Within the support library, in the namespace
lib::diff, there is a collection of loosely coupled tools
known as »the diff framework«. It revolves around generic representation and handling of structural differences.
Beyond some rather general assumptions, to avoid stipulating the usage of specific data elements or containers,
the framework is kept generic, cast in terms of elements, sequences and strategies
for access, indexing and traversal.
the atomic unit treated in diff detection, representation and application.
Elements are considered to be
lightweight copyable values
bearing distinct identity
unique as far as concerned
data is delivered in the form of a sequence, which might or might not be ordered, but in any case will be traversed once only.
the changes necessary to transform an input sequence (“old sequence”) into a desired target sequence (“new sequence”)
differences are spelled out in linearised form: as a sequence of constant-size diff actions, called »diff verbs«
a single element within a diff. Diff verbs are conceived as operations, which, when applied consuming the input sequence, will produce the target sequence of the diff.
the process of consuming a diff (sequence of diff verbs), with the goal to produce some effect at the target of diff application. Typically we want to apply a diff to a data sequence, to mutate it into a new shape, conforming with the shape of the diff’s “target sequence”
a facility producing a diff, which is a sequence of diff verbs. Typically, a diff generator works lazily, demand driven.
special kind of diff generator, which takes two data sequences as input: an “old sequence” and a “new sequence”. The diff detector traverses and compares these sequences to produce a diff, which describes the steps necessary to transform the “old” shape into the “new” shape of the data.
While in general this is a well studied subject, in Lumiera we’ll confine ourselves to a very specific flavour of diff handling: we rely on elementary atomic units with well established object identity. And in addition, within the scope of one coherent diff handling task, we require those elements to be unique. The purpose of this design decision is to segregate the notorious matching problem and treat diff handling in isolation.
Effectively this means that, for any given element, there can be at most one matching
counterpart in the other sequence, and the presence of such can be detected by using an index.
In fact, we retrieve an index for every sequence involved into the diff detection task;
this is our trade-off for simplicity in the diff detection algorithm.
[traditionally, diff detection schemes, especially those geared at text diff detection, engage into great lengths of producing an “optimal” diff, which effectively means to build specifically tuned pattern or decision tables, from which the final diff can then be pulled or interpreted. We acknowledge that in our case building a lookup table index with additional annotations can be as bad as O(n2) and worse; we might well be able to do better, but likely for the price of turning the algorithm into some kind of mental challenge.]
In case this turns out as a performance problem, we might consider integrating the index maintenance into the data structure to be diffed, which shifts the additional impact of indexing onto the data population phase.
[in the general tree diff case this is far from trivial, since we need an self-contained element index for every node, and we need the ability to take a snapshot of the “old” state before mutating a node into “new” shape]
By using the indices of the old and the new sequence, we are able to classify each element:
elements only present in the new sequence are treated as inserts
elements only present in the old sequence are treated as deletes
elements present in both sequences form the permutation
We consume both the old and the new sequence synchronously, while emitting the diff sequence.
The diff describes a sequence of operations, which, when applied, consume a sequence congruent to the old sequence, while emitting a sequence congruent to the new sequence. We use the following list diff language here:
insert the given argument element
elm at the current processing position
into the target sequence. This operation allows to inject new data
delete the next element
elm at current position.
For sake of verification, the element to be deleted is also included as argument (redundancy).
accepts the next element at current position into the resulting altered sequence. The element is given redundantly as argument.
effect a re-ordering of the target list contents. This verb requires to search for the (next respective single occurrence of the) given element in the remainder of the datastructure, to bring it forward and insert it as the next element.
processing hint, emitted at the position where an element previously fetched by
find verb happened to sit in the old order. This allows an optimising
implementation to “fetch” a copy and just drop or skip the original,
thereby avoiding to shift other elements.
Since inserts and deletes can be detected and emitted right at the processing frontier, for the remaining theoretical discussion, we consider the insert / delete part filtered away conceptually, and concentrate on generating the permutation part.
Permutation handling turns out to be the turning point and tricky part of diff generation;
the handling of new and missing members can be built on top, once we manage to describe a
permutation diff. If we thus consider — in theory — the inserts and deletes to be
filtered away, what remains is a permutation of index numbers to cause the re-ordering.
We may describe this re-ordering by the index numbers in the new sequence, given in the order
of the old sequence. For such a re-ordering permutation, the theoretically optimal result
can be achieved by Cycle Sort in linear time.
[ assuming random access by index is possible, Cycle Sort walks the sequence once. Whenever encountering an element out-of order, i.e. new postion != current position, we leap to the indicated new position, which necessarily will be out-of-order too, so we can leap from there to the next indicated position, until we jump back to the original position eventually. Such a closed cycle can then just be rotated into proper position. Each permutation can be decomposed into a finite number of closed cycles, which means, after rotating all cycles out of order, the permutation will be sorted completely.]
Starting from such “cycle rotations”, we possibly could work out a minimal set of moving operations.
But there is a twist: Our design avoids using index numbers, since we aim at stream processing
of diffs. We do not want to communicate index numbers to the consumer of the diff; rather we
want to communicate reference elements with our diff verbs. Thus we prefer the most simplistic
processing mechanism, which happens to be some variation of Insertion Sort.
[to support this choice, Insertion Sort — in spite of being O(n2) — turns out to be the best choice at sorting small data sets for reasons of cache locality; even typical Quicksort implementations switch to insertion sorting of small subsets for performance reasons]
This is the purpose of our
find verb: to extract some element known to be out of order, and
insert it at the current position.
Based on these choices, we’re able to consume two permutations of the same sequence simultaneously,
pick verbs to describe the re-ordering. Consider the two sequences
split into an already-processed part, and a part still-to-be-processed.
Matters are arranged such, that, in the to-be-processed part, each element appearing at the front of the “new” sequence can be consumed right away.
Now, to arrive at that invariant, we use indices to determine
if the element at head of the old sequence is not present in the new sequence, which means it has to be deleted
while an element appearing at head of the new sequence but not present in the old sequence needs to be inserted
and especially an element known to be present in both sequences, appearing at the head of the new sequence but non-matching at the old sequence prompts us to fetch the right element from further down in the sequence and insert it a current position
after that, the invariant is (re)established and we can consume the element and move the processing point forward.
For generating the diff description, we need index tables of the “old” and the “new” sequence,
which causes a O(n log n) complexity and storage in the order of the sequence length. Application
of the diff is quadratic, due to the find-and-insert passes.
[In the theoretical treatment of diff problems it is common to introduce a distance metric to describe how far apart the two sequences are in terms of atomic changes. This helps to make the quadratic (or worse) complexity of such algorithms look better: if we know the sequences are close, the nested sub-scans will be shorter than the whole sequence (with n·d < n2).
However, since our goal is to support permutations and we have to deal with arbitrary sequences, such an argument looks somewhat pointless. Let’s face it, structural diff computation is expensive; the only way to keep matters under control is to keep the local sequences short, which prompts us to exploit structural knowledge instead of comparing the entire data as flat sequence]
The handling of list differences can be used as prototype to build a description of structural changes in hierarchical data: traverse the structure and account for each element and each change. Such a description of changes won’t be optimal though. What appears as a insertion or deletion locally, might indeed be just the result of rearranging subtrees as a whole. The tree diff problem in this general form is known to be a rather tough challenge. But our goals are different here. Lumiera relies on a »External Tree Description« for symbolic representation of hierarchically structured elements, without actually implementing them. The purpose of this “external” description is to largely remove the need for a central data model to work against. A symbolic diff message allows to propagate data and structure changes, without even using the same data representation at both ends.
For this to work, we need some very generic meta representation. This can be a textual representation (e.g. JSON) — but within the application it seems more appropriate to use an abstracted and unspecific typed data representation, akin to “JSON with typed language data”. It can be considered symbolic, insofar it isn’t the data, it refers to it. For this approach to work, we need the following parts:
a generic node, which has an identity and some payload data. This
GenNode is treated as elementary value.
a record made from a collection of generic nodes, to take on the abstracted role of an object. Such a
Record<GenNode> is a sequence of nodes, partitioned in two scopes: the (named) attributes and the children.
together these elements form an essentially recursive structure: The record is comprised of nodes and the nodes might, besides elementary values, also carry records.
and finally we need an identification scheme, allowing to produce named and unnamed yet unique identities, also including opaque type information.
Type information is deliberately kept opaque, to prevent switch-on-type. We always presume synced (similar) data structures on both ends of the collaboration, where the partners share common knowledge about types and structure. Changes are indicated and propagated, not probed.
Exploiting the fact that
Record<GenNode> is essentially a sequence, we’re able to build the description
of structure changes as an extension layer on top of our linearised diff language format. We introduce
a bracketing construct to open and close sub scopes. Within each scope, the verbs of our list diff
language are deployed, just now with a
GenNode as payload. This yields the following tree diff language
prompts to insert the given argument element at the current processing position into the target sequence. This operation allows to inject new data
requires to delete the next element at current position.
[The payload of this and all the following verbs is a
GenNode, but only the ID part matters. This allows to
send a special ref element over the wire instead of having to send a full subtree, for
obvious performance reasons.]
For sake of verification, the ID of the argument payload is required to match the ID of the element about to be discarded.
just accepts the next element at current position into the resulting altered sequence. Again, the ID of the argument has to match the ID of the element to be picked, for sake of verification.
change the order of the target scope contents: this verb requires to search ahead for the (next respective single occurrence of the) given element further down into the remainder of the current record scope (but not into nested child scopes). The designated element is to be retrieved and inserted as the next element at current position.
this is a mere processing hint, emitted at the position where an element
previously extracted by a
find(ID) verb happened to sit within the old order.
pick existing elements up to the designated point.
As a special notation,
after(Ref::ATTRIBUTES) allows to fast forward
to the first child element, while
after(Ref::END) means to accept
all of the existing data contents as-is (possibly to append further
elements beyond that point).
bracketing construct to open a nested sub scope.
The element designated by the ID of the argument needs to be a record
(“nested child object”). Moreover, this element must have been
mentioned with the preceding diff verbs at that level, which means
that the element as such must already be present in the altered
target structure. The
mut(ID) verb then opens the designated nested
record for diff handling, and all subsequent diff verbs are to be
interpreted relative to this scope, until the corresponding
emu(ID) verb is encountered. As a special notation, right
after handling an element with the list diff verbs (i.e.
find), it is allowed immediately to open the
nested scope with
[this shortcut circumvents the problem that it is sometimes difficult to know the precise ID for unnamed children just created at the given scope. We acknowledge that diff messages will sometimes be hand-writtten, especially to populate a target data structure without knowing its implementation. The whole diff language is designed to be human readable.]
closing bracketing construct and counterpart to
mut(ID). This verb
must be given precisely at the end of the nested scope.
[it is not allowed to “return” from the middle of a scope, for sake of sanity. The diff messages transport a certain degree of redundancy to detect when the data structure at target does no longer conform to the assumptions made at the generation side.]
At this point, this child scope is left and the parent scope with all existing diff state is popped from an internal stack.