From 2dbaba677940b6b321b90a2feec36b0dddec0826 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Th=C3=A9ophile=20Bastian?= <contact@tobast.fr>
Date: Tue, 22 Aug 2017 16:43:33 +0200
Subject: [PATCH] Ullmann -- some more

---
 report/report.tex | 56 +++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 47 insertions(+), 9 deletions(-)

diff --git a/report/report.tex b/report/report.tex
index 7e8f329..08ca616 100644
--- a/report/report.tex
+++ b/report/report.tex
@@ -377,7 +377,7 @@ out to be still too slow in real-world cases --- which looks unlikely ---,
 would be to try to multithread this computation.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Group equality}
+\section{Group equality}\label{sec:group_equality}
 
 Given two circuit group gates, the task of group equality is to determine
 whether the two groups are structurally equivalent, as discussed above.
@@ -517,17 +517,55 @@ The refining function is detailed in Figure~\ref{alg:ullmann_refine}.
     \caption{Ullmann's refining function}\label{alg:ullmann_refine}
 \end{figure}
 
+\subsection{Ullmann for \emph{isomatch}}
+
+\paragraph{Graph used.} Our circuit is not actually a graph just as-is: indeed,
+a wire can be connected to multiple circuits (multiple gates' inputs, or even
+multiple gates' outputs when using tristate circuits). This could be transposed
+into a graph with $\frac{n(n-1)}{2}$ edges (the complete subgraph) for this
+particular wire.  Though, internally, a wire is better represented as a vertex
+itself, with $n$ edges linking it to the connected gates. This representation
+is also used in Ullmann's implementation, leading to a permutation matrix of
+$\left(\card{\text{needle gates}} + \card{\text{needle wires}}\right) \times
+\left(\card{\text{haystack gates}} + \card{\text{haystack wires}}\right)$.
+
+\paragraph{Final result.} Once a result (\ie{} a correct permutation) is
+obtained, we further need to check it is actually a solution of our problem.
+Indeed, while the structure is guaranteed by the algorithm to be the same, we
+still need to check that every circuit is equal to its matched one, through the
+procedure described in Section~\ref{sec:group_equality}. So far, only the
+equality of signatures was checked. We only need to check the circuits, as the
+wires are necessarily actually matching.
+
+\paragraph{Non-overlapping results.} We want our results to be non-overlapping
+(because we won't be able to perform a search-and-replace if it is not the
+case). Whenever two potential results are conflicting, an arbitrary one of the
+two can be returned (a human user is operating the software and can make a
+narrower search if needed).
+
+To match this specification, we must keep track of the circuits that are
+already included in a match. We also cannot include an ancestor of a circuit
+that was included in a match in another match (though this is not possible,
+because the needle can't be included in itself, but signature collisions could
+occur).
+
 \subsection{Implementation optimisations}
 
+\paragraph{Initial permutation matrix.} The matrix is first filled according to
+the signatures matches. It is then refined a bit more, by making sure that for
+every match, every potentially matching gate has the same ``wire kinds''.
+Indeed, a gate needle's wire must have at least the same inbound adjacent
+signatures as its matching haystack wire, and same goes for outbound adjacent
+signatures. Thus, two circuits cannot be matched if this condition is not
+respected for each pair of corresponding wires of those circuits, and their
+corresponding cell in the permutation matrix can be nulled.
 
-The matrix is first filled according to the signatures matches. It is then
-refined a bit more, by making sure that for every match, every potentially
-matching gate has the same ``wire kinds''. Indeed, a gate needle's wire must
-have at least the same inbound adjacent signatures as its matching haystack
-wire, and same goes for outbound adjacent signatures. Thus, two circuits cannot
-be matched if this condition is not respected for each pair of corresponding
-wires of those circuits, and their corresponding cell in the permutation matrix
-can be nulled.
+\paragraph{Pre-check.} The needle will, in most cases, not be found at all in
+a given hierarchy group of the haystack. To avoid wasting computation time, we
+first check that every signature present in the needle is present at least as
+many times in the haystack. This simple check saved a lot of time.
+
+\todo{More stuff?}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Performance}