unordered_multiset’s API affects its big-O

An STL-style container’s performance can be dramatically affected by minor changes to the underlying data structure’s invariants, which in turn can be dramatically constrained by the container’s chosen API. Since the Boost.Unordered performance project has put std::unordered_foo back in the spotlight, I thought this would be a good week to talk about my favorite little-known STL trivia tidbit: std::unordered_multiset’s decision to support .equal_range dramatically affects its performance!


First some background on the original std::multiset, which was part of C++98’s STL. Both set and multiset are represented in memory as a sorted binary search tree (specifically, on all implementations I’m aware of, it’s a red-black tree); the only difference is that multiset is allowed to contain duplicates.

std::set<int> s = {33,11,44,11,55,99,33};
std::multiset<int> ms = {33,11,44,11,55,99,33};
assert(std::ranges::equal(s, std::array{11,33,44,55,99}));
assert(std::ranges::equal(ms, std::array{11,11,33,33,44,55,99}));

Because set and multiset are stored in sorted order, they have the “special skills” .lower_bound(key), .upper_bound(key), and .equal_range(key):

auto [lo, hi] = ms.equal_range(33);
assert(std::distance(lo, hi) == 2);
assert(std::count(lo, hi, 33) == 2);
assert(lo == ms.lower_bound(33));
assert(hi == ms.upper_bound(33));

When C++11 added std::unordered_set and std::unordered_multiset, part of the idea was that (API-wise) they should be drop-in replacements for the tree-based containers. The only difference is that each unordered container is represented in memory as a hash table: an array of “buckets,” each bucket being a linked list of elements with the same hash modulo bucket_count(). Since the order of the elements depends on the order of the buckets, and the order of the buckets depends on hashing, not less-than, the elements in an unordered container aren’t intrinsically stored in sorted order. On libc++, for example, I see this:

std::unordered_set<int> us = {33,11,44,11,55,99,33};
std::unordered_multiset<int> ums = {33,11,44,11,55,99,33};
assert(std::ranges::equal(us, std::array{55,99,44,11,33}));
assert(std::ranges::equal(ums, std::array{55,44,33,33,11,11,99}));

And on libstdc++:

assert(std::ranges::equal(us, std::array{99,55,44,11,33}));
assert(std::ranges::equal(ums, std::array{99,55,44,11,11,33,33}));

Because unordered_set and unordered_multiset aren’t stored sorted, they can’t possibly have .lower_bound and .upper_bound methods. But notice that the unordered containers do provide .equal_range as a drop-in replacement for the tree-based containers’ .equal_range!

auto [lo, hi] = ums.equal_range(33);
assert(std::distance(lo, hi) == 2);
assert(std::count(lo, hi, 33) == 2);

The existence of std::unordered_multiset::equal_range implies that every standards-conforming implementation of std::unordered_multiset must store duplicate elements adjacent to each other. No conforming implementation can ever store the sequence

assert(std::ranges::equal(ums, std::array{55,44,33,11,11,33,99}));

because then the range denoted by the iterator-pair ums.equal_range(33) would contain some elements that were not actually duplicates of 33. (By the way, when I say “a is a duplicate of b,” I mean ums.key_eq()(a,b); in general this is a stronger condition than simply hashing to the same value, but may be weaker than a == b if the container uses a custom comparator.)

Notice that ums.equal_range(33) is well-defined, but std::equal_range(ums.begin(), ums.end(), 33) would have undefined behavior, because the std::equal_range algorithm requires as a precondition that the sequence be sorted.

Consequences of the equal_range invariant

For unordered_multiset, the decision to provide .equal_range has some interesting consequences. We basically had two possible bookkeeping designs a priori:

  • Store duplicates adjacent to each other. equal_range is possible. insert is \(O(n)\) (because it must preserve the invariant). count and erase(key) are relatively faster.

  • Store duplicates willy-nilly. equal_range is impossible. insert is \(O(1)\). count and erase(key) are relatively slower (because they cannot assume that all the duplicates are adjacent).

The Standard, by mandating that equal_range must be part of the container’s API, ends up incidentally mandating a low-level invariant on the data structure, which in turn affects the performance of insert, erase, and count!

The Standard mandates some other details in more straightforward ways; for example, the obscure .max_load_factor() API, or the .bucket_count() getter.

Joaquín López Muñoz points out that since unique-key associative containers like set provide .find(key), it only stands to reason that associative containers that can contain duplicate keys, like multiset, ought to provide something along the lines of .find_all_duplicates(key). The Standard’s unordered_multiset::equal_range(key) fills this ecological niche in the API, but one could imagine an alternative specification compatible with willy-nilly bookkeeping, such as a member function that returns a pair of duplicate_iterators with an implementation akin to C++20’s filter_view::iterator. Peter Dimov further points out that duplicate_iterator could have different invalidation guarantees; for example, we might choose to guarantee that these iterators are never invalidated by the insertion or erasure of some unrelated key.

Just for kicks, I modified libc++’s unordered_multiset to eliminate equal_range and make the algorithmic changes to insert, count, and erase. (Here’s the patch.) Then, I wrote a few quick and dirty benchmarks. You can see the actual benchmark code here, but here’s a high-level idea of what each benchmark was testing:

// inserts (ctor)
std::unordered_multiset<int> mm(g_data.begin(), g_data.end());

// insert calls
for (int i : g_data) mm.insert(i);

// erase calls
for (int i : g_unique_values) mm.erase(i);

// counts (all found)
for (int i : g_unique_values) mm.count(i);

// counts (mostly absent)
for (int i : g_unique_values) mm.count(i + 1);

Here are the benchmark results on a workload of 370,000 ints. First I used a flat distribution — 37 copies each of 10,000 randomly distributed values. Then, I ran the same operations again with a more Zipf-like distribution, where the top 10 values appear 1000 times each and the bottom 9000 values appear only once each.

Benchmark description Before (Flat) After (Flat) Before (Zipf) After (Zipf)
370,000 inserts (ctor) 394ms 73ms 68ms 4.7ms
370,000 insert calls 382ms 53ms 67ms 4.1ms
10,000 erase calls 107ms 113ms 5.4ms 5.6ms
10,000 counts (all found) 30ms 25ms 0.6ms 0.9ms
10,000 counts (mostly absent) 0.3ms 0.3ms 0.3ms 0.3ms

Scale up the amount of data by a factor of ten, to 3,700,000 inserts, and the differences become wildly exaggerated. Remember that our patch doesn’t just make insert 5x or 10x faster; it makes insert \(O(n)\) times faster!

Benchmark description Before (Flat) After (Flat) Before (Zipf) After (Zipf)
3,700,000 inserts (ctor) 8735ms 1828ms 21345ms 475ms
3,700,000 insert calls 8404ms 1935ms 22528ms 408ms
100,000 erase calls 1782ms 1799ms 504ms 575ms
100,000 counts (all found) 413ms 378ms 80ms 109ms
100,000 counts (mostly absent) 7.8ms 9.0ms 7.9ms 9.8ms

The macro trends are as expected: a massive \(O(n)\)–to–\(O(1)\) speedup for insert, and a constant-factor slowdown for count. The results for erase and count are surprisingly equivocal. The only explanations I can think of are that noise in the non-\(O(n)\) part masked the signal; or that my new implementations happened to make the optimizer happier than libc++’s original implementations (for example, by using nd = nd->next instead of ++it to walk the bucket), and this affected the constant factor enough to cancel out some of the theoretical advantage. In the case of count on the flat distribution, we even get a reproducible speedup; I can’t account for this in any way except to blame the optimizer.


The most important takeaway here is that changes to a container’s API can have dramatic and surprising knock-on effects to the underlying data structure and its performance. I’m trying to distinguish here between on the one hand the container and its API (unordered_multiset, equal_range) and on the other hand the data structure and its invariants (the hash table of linked lists, the invariant that all duplicates are stored sequentially). In theory, it’s possible to imagine implementing the same container API using different underlying data structures. But in practice, type authors have to be very careful that when their choice of API constrains the data structure underneath, those constraints aren’t unnatural or unduly pessimizing.

For example, above I mentioned that we might provide a .find_all_duplicates(key) that returns a pair of duplicate_iterators, which we might choose to guarantee are never invalidated by the insertion of some unrelated key. That guarantee is an API decision that constrains our implementation of insert! Now, when insert rehashes the container, the relative order of each set of duplicates must be preserved; otherwise someone iterating over a set of duplicates could enter an infinite loop. If we choose an API that doesn’t give that guarantee about invalidation, then we have more choices about how to implement rehashing.

In the specific case of unordered_multiset, I won’t even go so far as to say that C++11 necessarily made the wrong choice. Sure, we could have made unordered_multiset::insert arbitrarily faster, but only at the cost of a constant-factor slowdown to unordered_multiset::count — and so C++11 made a defensible choice from the point of view of a program whose workload requires many counts in a static container.

Much more importantly, C++11 made the right choice from the point of view of a programmer who already uses the equal_range method of std::multiset today and wants to upgrade to std::unordered_multiset tomorrow without rewriting too much of their code.

But did C++11 necessarily make the right choice here — do such workloads and programmers exist in sufficient numbers to justify the performance cost to other programmers? I wouldn’t go so far as to say that, either!

Posted 2022-06-23