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### 270. Binary search requirements overly strict

Section: 25.8.4 [alg.binary.search] Status: CD1 Submitter: Matt Austern Opened: 2000-10-18 Last modified: 2016-01-28

Priority: Not Prioritized

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Duplicate of: 472

Discussion:

Each of the four binary search algorithms (lower_bound, upper_bound, equal_range, binary_search) has a form that allows the user to pass a comparison function object. According to 25.3, paragraph 2, that comparison function object has to be a strict weak ordering.

This requirement is slightly too strict. Suppose we are searching through a sequence containing objects of type X, where X is some large record with an integer key. We might reasonably want to look up a record by key, in which case we would want to write something like this:

        struct key_comp {
          bool operator()(const X& x, int n) const {
            return x.key() < n;
          }
        }

        std::lower_bound(first, last, 47, key_comp());

key_comp is not a strict weak ordering, but there is no reason to prohibit its use in lower_bound.

There's no difficulty in implementing lower_bound so that it allows the use of something like key_comp. (It will probably work unless an implementor takes special pains to forbid it.) What's difficult is formulating language in the standard to specify what kind of comparison function is acceptable. We need a notion that's slightly more general than that of a strict weak ordering, one that can encompass a comparison function that involves different types. Expressing that notion may be complicated.

Additional questions raised at the Toronto meeting:

- Do we really want to specify what ordering the implementor must use when calling the function object? The standard gives specific expressions when describing these algorithms, but it also says that other expressions (with different argument order) are equivalent.
- If we are specifying ordering, note that the standard uses both orderings when describing `equal_range`.
- Are we talking about requiring these algorithms to work properly when passed a binary function object whose two argument types are not the same, or are we talking about requirements when they are passed a binary function object with several overloaded versions of `operator()`?
- The definition of a strict weak ordering does not appear to give any guidance on issues of overloading; it only discusses expressions, and all of the values in these expressions are of the same type. Some clarification would seem to be in order.

Additional discussion from Copenhagen:

- It was generally agreed that there is a real defect here: if the predicate is merely required to be a Strict Weak Ordering, then it's possible to pass in a function object with an overloaded operator(), where the version that's actually called does something completely inappropriate. (Such as returning a random value.)
- An alternative formulation was presented in a paper distributed by David Abrahams at the meeting, "Binary Search with Heterogeneous Comparison", J16-01/0027 = WG21 N1313: Instead of viewing the predicate as a Strict Weak Ordering acting on a sorted sequence, view the predicate/value pair as something that partitions a sequence. This is almost equivalent to saying that we should view binary search as if we are given a unary predicate and a sequence, such that f(*p) is true for all p below a specific point and false for all p above it. The proposed resolution is based on that alternative formulation.

Proposed resolution:

Change 25.3 [lib.alg.sorting] paragraph 3 from:

> 3 For all algorithms that take Compare, there is a version that uses operator< instead. That is, comp(*i, *j) != false defaults to *i < *j != false. For the algorithms to work correctly, comp has to induce a strict weak ordering on the values.

to:

> 3 For all algorithms that take Compare, there is a version that uses operator< instead. That is, comp(*i, *j) != false defaults to *i < *j != false. For algorithms other than those described in lib.alg.binary.search (25.3.3) to work correctly, comp has to induce a strict weak ordering on the values.

Add the following paragraph after 25.3 [lib.alg.sorting] paragraph 5:

> -6- A sequence [start, finish) is partitioned with respect to an expression f(e) if there exists an integer n such that for all 0 <= i < distance(start, finish), f(*(begin+i)) is true if and only if i < n.

Change 25.3.3 [lib.alg.binary.search] paragraph 1 from:

> -1- All of the algorithms in this section are versions of binary search and assume that the sequence being searched is in order according to the implied or explicit comparison function. They work on non-random access iterators minimizing the number of comparisons, which will be logarithmic for all types of iterators. They are especially appropriate for random access iterators, because these algorithms do a logarithmic number of steps through the data structure. For non-random access iterators they execute a linear number of steps.

to:

> -1- All of the algorithms in this section are versions of binary search and assume that the sequence being searched is partitioned with respect to an expression formed by binding the search key to an argument of the implied or explicit comparison function. They work on non-random access iterators minimizing the number of comparisons, which will be logarithmic for all types of iterators. They are especially appropriate for random access iterators, because these algorithms do a logarithmic number of steps through the data structure. For non-random access iterators they execute a linear number of steps.

Change 25.3.3.1 [lib.lower.bound] paragraph 1 from:

> -1- Requires: Type T is LessThanComparable (lib.lessthancomparable).

to:

> -1- Requires: The elements e of [first, last) are partitioned with respect to the expression e < value or comp(e, value)

Remove 25.3.3.1 [lib.lower.bound] paragraph 2:

> -2- Effects: Finds the first position into which value can be inserted without violating the ordering.

Change 25.3.3.2 [lib.upper.bound] paragraph 1 from:

> -1- Requires: Type T is LessThanComparable (lib.lessthancomparable).

to:

> -1- Requires: The elements e of [first, last) are partitioned with respect to the expression !(value < e) or !comp(value, e)

Remove 25.3.3.2 [lib.upper.bound] paragraph 2:

> -2- Effects: Finds the furthermost position into which value can be inserted without violating the ordering.

Change 25.3.3.3 [lib.equal.range] paragraph 1 from:

> -1- Requires: Type T is LessThanComparable (lib.lessthancomparable).

to:

> -1- Requires: The elements e of [first, last) are partitioned with respect to the expressions e < value and !(value < e) or comp(e, value) and !comp(value, e). Also, for all elements e of [first, last), e < value implies !(value < e) or comp(e, value) implies !comp(value, e)

Change 25.3.3.3 [lib.equal.range] paragraph 2 from:

> -2- Effects: Finds the largest subrange [i, j) such that the value can be inserted at any iterator k in it without violating the ordering. k satisfies the corresponding conditions: !(*k < value) && !(value < *k) or comp(*k, value) == false && comp(value, *k) == false.

to:

       -2- Returns:
             make_pair(lower_bound(first, last, value),
                       upper_bound(first, last, value))
           or
             make_pair(lower_bound(first, last, value, comp),
                       upper_bound(first, last, value, comp))

Change 25.3.3.3 [lib.binary.search] paragraph 1 from:

> -1- Requires: Type T is LessThanComparable (lib.lessthancomparable).

to:

> -1- Requires: The elements e of [first, last) are partitioned with respect to the expressions e < value and !(value < e) or comp(e, value) and !comp(value, e). Also, for all elements e of [first, last), e < value implies !(value < e) or comp(e, value) implies !comp(value, e)

[Copenhagen: Dave Abrahams provided this wording]

[Redmond: Minor changes in wording. (Removed "non-negative", and changed the "other than those described in" wording.) Also, the LWG decided to accept the "optional" part.]

Rationale:

The proposed resolution reinterprets binary search. Instead of thinking about searching for a value in a sorted range, we view that as an important special case of a more general algorithm: searching for the partition point in a partitioned range.

We also add a guarantee that the old wording did not: we ensure that the upper bound is no earlier than the lower bound, that the pair returned by equal_range is a valid range, and that the first part of that pair is the lower bound.