mutable poset: a data structure for asymptotic expressions

[dkrenn]

The aim of this ticket is to implement a data structure which is to be used within an asymptotic expression (see Meta-Ticket #17601). This mutable poset stores elements together with its successors and predecessors. Those data is updated when a new element is inserted or an element is removed. It offers a couple of efficient routines for manipulations (all of which will be needed for the asymptotic expression).

CC: @behackl @cheuberg @jm58660

Component: asymptotic expansions

Author: Daniel Krenn

Branch/Commit: a0b3d7b

Reviewer: Benjamin Hackl, Clemens Heuberger

Issue created by migration from https://trac.sagemath.org/ticket/17693

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from f7bce7e to 3c69f6b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

34e34dc	Trac #17693, comment 36, 19--23: improve docstring of covers (former `_search_covers_`)
490c223	Trac #17693, comment 36, 28--31: document role of self
3de0a1d	Trac #17693, comment 36, 33: explicitly mention merge and can_merge at top of docstring
485b46f	Trac #17693, comment 36, 34: raise exceptions when merge is not possible
d5d6bec	check=False in MutablePoset.merge since checked already before
6da3db1	Trac #17693, comment 36, 37: extend doc to mention more explicitly that merge is not allowed to change keys
f1e9f9e	Trac #17693, comment 36, 38: clarify parameter key
ae0cad9	Trac #17693, comment 36, 60: mention motivation asymptoic expansions in merge method
ec15597	Trac #17693: alternative implementation add
3c69f6b	Merge remote-tracking branch 'origin/u/cheuberg/asy/mutable-poset-add' into t/17693/asy/mutable-poset

[dkrenn]

comment:39

Replying to @cheuberg:

I read the documentation and the code. I pushed a few reviewer commits.

Cross-reviewed. One commit added.

I have a number of rather minor issues with the documentation which I ask you to fix in order facilitate future maintainance of the code. In three instances, I propose alternative implementations, which might be more readable or slightly more efficient.

Thanks; see my comments below.

1. Please add .. seealso:: blocks between corresponding methods of shell and poset as well as between the accessor methods (keys, elements, shells)

Done.

2. there is an empty "Introduction" section at the beginning of the module documentation, the next heading "Example" is on the same level.

Deleted.

3. this module was written as part of the asymptotic expansion effort. In contrast to the asymptotic expansion, it does not have the "experimental" decorator. I'd feel more comfortable having it, at least until a larger portion of the asymptotic expansions have been merged.

Activated it; it seems (for some (to me) unknown reason) the warning appears now in every doctest and not only once in the file. Thus, deactivated it again; and I am for keeping it this way.

4. MutablePoset.__init__ accepts a list of elements. This could be used in several doctests (in particular, in the set operations) as an abbreviation.

Good idea. Done.

5. MutablePosetShell.__init__: shall we check that element is not None? Otherwise, handling of special elements would probably be affected.

MutablePosetShell is typically used by the MutablePoset; and there, adding a None-element is forbidden (see :meth:add).

6. MutablePosetShell.key: I do not understand the sentence "The element is converted by the poset to the key".

Rewritten.

7. MutablePosetShell.key: I am surprised that the key is not cached. I imagine that many comparisons will be needed in the lifetime of a MutablePoset, and in every case, the property key has to be resolved, which calls the property poset, which calls get_key of the poset.

Now it is cached (see also follow up ticket #19281).

8. MutablePosetShell.__repr__: The note box in the docstring is surprising at this point: reading the source file from the top, this is the first point where the representation of the data is explained, after guessing it from is_special, is_oo, is_null above. I think that it would be better to document these conventions closer to the top, perhaps in the __init__ method or perhaps only as a comment in the source code.

Added a note in the class description.

9. MutablePosetShell.__repr__: The code replicates the behaviour of is_null and is_oo. As the __repr__ method is hardly time critical, I'd prefer using is_null and is_oo, here and thus hiding the internal convention.

Rewritten.

10. MutablePosetShell.le: the note box could be simplified by suppressing implementation details and speaking about the special elements.

Done.

11. MutablePosetShell.le: right <= left: neither right nor left are defined here.

Rewritten.

12. MutablePosetShell.le: the part if other.element is None could be simplified to return not other.successors() as self is already known not to be special here.

True; rewritten.

13. MutablePosetShell.le: If this is time critical, self._predecessors_ could be used instead of self.predecessors().

Changed.

14. MutablePosetShell.eq: note box, see above.

Simplified.

15. MutablePosetShell.eq: emphasize that elements with equal keys are considered equal? Currently, this information is somewhat hidden in the note box which at first glance only seems to explain handling of special elements.

Rewritten note box.

16. MutablePosetShell._copy_all_linked_: Short description: "Return a copy of all shells" does not correspond to the actual return value, which is only the copy of this shell.

Rewritten.

17. MutablePosetShell._copy_all_linked_: the interplay between this method and MutablePoset._copy_shells_ is not well documented: in particular, poset in _copy_all_linked_ is only used for setting the containing poset, but not for actual inserting into this poset.

Extended description of parameter poset.

18. MutablePosetShell._copy_all_linked_: I do not understand why you test oo == P.oo: I think that oo is an element of the new poset Q.

oo == P.oo tests that Q.oo is mapped to P.oo.

So oo is Q.oo would be more interesting?

oo is Q.oo is False since oo is not in Q, but just a copy of the oo in P with parent-poset Q. Inserting this into Q is done in _copy_shells_.

The current test demonstrates that the poset is not used in comparison, so that would rather belong to .eq?

I do not understand what you mean (but it is already late...).

19. MutablePosetShell._search_covers_: what is the role of self in this method? It trivially influences the return value, but what else?

_search_covers_ does not exist anymore (see 27).

20. MutablePosetShell._search_covers_: The explanation of the return value is unclear to me. Is the sentence "Note that..." meant to be a complete description? Does it change if reverse is set?

_search_covers_ does not exist anymore (see 27).

21. MutablePosetShell._search_covers_: I think that the notions of "upper cover" and "lower cover" need a brief definition; I did not find them in Wikipedia and sage.combinat.posets.posets.FinitePoset.upper_covers defines the notion.

_search_covers_ does not exist anymore (see 27). Incooperated in covers.

22. MutablePosetShell._search_covers_: According to Wikipedia, a cover is what is called an upper cover here. This is in contrast to the default behaviour here.

_search_covers_ does not exist anymore (see 27).

23. MutablePosetShell._search_covers_: what is the difference between this method and .predecessors? Is the shell necessarily an element of the poset? If not, then there should be a doctest covering this case.

_search_covers_ does not exist anymore (see 27). Incooperated in covers.

24. MutablePosetShell.covers: "originate" is somewhat foreign to the description of the poset.

Rewritten.

25. MutablePosetShell.covers: "which are at most the given shell": should that be "which are less than the given shell"?

Rewritten.

26. MutablePosetShell.covers see comments on MutablePosetShell._search_covers_.

Rewritten.

27. MutablePosetShell.covers: I think that the following would be an equivalent implementation of this method, which does not need a helper function with side effects.

        if self == shell:
            return set()
        covers = set().union(*(e.covers(shell, reverse)
                               for e in self.successors(reverse)
                               if e.le(shell, reverse)))
        if covers:
            return covers
        else:
            return set([self])

(pushed as branch u/cheuberg/asy/mutable-poset-cover)

Merged and minor simplification.

28. MutablePosetShell._iter_depth_first_visit_: document the role of self in this method (only shells >= this shell are visited).

Documented.

29. MutablePosetShell.iter_depth_first: document the role of self in this method (only shells >= this shell are visited).

Documented.

30. MutablePosetShell._iter_topological_visit_: document the role of self in this method (only shells <= this shell are visited, emphasize the contrast to depth first.).

Documented.

31. MutablePosetShell.iter_topological_first: document the role of self in this method (only shells <= this shell are visited, emphasize the contrast to depth first.).

Documented.

32. MutablePosetShell.iter_topological_sort, last example: function C is defined, but not used.

Deleted.

33. MutablePosetShell.merge: explain what merge is, in particular, that this is defined when constructing the poset.

Documented.

34. MutablePosetShell.merge: I am surprised that check=True leads to a silent failure instead of an exception and that poset._merge_ is None does not raise an exception. Document and test this behaviour?

Behavior changed.

35. MutablePosetShell.merge: what is the difference between poset.remove and poset.discard?

See Python's set.

36. MutablePosetShell.merge: document that the merge function resulting in None leads to deletion of the element.

Done.

37. MutablePosetShell.merge: document that the merge function must not change the position of the element in the poset.

Mentioned this more explicitly in docs

38. MutablePoset.__init__: Clarify the role of key: in particular, key does not only influence sorting, but that no two elements are allowed to have the same key.

Done.

39. MutablePoset.__len__: Clarify that null and oo are not counted.

Done.

40. MutablePoset.shells_topological: The sentence "If this is None, no sorting according to the representation string occurs." is unclear to me, in particular the role of the representation string.

Rewritten.

41. MutablePoset.keys_topological: Due to the chosen key, some of the elements added to the poset are ignored. I do not know why this is demonstrated here.

Removed.

42. MutablePoset.repr: INPUT section is incomplete.

Completed.

43. MutablePoset.repr_full: INPUT section is incomplete.

Completed.

44. MutablePoset.add: the loop for reverse in (False, True) simplifies in the second iteration: as all pairs of elements (s, l) with new covering s and l covering new have already been broken up while reverse=False. Thus it might be more straightforward to do it only once, not using reverse at all and saving a few intersections:

        new._predecessors_ = self.null.covers(new, reverse=False)
        new._successors_ = self.oo.covers(new, reverse=True)

        for s in new.predecessors():
            for l in s.successors().intersection(new.successors()):
                l.predecessors().remove(s)
                s.successors().remove(l)
            s.successors().add(new)
        for l in new.successors():
            l.predecessors().add(new)

(pushed as branch u/cheuberg/asy/mutable-poset-add)

Merged and cross-reviewed....ok.

45. MutablePoset.remove: the loop over all (not necessarily direct) successors via the depth first search iterator seems to be rather inefficient, especially if the poset is large. I propose an alternative based on the order relation and not based on the data structure:

        for upper in shell.successors():
            upper.predecessors().remove(shell)

        for lower in shell.predecessors():
            lower.successors().remove(shell)
            for upper in shell.successors():
                if not any(s <= upper 
                           for s in lower.successors()):
                    lower.successors().add(upper)
                    upper.predecessors().add(lower)

(pushed as branch u/cheuberg/asy/mutable-poset-remove)

This needs comparisons of the elements, but one of the main ideas of this data structure is that comparisions are only needed for inserting an element into the poset and none needed once it is inside. Thus I am for keeping the current solution. However, I am fine with introducing (now or at any point later) an algorithm option which makes it possible to select different removing algorithms depending on the situation.

46. MutablePoset.remove, MutablePoset.discard: add "see also blocks", but also an explanation of the difference between discard and remove. I guess that both methods are available due to some compatibility concern (in that case, please add a remark in the code or the docstring). Otherwise, I'd suggest removing discard, removing the parameter raise_key_error and let the user catch the exception.

Python's set has both, remove and discard. Thus this mutable poset should have it as well. Note-block added. Seealso added.

47. MutablePoset.pop: remove key=lambda c: -c from this example as it is not pertinent to pop.

Done.

48. MutablePoset.pop: mention that special elements cannot be popped.

Done.

49. MutablePoset.pop: IMHO you can remove the first four line (the try: block deleting kwargs['include_special']) because setting kwargs['include_special'] = False should result in the same result.

Deleted.

50. MutablePoset.union: The doctest P.union(P, Q, Q, P) neither fits the description "a poset" or "an iterable of future elements".

Rewritten INPUT-block

51. MutablePoset.union: the word "union" sounds symmetric, however, due to keys and merge functions, it might not be commutative. The note box is not helpful for me.

Note added/changed.

52. MutablePoset.union: the TODO box from union_update also applies here.

Copied.

53. MutablePoset.union_update: why is the semantics of other different w.r.t union, but has the same description? It would seem more logical to me if union would simply call copy and then pass its argument to update_union and let the latter function sort out the iterators?

Indeed. Code rewritten.

54. MutablePoset.difference and MutablePoset.difference_update: see comments on union and union_update

Done.

55. MutablePoset.intersection and MutablePoset.intersection_update: see comments on union and union_update

Done.

56. MutablePoset.intersection_update: how can the AttributeError occur? After all, self is a MutablePoset, so the method keys will be available unless something very strange happens ...

True. Code simplified.

57. MutablePoset.merge: documentation of reverse: what is the default direction?

Documented.

58. MutablePoset.merge: add doctest for RuntimeError.

Added.

59. MutablePoset.merge: document that can_merge is applied in the sense of the condition of depth first iteration, i.e., once can_merge fails, the successors are no longer tested. This is some kind of monotonicity condition on can_merge.

Added note.

60. MutablePoset.merge: it should be mentioned somewhere that this (at first sight strange) merge magic is motivated by asymptotic expansions.

Mentioned.

61. MutablePoset.map: IMHO the example does alter the keys, because the key was the identity map here.

Example changed.

62. MutablePoset.mapping: does the map have to be key preserving as in the method map? Is this method actually needed, see also the recent discussion on inplace vs. non-inplace operations on sage-devel?

Added a note on key-order preservation.

I think having MutablePoset.mapping is useful (from a ui point of view) since one would not naturally thinks that the copy method has a parameter for applying a mapping function.

[cheuberg]

Changed branch from u/dkrenn/asy/mutable-poset to u/cheuberg/asy/mutable-poset

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

db042e0	Trac #17693: fix ReSt and hyperlink issues
bbe02ef	Trac #17693: minor language issues and typos
4deddbd	Trac #17693: remove comment on non-commutativity for difference and update_difference

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 3c69f6b to 4deddbd

[cheuberg]

comment:43

Thank you for your changes. I added three commits. See my remaining comments below.

Replying to @dkrenn:

Replying to @cheuberg:

3. this module was written as part of the asymptotic expansion effort. In contrast to the asymptotic expansion, it does not have the "experimental" decorator. I'd feel more comfortable having it, at least until a larger portion of the asymptotic expansions have been merged.

Activated it; it seems (for some (to me) unknown reason) the warning appears now in every doctest and not only once in the file. Thus, deactivated it again; and I am for keeping it this way.

ok.

7. MutablePosetShell.key: I am surprised that the key is not cached. I imagine that many comparisons will be needed in the lifetime of a MutablePoset, and in every case, the property key has to be resolved, which calls the property poset, which calls get_key of the poset.

Now it is cached (see also follow up ticket #19281).

I rather thought about calling key in __init__ as I guess that the key will be needed at least once in the lifetime of every MutablePosetShell.

18. MutablePosetShell._copy_all_linked_: I do not understand why you test oo == P.oo: I think that oo is an element of the new poset Q.

oo == P.oo tests that Q.oo is mapped to P.oo.

wouldn't oo.is_oo do it without comparing shells of different posets, which might be confusing?

So oo is Q.oo would be more interesting?

oo is Q.oo is False since oo is not in Q, but just a copy of the oo in P with parent-poset Q.

that would be good to test (and comment on).

Inserting this into Q is done in _copy_shells_.

The current test demonstrates that the poset is not used in comparison, so that would rather belong to .eq?

I do not understand what you mean (but it is already late...).

I mean that shells e and f might be equal even if they do belong to different posets; this might be an interesting doctest or example for equality.

22. MutablePosetShell._search_covers_: According to Wikipedia, a cover is what is called an upper cover here. This is in contrast to the default behaviour here.

_search_covers_ does not exist anymore (see 27).

This does not solve the problem.

What about renaming the method covers to lower_covers (and adapting the docstring slightly, removing "upper " from the one-sentence-description as well as from the definition?) For symmetry, a method upper_covers would be nice.

45. MutablePoset.remove: the loop over all (not necessarily direct) successors via the depth first search iterator seems to be rather inefficient, especially if the poset is large. I propose an alternative based on the order relation and not based on the data structure:

        for upper in shell.successors():
            upper.predecessors().remove(shell)

        for lower in shell.predecessors():
            lower.successors().remove(shell)
            for upper in shell.successors():
                if not any(s <= upper 
                           for s in lower.successors()):
                    lower.successors().add(upper)
                    upper.predecessors().add(lower)

(pushed as branch u/cheuberg/asy/mutable-poset-remove)

This needs comparisons of the elements, but one of the main ideas of this data structure is that comparisions are only needed for inserting an element into the poset and none needed once it is inside. Thus I am for keeping the current solution. However, I am fine with introducing (now or at any point later) an algorithm option which makes it possible to select different removing algorithms depending on the situation.

This should be discussed when more benchmarking results are available. I opened #19300 for that.

54. MutablePoset.difference and MutablePoset.difference_update: see comments on union and union_update

Done.

removed comment on non-commutativity because difference is non-commutative by definition.

55. MutablePoset.intersection and MutablePoset.intersection_update: see comments on union and union_update

Done.

Remove comment on non-commutativity? merge does not play a role here, and keys might be covered in the previous sentence?

63. MutablePoset.update_union: "Due to keys and a merge function... this operation might not be commutative": This method is non-commutative by definition, as self is changed and other is not. So perhaps: "left.update_union(right)" and "right.update_union(left) might result in different posets"? The same for other update_... methods where non-commutativity might be surprising.

[dkrenn]

Changed branch from u/cheuberg/asy/mutable-poset to u/dkrenn/asy/mutable-poset

[dkrenn]

comment:45

Replying to @cheuberg:

Thank you for your changes. I added three commits. See my remaining comments below.

Cross-review...ok.

7. MutablePosetShell.key: I am surprised that the key is not cached. I imagine that many comparisons will be needed in the lifetime of a MutablePoset, and in every case, the property key has to be resolved, which calls the property poset, which calls get_key of the poset.

Now it is cached (see also follow up ticket #19281).

I rather thought about calling key in __init__ as I guess that the key will be needed at least once in the lifetime of every MutablePosetShell.

Oh...yes, I agree; changed.

18. MutablePosetShell._copy_all_linked_: I do not understand why you test oo == P.oo: I think that oo is an element of the new poset Q.

oo == P.oo tests that Q.oo is mapped to P.oo.

wouldn't oo.is_oo do it without comparing shells of different posets, which might be confusing?

True. Changed.

So oo is Q.oo would be more interesting?

oo is Q.oo is False since oo is not in Q, but just a copy of the oo in P with parent-poset Q.

that would be good to test (and comment on).

Inserted.

Inserting this into Q is done in _copy_shells_.

The current test demonstrates that the poset is not used in comparison, so that would rather belong to .eq?

I do not understand what you mean (but it is already late...).

I mean that shells e and f might be equal even if they do belong to different posets; this might be an interesting doctest or example for equality.

Now I understand what you mean. Inserted a doctest there.

22. MutablePosetShell._search_covers_: According to Wikipedia, a cover is what is called an upper cover here. This is in contrast to the default behaviour here.

_search_covers_ does not exist anymore (see 27).

This does not solve the problem.

What about renaming the method covers to lower_covers (and adapting the docstring slightly, removing "upper " from the one-sentence-description as well as from the definition?) For symmetry, a method upper_covers would be nice.

Good idea; changed and inserted upper_covers.

55. MutablePoset.intersection and MutablePoset.intersection_update: see comments on union and union_update

Done.

Remove comment on non-commutativity? merge does not play a role here, and keys might be covered in the previous sentence?

Removed.

63. MutablePoset.update_union: "Due to keys and a merge function... this operation might not be commutative": This method is non-commutative by definition, as self is changed and other is not. So perhaps: "left.update_union(right)" and "right.update_union(left) might result in different posets"? The same for other update_... methods where non-commutativity might be surprising.

Changed.

---

New commits:

f7bd83a	Trac #17693, comment 43, 7: do caching of key in __init__
a554d69	rac #17693, comment 43, 18: use .is_oo to test for oo in doctest
3ac7ed1	rac #17693, comment 43, 18: add a doctest "oo is Q.oo"
2f675b3	Trac #17693, comment 43, 18: add a doctest in eq (comparing elements in different posets)
f064a3b	Trac #17693, comment 43, 22: covers --> lower_covers, upper_covers
bea925c	Trac #17693, comment 43, 55: remove comment on non-commutativity
4e73b45	Trac #17693, comment 43, 63: extend/rewrite non-commutativity note

[dkrenn]

Changed commit from 4deddbd to 4e73b45

[cheuberg]

Changed branch from u/dkrenn/asy/mutable-poset to u/cheuberg/asy/mutable-poset

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 4e73b45 to a0b3d7b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

e8d8d2c	Trac #17693: typos
4b7f929	Trac #17693: fix two copy&paste errors
a0b3d7b	Trac #17693: ReSt improvement

[cheuberg]

comment:49

Added three final reviewer commits.

Doctests pass, documentation builds, documentation seems to be fine, code seems to be fine.

[dkrenn]

comment:50

Replying to @cheuberg:

Added three final reviewer commits.

Thanks; checked.

Doctests pass, documentation builds, documentation seems to be fine, code seems to be fine.

:)

[vbraun]

Changed branch from u/cheuberg/asy/mutable-poset to a0b3d7b