This proposal defines *immutable* data structures for queues, sets, and maps. A structure is immutable when all its operations leave the structure unchanged. Note that none of the procedures specified here ends with a !.

Immutable structures are sometimes called *persistent* and are closely related to *pure-functional* (a.k.a. *pure*) structures. The availability of immutable data structures facilitates writing efficient programs in the pure-functional style.

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.

Deques, sets, and maps are disjoint from all other Scheme types, and deques are disjoint from sets and maps. It is unspecified whether sets and maps are disjoint.

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. A pair of lists is also a suitable implementation.

(ideque [ *element* ...])

Returns a deque containing the *elements*. 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-tabulate *n proc*)

Invokes *proc* successively on the integers 0 through *n* - 1 inclusive, and returns a deque containing the results in the given order. Takes O(n) time.

(ideque-unfold *stop? mapper successor seed*)

Invokes the predicate *stop?* on *seed*. If it returns false, generate the next result by applying *mapper* to *seed*, generate the next seed by applying *successor* to *seed*, and repeat this algorithm with the new seed. If *stop?* returns true, return a deque containing the results in order of accumulation. Takes O(n) time.

(ideque-unfold-right *stop? mapper successor seed*)

Invokes the predicate *stop?* on *seed*. If it returns false, generate the next result by applying *mapper* to *seed*, generate the next seed by applying *successor* to *seed*, and repeat the algorithm with the new seed. If *stop?* returns true, return a deque containing the results in reverse order of accumulation. Takes O(n) time.

(ideque? *x*)

Returns #t if *x* is an ideque, and #f otherwise. Takes O(1) time.

(ideque-empty? *deque*)

Returns #t if *deque* contains zero elements, and #f otherwise. Takes O(1) time.

(ideque-front *deque*)

(ideque-back *deque*)

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

(ideque-remove-front *deque*)

(ideque-remove-back *deque*)

Returns a deque with the front/back element of *deque* removed. It is an error for *deque* to be empty. Each takes O(1) time.

(ideque-add-front *deque obj*)

(ideque-add-back *deque obj*)

Returns a deque with *obj* pushed to the front/back of *deque*. Each takes O(1) time.

(ideque-take *deque n*)

(ideque-take-right *deque n*)

Returns a deque containing the first/last *n* elements of *deque*. Takes O(n) time.

(ideque-drop *deque n*)

(ideque-drop-right *deque n*)

Returns a deque containing all but the first/last *n* elements of *deque*. Takes O(n) time.

(ideque-split-at *deque n*)

Returns two values, the results of (ideque-take *deque n*) and (ideque-drop *deque n*) respectively. Takes O(n) time.

(ideque-length *deque*)

Returns the length of *deque* as an exact integer. May take O(n) time, though O(1) is optimal.

(ideque-append *deque* ...)

Returns a 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.

(ideque-concatenate *list-of-deques*)

Returns a deque with the contents of the first deque in *list-of-deques* followed by the others. This is provided for Schemes in which the number of arguments which can be passed to apply is limited. 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.

(ideque-reverse *deque*)

Returns a deque containing the elements of *deque* in reverse order. Takes O(n) time.

(ideque-count *pred deque*)

Returns the number of elements of *deque* which satisfy *pred* as an exact integer. Takes O(n) time.

(ideque-delete *pred deque*)

Returns a deque containing the elements of *deque*, except for those which satisfy *pred*. Takes O(n) time.

(ideque-zip *deque* ...)

Returns a deque of lists (not deques) each of which contains the corresponding element of the argument deques in the order specified. Processing stops when all the elements of any deque have been seen. Takes O(kn) time, where *k* is the number of deques and *n* is the number of elements involved.

(ideque-map *proc deque*)

Applies *proc* to each element of *deque* and returns a deque containing the results in order. Takes O(n) time.

(ideque-for-each *proc deque*)

Applies *proc* to each element of *deque* in order and returns an unspecified result. Takes O(n) time.

(ideque-fold *proc nil deque*)

(ideque-fold-right *proc nil deque*)

Invokes *proc* on each member of *deque* in order/reverse order, passing the result of the previous invocation as a second argument. For the first invocation, *nil* is used as the second argument. Returns the result of the last invocation, or *nil* if there was no invocation. Takes O(n) time.

(ideque-append-map *proc deque*)

Applies *proc* to each element of *deque*. It is an error if the result is not a list. Returns a deque containing the elements of the lists in order. Takes O(n) time.

(ideque-filter *pred deque*)

(ideque-remove *pred deque*)

Returns a deque which contains the elements of *deque* that do/do not satisfy *pred*. Takes O(n) time.

(ideque-partition *proc deque*)

(ideque-find *pred deque failure*)

(ideque-find-right *pred deque failure*)

Returns the first/last element of *deque* that satisfies *pred*. If there is no such element, invokes the thunk *failure* and returns what it returns. Takes O(n) time.

(ideque-take-while *pred deque*)

(ideque-take-while-right *pred deque*)

Returns a deque containing the longest initial/final prefix of elements in *deque* all of which satisfy *pred*. Takes O(n) time.

(ideque-drop-while *pred deque*)

(ideque-drop-while-right *proc deque*)

Returns a deque which omits the longest initial/final prefix of elements in *deque* all of which satisfy *pred*, but includes all other elements of *deque*. Takes O(n) time.

(ideque-span *pred deque*)

(ideque-break *pred deque*)

Returns two values, the initial prefix of the elements of *deque* which do/do not satisfy *pred*, and the remaining elements. Takes O(n) time.

(ideque-any? *pred deque*)

(ideque-every? *pred deque*)

Invokes *pred* on the elements of *deque* in order until one of them returns a true/false value, which is then returned. If there are no such elements, returns #f/#t. Takes O(n) time.

(list->ideque *list*)

(ideque->list *deque*)

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.

ideque-comparator

A comparator suitable for comparing ideques. It does not provide comparison procedures, as there is no ordering between ideques.

A sorted set data structure stores a finite collection of unique elements with a defined comparator.

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.

If two elements are to be inserted into a set that are equal in the sense of the set's comparator but are not eqv?, the first to be specified or generated prevails.

(iset *comparator* [ *element* ...])

Returns a set containing elements element..., where *comparator* provides the criterion of identity and ordering. Takes O(n log n) time.

(iset-tabulate *n proc*)

Invokes *proc* successively on the integers 0 through *n* - 1 inclusive, and returns a set containing the results. Takes O(n) time.

(iset-unfold *stop? mapper successor seed*)

Invokes the predicate *stop?* on *seed*. If it returns false, generate the next result by applying *mapper* to *seed*, generate the next seed by applying *successor* to *seed*, and repeat this algorithm with the new seed. If *stop?* returns true, return a set containing the results. Takes O(n) time.

(iset? *obj*)

Returns #t if *obj* is a set, and #f otherwise. Takes O(1) time.

(iset-empty? *set*)

Returns #t if *set* contains zero elements, and #f otherwise. Takes O(1) time.

(iset-member? *set obj*)

Returns #t if *set* contains *obj*, and #f 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.

(iset-comparator *set*)

Returns the comparator of *set*. Takes O(1) time.

(iset-adjoin *set obj*)

Returns a set which contains the elements of *set* and *obj* as well. If there is already an element of *set* that is equal (in the sense of the comparator) to *obj*, the existing element of *set* prevails. Takes O(log n) time.

(iset-adjoin-all *set list*)

Returns a set which contains the elements of *set* and the elements of *list* as well. It is an error if the elements of *list* are not increasing in the sense of the comparator of *set*. If there is already an element of *set* which is equal (in the sense of the comparator) to an element of *list*, the element of *set* prevails. Takes O(k log n) time, where *k* is the length of *list*.

(iset-replace *set obj*)

Returns a set which contains the elements of *set* and *obj* as well. If there is already an element of *set* that is equal (in the sense of the comparator) to *obj*, *obj* prevails. Takes O(log n) time.

(iset-delete *set obj*)

Returns a set which contains the elements of *set* with the exception of *obj*, if present. If there is already an element of *set* that is equal (in the sense of the comparator) to *obj*, the element of *set* prevails. Takes O(log n) time.

(iset-delete-all *set list*)

Returns a set which contains the elements of *set*, excluding the elements of *list*. It is an error if the elements of *list* are not increasing in the sense of the comparator of *set*. Takes O(k log n) time, where *k* is the length of *list*.

(iset-predecessor *set obj failure*)

(iset-successor *set obj failure*)

Returns the element that immediately precedes/succeeds *obj* in *set's* ordering. If no such element exists because *obj* is the minimum/maximum element, or because *set* is empty, returns the result of invoking the thunk *failure*. Takes O(log n) time.

(iset-search *set obj failure success*)

A continuation-based universal update procedure. Attempts to find an element in *set* equal (in the sense of the comparator) to *obj*. When such an element is found, iset-search calls *(success match update remove)*. The *success* procedure either tail-calls * (update new-element ret)* to replace the matched element with the new element, or else tail-calls *(remove ret)* to remove the matched element from *set*.

When no such match is found, iset-search calls *(failure insert ignore)*, which either tail-calls *(insert ret)* to insert *obj* into *set*, or else tail-calls *(ignore ret)* .`

In all cases, iset-search returns two values, a set reflecting the indicated modification (if any) and the value *ret* 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-size *set*)

Returns the size of *set* as an exact integer. May take O(n) time, though O(1) is optimal.

(iset-find *set obj failure*)

Returns the element equal (in the sense of the comparator of *set*) to *obj* in *set*, or the result of invoking the thunk *failure* if no such element exists. Takes O(log n) time.

(iset-count *pred set*)

Returns the number of elements in *set* which satisfy *pred* as an exact integer. Takes O(n) time.

(iset-any? *pred set*)

(iset-every? *pred set*)

Invokes *pred* on the elements of *set* until one of them returns a true/false value, which is then returned. If there are no such elements, returns #f/#t. Takes O(n) time.

(iset-range= *set obj*)

(iset-range< *set obj*)

(iset-range> *set obj*)

(iset-range<= *set obj*)

(iset-range>= *set obj*)

Returns a set containing only the elements of set that are equal to / less than / greater than / less than or equal to / greater than or equal to *obj*. 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.

(iset-between *set least include-least most include-most*)

Returns a set containing the elements of *set* in the interval between *least*and *most*. If *include-least/include-most* is true then the result includes an element 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-filter *pred set*)

(iset-remove *pred set*)

Returns a set containing only those elements x for which (predicate? x) returns true/false. Takes O(n log n) time; O(n) is optimal.

(iset-partition*pred set*)

Returns two values, (iset-filter *pred set*) and (iset-remove *pred set*) respectively.

(iset-fold *proc nil 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* [ *comparator* ])

Returns a set containing the result of invoking *proc* on every element in *set*. It is an error unless *proc* is a *monotone* unary procedure that preserves the order of set elements. Observe that mapping a set of unique elements with a monotone function yields a set of unique elements, so element uniqueness follows from the monotonicity assumption. If *comparator* is given, it is the comparator of the result; otherwise the result uses the same comparator as *set*. Takes O(n) time.

(iset-map*proc set* [ *comparator* [ *merger* ] ])

Like iset-map/monotone, except that *proc* is not required to be monotone. The merger procedure is used to select among any duplicate elements (in the sense of the comparator of *set*) that might be created; it returns the value to be used; if absent, the element chosen is implementation-specific. Takes O(n log n) time.

(iset-for-each *proc set*)

Invokes '''proc'' on every ''element'' in ''set''. The result is unspecified. Takes O(n) time.

Note: The following three predicates do not obey the trichotomy law and therefore do not constitute a total order on sets.

(iset=? *set1 set2* ...)

Returns #t if each *set* contains the same elements, and #f otherwise.

(iset<? ''set1 set2'' ...)`

Returns #t if each *set* other than the last is a proper subset of the following set, and #f otherwise.

(iset>? ''set1 set2'' ...)`

Returns #t if each *set* other than the last is a proper superset of the following set, and #f otherwise.

(iset<=? ''set1 set2'' ...)`

Returns #t if each *set* other than the last is a subset of the following set, and #f otherwise.

(iset>=? ''set1 set2'' ...)`

Returns #t if each *set* other than the last is a superset of the following set, and #f otherwise.

(iset->list *set*)

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

(ordered-list->iset comparator list)

Returns a set containing the elements of *list* and using *comparator*. 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 *comparator list [ *merger'' ])

Returns a set containing the elements of *list* and using *comparator*. It is an error if list is not a proper list, but it may contain duplicates and need not be in order. The merger procedure is used to select among any duplicate elements (in the sense of the comparator of *set*) that might be created; it accepts the existing and new elements and returns the value to be used. Takes O(n log n) time.

(iset-union *set* ... )

(iset-intersection *set* ... )

(iset-difference *set* ... )

(iset-xor *set _{1} set_{2}*)

Returns a set containing the union/intersection/difference/symmetric difference of the arguments. All the arguments must be sets sharing an equivalent comparator. The set operator is applied in a left-associative order. If an element is found in more than one set, the first set in the argument list prevails. May take O(kn log n) time, where k is the number of sets and n is the number of elements involved, though O(kn) time is optimal.

A map data structure stores key-value associations from a map of keys with a comparator to 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 may implement a distinct data structure specific to maps.

If two associations are to be inserted into a map that are equal in the sense of the map's comparator but are not eqv?, the first to be specified or generated prevails.

(imap *comparator* ( *key value* ...])

Returns a map using*comparator*. For each pair of arguments, an association is added to the map with *key* as its key and *value* as its value. Takes O(n log n) time.

(imap-tabulate *n proc*)

Invokes *proc* successively on the integers 0 through *n* - 1 inclusive, and returns a map containing the results. Takes O(n) time.

(imap-unfold *stop? mapper successor seed*)

Invokes the predicate *stop?* on *seed*. If it returns false, generate the next result by applying *mapper* to *seed*, generate the next seed by applying *successor* to *seed*, and repeat this algorithm with the new seed. If *stop?* returns true, return a map containing the results. Takes O(n) time.

(imap? *obj*)

Returns #t if *obj* is a map, and #f otherwise. Takes O(1) time.

(imap-empty? *map*)

Returns #t if *map* contains zero associations, and #f otherwise. Takes O(1) time.

(imap-contains? *map obj*)

Returns #t if *map* contains *obj* as a key, and #f otherwise. Takes O(log n) time.

(imap-min *map*)

(imap-max *map*)

Returns the least/greatest key of *map*. It is an error for*map* to be empty. Takes O(log n) time; O(1) is optimal.

(imap-comparator *map*)

Returns the comparator of *map*. Takes O(1) time.

(imap-add *map key value*)

Returns a map which contains the associations of *map* and an association with *key* and *value* as well. If there is already an association of *map* whose key is equal (in the sense of the comparator) to *key*, the existing key prevails. Takes O(log n) time.

(imap-add-all *map key-list value-list*)

Returns a map which contains the associations of *map* and associations constructed from the corresponding associations of *key-list* and *value-list* as well. It is an error if the associations of *key-list* are not increasing in the sense of the comparator of *map*. If there is already an association of *map* which is equal (in the sense of the comparator) to an association of *list*, the key of *map* prevails. Takes O(k log n) time, where *k* is the length of *list*.

(imap-replace *map key value*)

Returns a map which contains the associations of *map* and an association with *key* and *value* as well. If there is already an association of *map* whose key is equal (in the sense of the comparator) to *key*, *key* prevails. Takes O(log n) time.

(imap-delete *map key*)

Returns a map which contains the associations of *map* with the exception of any association whose key is *key*. If there is already an association of *map* whose key is equal (in the sense of the comparator) to *key*, the existing key prevails. Takes O(log n) time.

(imap-delete-all *map key-list*)

Returns a map which contains the associations of *map*, excluding any associations whose keys appear in *key-list*. It is an error if the associations of *list* are not increasing in the sense of the comparator of *map*. Takes O(k log n) time, where *k* is the length of *key-list*.

(imap-predecessor *map obj failure*)

(imap-successor *map obj failure*)

Returns the key that immediately precedes/succeeds obj in map's ordering. If no such association exists because *obj* is the minimum/maximum key, or because *map* is empty, returns the result of invoking the thunk *failure*. Takes O(log n) time.

(imap-search *map obj failure success*)

A continuation-based universal update procedure. Attempts to find an association in *map* whose key is equal (in the sense of the comparator) to *obj*. When such an association is found, imap-search calls *(success key update remove)*. The *success* procedure either tail-calls * (update new-value ret)* to return a map that associates the matched key with *new-value*, or else tail-calls *(remove ret)* to remove the matched association from *map*.

When no such association is found, imap-search calls *(failure insert ignore)*, which either tail-calls *(insert new-value ret)* to insert an association whose key is *obj* and whose value is *new-value* into *map*, or else tail-calls *(ignore ret)* .`

In all cases, imap-search returns two values, a map reflecting the indicated modification (if any) and the value *ret* 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-size *map*)

Returns the size of *map* as an exact integer. May take O(n) time, though O(1) is optimal.

(imap-find *map pred failure*)

For each association of *map*, invoke *proc* on its key and value. If *proc* returns true on a value, then return that value. If all the calls to *proc* return #f, return the result of invoking the thunk *failure*. Takes O(log n) time.

(imap-count *pred map*)

Returns the number of associations in *map* which satisfy *pred* as an exact integer. Takes O(n) time.

(imap-any *pred map*)

(imap-every *pred map*)

Invokes *pred* on the associations of *map* in order until one of them returns a true/false value, which is then returned. If there are no such associations, returns #f/#t. Takes O(n) time.

(imap-range= *map obj*)

(imap-range< *map obj*)

(imap-range> *map obj*)

(imap-range<= *map obj*)

(imap-range>= *map obj*)

Returns a map containing only the associations of map that are equal to / less than / greater than / less than or equal to / greater than or equal to *obj*. Takes O(k log k + log n) time, where n is the number of associations in the map and k is the number of associations returned; O(k + log n) is optimal.

Note that since map associations are unique, imap-range= returns a map of at most one association.

(imap-between *map least include-least most include-most*)

Returns a map containing the associations of *map* in the interval between *least*and *most*. If *include-least/include-most* is true then the result includes an association 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 associations in the map and k is the number of associations returned; O(k + log n) is optimal.

(imap-filter *pred map*)

(imap-remove *pred map*)

Returns a map containing only those associations x for which (predicate? x) returns true/false. Takes O(n log n) time; O(n) is optimal.

(imap-partition*pred map*)

Returns two values, (imap-filter *pred map*) and (imap-remove *pred map*) respectively.

(imap-fold *proc nil map*)

The fundamental map iterator. Equivalent to, but may be more efficient than, (fold *proc base* (imap->ordered-list *map*)). Takes O(n) time.

(imap-map/monotone *proc map* [ *comparator* ])

Returns a map containing the result of invoking *proc* on every association in *map*. It is an error unless *proc* is a *monotone* unary procedure that preserves the order of map associations. Observe that mapping a map of unique associations with a monotone function yields a map of unique associations, so association uniqueness follows from the monotonicity assumption. If *comparator* is given, it is the comparator of the result; otherwise the result uses the same comparator as *map*. Takes O(n) time.

(imap-map*proc map* [ *comparator* [ *merger* ] ])

Like imap-map/monotone, except that *proc* is not required to be monotone. The merger procedure is used to select among any duplicate associations (in the sense of the comparator of *map*) that might be created; it returns the value to be used; if absent, the association chosen is implementation-specific. Takes O(n log n) time.

(imap-for-each *proc map*)

Invokes '''proc'' on every ''association'' in ''map''. The result is unspecified. Takes O(n) time.

Note: The following three predicates do not obey the trichotomy law and therefore do not constitute a total order on sets.

(imap=? *set1 set2* ...)

Returns #t if each *map* contains the same associations, and #f otherwise.

(imap<? ''set1 set2'' ...)`

Returns #t if each *map* other than the last is a proper subset of the following map, and #f otherwise.

(imap>? ''set1 set2'' ...)`

Returns #t if each *map* other than the last is a proper superset of the following map, and #f otherwise.

(imap<=? ''set1 set2'' ...)`

Returns #t if each *map* other than the last is a subset of the following map, and #f otherwise.

(imap>=? ''set1 set2'' ...)`

Returns #t if each *map* other than the last is a superset of the following map, and #f otherwise.

(imap-keys *imap*)

(imap-values *imap*)

Returns a list of the keys/values of *imap* in increasing order.

(imap-entries *imap*)

Returns two values, lists of the keys and values of *imap* in increasing order.

(imap->alist *map*)

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

(ordered-alist->imap comparator list)

Returns a map containing the associations of *list* and using *comparator*. It is an error for *alist* to be anything other than an alist in increasing order. Takes O(n log n) time; O(n) is optimal.

(alist->imap *comparator list [ *merger'' ])

Returns a map containing the associations of *alist* and using *comparator*.It is an error unless *alist* is a proper association list, but it may contain duplicates and need not be in order. The *merger* procedure is used to select among any duplicate keys (in the sense of the comparator of *map*) that might be created; it accepts the existing and new keys and returns the key to be used. Takes O(n log n) time.

(imap-union *map* ... )

(imap-intersection *map* ... )

(imap-difference *map* ... )

(imap-xor *map _{1} map_{2}*)

Returns a map containing the union/intersection/difference/symmetric difference of the arguments. All the arguments must be sets sharing an equivalent comparator. The map operator is applied in a left-associative order. If an association is found in more than one set, the first map in the argument list prevails. May take O(kn log n) time, where k is the number of sets and n is the number of associations involved, though O(kn) time is optimal.