text:
\section{Introduction}\label{sec:introduction}
In this paper we introduce a technique for compressing binary decision
diagrams for those cases where random access to the compressed
representation is not needed. We will use the term \emph{offline} to
describe a BDD stored in such a manner that it no longer allows random
access to its structure without decompressing the BDD first. The two
primary areas in which decision diagrams are used in practice are
verification and configuration. In both of these areas it is sometimes
important to store binary decision diagrams offline using as little
space as possible. Primarily the need for such compression arises when
it is necessary to transmit binary decision diagrams across
communication channels with limited bandwidth. In the area of
verification this need arises for example when using a networked
cluster of computers to perform a distributed compilation of a binary
decision diagram \cite{distributed_verification}. A similar exchange of
BDD data takes place in \emph{distributed configuration} as described
in \cite{distributed_config}. In such approaches the fact that the
network bandwidth is much lower than the memory bandwidth can become a
major bottleneck as computers stall waiting to receive data to
process. Transmitting the binary decision diagrams in a compressed
representation can help alleviate this problem. The same problem
arises in standard interactive configuration. Consider for example a
web-based interactive configurator that uses BDDs to store the set of
valid configurations. It must either transmit the BDD storing the
configuration data to the customers computer or perform all
computations on the server leading to a network delay each time the
user makes a choice during the configuration. In order to reduce the
required bandwidth as well as lower the load time in the first case, it
is benefical to transmit the BDD in a compressed format.

The paper is organized as follows: In Section \ref{sec:preliminaries}
we present the neccessary notation and definitions. In Section
\ref{sec:compression} we present our compression scheme. Finally, in
Section \ref{sec:experiments} we show the compression attained by
applying our compression scheme on different instances.

\subsection{Related work}
Most of the work done on compressing decision diagrams aim to achieve
large reductions in size, while maintaining random access, by means of
better variable orderings \cite{fanin}\cite{exact} or modifications to
the decision diagram data structure \cite{embedded}. Such work is
mostly orthogonal to the aim of this paper, as the compression
strategy we present easily can be adapted for use with the variations
over basic BDDs.

The only previous work we are aware of for compressing BDDs for
offline storage is the work by Starkey and
Bryant\cite{bdd_media_compression} and the follow-up paper by Mateu
and Prades-Nebot\cite{bdd_image_compression} which both describes
techniques for image compression using BDDs. The latter of these
includes a non-trivial encoding algorithm for storing the BDD
offline. Finally Kieffer et.al\cite{universal_bdd_compression} gives
theoretical results for using BDDs for general data compression
including a technique for storing BDDs. After presenting our own
technique, we present empirical results comparing the new encoder with
the encoders from \cite{bdd_image_compression} and
\cite{universal_bdd_compression}.

\section{Preliminaries}\label{sec:preliminaries}
\begin{definition}[Ordered Binary Decision Diagram]\cite{BRYANT-BDD}
  An ordered binary decision diagram on $n$ binary variables $X = \{
  x_1, \ldots, x_n \}$ is a layered directed acyclic graph $G(V,E)$
  with $n+1$ layers (some of which may be empty) and exactly one
  root. We use $l(u)$ to denote the layer in which the node $u$
  resides. In addition the following properties must be satisfied:
\begin{itemize}
\item If $|V| \ne 1$, there are exactly two nodes in layer
  $n+1$. These nodes have no outgoing edges and are denoted the
  1-terminal and the 0-terminal. If $|V| = 1$ the layer will either
  contain the 1-terminal or the 0-terminal.
\item All nodes in layer $1$ to $n$ have exactly two outgoing
  edges, denoted the \emph{low} and \emph{high} edge respectively.  We
  use $low(u)$ and $high(u)$ to denote the end-point of the low and
  high edge of $u$ respectively.
\item For any edge $(u,v) \in E$ it is the case that $l(u) < l(v)$
\end{itemize}
\end{definition}

We use $E_{low}$ and $E_{high}$ to denote the set of low and high
edges respectively. An edge $(u,v)$ such that $l(u) + 1 < l(v)$ is
called a long edge and is said to skip layer $l(u) + 1$ to
$l(v)-1$. The length of an edge $(u,v)$ is defined as $l(v) - l(u)$.

\begin{definition}[Reduced ordered Binary Decision Diagram]
  An ordered Binary Decision Diagram is called \emph{reduced} iff for
  any two distinct nodes $u,v$ it holds that $low(u) \not= low(v) \lor
  high(u) \not = high(v)$ and further that $high(u)\not = low(u)$ for
  all nodes $u$.
\end{definition}

In the rest of this paper we will assume for all BDDs we are
considering that they are ordered and reduced.

\begin{definition}[Solution to a BDD]
  A complete assignment $\rho$ to $X$ is a solution to an
  BDD $G(V,E)$ iff there exists a path $P$ from the root in $G$ to
  the 1-terminal such that for every assignment $(x_i, b) \in \rho$,
  where $b \in \{ low,high \}$, there exists an edge $(u,v)$ in $P$
  such that one of the following holds:
\begin{itemize}
\item[\textbf{-}] $l(u) < i < l(v)$
\item[\textbf{-}] $l(u) = i$ and $(u,v) \in E_b$
\end{itemize}
\end{definition}

\begin{figure}
\subfigure[]
{\includegraphics[scale=0.44]{./pics/var3example-bdd-layers.eps}\label{fig:bdd}} \hfill
\subfigure[]
{\includegraphics[scale=0.44]{./pics/var3example-bdd-layers-backbone1.eps}\label{fig:spanning1}} \hfill
\subfigure[]
{\includegraphics[scale=0.44]{./pics/var3example-bdd-layers-backbone2.eps}\label{fig:spanning2}} \hfill
\caption{From left to right: A BDD and two different spanning trees on the BDD. Solid and dashed edges corresponds to
\emph{high} and \emph{low} edges respectively}
\end{figure}

\begin{example}
  Figure \ref{fig:bdd} contains a BDD on the binary variables $X
  = \{x_1,x_2,x_3,x_4\}$ which the solutions
  $\{(1,1,1,1),$ $(1,1,0,1),$ $(1,0,1,0),$ $(1,0,0,0),$ $(0,1,1,0),$
  $(0,1,0,1),$ $(0,0,1,0),$ $(0,0,0,1)\}$.
\end{example}

\begin{definition}[BFS order]\label{def:bfs_order}
  A BFS ordering $id_b: V \rightarrow \{1,\ldots,|V|\}$ of the nodes
  in a layered DAG $G(V,E)$ rooted in $r$ is the ordering of $V$ in
  the order they are visited by a BFS in the DAG starting at $r$ and
  traversing left edges prior to right edges.
\end{definition}

\begin{definition}[Layer order]\label{def:layer_order}
  A layer ordering $id_l: V \rightarrow \{1,\ldots,|V|\}$ of the nodes
  in a layered DAG $G(V,E)$ rooted in $r$ is the ordering of $V$ layer
  by layer in increasing order of the layer. Nodes at the same layer
  are ordered in the order that they are visited by a DFS in the DAG
  starting at $r$ and traversing left edges prior to right edges.
\end{definition}
We refer to $id_b(v)$ and $id_l(v)$ by ``the BFS id of $v$'' and ``the
layer id of $v$'' respectively. Note that if all edges in a layered
DAG has the same length then the ordering $id_l$ and $id_b$ will be
the same.

In our compression scheme we will make use of the following well-known
fact:
\begin{lemma}\label{lemma:succinct}
Every binary tree can be unambiguously encoded using 2 bits pr. node.
\end{lemma}
To achieve such an encoding each node $v$ is encoded using two
bits. The first bit and the second bit is true iff $v$ contains a left
and a right child respectively. In order to make decoding possible the
order in which the children of already decoded nodes appear in the
encoded data must be known. This can for example be ensured by
encoding the nodes in a DFS or BFS order with either left-first or
right-first traversal. As an example, the encoding of the nodes of the
binary tree in Figure \ref{fig:spanning2} in BFS order is
$(11,11,00,11,00,00,00)$.

\section{The Compression technique}\label{sec:compression}
Our compression technique can be summarized by the following steps:
\begin{enumerate}
\item Build a spanning tree on the BDD (Section \ref{sub:spanning}).
\item Encode edges in the spanning tree, using Lemma \ref{lemma:succinct}
\item Encode by one bit the order in which the two terminals appear in
  the spanning tree.
\item Encode the length of the edges in the spanning tree where
  necessary (Section \ref{sub:layerinfo}).
\item Encode the edges that are not in the spanning tree (Section
  \ref{sub:nontreeedges}).
\item Compress the resulting data using standard compression
techniques.
\end{enumerate}

The decoder starts by reverting step (6) by decompressing the data. It
then recreates the binary tree (1-2), restores the correct layer of
each node (4), and restores the remaining missing edges (5). Below we
give the details of each step.

\subsection{Building the spanning tree on the BDD}\label{sub:spanning}
\begin{definition}[Spanning Tree]
  A spanning tree $G^T(V^T,E^T)$ on a BDD $G(V,E)$ is a subgraph of
  $G$, for which $V^T = V$, and any two vertices are connected by
  exactly one path of edges in $E^T$. An edge is called a \emph{tree
    edge} if it is contained in the spanning tree and a \emph{nontree
    edge} otherwise.
\end{definition}

The most obvious way to construct a spanning tree on a graph is to use
DFS or BFS. In the case of a rooted DAG one can obtain a spanning tree
by, for each node $v$ except the root, adding a single edge with
endpoint in $v$ to the set of tree edges. Two examples of spanning
trees for the BDD in Figure \ref{fig:bdd} are shown in Figure
\ref{fig:spanning1} and \ref{fig:spanning2}.

In our encoder we will construct a spanning tree containing as few
long edges as possible. Hence when a node $v$ in the BDD has multiple
parents $u_1,\ldots,u_k$ and we have to choose one of the edges
$(u_1,v),\ldots,(u_k,v)$ to add to the spanning tree, we will always
choose the shortest possible edge, that is an edge $(u_i,v)$ where
$l(u_i) \geq l(u_1),\ldots,l(u_k)$. Ties are broken by choosing the
edge $(u,v)$ with the smallest $id_l(u)$. Note that the resulting
spanning tree is uniquely defined regardless of which order we process
nodes in. Using this construction we achieve a spanning tree with a
minimal number of long edges. This can easily be seen by noting that
precisely one ingoing edge must be chosen for each node, and
additionally that the choice of one edge can never exclude an edge to
another node from consideration.

\begin{example}
  The spanning tree in Figure \ref{fig:spanning1} contains three long
  edges, whereas the spanning tree in Figure \ref{fig:spanning2} only
  contains one. The latter of these would be the one constructed by
  our encoder upon compressing the BDD in Figure \ref{fig:bdd}. The
  single long edge in figure \ref{fig:spanning2} has to be included in
  the tree as it is the only possible way for the spanning tree to
  include the node in layer 1.
\end{example}

\subsection{Encoding the lengths of the tree edges}\label{sub:layerinfo}
The spanning tree is stored as a binary tree where all edges have the
same length. Since some of the edges in the spanning tree may
correspond to long edges in the BDD, the binary tree itself may not be
sufficient to reconstruct the layer information of the nodes during
decoding. In order to enable the decoder to deduce the correct layer
we therefore encode the location and the length of each long edge that
is included in the spanning tree. The location of a long edge $(u,v)$
is uniquely specified by the BFS order of the end point of the edge,
that is $id_b(v)$.

When encoding the location of the long edges
$(u_1,v_1),\ldots,(u_k,v_k)$ we will, instead of outputting the
integers $id_b(v_1), \ldots id_b(v_k)$, output a bitvector of length
$|V|$ for which entries $id_b(v_1), \ldots ,id_b(v_k)$ are true and
all other entries are false. Though this encoding is likely to require
more bits than encoding by listing the integers, the bitvector will be
compressed very efficiently when the standard compression is applied
in the final phase.

\begin{figure}
\begin{center}
\includegraphics[scale=0.46]{./pics/2intervals.eps}
\end{center}
\caption{ A spanning tree on a BDD. The black edges are tree edges and
  the gray edges are nontree edges. The nodes are labeled in layer
  order}
\label{fig:bdd+tree}
\end{figure}

\subsection{Encoding nontree edges}\label{sub:nontreeedges}
When the spanning tree and the layer information is encoded, we only
need to encode the nontree edges, that is, those edges in the BDD that
are not contained in the spanning tree.

We know that half of the edges in the BDD will be encoded as nontree
edges as it follows from the following observation:
\begin{observation}
  Let $G(V,E)$ be a BDD containing at least $3$ nodes. Then any
  spanning tree on $G$ will contain exactly $|E|/2+1$ edges
\end{observation}
\begin{proof}
  By the assumption that $|V| > 2$, it holds for any BDD $G(V,E)$ that
  $|E| = 2(|V|-2)$, since all nodes in a BDD except the two terminals
  have two children. Further any tree with $|V|$ nodes contains
  $|V|-1$ edges, which equals $|E|/2 + 1$ edges.
\end{proof}

As an example the construction in Figure \ref{fig:bdd+tree} contains
19 tree edges and 17 nontree edges.

Since every node except the terminals in the BDD has two children and
we have the spanning tree available with restored layer information,
we know the start point of each of the nontree edges that has to be
added to the spanning tree in order to reconstruct the BDD. Hence if
we encode the nontree edges in some fixed order according to their
start point then we only need to encode the end point of each of the
nontree edges. We will call the endpoint of any nontree edge that has
yet to be encoded an \emph{incomplete child}. By $S$, we will denote
the sequence of incomplete children in the order in which their
parents appear in the layer ordering of the nodes, that is $S =
(v_1,\ldots, v_k)$ for the nontree edges
$(u_1,v_1),\ldots,(u_k,v_k)$. By $id_l(S)$ we denote the corresponding
sequence of layer ids, that is $id_l(S) =
(id_l(v_1),\ldots,id_l(v_k))$.

We will now describe three encodings of nontree edges in Section
\ref{sub:largeindegree}, Section \ref{sub:smallindegree} and Section
\ref{sub:longforward} which we will combine to encode all the
nontree edges.

\subsubsection{Incomplete children with large in-degree}\label{sub:largeindegree}
In order for a standard compression technique to successfully compress
the sequence $id_l(S)$ it is important that the symbols in the
sequence appears with high frequency. We note that nodes with
in-degree $d$ will appear $d-1$ times in the sequence of nontree
edges. Hence by applying standard compression we will be able to
efficiently compress those nontree children that have a high in-degree
if they are separated from the nodes that have a low
in-degree. Therefore we split the sequence of nodes appearing as
incomplete children in $S$ into two disjoint subsequences $H$ and $L$,
the first containing those incomplete children that have an in-degree
larger than a specified threshold in their order of appearence in $S$,
the latter containing the rest. For example if the threshold is 3 then
$H=(19,19,19,19,19,19,19,19,20,20,20,20)$ for the BDD in Figure
\ref{fig:bdd+tree}. Based on $H$ we construct the sequence of integers
$S^H$ on the sequence of nodes $v_1, \ldots ,v_{|V|}$ in $S$ by
encoding $v_i \in H$ as $id_l(v_i)$ and $v_i \in L$ as $0$. By the
encoded $0$s we indicate the incomplete children that are not among
the incomplete children with high in-degree. Note that all integers
appearing in $S^H$ occurs with a high frequency and therefore will
compress efficiently using standard compression techniques. The
remaining incomplete children $L$, we code separately, as described in
the next two sections.

\subsubsection{Incomplete children with small in-degree}\label{sub:smallindegree}
Using the above encoding, we are left with a sequence of nontree
children $L$, with very few repetitions. When encoding this sequence
we will exploit the fact that the sequence of integers in $id_l(L)$
will in most instances tend to be increasing. Below we argue why this
is the case.

\begin{enumerate}
\item The first reason is that any node $u$, with an outgoing edge of length
$k$ is naturally restricted to the children occurring in layer
$l(u)+k$, and therefore to the range of $id_l$ indices of the nodes at
layer $l(u)+k$. Since most edges in a BDD are usually rather short
(except those to the terminals), this leads to a natural increasing
progression in $id_l(L)$.

\item Secondly, when examining a set of layers in a BDD it is very
common to see disjoint substructures. For example, in Figure
\ref{fig:bdd+tree}, we have two disjoint substructures induced by the
nodes $2,4,5,8,9,10$ and $3,6,7,11,12,13$. Given two disjoint
substructures then for any given layer $l$ let $I^l_1$ and $I^l_2$ be
the sequences of layer indices of the nodes in that layer from each of
them respectively. Assume for convinience that $I^l_1$ contains the
smallest index. Then it is the case that all indices in $I^l_1$ are
strictly smaller than all indices in $I^l_2$, and furthermore the same
applies to $I^j_1$ and $I^j_2$ for any other layer $j$. Since the
incomplete children of a layer are encoded according to the order of
their parents, this means that $I^i_1$ will always appear in $id_l(L)$
before $I^{i}_2$ for any layer $i$, helping to ensure an increasing
tendency in $id_l(L)$.

\item Thirdly, some possible nontree edges cannot exist, since had
  they in fact existed they would have been included in the spanning
  tree. This constraint on
  the nontree edges is stated in the following observation:

\begin{observation}\label{obs:illegaledges}
  For every nontree edge $(u,v)$ it holds that $$v \not \in \{v' \mid
  \exists (u',v') \in E^T: l(u') = l(u) \land id_l(u) < id_l(u')\}$$ In other
  words: assume that there is a nontree edge $(u,v)$ for which a right
  sibling $u'$ of $u$ has the child $v'$ in the spanning tree, then $v \ne
  v'$.
\end{observation}
\begin{proof}
  If $v = v'$ then since $u$ and $u'$ are on the same layer, and since
  $id_l(u) < id_l(u')$ the spanning tree would contain $(u,v)$ rather
  than $(u',v)$, which contradicts that $(u,v)$ is a nontree
  edge. This is due to the fact that when the spanning tree is
  constructed by a traversal of the nodes in the BDD in layer order.
\end{proof}

\begin{example}
  In Figure \ref{fig:bdd+tree} an incomplete child of the
  node $5$ can neither be $11, 12$ or $13$, since in that case the
  corresponding edge would be a tree edge, which contradict that the
  child is incomplete. On the other hand an incomplete child of $7$
  can be any of the nodes $8,\ldots,12$ since $7$ is positioned to the right of
  all its siblings.
\end{example}

What follows from Observation \ref{obs:illegaledges} is, roughly
stated, that incomplete children with parents in the ``left part'' of
a layer are bound to have one of the smaller layer ids in the layer,
whereas the incomplete children with parents in the ``right part'' of
the of a layer can have any layer id occurring in the layer.
\end{enumerate}

\noindent As a conclusion of three the reasons mentioned above about
why we expect the sequence of incomplete children to tend towards being
increasing, and as we have observed the increasing trend of $id(L)$ in
the instances we have tested on, we choose to exploit this fact by
encoding the sequence $id_l(L)$ by delta coding:

\begin{definition}[Delta Coding]
Consider any sequence of integers $(i_1, \ldots i_k) \in \mathbb Z^k$
  for any $k \in \mathbb N$. We define the delta coding of $(i_1,
  \ldots i_k)$ by $\Delta(i_1, \ldots i_k) = (i_1, i_2-i_1, i_3-i_2,
  \ldots, i_k-i_{k-1})$
\end{definition}
For instance if $id_l(L) = (9,12,14,15,16,17,18,20)$, then
$\Delta(id_l(L)) = (9,3,2,1,1,1,1,2)$. Standard compression will be
able to compress the latter sequence much better than the former
sequence.

\subsubsection{Long forward edges}\label{sub:longforward}
Using the encoding of Section \ref{sub:smallindegree} nontree long
edges will often be expensive to encode since they have an incomplete
child with an id that is a lot larger than for the short edges. Hence
they will often result in large deltas in the delta coding. The
following approach is an alternative way of encoding some of the long
edges. This technique is therefore applied prior to the technique in
Section \ref{sub:smallindegree} and only the remaining edges will be
coded using delta-coding.

A nontree edge $(u,v)$ is a \emph{forward edge} if $u$ is an ancestor
of $v$ in the spanning tree. Consider any forward edge $(u,v)$ in the
graph with length $k$. This edge can be unambiguously encoded by
$id_l(v)$ and the length of the edge, as $u$ if the ancestor of $v$ at
layer $l(v)-k$.

In order to know which nodes that are endpoints of forward edges we
label each node $v$ by the number of forward edges ending in $v$.
This will be $0$ for most nodes and very seldom be more than $1$,
ensuring a good compression of the labelling. After this is done we
encode the length of the forward edges. If there are very few long
edges it might not be worth the effort to write the labelling on the
edges. Hence we set a threshold on the number of forward edges that it
needed in order to make the encoding of these edges useful. If the
threshold is not exceeded all long forward edges are instead encoded
as described in Section \ref{sub:smallindegree}.

\subsection{An example}
As a final part in the description of our compression technique, we
show how our technique would compress the BDD in Figure
\ref{fig:bdd+tree}.
\begin{example}
  Consider the encoding of the BDD in Figure \ref{fig:bdd+tree}. We
  first use Lemma \ref{lemma:succinct} to encode the spanning tree as
  the bitstring (comma separated only to ease readability)
  $11,1111,11011101, 101010000000,010100,1100,0000$. As there are no
  long edges in the spanning tree we do not need to encode layer
  information, we will output $0$ to denote that the total number of
  layer information that is to be added is $0$. If we suppose a
  threshold of $5$ in the encoding of nontree edges with high indegree
  only the node $19$ will be encoded as an incomplete child with high
  indegree. This will be encoded as
  $0\;0\;19\;19\;19\;19\;0\;19\;0\;19\;0\;0\;0\;0\;0\;19\;0$

We are now left with two long forward edges of length 2, namely
$(14,20)$ and $(16,20)$.  To encode them we first specify which of the
remaining nontree edges that are long forward edges by a bitvector
$0000010010$ and the length $2,2$. Finally we encode the remain
nontree edges by $9\;12\;14\;15\;16\;17\;18\;20$ which in delta coding
will be $9\;3\;2\;1\;1\;1\;1\;2$.
\end{example}

\section{Experiments}\label{sec:experiments}
In this section we provide empirical results from compressing a large
set of BDDs from various sources using the new encoder described in
this paper and as well as the encoders from
\cite{bdd_image_compression} and \cite{universal_bdd_compression}. For
further comparison we also provide the results from a naive
encoder. The naive encoder outputs the size of each layer followed by
a list of children. This representation is very similar to the
in-memory representation of a BDD except that the layer information is
not stored for each node but rather implicitly using the layer sizes.

\subsection{Instances}
Many of the instances we show results for are taken from the
configuration library CLib \cite{clib}. As a BDD only allows binary
variables, additional steps must be taken in order to encode solutions
to problems containing variables with domains of size larger than
2. For each non-binary variable in a problem its customary to either
use a number of binary variables logarithmic in the size of the domain
of the variable and adjust the constraints accordingly or use one
variable for each domain value. These methods are known as
log-encoding\cite{yellowbook} and direct-encoding respectively. In the
instances we have tested with all those named with the suffix ``dir''
was compiled using direct encoding, while the remaining were build
using log-encoding. The instances fall into the following groups:

\paragraph{Product Configuration}
The instances in this group are all BDDs compiled for use with
standard interactive product configurators. For example the ``renault''
instance is a car configuration instance, and the others are various
PC configuration instances.

\paragraph{Power Supply Restoration}
These instances were compiled for use in configuring the restoration
of a power supply grid after a failure. As such they are also a type
of configuration instances.

\paragraph{Fault Trees}
These are instances built for use in reliability analysis using fault
trees.

\paragraph{Combinatorial}
The combinatorial group contains various ``toy'' chess problems of a
combinatorial nature. For example the classic problem of placing 8
queens on a chessboard without any piece being threatened is
represented by the instance ``8x8queen''. The ``5x27queens'' instance
models placing 5 queens on a 5x27 chess board.

\paragraph{Multipliers}
This group contains two BDDs both of which represent the value of the
middle bit in the output obtained by multiplying two groups of 10
input bits \cite{vlsi}. These are build mixing the input bits
(``mult-mix-10'') and separating the input bits (``mult-apart-10'').

\subsection{Post compression}
All the tested encoders create an encoding that is meant to be
subsequently compressed using standard entropy coding methods. In
\cite{bdd_image_compression} arithmetic coding is used while the
choice of entropy coding is not discussed in
\cite{universal_bdd_compression}. To avoid the empirical results being
affected by the choice of standard coding, we instead apply
LZMA\cite{lzma_sdk} to the output of all encoders to produce the final
encoding. Due to implementation details of this final compression
step, it is sometimes beneficial to produce the output that has to be
compressed on a byte level instead of a bit level. To ensure a fair
comparison the results stated for \cite{bdd_image_compression},
\cite{universal_bdd_compression} and the naive approach are obtained
by trying to output both on bit level and on byte level and
stating the best compression among the two results.  Our own encoder
was only tested outputting on a byte level.

\subsection{Conclusions}
From the empirical results shown in Figure \ref{fig:results} we can
immediately see that it is worthwhile to make use of a dedicated BDD
encoder, as the naive encoding, being only compressed by LZMA, is
outperformed with a factor of up to 20 on some instances. Furthermore
we can see that the encoder introduced in this paper is consistently
able to perform as well or better than the other encoders on all
tested instances. In particular the largest BDD in our test
(``complex-P3'') required about twice as much space when using either
of the two other dedicated encoders.

\paragraph{Instance properties}
For most of the instances included here it is the case that a very large
fraction (30\% to nearly 50\%) of the edges lead to the
zero-terminal. The exception to this are the multiplier instances,
``5x27queens'' and the ``rook'' instances. Slightly less expected is the
fact that it is quite rare for nodes other than the zero-terminal to
have a significant in-degree, this only occurring with any great
significance in ``5x27queens'' and to lesser extent in ``complex-P2''
and the multipliers. This means that in quite a few cases nearly all
of the non-tree edges are simply edges to the zero-terminal,
essentially turning $S^H$ into a bitvector, marking almost all the
edges as zero-terminal edges, allowing for very efficient
compression. This can be seen in the results where the ``5x27queens'',
the rook and the multiplier instances all turn out to compress less
efficiently. An additional important trend is that nodes which cannot
be reached by following a short edge from a parent are very rare,
meaning that our encoder in by far the most cases only need to provide
layer information for less than $1\%$ of the nodes, which is a
significant advantage over previous encoders.

\paragraph{Availability}
The Java source code used for these experiments (including a
command-line encoder and decoder for BDDs in the BuDDy \cite{buddy}
file format) is available along with all instances used in these
experiments at (URL removed for blind review).

\begin{figure}
\begin{center}
\footnotesize
\begin{tabular}{| c |  c |  c |  c | c | c|}
\hline
 \textbf{Name} & $|V|$ & \textbf{this paper}  & \cite{bdd_image_compression} &  \cite{universal_bdd_compression} & \textbf{Naive} \\
\hline
\multicolumn{6}{|c|}{\textbf{Product Configuration}} \\
\hline
renault          & 455798  & 0,90 & 126\% & 103\% & 402 \%\\
renault-dir      & 1392863 & 0,23 & 198\% & 214\% & 1352\%\\
pc-CP            & 16496   & 0,76 & 220\% & 209\% & 788 \%\\
pc               & 3467    & 2,19 & 224\% & 211\% & 436 \%\\
Big-PC           & 356696  & 0,38 & 334\% & 266\% & 1345\%\\
Big-PC-dir       & 1291600 & 0,17 & 260\% & 260\% & 2035\%\\
\hline
\multicolumn{6}{|c|}{\textbf{Power Supply Restoration}} \\
\hline
complex-P3       & 2812872 & 0,44 & 243\% & 202\% & 951 \% \\
complex-P2       & 163432  & 1,16 & 181\% & 167\% & 541 \%\\
1-6+22-32        & 20937   & 1,89 & 136\% & 154\% & 413 \%\\
1-6+22-32-dir    & 61944   & 0,99 & 135\% & 161\% & 606 \%\\

\hline
\multicolumn{6}{|c|}{\textbf{Fault Trees}} \\
\hline
isp9607          & 228706  & 0,63 & 389\% & 204\% & 873 \%\\
isp9605          & 4570    & 3,30 & 130\% & 145\% & 305 \%\\
chinese          & 3590    & 2,06 & 214\% & 160\% & 450 \%\\
\hline
\multicolumn{6}{|c|}{\textbf{Combinatorial}}\\
\hline
5x27queens     & 562764  & 4,33 & 108\% & 109\% & 204 \% \\
13x13rook        & 76808   & 3,56 & 210\% & 165\% & 311 \% \\
8x8rook          & 1339    & 6,03 & 140\% & 139\% & 277 \%\\
8x8queen-dir     & 2453    & 2,17 & 115\% & 178\% & 374 \%\\
8x8queen         &  879    & 4,29 & 114\% & 138\% & 332 \% \\
\hline
\multicolumn{6}{|c|}{ \textbf{Multipliers} }\\
\hline
mult-mix-10      & 42468   & 9,92*& 114\% & 107\% & 169 \%\\
mult-apart-10    & 31260   & 8,07*& 120\% & 124\% & 202 \%\\
\hline
\end{tabular}
\end{center}
\caption{The above table shows the name and size (in nodes) of each of
  the instances tested. The result of the new encoder, in bits per
  node, is then showed, followed by the relative results of the rest
  of the encoders. The * indicates results obtained from our encoder
  when delta coding is not used in the encoding of nontree edges.}
\label{fig:results}
\end{figure}

redpajama_set_name: RedPajamaArXiv


text:
Q: NginX Grok Pattern - Handling Backslashes or Brackets? I'm trying to nut out all _grokparsefailure's on my logstash box.
Seems the only two culprits are NGINX logs which trip up my NGINXACCESS pattern:
 %{IPORHOST:clientip} %{NGUSER:ident} %{NGUSER:auth} \[%{HTTPDATE:timestamp}\] "%{WORD:verb} %{URIPATHPARAM:request} HTTP/%{NUMBER:httpversion}" %{NUMBER:response} (?:%{NUMBER:bytes}|-) (?:"(?:%{URI:referrer}|-)"|%{QS:referrer}) %{QS:agent}

the following are two examples of message results that get tagged as grok fails.
172.31.0.2 - - [30/Jul/2015:15:10:49 +1000] "GET /web-app/[EXPAND] HTTP/1.1" 404 6432 "-" "Amazon CloudFront" "web-app.mydomain.com" "127.0.0.1"

172.31.0.2 - - [30/Jul/2015:14:13:52 +1000] "GET /web-app/show?wid=5540cfbc3asdf034ct=&domain=apptest.mydomain.com&ttl=\x5C%2230\x5C%22&filter_id=14026&unique_id=1 HTTP/1.1" 200 11400 "http://apptest.mydomain.com/"; "Amazon CloudFront" "apptest.mydomain.com" "127.0.0.1"

going through the grok debugger, the fail relates to %{URIPATHPARAM:request} hitting the brackets for [EXPAND] in the first example and the backslashes for the \x5C%2230\x5C%22 in the second. ie. if i remove [, ], or \ from the inputs then grok matches fine.
I can't seem to workout how to get the URIPATHPARAM grok filter to deal with those examples of brackets and backslash. Any ideas?

A: Generally I would recommend to use another pattern as @Alain suggested. If you still want to solve this with a more exact pattern you can use a grok field like this:
(?<request>(?:/[A-Za-z0-9$.+!*'(){}\[\]\\,~:;=&@#?%_\-]*)+)
(This is a mix of URIPATH and URIPARAM with backslashes and brackets.)
The entire grok pattern would look like this:
%{IPORHOST:clientip} %{NGUSER:ident} %{NGUSER:auth} \[%{HTTPDATE:timestamp}\] "%{WORD:verb} (?<request>(?:/[A-Za-z0-9$.+!*'(){}\[\]\\,~:;=&@#?%_\-]*)+) HTTP/%{NUMBER:httpversion}" %{NUMBER:response} (?:%{NUMBER:bytes}|-) (?:"(?:%{URI:referrer}|-)"|%{QS:referrer}) %{QS:agent}

This works for both of your given examples.
However, there is another issue with your inputs. The second one has a semicolon behind its referrer ("http://apptest.mydomain.com/";) which the first one has not. You'll have to take care of that.
So you need the pattern to have an optional semicolon (?:;|):
%{IPORHOST:clientip} %{NGUSER:ident} %{NGUSER:auth} \[%{HTTPDATE:timestamp}\] "%{WORD:verb} %{DATA:request} HTTP/%{NUMBER:httpversion}" %{NUMBER:response} (?:%{NUMBER:bytes}|-) (?:"(?:%{URI:referrer}|-)"|%{QS:referrer})(?:;|) %{QS:agent}

A: As you've discovered, URIPATH doesn't allow for brackets.  Since you have/want brackets, you'll need to use something else.  How about %{NOTSPACE}?

redpajama_set_name: RedPajamaStackExchange


This month Sammy has been obsessed with making her grandmother's crepes as a mid-afternoon snack and as a dessert. While enjoying her freshly made crepes with caramelised peach, yoghurt and local honey, she had begun to think about the pancake and its many variations not only within western cultures, but also its delightful eastern cousins.
The best pancakes are both sweet and savoury. Here are Sammy's top 5 delicious pancake discoveries from around the world.
1. Blend all ingredients together to form a smooth batter.
2. Heat a non-stick fry pan or crepe pan to medium high heat, once hot add ¼ tsp of butter to the pan and melt.
3. Pour in just enough batter to make one thin crepe. Tilt pan to spread batter evenly.
4. Once the top is cooked, allow approx. 1 min, then flip and cook for another 30 seconds.
5. Set aside and repeat with remaining batter. Crepes can be stored in fridge covered with cling film for 1 -2 days. Enjoy with desired filling.
America sure loves its pancakes with the dish being served in every diner across the country, and not to mention the endless list of pancake houses. These pancakes are perfectly fluffy and buttery. They are usually served with a minimum 3 pancakes that can range in size from a 10cm diameter to a whopping 20cm+. They are made extra fluffy by adding a bit of baking powder to the batter and are always served with a knob of butter and a very generous drizzle of maple or golden syrup. For those wanting a few more calories and a salty hit, grilled bacon rashers are added atop these fluffy masterpieces. Although delicious and a culinary icon, it's probably best to enjoy these pancakes sparingly due to their uber high fat content!
The Chinese spring onion pancake is slightly different to most other pancakes. There is no batter but rather a dough that is made and rested, before being rolled out into a pancake shape, then sprinkled with sliced spring onions. Next it's rolled and then turned into a snail shape before being flattened again, back into its original pancake form. The reason for this is to create layers on the dough and spring onions mix. That way when the pancake cooks in oil on the pan, the outside becomes flaky and crispy while the inside is deliciously soft.
This is a fantastic savoury pancake which I enjoyed in Bangkok on a trip last year. This dish takes the meaning of pancake to the next level by the amount of skill and many layers of flavour that it's made with. It's made with arrow root and rice flour, which is poured onto a flat plate to begin the pancake, then layered with par cooked mussels and a layer of beaten eggs. This is all topped off with bean sprouts and spring onions and accompanied with a sweet chilli dipping sauce.
These miniature fluffy pancakes with a spongy texture are popular at many festivals and food fairs. You'll first recognise the plates piled with these cute little pancakes and with a decent dusting of icing sugar. Half the fun of eating these pancakes is watching them being made in custom poffertjes fry pans. These custom pans look like reverse jumbo bubble wrap, made with very non-stick steel. The chef is usually a pro at pouring the perfect amount of batter into each round and then flipping them with a long wooden skewer. Then in no time you'll be enjoying your own plate piled full of little dutch pancakes. You can now purchase dutch pancake pans at good food stores and online.


text:
Doomsday Scenarios Are as Harmful as Climate Change Denial
Jul 13, 2017 3:30AM EDT Home
By Michael E. Mann, Susan Joy Hassol and Tom Toles
It is easy to understand why advocates for climate action have become somewhat dispirited in recent months. In the space of less than a year, we've seen the U.S. go from playing a leading role in international climate negotiations to now being the only nation in the world to renege on its commitment to the 2015 Paris climate accord.
It is in this environment of defeat and despair that we've witnessed a dramatic rise in the prominence of climate doomism—commentary that portrays climate change not just as a threat that requires an urgent response but also as an essentially lost cause, a hopeless fight. Some of the more egregious examples can be found among fringe characters such as ecologist Guy McPherson—a doomist cult hero who insists that exponential climate change likely will render human beings and all other species extinct within 10 years.
Climate Doomsday Essay 'Is Exactly What We Don't Need' https://t.co/rPhsJqs3yG @greenpeaceusa @SierraClub @foe_us @MichaelEMann @350
— EcoWatch (@EcoWatch) July 13, 2017
Such rhetoric is in many ways as pernicious as outright climate change denial, for it leads us down the same path of inaction. Whether climate change is a hoax (as President Trump has asserted) or beyond our control (as McPherson insists), there would obviously be no reason to cut carbon emissions.
Doomist narratives, albeit of a more nuanced and subtle variety, are now starting to appear in respected, mainstream venues, written by otherwise able and thoughtful journalists. In this vein comes a recent New York Magazine article The Uninhabitable Earth by David Wallace-Wells.
It is important to be up front about the risks of unmitigated climate change, and it is critical to keep in mind the potential for unpleasant surprises and worst-case scenarios, the so-called fat tail of risk. It is, moreover, appropriate to criticize those who understate the risks. But there is also a danger in overstatement that presents the problem as unsolvable and future outcomes as inevitable.
The New York magazine article paints an overly bleak picture, arguing that climate change could render the Earth uninhabitable by the end of this century. Its opening story about the "flooding" of a seed vault in Norway leaves out that one of the vault's creators told NPR "there was really no flood." It exaggerates the near-term threat of climate "feedbacks" involving the release of frozen methane. It mischaracterizes one recent study as demonstrating that the globe is warming "more than twice as fast as scientists had thought," when in fact the study in question simply showed that one dataset that had tended to show less warming than other datasets has now been brought in line with the others after some problems were corrected for. The warming of the globe is progressing as models predicted. And that is plenty bad enough.
The evidence that climate change is a serious challenge that we must tackle now is very clear. There is no need to overstate it, particularly when it feeds a paralyzing narrative of doom and hopelessness. Some seem to think that people need to be shocked and frightened to get them to engage with climate change. But research shows that the most motivating emotions are worry, interest and hope. Importantly, fear does not motivate, and appealing to it is often counter-productive as it tends to distance people from the problem, leading them to disengage, doubt and even dismiss it.
It is important to communicate both the threat and the opportunity in the climate challenge. Those paying attention are worried, and should be, but there are also reasons for hope. The active engagement of many cities, states and corporations, and the commitments of virtually every nation (minus one) is a very hopeful sign. The rapid movement in the global energy market towards cleaner options is another. Experts are laying out pathways to avoid disastrous levels of climate change and clearly expressing the urgency of action. There is still time to avoid the worst outcomes, if we act boldly now, not out of fear, but out of confidence that the future is largely in our hands.
Michael E. Mann is distinguished professor of atmospheric science at Pennsylvania State University and director of the Penn State Earth System Science Center. Susan Joy Hassol is the director of Climate Communication LLC. Tom Toles is the editorial cartoonist for The Post.
climate deniermichael mann
Former Exxon CEO Lee Raymond Quits JPMorgan Board
DC and Minnesota Sue Fossil Fuel Giants for Misleading Public on Climate Change
U.S. Ranks 24th in New Environmental Performance Index, Near Bottom of Developed Countries

redpajama_set_name: RedPajamaCommonCrawl


Quick Links: New Taco guy here. Wanting to learn how to upgrade my Gen3 Tacoma Hello. My wife and I purchased a Toyota Tundra Limited 4.7 V8 KingTacoRey's Super-White '15 Commuter Build New to forum Hello from Las Vegas!
Discussion in 'New Members' started by rallyandbosox, Jan 26, 2008.
My name is Jay, I've been posting on TN, CT, and AZToyz but hadn't heard about TW until now.
I own an '07 Speedway Blue DC 4x4 TRD Sport. Mods so far are: BFG A/Ts, Grille Guard, 3" Revtek Spacers, 2" Toytec AALs, Fog lights, and Debaged.
I am in Arizona, but from Massachusetts.
Welcome to TW, sounds like you have plenty of of knowledge to share, so do you live in a major metro or someplace else ?
Thank you. I live in Tempe, which is metropolitan Phoenix.


text:
Vitamin D and omega-3 fatty acid supplements in children with autism spectrum disorder: a study protocol for a factorial randomised, double-blind, placebo-controlled trial
Hajar Mazahery1,
Cathryn Conlon1,
Kathryn L. Beck1,
Marlena C. Kruger1,
Welma Stonehouse2,
Carlos A. Camargo Jr.3,
Barbara J. Meyer4,
Bobby Tsang5,
Owen Mugridge1 &
Pamela R. von Hurst1
There is strong mechanistic evidence to suggest that vitamin D and omega-3 long chain polyunsaturated fatty acids (n-3 LCPUFAs), specifically docosahexaenoic acid (DHA), have the potential to significantly improve the symptoms of autism spectrum disorder (ASD). However, there are no trials that have measured the effect of both vitamin D and n-3 LCPUFA supplementation on autism severity symptoms. The objective of this 2 × 2 factorial trial is to investigate the effect of vitamin D, n-3 LCPUFAs or a combination of both on core symptoms of ASD.
Children with ASD living in New Zealand (n = 168 children) will be randomised to one of four treatments daily: vitamin D (2000 IU), n-3 LCPUFAs (722 mg DHA), vitamin D (2000 IU) + n-3 LCPUFAs (722 mg DHA) or placebo for 12 months. All researchers, participants and their caregivers will be blinded until the data analysis is completed, and randomisation of the active/placebo capsules and allocation will be fully concealed from all mentioned parties. The primary outcome measures are the change in social-communicative functioning, sensory processing issues and problem behaviours between baseline and 12 months. A secondary outcome measure is the effect on gastrointestinal symptoms. Baseline data will be used to assess and correct basic nutritional deficiencies prior to treatment allocation. For safety measures, serum 25-hydroxyvitamin D 25(OH)D and calcium will be monitored at baseline, 6 and 12 months, and weekly compliance and gastrointestinal symptom diaries will be completed by caregivers throughout the study period.
To our knowledge there are no randomised controlled trials assessing the effects of both vitamin D and DHA supplementation on core symptoms of ASD. If it is shown that either vitamin D, DHA or both are effective, the trial would reveal a non-invasive approach to managing ASD symptoms.
Australian New Zealand Clinical Trial Registry, ACTRN12615000144516. Registered on 16 February 2015.
Autism spectrum disorder (ASD) is a neurodevelopmental disorder usually diagnosed when developmental, educational and social demands increase [1]. ASD is believed to affect 1 % of the New Zealand population [1]. Diagnostic criteria for ASD include delays or difficulties in sociocommunicative functioning, restricted and repetitive behaviours/interests, sensory issues and aberrant behaviours [1, 2]. ASD is also associated with medical conditions such as gastrointestinal problems [1–5]. The clinical symptoms of individuals with ASD vary widely [5–7], suggesting that it is multi-factorial in nature.
It is generally agreed that both genetic and environmental factors contribute to the development of ASD. The high heritability of ASD has been shown by twin and familial studies [8, 9]. However, it has been reported that only 30 % of ASD cases are clearly associated with a syndrome or genetic markers leaving the aetiology of most cases without explanation [10].
Mechanistic evidence, as well as a scattering of ecological and cross-sectional studies, suggests that vitamin D may play an important role in the aetiology of ASD. Vitamin D receptors and 1α-hydroxylase have been identified in different regions of the brain and sensing neurons [11–13]. The active form of vitamin D has been shown to have an important role in the neuronal differentiation, structure, function and connectivity of the developing brain [14]. Vitamin D response elements have been identified on genes involved in serotonin and oxytocin synthesis [15]. Lower levels of plasma oxytocin [16] and abnormal serotonin concentrations in the brain and tissues outside the blood–brain barrier have been shown in populations with ASD [17, 18]. Oxytocin and serotonin have been implicated in modulating social behaviour [19, 20].
The serum level of 25-hydroxyvitamin D (25(OH)D), the best available marker of vitamin D status [21, 22], has been shown to be significantly lower in autistic individuals than in their healthy counterparts [23, 24]. Similarly, higher prevalence of ASD has been reported at higher latitudes and in individuals exposed to lower UVB radiation levels [24, 25]. In adults with severe autism living in a community centre in Italy, problem behaviours significantly increased during spring and decreased during autumn [26]. Depletion of vitamin D in body stores by the end of winter and early spring seasons (due to lack of sun exposure) may have exacerbated the symptoms of autism and increased problem behaviours observed in this study.
The potential role of vitamin D deficiency in autism has received surprisingly little attention. While a few case studies have reported beneficial effects of vitamin D supplementation on autistic core symptoms [27], no randomised, placebo-controlled trial with vitamin D supplementation has been conducted to date [28]. Jia et al. [27] reported that shifting serum 25(OH)D concentration in a child with ASD from 31 to 203 nmol/L after 2 months of high-dose vitamin D supplementation (150,000 international units (IU) per month administered intramuscularly plus 400 IU per day administered orally) improved autistic core symptoms. Although other trials investigating the effect of multivitamin/mineral supplements containing low doses of vitamin D on autism symptoms have provided promising results [29, 30], the individual effect of each nutrient cannot be determined from these studies.
Omega-3 long chain polyunsaturated fatty acids (n-3 LCPUFAs) also have the potential to positively affect children with ASD. These n-3 LCPUFAs, mainly DHA, are necessary for normal development and functioning of the brain and auditory and visual processing system [31–33]. Long-term DHA depletion results in significant losses in brain DHA with consequent loss of brain function [34]. Evidence shows that children with ASD have an increased omega-6 to omega-3 ratio in blood and low blood concentrations of n-3 LCPUFAs, which could be due to either low dietary intake or differences in fatty acid metabolism and incorporation into cellular membranes of children with ASD [35–37].
Reports on the benefits of n-3 LCPUFAs in treating ASD are inconclusive. There are, to our knowledge, only four randomised, placebo-controlled trials [38–41], three of which are small pilot studies. Bent et al. [40] and Amminger et al. [39] found that omega-3 supplementation was superior over a placebo (12 and 6 weeks, respectively) for reducing symptoms of hyperactivity and stereotypic behaviour in children with ASD. However, more recent studies have found that supplementation with n-3 LCPUFAs for 6 months had no beneficial effect on core symptom domains of ASD in children aged 2 to 5 years (n = 38) [41] and 3 to 10 years (n = 48) [38]. However, these studies are limited by their low participant numbers and short treatment periods.
In addition to these studies on the nutrients' effects when given individually, there are speculations that vitamin D and n-3 LCPUFAs may improve ASD symptoms because of their shared functions and each nutrient-specific role that complements the other nutrient's functions [42, 43]. Both nutrients are powerful anti-inflammatory agents, immune modulators and neuroprotectors [42]. Furthermore, evidence suggests that while vitamin D regulates serotonin synthesis, omega-3 fatty acids increase serotonin release and membrane fluidity and thus increase serotonin accessibility [43]. ASD is associated with increased inflammation, oxidative stress, immune dysregulation and/or mitochondrial dysfunction in brain regions that are involved in social behaviour, sensory and motor coordination, memory, speech and auditory processing, and also with neurotransmitter dysregulation [17, 18, 44].
Gastrointestinal problems have also been reported to be common in children with ASD. Compared to their typically developing siblings (12 %), autistic children have more gastrointestinal symptoms (42 %) [3–5]. Gastrointestinal problems may relate to abnormal gut flora [45], decreased activity of digestive enzymes [46] or increased intestinal permeability [47]. Vitamin D deficiency has been implicated in the pathophysiology of some gastrointestinal diseases, including inflammatory bowel disease [48], and thus might play a role in ASD-related gastrointestinal problems. Likewise, n-3 LCPUFAs have been shown to reduce symptoms of ulcerative colitis [49], to support epithelial integrity in vitro [50], and to alter the gut microbiota composition of both neurodevelopmentally normal and early life-stressed animals [51].
Unusual eating habits, a risk factor for nutrient deficiencies, are common in ASD [4]. Inadequate intakes of magnesium, zinc, folate, vitamins A, E, B12, K and D, as well as low intake of foods rich in n-3 LCPUFAs, have been reported in children with autism [52–58]. However, a broad picture of the nutritional status of affected children in New Zealand is lacking.
Both vitamin D and omega-3 status, defined as omega-3 index (red blood cell (RBC) DHA + eicosapentaenoic acid (EPA)), will be low in children with ASD at baseline (25(OH)D <75 nmol/L [59] and omega-3 index of approximately 4–6 % [60, 61])
Improving either vitamin D or omega-3 status with supplementation will reduce the severity of ASD symptoms in children with ASD
Combined vitamin D and n-3 LCPUFA supplementation will be more effective than either supplement alone or placebo in reducing the severity of ASD symptoms in children with ASD
To establish the vitamin D and RBC fatty acid status of children with ASD living in Auckland, New Zealand
To investigate the effect of improving either vitamin D or omega-3 status in reducing the symptoms of ASD including sociocommunicative functioning, sensory issues and aberrant behaviours (primary outcomes) and gastrointestinal symptoms (secondary outcome)
To establish the effectiveness of supplementation with combined vitamin D and n-3 LCPUFAs in reducing the symptoms of ASD including sociocommunicative functioning, sensory issues and aberrant behaviours (primary outcomes) and gastrointestinal symptoms (secondary outcome)
This study consists of two stages: stage 1 will include recruitment and screening, while stage 2 is a vitamin D and n-3 LCPUFA randomised, double-blind, placebo-controlled trial (Fig. 1). The duration of the intervention is 12 months.
Schematic diagram of study design. 1Blood biomarkers: 25-hydroxyvitamin D (25(OH)D), red blood cell (RBC) fatty acids, calcium, albumin, iron studies, vitamin B12, folate, full blood count and vitamin A. Questionnaires: sociodemographics, medical history, eating/mealtime behaviours and food diary. 2Questionnaires: sociocommunicative functioning, sensory problems and aberrant behaviours (primary outcomes); gastrointestinal symptoms (secondary outcome), sun exposure and skin colour. Anthropometry: weight and height. 3Blood biomarkers: 25(OH)D, RBC fatty acids, calcium and albumin. Questionnaires: sociocommunicative functioning, sensory problems, aberrant behaviours (primary outcomes) and eating/mealtime behaviours and diet quality. Anthropometry: weight and height. 4Blood biomarkers: 25(OH)D, calcium and albumin. *Gastrointestinal symptoms (secondary outcome) and medication/supplement use/incidence of adverse events/supplement compliance will be monitored throughout the study period
Stage 1 will provide the opportunity for a comprehensive description of the study population with respect to nutritional status (biochemical indices and dietary intake), demographics and medical history. Stage 2 will demonstrate the efficacy of supplementation with vitamin D, n-3 LCPUFAs or both on reducing ASD symptoms.
This study is a collaboration between Massey University and the Waitemata District Health Board (WDHB), New Zealand. Caregivers of children who meet the criteria for the study will be approached in the first instance by the WDHB Developmental Coordinators.
We calculated that 42 participants (a minimum of 34 participants, and allowing for a 20-% potential dropout rate) would be required for each arm of the trial to demonstrate a clinically significant difference at 80 % power and 5 % statistical significance. Power calculations were based on a 17-unit difference between supplemented groups and placebo in change from baseline to endpoint on the Social Responsiveness Scale (SRS) total score [62], on a mean SRS and standard deviation of 105 and 24.7 units in untreated children with ASD, respectively (from our 2015 pilot study, unpublished). The sample size was calculated using the formula below [63]:
$$ N= 2{\alpha}^2K/{\left({\mu}_2\hbox{-} {\mu}_1\right)}^2 $$
where N is the sample size required per group, α is the SD, K is the constant (7.9 denotes 80 % power and 5 % significance), and (μ2 − μ1) is the difference in SRS total score between groups.
To ensure that the study is adequately powered, a blinded interim analysis at >50 % of the initially planned enrolment will be performed by an independent third party to estimate the variance for potential sample size increase.
Children will be eligible for this study if they are aged between 2.5 and 8 years, have a medical diagnosis of ASD confirmed by both a developmental paediatrician in accordance with the criteria listed in the Diagnostic and Statistical Manual of Mental Disorders, version five (DSM-5) [2], and onset of symptoms after 18 months of age. The lower limit of 2.5 years has been chosen based on the age criteria of the psychological assessment tools, and the upper limit of 8 years has been chosen to avoid the confounding effect of behavioural changes associated with pubertal stage. The caregiver's proficiency in English is a requirement (due to the nature of outcome assessment tools). Volunteers are excluded if they were diagnosed as having developmental delay since birth.
Additional inclusion criteria for the trial are: liver function within the normal range (albumin 34–48 g/L) and serum 25(OH)D <75 + 10 nmol/L if they enter the trial in winter and <105 nmol/L + 10 nmol/L if they enter the trial in summer. A 10-nmol/L variation was chosen because of the potential assay variability [64]. We have applied two different cut-off points for exclusion because there is a large seasonal variation in serum 25(OH)D concentrations in New Zealand ranging from 30 nmol/L [65] to 44 nmol/L [66].
The study will take place in Auckland, New Zealand. Non-fasted blood samples will be collected at the North Shore or Waitakere Hospitals in Auckland, New Zealand. Questionnaires and anthropometry will be undertaken at Massey University, Auckland, New Zealand. Auckland is New Zealand's largest city with a population of just over one million. It has been estimated that approximately 13,000 (32.5 %) individuals with ASD reside in the Greater Auckland region [1].
Stage 1 – screening
Screening and recruitment (stage 1 of the study) will take place over a 24-month period, which commenced in January 2015. Children who meet the initial inclusion criteria will have a blood draw and will be screened for nutritional deficiencies. See Tables 1 and 2 for outcome measures, testing methods, and schedule of enrolment, intervention, and assessment, respectively.
Table 1 Summary of the study outcome measures and methods
Table 2 Schedule of enrolment, intervention, and assessment
Pre-intervention preparation
Prior to randomisation and inclusion in the trial, vitamin D, iron and vitamin B12 deficiencies will be addressed. Refer to Table 3 for a list of nutritional deficiencies and the management strategies applied in this trial.
Table 3 Nutritional deficiencies and their management strategies prior to entering the intervention trial
Stage 2 – vitamin D and n-3 LCPUFA intervention
The intervention consists of 2000 IU of vitamin D3 per day, 722 mg of DHA per day, 2000 IU of vitamin D3 plus 722 mg of DHA per day or placebo, in the form of four oral capsules. The treatment materials will be delivered in 750-mg gel capsules with a tear-off nozzle manufactured and supplied by Douglas Nutrition Ltd., Auckland, New Zealand. The study capsules, vitamin D, n-3 LCPUFAs and placebo are identical in appearance and all are tasteless and colourless. The child is required to consume the contents of four capsules per day mixed into their food of preference or by oral administration by syringe. Refer to Table 4 for the total daily intake and contents of each capsule. Throughout the study period, children are allowed to have any therapy or medication for autism as well as any supplement provided that it does not contain vitamin D or omega-3.
Table 4 Total daily intake of vitamin D and omega-3 long chain polyunsaturated fatty acids (n-3 LCPUFAs) and the contents of each capsule, vitamin D, n-3 LCPUFAs and placebo
Supplementation with 2000 IU vitamin D3 has been shown to be a safe dose in infants [67, 68], and the French Society of Paediatrics recommends 1000 to 1200 IU per day in breast-fed infants [69]. The body mass of 2.5–8 year-olds in our trial will be considerably greater than infants', which should further reduce the risk of adverse events. Furthermore, 2000 IU per day is less than the safe upper limit of 2500 and 3000 IU/day suggested by the IOM for 1–3 and 4–8 year age groups, respectively [70].
The DHA dose of 722 mg/day is physiologically relevant and achievable through diet (equivalent to approximately three servings of fatty fish per week) and is comparable to doses used in other trials in children investigating the effects of n-3 LCPUFAs on behaviour and learning [71]. No side-effects have been reported in children with a dose of 600 mg DHA/day [72].
Randomisation, blinding and concealed allocation
Children will be randomly allocated to one of four groups, having been stratified for age and severity of ASD. Randomisation of the active/placebo capsules, the randomised sequence list, and group assignment will be fully concealed from the researchers, children and caregivers for the entire study, including the data analysis. A third party not involved in any aspects of the study will generate a random block design in blocks of four and eight using Randomization.com (http://www.randomization.com/). The third party will allocate a treatment code to a child once their eligibility for the intervention is confirmed and the caregiver's consent is received. In an emergency situation where breaking of the study blind will be required, plans will be in place for the principal investigator to contact the third party responsible for randomisation to reveal the treatment assignment for a given participant.
Participants (caregivers and children) will attend the Human Nutrition Research Unit (HNRU) at Massey University on two occasions: baseline and 12 months. Once recruited into stage 2 and prior to being given a 4-month supply of supplements, caregivers will complete some questionnaires on core symptoms of ASD (sociocommunicative functioning and sensory issues) and aberrant behaviours as the study primary outcome measures, and on gastrointestinal symptoms as the secondary outcome measure. Once the intervention is completed (12 months), baseline assessments of core symptoms of ASD and aberrant behaviours completed by caregivers will be repeated. Caregivers will also complete weekly gastrointestinal symptom diaries over the study period.
Further information will be collected to describe the study population characteristics at the baseline. This information includes eating/mealtime behaviours, food diaries, sun exposure, skin colour and anthropometry (weight and height). At the final visit, baseline assessment of eating/mealtime behaviours will be repeated and children's weight/height will be measured. Refer to Table 1 for outcome measures and testing methods.
Blood sampling and analysis
Children will have their blood samples drawn on three occasions, baseline (stage 1 – screening), 6 months (safety measures) and 12 months (endpoint). Refer to Table 1 for nutritional biomarkers and testing methods. The non-fasted blood samples will be collected under the supervision of paediatric staff and processed at North Shore or Waitakere Hospitals of WDHB laboratory service. Nutritional biomarkers will be assayed from a venous blood sample. These include the following: 25(OH)D, RBC fatty acids, calcium, albumin, iron studies, vitamin B12 and folate, full blood count and vitamin A. With the exception of RBC fatty acids and vitamin A, all biomarkers will be analysed at North Shore Hospital. RBC fatty acids will be analysed at the University of Wollongong, NSW, Australia, and vitamin A in a laboratory at Massey University, Auckland, New Zealand.
The primary outcome measures are psychological assessments of core symptoms of ASD and co-occurring problem behaviours which are detailed in Table 1. The secondary outcome measure is the assessment of gastrointestinal problems. Standardised instructions will be given to all caregivers on how to complete the questionnaires during their visit to the HNRU. Before the participant departs the HNRU researchers check all answers for completeness.
Social Responsiveness Scale™, Second Edition (SRS-2) [73]: the SRS-2 versions specific for age groups 2.5–4.5 and 4.5 through 18 years will be used. SRS-2 identifies social impairment associated with ASDs and quantifies its severity in the domains of social awareness, social information processing, reciprocal social communication, social anxiety/avoidance, and stereotypic behaviour/restricted interests [73]. The clinical validity and sensitivity of SRS-2 has been determined in populations with ASD [74].
Sensory Processing Measures™ (SPM) [75]: the SPM versions specific for age groups 2–5 and 5–12 years will be used. SPM assesses sensory processing, planning and ideas (praxis) and social participation in children. The scales measure social participation, vision, hearing, touch, body awareness (proprioception), balance and motion (vestibular function) and planning and ideas (praxis) [75]. This tool has been standardised, validated and used in children with ASD [76, 77].
Aberrant Behaviour Checklist (ABC) [78]: the ABC measures the variety of behaviour problems, namely irritability, social withdrawal, stereotypic behaviour, hyperactivity and inappropriate speech [79]. It has been validated in children with ASD [80] and has been widely used in treatment outcome studies of ASD [40].
Gastrointestinal symptoms questionnaire: the gastrointestinal symptoms questionnaire is designed specifically for this study and includes gastrointestinal signs and symptoms most commonly reported in children with ASD [81, 82]. Caregivers of study participants will be provided with weekly online diaries and will be asked to record if the child had constipation, diarrhoea, flatulence, abdominal pain, vomiting/nausea, abdominal distension, unexplained daytime irritability or/and unexplained night-time awakening during the past week, and if so how many times these occurred. The gastrointestinal symptoms questionnaire has not been validated, but includes questions on gastrointestinal symptoms most widely reported in ASD populations, or behaviours that might be a consequence of gastrointestinal symptoms. We are not aware of a validated instrument for assessing gastrointestinal symptoms in autistic children, and certainly not in New Zealand.
Dietary Index of Children's Eating (DICE): the DICE is a simple diet-quality assessment tool which has been developed by the research team, based on the New Zealand Ministry of Health Food and Nutrition Guidelines for Healthy Children and Young People [83]. This tool has been validated in a cohort of New Zealand healthy children aged 2 to 8 years (unpublished), and will be validated in ASD children in this study against the 4-day estimated food diary and biochemical markers.
Behavioural Paediatrics Feeding Assessment Scale (BPFAS): the BPFAS is a simple assessment tool which measures a child's mealtime behaviour and parents' attitudes and behaviours. It is a 35-item scale questionnaire comprising 25-likert scale (5 points) questions about child behaviour and 10 dichotomous questions about parents' attitudes and behaviours. Cut-off scores for the BPFAS have recently been established [84]. The BPFAS is the most reliable parent-administered feeding questionnaire, with good internal validity and test-retest reliability [85].
Four-day estimated food diary: dietary intake data will be collected by a 4-day estimated food diary, including one weekend day. Instructions on how to accurately complete the food diary will be provided with the food diary. Participants will be given a free-post, pre-addressed envelope for the return of the booklet. Average macro and micronutrient intake will be assessed over four reported days using FoodWorks Professional Edition 7 (Xryis Software, Brisbane, QLD, Australia, 2012).
In the present study, a 4-day food diary was chosen because of the high respondent burden and time-consuming characteristics of longer food diaries such as a 7-day food diary. Because of the high within-person variation in nutrient intakes, it is recommended to record dietary intakes over a longer period of time to have a highly accurate estimate of intake. However, if a 4-day food diary covers different days randomly, it can provide accurate estimates of dietary intake [86].
Information regarding sun exposure and skin colour, nutritional supplements and any special dietary regimens followed will be collected by questionnaires specifically designed for the purpose of the current study. The sun exposure questionnaire includes a question on caregiver's beliefs and attitudes toward sun exposure as well as questions on country and city of residency in pregnancy, season of birth and child's sensitivity to temperature and light extremes.
Compliance to medication/adherence to study protocol
Caregivers will receive weekly emails containing a link to the online compliance and gastrointestinal issues diary and a tip/fact about autism and nutritional matters. Caregivers will be contacted by telephone at 1, 3, 6 and 9 months for morale purposes and to encourage compliance. Caregivers also will receive quarterly trial newsletters. The newsletters will include an update on the study, generic topics about ASD, a caregiver's experience in relation to that topic, and entertainments/competitions for the study children.
New intervention materials will be sent out to participants every 4 months and caregivers will be asked to place the bottles from the previous months aside, with any unused capsules in them, to be returned at their next visit (6 or 12 months) at which stage unused capsules will be counted and recorded. Compliance to treatment will be analysed by counting each participant's remaining supplements once they have completed the intervention.
All participants will be recalled at 6 months for a blood test to check for hypervitaminosis D (serum 25(OH)D >225 nmol/L) and hypercalcaemia (serum Ca >2.7 mmol/L). Results will be checked by a third party who is unblinded (but has no involvement in analysis of results). The third party will de-identify blood test results and send them to the trial paediatrician and senior investigator for review and recommendation if the child's serum 25(OH)D or calcium level is above the safe upper limit. If 25(OH)D concentrations are at or approaching 225 nmol/L, or if hypercalcaemia is present, dose administration will be adjusted.
Compliance and gastrointestinal symptom diaries will be monitored on a weekly basis and all side effects will be recorded in the adverse events log to keep a track of the child's general health and behavioural reactions. In the case of any adverse events, the child's health will be monitored more closely for three to four consecutive weeks, and if the adverse event persists, the reports will be referred to the trial paediatrician and senior investigator for further investigation.
Dissemination of results
Following the receipt and analysis of the food diary and the completion of the biochemical assays at screening, each participant will receive a feedback form. Anthropometric measurements and blood results (iron studies, vitamin B12, folate and full blood count) will also be included. Once recruitment into the trial is completed, participants who have not proceeded into the trial will receive notification of their vitamin D and RBC fatty acid levels.
On completion of the trial, participants will be informed of their baseline and end vitamin D and RBC fatty acid status, and whether they were taking the active or placebo dose. They will also receive a summary of psychological assessment outcomes of SRS-2 and SPM.
Participants and other stakeholders (such as health professionals, district health boards, primary health organisations, ASD support groups) will be given access to the study's findings. Results will be presented at scientific conferences both nationally and internationally, prepared for publication in peer-reviewed journals, and circulated to the media.
Data handling and statistical analysis
Name and address details will be maintained in Microsoft Excel. Check boxes will record the progress of a participant through the study. All other data will be entered into a single Microsoft Excel spreadsheet with participants identified only by their unique Subject Number. Scorings of the questionnaires will be double-checked by the psychologist and the researcher. All entries will be double-checked by another member of the research team. All documents will be stored safely under confidential conditions and archived for 5 years.
Statistical analysis will be performed using IBM SPSS version 21.0 (IBM Corp; released 2012. IBM SPSS Statistics for Windows Version 21.0. Armonk, NY, USA). Before commencement of statistical analysis the data will be cleaned and checked for coding errors. The data will be checked for plausibility by randomly checking the accuracy and completeness and verifying against source data. The variables will be tested for normality using the Kolmogorov-Smirnov test, the Shapiro-Wilk test and normality plots. Non-normally distributed data will be transformed into approximate normal distributions by logarithmic transformations. The data will be reported appropriately as mean (standard deviation) for normally distributed data; transformed data will be back-transformed from summary statistics into geometric mean (95 % CI), non-normally distributed data will be described as median (25th, 75th percentiles) and categorical data as frequencies.
Baseline characteristics of participants will be compared among groups using analysis of variance (ANOVA) for parametric data and the Kruskal-Wallis test for non-parametric data. The primary analysis, comparing the effects of treatment on symptoms of autism over 12 months, will be conducted using a generalised linear mixed-models procedure. Treatments and time will be included as fixed effects and the interactions between interventions and time will be tested. If significant main effects or interaction effects are observed, post-hoc analysis with Bonferroni adjustments will be performed. Potential confounding factors and effect modifiers (e.g. baseline 25(OH)D and RBC fatty acids, symptoms of autism at baseline, age and gender) will be investigated within the model. Logistic regression will be used to test the multiplicative interaction. Rothman's synergy index, which would be equal to unity under additivity, and less than unity when suggesting antagonism, will be utilised to examine the postulated interaction effect of vitamin D and omega-3 LCPUFAs on core symptoms of ASD.
The secondary analysis, comparing the effects of treatment on gastrointestinal symptoms over 12 months, will be conducted using the same procedure. Potential confounding factors and effect modifiers (e.g. baseline diagnosis of any gastrointestinal symptoms and medication/supplement use) will be investigated within the model.
Differences between participants who complete and withdraw from the trial will be analysed using an independent t test or the Mann–Whitney test for continuous variables (e.g. age) and chi-square for categorical variables (e.g. gender).
Associations between severity of ASD symptoms and nutritional status (specifically, vitamin D, RBC fatty acids, iron and vitamin B12) at baseline will be assessed using regression analysis. The information will be used to assess treatment response according to tertiles of nutritional status showing significant relationship with severity of ASD.
Both intention-to-treat and per-protocol analyses will be utilised, though the primary method of analysis will be intention-to-treat. Statistical significance will be based on two-tailed tests with P <0.05 considered significant.
ASD is a life-long, disabling condition that is associated with deficits in social-communicative functioning, stereotypic behaviour and many behavioural and medical conditions including gastrointestinal symptoms [1, 2, 5–7]. The main purpose of this study is to measure the effect of vitamin D, n-3 LCPUFAs or a combination of both on the symptoms of ASD in affected children. There is a widespread interest in the mechanistic role of vitamin D and n-3 LCPUFAs in brain development and function, with some supportive clinical and epidemiologic studies. However, the effect of supplementing these nutrients on ASD pathogenesis and progression is not known. We anticipate that this trial will provide important insights into this causality of reported associations. As far as we are aware, no other randomised, double-blind, placebo-controlled trial has investigated the effects of vitamin D on symptoms of ASD, and the few trials that have been conducted with n-3 LCPUFAs [38–41] have been limited by small samples sizes and short trial duration and have shown conflicting results.
The strength of this project lies in its design: part 1 has been designed to provide insight into the nutritional status of children with ASD in New Zealand. Part 2 has been designed using a 'Criterion Standard' approach (randomised, double-blind, placebo-controlled trial) to investigate the effect of supplementation. The design minimises the effect of potential confounding factors by correcting some nutritional deficiencies prior to the trial entry and taking into account the effect of confounders and covariates on ASD symptoms over time. Our sample size and trial duration will also ensure an adequate power to detect clinically and statistically significant results.
If this trial is able to identify nutritional interventions that can make even a small difference to the lives of children with ASD by reducing their symptoms, the benefits will be considerable in terms of social and emotional well-being and educational achievements. Furthermore, it will also reduce the emotional, physical and financial strains among families or caregivers of autistic children and the wider societal networks. The potential benefits of the current study go beyond New Zealand and will affect all regions where ASD exists, and both vitamin D and omega-3 status below the optimal level is highly prevalent – the general trend of the most regions worldwide.
At the time of manuscript submission, the trial was recruiting participants.
25(OH)D, 25-hydroxyvitamin D; ABC, Aberrant Behaviour checklist; ASD, autism spectrum disorder; BPFAS, Behavioural Paediatrics Feeding Assessment Scale; CI, confidence interval; DHA, docosahexaenoic acid; DICE, Dietary Index of Children's Eating; EPA, eicosapentaenoic acid; HNRU, Human Nutrition Research Unit; IU, international unit; n-3 LCPUFAs, omega-3 long chain polyunsaturated fatty acids; RBC, red blood cell; SPM, Sensory Processing Measures; SRS, Social Responsiveness Scale; WDHB, Waitemata District Health Board
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This study was partially funded by Massey University Strategic Innovation Fund. Additional support was provided by Douglas Nutrition Pty. Ltd., who supplied the active supplement and the placebo. We are extremely grateful to all the families who took and will take part in this study, the WDHB, Autism New Zealand and Children Autism Foundation for their help in recruiting them, and the whole VIDOMA team, which includes volunteers, phlebotomists, psychologists, a paediatrician, research scientists and a trial manager.
PRvH conceived and designed the study, acquired funding and ethics approval, and has supervised the study. HM drafted and wrote the manuscript, designed sun exposure and gastrointestinal symptom questionnaires and study newsletters, has coordinated recruitment, participant management, and data collection. OM has coordinated recruitment, participant management, and data collection. CC has supervised the study. KLB has supervised the study. MK has supervised the study. WS advised on the omega-3 fatty acid section and statistical analysis. CACJr advised on the vitamin D section. BM advised on the RBC fatty acids laboratory protocol. BT advised on the autism section. All authors read and approved the final manuscript.
Funding and ethics
Partial funding for the study was provided by Massey University Strategic Innovation Fund, Massey University, New Zealand to cover the pilot study in 2015. Additional support has been provided by Douglas Nutrition, Pty. Ltd., NZ who are supplying the active supplement and identical-appearing placebo, but who have no input into study design, implementation, data management, statistical analysis or reporting of results.
Ethical approval was granted by Health and Disability Ethics Committees, NZ, Reference No. 14/NTA/113. Caregivers will sign informed consent forms (main and ancillary studies) for participation in this study that is collected by the research coordinator (OM) or the investigator (HM) (see Additional files 1, 2 and 3).
The trial has been registered with the Australian New Zealand Clinical Trial Registry, ACTRN12615000144516.
The trial will be reported according to the Standard Protocol Items: Recommendations for Intervention Trials (SPIRIT) (see Additional file 4).
Institute of Food Science and Technology – School of Food and Nutrition, Massey University, Auckland, New Zealand
Hajar Mazahery, Cathryn Conlon, Kathryn L. Beck, Marlena C. Kruger, Owen Mugridge & Pamela R. von Hurst
Commonwealth Scientific Industrial Research Organisation (CSIRO) Food, Nutrition and Bioproducts, Adelaide, SA, Australia
Welma Stonehouse
Department of Emergency Medicine, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA
Carlos A. Camargo Jr.
School of Medicine, University of Wollongong, Illawarra, NSW, 2522, Australia
Barbara J. Meyer
North Shore Hospital, Waitemata District Health Board, Auckland, New Zealand
Bobby Tsang
Hajar Mazahery
Cathryn Conlon
Kathryn L. Beck
Marlena C. Kruger
Owen Mugridge
Pamela R. von Hurst
Correspondence to Pamela R. von Hurst.
Information Sheet. (PDF 319 kb)
Consent Form-Main. (DOC 242 kb)
Consent Form-Storage of Blood Sample. (DOC 244 kb)
SPIRIT Checklist. (DOCX 50 kb)
Mazahery, H., Conlon, C., Beck, K.L. et al. Vitamin D and omega-3 fatty acid supplements in children with autism spectrum disorder: a study protocol for a factorial randomised, double-blind, placebo-controlled trial. Trials 17, 295 (2016). https://doi.org/10.1186/s13063-016-1428-8


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