Rust is missing an advanced heap
Gankra opened this issue · comments
One that lets you modify the priority of keys.
My academic background tells me that a pairing_heap is excellent, but I've also heard good things of a meldable heap?
Of note is a nasty API issue around how you specify that a key is to be reprioritized.
My intuition is that push returns a token, and then you increase_key(&token, new_weight) or whatever. The token is a pointer to the node that contains the element, but this has a fundamental memory safety issue (the element associated with the token may have been popped).
One solution is to just make decrease/increase unsafe.
The other is to use Rcs internally, and have tokens be a Weak.
Neither is super great, honestly.
Note that I posit this is impossible to overcome. This is a fundamental issue of data trust, and no mechanism I am aware of could possibly prevent having to validate the tokens (and in turn, maintaining metadata to make it possible to validate the tokens).
Perhaps there's a fundamentally different approach, though?
The pairing heap could have an intrusive map, so you just do lookup with actual keys, but this is definitely way more overhead than necessary.
Note that this kind of heap is generally an optimization to begin with, so too much overhead negates even using it.
I think a way of handling this is to allocate the elements of the queue "manually" by just storing them in a big vec, and refer to them using indices internally. The elements are allocated in a big vec (which can grow dynamically, indices remain valid!). Each "slot" of this array also has a u64 "version number". Whenever an element is popped, the slot in the big array is cleared, and the corresponding version number is incremented. Tokens contain an index to a slot in the element array, as well as the current version number. decrease_key can thus fail, if the token refers to un "unallocated" element in the array, or if the version number isn't correct.
Note that this is essentially just implementing manual memory control over an array and using indices, but the upside is that it needs no unsafe code an won't corrupt the global program heap or the runtime or anything. At worst the data structure fails. But yes, this does mean runtime validation of tokens.
What are the required complexities of the desired operations? In the persistent data structure world, I use a priority search queue when needing to modify priorities, but that not may be best for Rust.
It seems like this is a good use case for just making the data structure intrusive (as opposed to storing extra intrusive metadata). Then removal from the structure doesn't actually free the associated memory.
Would a BTreeMap work here? You get logarithmic insert, delete, update, and find_min. It's also cache friendly.
It would be nice to see a comparison of heap performance on modern hardware, but I'm having trouble finding one.
For reference, the persistent tree-based priority search queue:
- insert is log
- delete is log
- modify is log
- adjust priority is log
- lookup is log
- find min priority is constant
- delete min priority is log
- build is linear
Algorithmic complexity is notably total nonsense for high-performance heaps.
A fibonacci is the theoretical best heap: https://en.wikipedia.org/wiki/Fibonacci_heap
Find-minimum is O(1) amortized time.[1] Operations insert, decrease key, and merge (union) work in constant amortized time.[2] Operations delete and delete minimum work in O(log n) amortized time.[2] This means that starting from an empty data structure, any sequence of a operations from the first group and b operations from the second group would take O(a + b log n) time. In a binomial heap such a sequence of operations would take O((a + b) log n) time. A Fibonacci heap is thus better than a binomial heap when b is asymptotically smaller than a.
A pairing heap meanwhile both has non-tight bounds, and the lower bounds it does have make it theoretically worse than a fibonacci heap. However in practice the hidden constants on a fibonacci heap are attrocious, and a pairing heap is supposed to be better.
So mostly I care about actual perf. a simple benchmark would be "do dijkstras as fast as possible".
Yes, I only care about actual performance also. Ironically, I just tweeted today on the pointlessness of FIbonacci heaps :-) https://twitter.com/franklinchen/status/644894021144461312
IME a D-ary heap organized as a complete D-ary tree encoded in an array (implemented like a binary heap, but with better cache locality.. so child pointers are implicit etc.) beats all of the fancy heaps in practice. Benchmark to find the best D (possibly depends on element size).
It doesn't have the best time complexity for many operations, but the constant factors are pretty minimal so you'd have to be doing something pretty out of the ordinary for it to be beaten by more complex option.
@ssylvan How does one provide decrease_key that isn't O(n) (literally a linear search for the element) for such a data structure?
Perhaps unrelated, but I have a binary heap implementation in Java (unfortunately belonging to my employer, so I can't show code) that allows to change the first element without allocation, re-shuffling the heap as necessary (I use this for a fast "first N out of X" where X ≫ N. It uses a bounded array for data storage.
This interface could be made quite elegant in rust, e.g. fn poke<F, T, U>(self, f: F) -> Option<U> where F: FnOnce(&mut T) -> U. This nicely complements peek, so there are no problems with reordering.
@gankro Reduce priority value, then swap with parent if necessary to maintain heap invariant (repeat until you're at root, or the element is in the right place). That's O(log_d n), where d is the branching factor (i.e. higher branching factor means you don't need to do a lot of work for even huge heaps.. e.g. a billion elements for d=8 is only 10 levels deep).
@ssylvan Note that the constant factors grow with the branching factor, also as your d approaches n, our operations become essentially linear.
@ssylvan I don't understand how you're finding the element. External users of the data structure can't just have pointers/indices, because those are constantly shifting without a node-based structure.
@gankro You'd need to have some kind of "stable handle" just like in the other heaps. The handle would just have an index into the array (and possibly a version number, for validation). The heap element in turn would have a pointer (or index) to the stable handle and update where the handle points whenever the element moves. This is all O(1).
@llogiq Yes. Greater branching factor means a faster decrease_key (because fewer levels), but more expensive delete_min, etc.
That said, the actual operation you care about is cache misses, not comparisons, so you can pick a branching factor equal to cache_line_size/sizeof(T) without really affecting constant factors.
A container that provides a stable handles to the elements stored inside is
building block of many data structures, not only priority queues. In C++ it is
quite typical to use a std::list to that effect, as it guarantees iterators
that are valid until erase. For example boost.heap uses std::list to
provide additional level of indirection required for implicit d-ary heaps that
permit modifications of element priority.
It is interesting to ask then what kind of data structure could play that role
in Rust, while retaining memory safety guarantees at relatively low cost.
Such a container would not be required to detect other handle misuse as long as
it does not cause memory unsafety, for example:
- Handle referring to an element that has already removed, but storage is
reused by another element. - Handle originating from a different container.
Such a container should in general provide an interface supporting following
operations:
impl<T> Container<T> {
...
pub fn push(&mut self, elem: T) -> Handle;
pub fn remove(&mut self, handle: Handle) -> T;
pub fn get(&self, handle: Handle) -> &T;
pub fn get_mut(&mut self, handle: Handle) -> &mut T;
...
}Looking at existing design space, there seem to be two major approaches
possible. First, maps in general. Not necessarily a great fit, due to
additional overhead, but simple ones like array with integer indexes as keys
could be considered. Second, pointers with referencing counting. I expand on
both below, but also wonder what further alternatives and improvements are
possible?
Array with indexes as handles
The simplest design would be to store elements inside a vector and make handles
from indexes. To efficiently find free space for next element, vacant entries
could be linked into a free-list.
pub struct Container<T> {
data: Vec<Entry<T>>,
/// Head of free-list.
free: usize,
/// Number of occupied entries.
size: usize,
}
enum Entry<T> {
/// Empty space, with an index of next free entry.
Vacant(usize)
/// Space used by an element.
Occupied(T)
}
pub struct Handle(usize)To validate an untrusted handle you would:
- Check that a handle index is within vector bounds.
- Check that the entry at given index is occupied.
This is essentially a design used by Carl's slab allocator, and what Sebastian suggested
earlier in the thread.
Note that this only validates handle as far as memory safety is required. It
does not check for handles that corresponds to entries that have been removed
and replaced with something else in the meantime, nor does it check for a
handle being used with a different container altogether.
Reference counted pointers
Another idea is to used reference counted pointers as suggested by Gankro
initially. Weak pointer is stashed in a handle and strong one in the container.
For example, a doubly linked list with reference counted nodes:
pub struct Container<T> {
head / tail: ... linked list of nodes ...
nonce: Box<Nonce>,
}
pub struct Handle<T> {
elem: Weak<Node<T>>
nonce: *const Nonce
}To validate an untrusted handle you would:
- Check that a handle is from this container by comparing address of a nonce.
- Check that element is still valid by upgrading weak pointer to strong one.
The check that handle is from current container seems to be crucial if you want
to expose mutable references to elements in API. Otherwise, multiple mutable
references could be formed by using a valid handle with multiple unrelated
containers. Note that nonce doesn't need to be a reference counted pointer,
because even though a nonce could be reused after container is destroyed, the
element pointers never outlive the nonce.
This addresses the case of elements already removed from list, case of handle
used with different container, and in general as far as I can see this approach
actually solves all possible misuse involving handles. Though, it does require
storing two pointers in each handle, one of which is ref counted.
So I wrote https://github.com/apopiak/hollow_heap and I'd like to get a code review/know how to polish it so it might be a contender for contain-rs.