# Sorted Associative Container

Category: containers | Component type: concept

### Description

A Sorted Associative Container is a type of Associative Container. Sorted Associative Containers use an ordering relation on their keys; two keys are considered to be equivalent if neither one is less than the other. (If the ordering relation is case-insensitive string comparison, for example, then the keys "abcde" and "aBcDe" are equivalent.)

Sorted Associative Containers guarantee that the complexity for most operations is never worse than logarithmic [1], and they also guarantee that their elements are always sorted in ascending order by key.

### Refinement of

Reversible Container, Associative Container

### Associated types

Two new types are introduced, in addition to the types defined in the Associative Container and Reversible Container requirements.

`X::key_compare` | The type of a Strict Weak Ordering used to compare keys. Its argument type must be `X::key_type`.
`X::value_compare` | The type of a Strict Weak Ordering used to compare values. Its argument type must be `X::value_type`, and it compares two objects of `value_type` by passing the keys associated with those objects to a function object of type `key_compare`.

### Notation

`X` | A type that is a model of Sorted Associative Container
`a` | Object of type `X`
`t` | Object of type `X::value_type`
`k` | Object of type `X::key_type`
`p`, `q` | Object of type `X::iterator`
`c` | Object of type `X::key_compare`

### Definitions

### Valid expressions

In addition to the expressions defined in Associative Container and Reversible Container, the following expressions must be valid.

Name | Expression | Type requirements | Return type
--- | --- | --- | ---
Default constructor |
    X()
    X a;
Constructor with compare |
    X(c)
    X a(c);
Key comparison | `a.key_comp()` |  | `X::key_compare`
Value comparison | `a::value_compare()` |  | `X::value_compare`
Lower bound | `a.lower_bound(k)` |  | `iterator` if `a` is mutable, otherwise `const_iterator`.
Upper bound | `a.upper_bound(k)` |  | `iterator` if `a` is mutable, otherwise `const_iterator`.
Equal range | `a.equal_range(k)` |  | `pair<iterator, iterator>` if `a` is mutable, otherwise `pair<const_iterator, const_iterator>`.

### Expression semantics

Name | Expression | Precondition | Semantics | Postcondition
--- | --- | --- | --- | ---
Default constructor |
    X()
    X a;
|  | Creates an empty container, using `key_compare()` as the comparison object. | The size of the container is `0`.
Constructor with compare |
    X(c)
    X a(c);
|  | Creates an empty container, using `c` as the comparison object. | The size of the container is `0`. `key_comp()` returns a function object that is equivalent to `c`.
Key comparison | `a.key_comp()` |  | Returns the key comparison object used by `a`. |
Value comparison | `a::value_compare()` |  | Returns the value comparison object used by `a`. | If `t1` and `t2` are objects of type `value_type`, and `k1` and `k2` are the keys associated with them, then `a.value_comp()(t1, t2)` is equivalent to `a.key_comp()(k1, k2)`.
Lower bound | `a.lower_bound(k)` |  | Returns an iterator pointing to the first element whose key is not less than `k`. Returns `a.end()` if no such element exists. | If `a` contains any elements that have the same key as `k`, then the return value of `lower_bound` points to the first such element.
Upper bound | `a.upper_bound(k)` |  | Returns an iterator pointing to the first element whose key is greater than `k`. Returns `a.end()` if no such element exists. | If `a` contains any elements that have the same key as `k`, then the return value of `upper_bound` points to one past the last such element.
Equal range | `a.equal_range(k)` |  | Returns a pair whose first element is `a.lower_bound(k)` and whose second element is `a.upper_bound(k)`. |

### Complexity guarantees

`key_comp()` and `value_comp()` are constant time.

Erase element is constant time.

Erase key is `O(log(size()) + count(k))`. [1]

Erase range is `O(log(size()) + N)`, where `N` is the length of the range. [1]

Find is logarithmic. [1]

Count is `O(log(size()) + count(k))`. [1]

Lower bound, upper bound, and equal range are logarithmic. [1]

### Invariants

Definition of `value_comp` | If `t1` and `t2` are objects of type `X::value_type` and `k1` and `k2` are the keys associated with those objects, then `a.value_comp()` returns a function object such that `a.value_comp()(t1, t2)` is equivalent to `a.key_comp()(k1, k2)`.
Ascending order | The elements in a Sorted Associative Container are always arranged in ascending order by key. That is, if `a` is a Sorted Associative Container, then `is_sorted(a.begin(), a.end(), a.value_comp())` is always `true`.

### Models

- `set`
- `multiset`
- `map`
- `multimap`

### Notes

[1] This is a much stronger guarantee than the one provided by Associative Container. The guarantees in Associative Container only apply to average complexity; worst case complexity is allowed to be greater. Sorted Associative Container, however, provides an upper limit on worst case complexity.

[2] This definition is consistent with the semantics described in Associative Container. It is a stronger condition, though: if `a` contains no elements with the key `k`, then `a.equal_range(k)` returns an empty range that indicates the position where those elements would be if they did exist. The Associative Container requirements, however, merely state that the return value is an arbitrary empty range.

### See also

Associative Container, Hashed Associative Container

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