n log n) time; O(n) is optimal.

(list->iset precedes? list)Returns a set containing the elements of list}}} and using order predicate {{{precedes?}}}. {{{list}}} must be a proper list, but may contain duplicates and need not be in order. When a subset of input elements are equal according to {{{precedes?}}}, only the first such element in {{{list is a member of the returned set. Takes O(n log n) time.

These procedures compare x}}} to {{{y}}} according to {{{set}}}'s order predicate and the conventional meaning of {{{<}}}, {{{=}}}, and {{{>.

A map data structure, as specified here, stores key-value associations between a set of keys with an order predicate and arbitrary value objects. It is an alternative to an association list or hash table, which also store key-value associations, but with different key constraints and efficiency guarantees.

In the same way that a list of key-value dotted pairs can implement an association list, a set of key-value dotted pairs can implement a map. Implementations may use this approach, or implement a distinct data structure specific to maps.

The behavior and efficiency of these operations is the same as the analogous set procedures.

The behavior and efficiency of these operations is the same as the analogous set procedures.

The behavior and efficiency of these operations is the same as the analogous set procedures.

(imap-min-value map) (imap-max-value map)Returns the value associated with (imap-min-key map)}}} and {{{(imap-max-key map) respectively. Takes O(log n) time; O(1) is optimal.

A continuation-based universal update procedure. Attempts to find a key key-match}}} equal to {{{key}}} in {{{set}}}. When such a {{{key-match is found, calls

(present-proc key-match value update remove)which must either tail-call

(update new-key new-value ret)to replace key}}} with {{{new-key}}} and the corresponding value with {{{new-value, or tail-call

(remove ret)to remove key-match}}} and its corresponding value from {{{map}}}. In either case {{{ret}}} will be returned from the enclosing {{{imap-update call.

When no such key-match}}} is found, {{{imap-update calls

(absent-proc insert ignore)which should either tail-call

(insert new-key value ret)to insert new-key}}} and {{{value}}} into {{{map, or else tail-call

(ignore ret)to leave map}}} unchanged. Again, {{{ret}}} will be returned from the enclosing {{{imap-update call.

In any case, imap-update returns

(values new-map ret)where new-map}}} is a new map reflecting the indicated modification, if any; and {{{ret is the return value produced by one of the continuations. It runs in O(log n) time.

(This procedure is based on an analogous procedure for hash tables suggested by Alexey Radul and attributed to Taylor Campbell.)

(imap-find map key [absent-thunk])Returns the value associated with key}}} in {{{map}}}, or the result of evaluating {{{(absent-thunk)}}} if no such value exists. If {{{absent-thunk}}} is unspecified then a default procedure that always returns {{{#false is used. Takes O(log n) time.

(imap-include map key value) (imap-exclude map key)Return a new set that certainly does/doesn't include key}}} and the associated {{{value}}}. {{{imap-include}}} will replace any prior association to {{{key}}} that might exist in {{{map. Each operation takes O(log n) time.

(imap-key-predecessor map key [absent-thunk]) (imap-key-successor map key [absent-thunk])The behavior and efficiency of these operations is the same as the analogous set procedures.

Returns a new map containing the associations of map}}} whose keys are in the interval between {{{least}}} and {{{most}}}. If {{{include-least}}}/{{{include-most}}} is non-{{{#false}}} then the result includes associations with keys equal to {{{least}}}/{{{most respectively; otherwise those associations are not included. Takes O(k log k + log n) time, where n is the number of elements in the set and k is the number of elements returned; O(k + log n) is optimal.

(imap-range< map key) (imap-range<= map key) (imap-range= map key) (imap-range>= map key) (imap-range> map key)The behavior and efficiency of these operations is the same as the analogous set procedures.

Returns a new set containing only those associations for which (proc key value)}}} returns non-{{{#false. Takes O(n log n) time; O(n) is optimal.

(imap-fold proc base map)The fundamental map iterator. Calls

(proc key value accum)successively to accumulate a value initialized to base. Takes O(n) time.

(imap-map-values proc map)Returns a new map where each value is replaced with the result of evaluating

(proc key value)for that value's key. Takes O(n) time.

(imap-map/monotone proc map [precedes?])Returns a new map containing the association values returned by calls to

(proc key value)for each key-value association in map}}}. {{{proc}}} must return {{{(values new-key new-value)}}}. The key mapping must be a ''monotone,'' preserving the order and uniqueness of keys. The value mapping need not be monotone. If {{{precedes?}}} is given, it is the key order predicate of the new map; otherwise the new map uses the same order predicate as {{{map. Takes O(n) time.

(imap-map proc map [precedes?])As imap-map/monotone}}}, except the {{{proc}}} is not required to be monotone. When multiple keys are mapped to the same {{{new-key, only the association originating from the least pre-mapped key is retained. Takes O(n log n) time.

Returns an association list containing the associations of map in increasing order by key. Takes O(n) time.

(imap-keys map)Returns a set containing the keys of map. Takes O(n log n) time; O(n) is optimal.

(imap-values map)Returns a list containing the values of map in increasing order by key. Takes O(n) time.

(ordered-alist->imap precedes? alist)Returns a map containing the associations of alist}}} and using order predicate {{{precedes?}}}. It is an error for {{{list to be anything other than a proper association list in increasing order by key. Takes O(n log n) time; O(n) is optimal.

(alist->imap precedes? alist)The behavior and efficiency of these operations is the same as the analogous set procedures.

These procedures compare key x}}} to key {{{y}}} according to {{{set}}}'s order predicate and the conventional meaning of {{{<}}}, {{{=}}}, and {{{>.

- Should there be an explicitly-immutable pair and list?
- "imap" is a name clash with the IMAP email protocol. Is this a dealbreaker?