The type of keys.

The type of values.

The type of maps. A map can be viewed as a set of pairs (x, v) of a key x and a value v. When a pair (x, v) exists in the map m, we say that the key x is present with value v in the map m. At all times, a map m contains at most one key of each equivalence class: that is, mem m x and mem m y and equiv x y imply x = y.

t is a synonym for map.

create() creates a fresh empty map.

Time complexity: O(1).

copy m returns a new map whose key-value bindings are those of m.

Time complexity: O(c), where c is the capacity of the map m.

If x or some equivalent key is present in the map m, then add_if_absent m x v has no effect and returns false. Otherwise, add_if_absent m x v inserts the key x with value v into the map m and returns true.

Thus, add_if_absent m x v returns true if and only if the cardinality of the map m increases as a result of this operation.

If necessary, the capacity of the map m is increased.

Time complexity: the cost of an insertion operation is often O(1); however, if the capacity of the map must be increased, it is O(n). Because this costly event is infrequent, the amortized complexity of insertion is O(\log n).

If some key that is equivalent to x is present in the map m, then replace m x v removes this pre-existing key and its value, inserts the key x with value v into the map m, and returns false. Otherwise, replace m x v inserts the key x with value v into the map m and returns true.

Thus, replace m x v returns true if and only if the cardinality of the map m increases as a result of this operation.

If necessary, the capacity of the map m is increased.

Time complexity: the cost of an insertion operation is often O(1); however, if the capacity of the map must be increased, it is O(n). Because this costly event is infrequent, the amortized complexity of insertion is O(\log n).

mem m x determines whether the key x, or some key y that is equivalent to x, is present in the map m.

Time complexity: O(1).

find_key m x determines whether some key y that is equivalent to x is present in the map m. If so, y is returned. Otherwise, Not_found is raised.

Time complexity: O(1).

find_value m x determines whether some key y that is equivalent to x is present with value v in the map m. If so, v is returned. Otherwise, Not_found is raised.

Time complexity: O(1).

find is a synonym for find_value.

If the map m has nonzero cardinality, then choose m returns a key that is present in the map m. This key is chosen at random. Otherwise, choose m raises Not_found.

choose invokes Random.int. Two successive calls to choose m can return different results.

Time complexity: O(c) in the worst case and O(c/n) in expectation, where c is the capacity of the map m and n is its cardinality.

If the occupancy rate n/c remains above a certain fixed threshold, then these bounds can be written under the form O(n) in the worst case and O(1) in expectation.

If choose is used in a loop where entries are repeatedly removed then it is recommended to repeatedly call tighten so as to maintain a high occupancy rate.

find_key_else_add m x determines whether some key y that is equivalent to x is present in the map m. If so, y is returned. Otherwise, the key x with value v is inserted into the map m, and Not_found is raised.

find_key_else_add m x v is equivalent to try find_key m x v with Not_found -> ignore (add_if_absent m x v); raise Not_found.

Time complexity: the cost of an insertion operation is often O(1); however, if the capacity of the map must be increased, it is O(n). Because this costly event is infrequent, the amortized complexity of insertion is O(\log n).

find_value_else_add m x determines whether some key y that is equivalent to x is present in the map m with value v. If so, v is returned. Otherwise, the key x with value v is inserted into the map m, and Not_found is raised.

find_value_else_add m x v is equivalent to try find_value m x v with Not_found -> ignore (add_if_absent m x v); raise Not_found.

Time complexity: the cost of an insertion operation is often O(1); however, if the capacity of the map must be increased, it is O(n). Because this costly event is infrequent, the amortized complexity of insertion is O(\log n).

If some key y that is equivalent to x is present in the map m, then remove m x removes y from the map m. Otherwise, nothing happens.

Time complexity: O(1).

If some key y that is equivalent to x is present in the map m, then find_key_and_remove m x removes y from the map m and returns y. Otherwise, the map m is unaffected, and Not_found is raised.

Time complexity: O(1).

If some key y that is equivalent to x is present with value v in the map m, then find_value_and_remove m x removes y from the map m and returns v. Otherwise, the map m is unaffected, and Not_found is raised.

Time complexity: O(1).

foreach_key f m applies the user-supplied function f in turn to each key x in the map m. The function f must not modify the map m: that is, no key-value pairs can be inserted or deleted while iteration is ongoing.

Time complexity: O(c), where c is the capacity of the map m.

foreach_key_value f m applies the user-supplied function f in turn to each pair of a key x and value v in the map m. The function f must not modify the map m: that is, no key-value pairs can be inserted or deleted while iteration is ongoing.

Time complexity: O(c), where c is the capacity of the map m.

iter is a synonym for foreach_key_value.

cardinal m returns the cardinality of the map m, that is, the number of inhabitants of this map.

Time complexity: O(1).

is_empty m is equivalent to cardinal m = 0.

Time complexity: O(1).

clear m empties the map m. The internal data arrays are retained, and are erased. Thus, the capacity of the map m is unchanged.

Time complexity: O(c), where c is the capacity of the map m.

reset m empties the map m. The internal data arrays are abandoned. Thus, the capacity of the map m is reset to a small constant.

Time complexity: O(1).

tighten m decreases the capacity of the map m, if necessary and if possible, so as to ensure that the occupancy rate n/c is high enough. It guarantees either c = O(1), which means that the capacity is below a certain constant, or c = O(n), which means that the occupancy rate is above a certain constant.

Time complexity: O(c), where c is the capacity of the set s.

In the case where there is nothing to do, tighten has constant cost. Thus, the amortized complexity of a call to tighten, in a loop where entries are repeatedly removed, is O(\log n).

cleanup m invokes tighten m and eliminates the tombstones that earlier deletion operations may have created in the internal data array. This can speed up future insertions and lookups.

Time complexity: O(c), where c is the capacity of the map m.

show show_key show_value m returns a textual representation of the map m. The user-supplied functions show_key and show_value are used to obtain textual representations of keys and values.

Time complexity: O(c), where c is the capacity of the map m.

capacity m returns the current capacity of the map m, that is, the current size of its internal data arrays.

Time complexity: O(1).

occupation m returns the current occupation of the map m, that is, the number of occupied entries in its internal data arrays. This number may be greater than cardinal m.

Time complexity: O(1).

Assume that the key x is present in the map m. We say that this key has search length k if the function call mem m x requires reading k+1 successive slots in the internal data array of the map m. In the best case, a key has search length 0. If there are collisions, then some keys have search length greater than 0.

A present-key histogram for the map m is a finite association map that maps a natural integer k to the number in keys of the map m that have search length k. The cardinality of this histogram is n, the cardinality of the map m.

The average search length should be a good a predictor of the cost of searching for a key that is present in the map.

We say that the slot at index i in an internal data array has insertion length k if finding the first empty slot, beginning at index i, requires reading k+1 successive slots. An empty slot has insertion length 0. A nonempty slot has insertion length greater than 0.

An absent-key histogram for the map m is a finite association map that maps a natural integer k to the number of slots in the data array of the map m that have insertion length k. The cardinality of this histogram is c, the capacity of the map m.

The average insertion length should be a good a predictor of the cost of inserting a key that is not present in the map.

present_key_histogram m returns a present-key histogram for the map m.

Time complexity: O(c\log c), where c is the capacity of the map m.

absent_key_histogram m returns an absent-key histogram for the map m.

Time complexity: O(c \log c), where c is the capacity of the set s.

average h returns the average value of the histogram h.

Time complexity: O(n), where n is the cardinality of the histogram h.

statistics m returns a string of information about the map m. This information includes the cardinality, capacity, occupancy rate, average search length, present-key histogram, average insertion length, and absent-key histogram.

Time complexity: O(c \log c), where c is the capacity of the map m.