From d4c10896b223ee1932c185b46a7a1dfe125fb482 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Th=C3=A9ophile=20Bastian?= <contact@tobast.fr>
Date: Sat, 19 Aug 2017 23:25:21 +0200
Subject: [PATCH] General approach

---
 common/refs.bib   | 12 ++++++++++++
 report/report.tex | 42 ++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 54 insertions(+)

diff --git a/common/refs.bib b/common/refs.bib
index 6a012ae..ded02f9 100644
--- a/common/refs.bib
+++ b/common/refs.bib
@@ -75,3 +75,15 @@
   booktitle={Int. Workshop on Designing Correct Circuits},
   year={2006}
 }
+
+@article{ullmann1976algorithm,
+  title={An algorithm for subgraph isomorphism},
+  author={Ullmann, Julian R},
+  journal={Journal of the ACM (JACM)},
+  volume={23},
+  number={1},
+  pages={31--42},
+  year={1976},
+  publisher={ACM}
+}
+
diff --git a/report/report.tex b/report/report.tex
index c52149d..a52a981 100644
--- a/report/report.tex
+++ b/report/report.tex
@@ -227,9 +227,51 @@ matching operation will be executed often, and often multiple times in a row.
 It must then remain fast enough for the human not to lose too much time, and
 eventually lose patience.
 
+\todo{Mention clean codebase somewhere}
+
+\todo{Mention VossII somewhere}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{General approach}
 
+The global strategy used to solve efficiently the problem can be broken down to
+three main parts.
+
+\paragraph{Signatures.} The initial idea to make the computation fast is to
+aggregate the inner data of a gate --- be it a leaf gate or a group --- in a
+kind of hash, a 32 bits unsigned integer. This approach is directly inspired
+from what was done in fl, back at Intel. This hash must be easy to compute,
+and must be based only on the structure of the graph --- that is, must be
+entirely oblivious of the labels given, the order in which the circuit is
+described, the order in which different circuits are plugged on a wire, \ldots.
+The signature equality, moreover, must be sound; that is, two signatures must
+necessarily be equal if the circuits are indeed equal.
+
+This makes it possible to rule out quickly whether two circuits are candidates
+for a match or not, and run the costy actual equality algorithm on fewer gates.
+
+
+\paragraph{Group equality.} The group equality algorithm is a standard
+backtracking algorithm. It tries to build a match between the graphs by
+trying the diverse permutations of elements with the same signature. It can
+also communicate with the signing part, to request a more precise (but slightly
+slower to compute) signature when it has too many permutations to try.
+
+This part could be enhanced, but does not slow down the algorithm on the tested
+examples, so I focused on other parts.
+
+
+\paragraph{Pattern matching.} This part is the one responsible to answer
+queries for occurrences of a sub-circuit in a circuit. It uses extensively the
+signatures to determine whether two circuits could be a match or not before
+spending too much time actually finding matches, but cannot rely on it as
+heavily as group equality, since only the first level of precision is
+applicable here (detailed later).
+
+This part mostly consists in an implementation of Ullmann's algorithm for
+subgraph isomorphism~\cite{ullmann1976algorithm}, a well-known algorithm for
+this problem, that uses the specificities of the graph to be a little faster.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Signatures}
 \todo{}