Add code quality subsection
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@ -53,8 +53,8 @@
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transformations to a circuit.
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This problem turns out to be more or less the \emph{subgraph isomorphism
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problem}, which is NP-complete, and must nevertheless be solved fast on
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processor-sized circuits on this particular case.
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problem}, which is NP-complete, and must nevertheless be solved efficiently
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on processor-sized circuits on this particular case.
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During my internship, I developed a C++ library to perform this task that
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will be integrated in VossII, based on a few well-known algorithms as well
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@ -211,7 +211,7 @@ available on GitHub:
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\url{https://github.com/tobast/circuit-isomatch/}
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\end{center}
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\subsection{Objective}
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\subsection{Problems}
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More precisely, the problems that \emph{isomatch} must solve are the following.
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@ -247,6 +247,39 @@ to be NP-complete~\cite{cook1971complexity}. Even though a few algorithms
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is nevertheless necessary to implement them the right way, and with the right
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heuristics, to get the desired efficiency for the given problem.
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\subsection{Code quality}
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Another prominent objective was to keep the codebase as clean as possible.
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Indeed, this code will probably have to be maintained for quite some time, and
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most probably by other people than me. This means that the code and all its
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surroundings must be really clean, readable and reusable. I tried to put a lot
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of effort in making the code idiomatic and easy to use, through \eg{} the
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implementation of iterators over my data structures when needed, idiomatic
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C++14, etc.
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This also means that the code has to be well-documented: the git history had to
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be kept clean and understandable, and a clean documentation can be generated
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from the code, using \texttt{doxygen}. The latest documentation is also
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compiled as HTML pages here:
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\begin{center}
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\raisebox{-0.4\height}{
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\includegraphics[height=2.3em]{../common/docs.png}}
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\hspace{1em}
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\url{https://tobast.fr/m1/isomatch}
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\end{center}
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Since the code is C++, it is also very prone to diverse bugs. While I did not
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took the time to integrate unit tests --- which would have been a great
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addition ---, I used a sequence of test that can be run using \lstc{make
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test}, and tests a lot of features of isomatch.
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The code is also tested regularly and on a wide variety of cases with
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\lstbash{valgrind} to ensure that there are no memory errors ---
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use-after-free, unallocated memory, memory leaks, bad pointer
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arithmetics,~\ldots In every tested case, strictly no memory is lost, and no
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invalid read was reported.
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\subsection{Sought efficiency}
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The goal of \textit{isomatch} is to be applied to large circuits on-the-fly,
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@ -257,10 +290,6 @@ matching operations will be executed quite often, and often multiple times in a
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row. It must then remain fast enough for the human not to lose too much time,
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and eventually lose patience.
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\todo{Mention clean codebase somewhere}
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\todo{Mention VossII somewhere}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{General approach}
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@ -268,7 +297,7 @@ The global strategy used to solve efficiently the problem can be broken down to
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three main parts.
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\paragraph{Signatures.} The initial idea to make the computation fast is to
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aggregate the inner data of a gate --- be it a leaf gate or a group --- in a
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aggregate the inner data of a gate~---~be it a leaf gate or a group~---~in a
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kind of hash, a 64 bits unsigned integer. This approach is directly inspired
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from what was done in fl, back at Intel. This hash must be easy to compute,
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and must be based only on the structure of the graph --- that is, must be
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@ -307,8 +336,10 @@ this problem, that uses the specificities of the graph to be a little faster.
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The signature is computed as a simple hash of the element, and is defined for
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every type of expression and circuit. It could probably be enhanced with a bit
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more work to cover more uniformly the hash space, but no collision was observed
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on the examples tested.
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more work to cover more uniformly the hash space, but no illegitimate collision
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(that is, a collision that could be avoided with a better hash function, as
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opposed to collisions due to an equal local graph structure) was observed on
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the examples tested.
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\paragraph{Signature constants.} Signature constants are used all around the
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signing process, and is a 5-tuple $\sigconst{} = (a, x_l, x_h, d_l, d_h)$ of 32
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@ -316,22 +347,24 @@ bits unsigned numbers. All of $x_l$, $x_h$, $d_l$ and $d_h$ are picked as prime
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numbers between $10^8$ and $10^9$ (which just fits in a 32 bits unsigned
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integer); while $a$ is a random integer uniformly picked between $2^{16}$ and
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$2^{32}$. These constants are generated by a small python script,
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\path{util/primegen/pickPrimes.py}.
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\path{util/primegen/pickPrimes.py} in the repository.
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Those constants are used to produce a 64 bits unsigned value out of another 64
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bits unsigned value, called $v$ thereafter, through an operator $\sigop$,
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computed as follows (with all computations done on 64 bits unsigned integers).
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\vspace{1em}
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\begin{algorithmic}
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\Function{$\sigop$}{$\sigconst{}, v$}
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\State{} $out1 \gets (v + a) \cdot x_l$
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\State{} $v_h \gets (v \lsr 32) \xor (out1 \lsr 32)$
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\State{} $low \gets out1 \,\%\, d_l$
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\State{} $high \gets \left((v_h + a) \cdot x_h \right) \%\, d_h$
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\State{} \Return{} $low + 2^{32} \cdot high$
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\EndFunction{}
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\end{algorithmic}
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\begin{center}
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\begin{algorithmic}
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\Function{$\sigop$}{$\sigconst{}, v$}
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\State{} $out1 \gets (v + a) \cdot x_l$
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\State{} $v_h \gets (v \lsr 32) \xor (out1 \lsr 32)$
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\State{} $low \gets out1 \,\%\, d_l$
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\State{} $high \gets \left((v_h + a) \cdot x_h \right) \%\, d_h$
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\State{} \Return{} $low + 2^{32} \cdot high$
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\EndFunction{}
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\end{algorithmic}
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\end{center}
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\paragraph{Expressions.} Each type of expression (or, in the case of
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expression with operator, each type of operator) has its signature constant,
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@ -358,7 +391,7 @@ capture at all the \emph{structure} of the graph. An information we can capture
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without breaking the signature's independence towards the order of description
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of the graph, is the set of its neighbours. Yet, we cannot ``label'' the gates
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without breaking this rule; thus, we represent the set of neighbours by the set
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of our \emph{neighbours' signatures}.
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of the \emph{neighbours' signatures}.
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At this point, we can define the \emph{signature of order $n$} ($n \in
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\natset$) of a circuit $C$ as follows:
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@ -375,13 +408,13 @@ At this point, we can define the \emph{signature of order $n$} ($n \in
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The ``IO adjacency'' term is an additional term in the signatures of order
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above $0$, indicating what input and output pins of the circuit group
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containing the current gate are adjacent to it.
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containing the current gate are adjacent to it. Adding this information to the
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signature was necessary, since a lot of gates can be signed differently using
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this information (see Corner cases in Section~\ref{ssec:corner_cases}).
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The default order of signature used in all computations, unless more is useful,
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is 2, after a few benchmarks.
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\todo{explain range of $n$}
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\paragraph{Efficiency.} Every circuit memoizes all it can concerning its
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signature: the inner signature, the IO adjacency, the signatures of order $n$
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already computed, etc.
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@ -400,8 +433,10 @@ or its children are modified. A memoized data is always stored alongside with a
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timestamp of computation, which invalidates a previous result when needed.
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One possible path of investigation for future work, if the computation turns
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out to be still too slow in real-world cases --- which looks unlikely ---,
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would be to try to multithread this computation.
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out to be still too slow in real-world cases --- which looks unlikely, unless
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fl's substitution is run on a regular basis for a huge number of cases using
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\eg{} a crontab for automated testing ---, would be to try to multithread this
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computation.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Group equality}\label{sec:group_equality}
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@ -428,7 +463,7 @@ the number of permutations examined to no more than $4$ in studied cases.
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Once a permutation is judged worth to be examined, the group equality is run
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recursively on all its matched gates. If this step succeeds, the graph
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structure is then checked. If both steps succeeds, the permutation is correct
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structure is then checked. If both steps succeed, the permutation is correct
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and an isomorphism has been found; if not, we move on to the next permutation.
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\todo{Anything more to tell here?}
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@ -449,7 +484,10 @@ was first described by Julian R Ullmann in 1976~\cite{ullmann1976algorithm}.
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Another, more recent algorithm to deal with this problem is Luigi P Cordella's
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VF2 algorithm~\cite{cordella2004sub}, published in 2004. This algorithm is
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mostly Ullmann's algorithm, transcribed in a recursive writing, with the
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addition of five heuristics. \qtodo{Why not use it then?}
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addition of five heuristics. I originally planned to implement both algorithms
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and benchmark both, but had no time to do so in the end; though, Ullmann with
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the few additional heuristics applicable in our very specific case turned out
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to be fast enough.
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Ullmann is a widely used and fast algorithm for this problem. It makes an
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extensive use of adjacency matrix description of the graph, and the initial
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@ -461,8 +499,9 @@ matrix. Each $1$ in a cell $(i, j)$ indicates that the $i$-th needle part is a
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possible match with the $j$-th haystack part. This matrix is called $perm$
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thereafter.
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The algorithm, left apart the \textsc{refine} function (detailed just after),
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is described in Figure~\ref{alg:ullmann}.
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The algorithm, left apart the \textsc{refine} function, which is detailed just
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after and can be omitted for a (way) slower version of the algorithm, is
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described in Figure~\ref{alg:ullmann}.
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\begin{figure}[h]
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\begin{algorithmic}
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@ -505,7 +544,7 @@ is described in Figure~\ref{alg:ullmann}.
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The refining process is the actual keystone of the algorithm. It is the
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mechanism allowing the algorithm to cut down many exploration branches, by
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removing ones from the matrix.
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changing ones to zeroes in the matrix being built.
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The idea is that a match between a needle's vertex $i$ and a haystack's vertex
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$j$ is only possible if, for each neighbour $k$ of $i$, $j$ has a neighbour
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@ -583,7 +622,12 @@ occur).
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\subsection{Implementation optimisations}
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\paragraph{Initial permutation matrix.} The matrix is first filled according to
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the signatures matches. It is then refined a bit more, by making sure that for
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the signatures matches. Note that only signatures of order 0 --- \ie{} the
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inner data of a vertex --- can be used here: indeed, we cannot rely on the
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context here, since there can be some context in the haystack that is absent
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from the needle, and we cannot check for ``context inclusion'' with our
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definition of signatures: \emph{all} the context must be exactly the same for
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two signatures to match. It is then refined a bit more, by making sure that for
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every match, every potentially matching gate has the same ``wire kinds''.
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Indeed, a gate needle's wire must have at least the same inbound adjacent
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signatures as its matching haystack wire, and same goes for outbound adjacent
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@ -665,6 +709,13 @@ for a single run) and measured by the command \texttt{time}.
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\end{tikzpicture}
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\end{center}
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The computation time is more or less linear in in the level of signature
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required, which is coherent with the implementation. In practice, only small
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portions of a circuit will be signed with a high order, which means that we can
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afford really high order signatures (\eg{} 40 or 50, which already means that
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the diameter of the group is 40 or 50) without having a real impact on the
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computation time.
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\paragraph{Equality.} To test the circuit group equality, a small piece of
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code takes a circuit, scrambles it as much as possible
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@ -680,13 +731,15 @@ considerably speeding it up: the same program proving only one way takes about
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\subsection{Corner cases}
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\subsection{Corner cases}\label{ssec:corner_cases}
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There were a few observed cases where the algorithm tends to be slower on
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certain configurations.
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certain configurations, and a few other such cases that could be fixed.
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\todo{More corner cases}
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\todo{Corner case: io pins, io adjacency}
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\paragraph{Split/merge trees.} A common pattern that tends to slow down the
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algorithm is split/merge trees. Those patterns occur when one wants to merge
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$n$ one bit wires into a single $n$ bits wire, or the other way around.
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@ -712,6 +765,12 @@ nodes on the layer below cannot be freely exchanged.
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\todo{Figure describing the problem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Conclusion}
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\todo{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\printbibliography{}
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