text:
\section{Introduction: reconstructions from unorganized clouds}
\label{sec:intro}

Potential molecules for new colloidal products are tested by simulations that produce unorganized finite clouds of points (one point per molecule in Fig.~\ref{fig:cylindrical+branched}).
Molecules tend to form clusters (called {\em micelles}) whose shapes (degrees of branching, edge-lengths) affect physical properties of colloidal products, e.g. their viscosity.

These 3D micelles can have complicated branched shapes as in Fig.~\ref{fig:Christmas_outputs} and are visually analyzed by human experts who struggle to make reliable measurements quickly.
To substantially speed-up the discovery of new molecules, we propose a new Approximate Skeleton $\mathrm{ASk}(C)$ to solve the following problem.
\smallskip

\noindent
\textbf{The tree reconstruction problem}.
Given a point cloud $C\subset\mathbb R^m$ and an error $\varepsilon$, design a fast algorithm to build a straight-line tree $T\subset\mathbb R^m$ (see Definition~\ref{dfn:graph}) that has a minimum number of vertices and whose $\varepsilon$-offset (neighborhood) covers $C$.
\medskip

The first (combinatorial) guarantee is for the number of vertices in $\mathrm{ASk}(C)$, which is close to the minimum number in an optimal tree for a given approximation error by factor 2, see Theorem~\ref{thm:size}.
The second (geometric) guarantee about a near linear time for building $\mathrm{ASk}(C)$ is the number of points $n$ in $C$, see Corollary~\ref{cor:time}.
\smallskip

\begin{figure}[h]
\includegraphics[width=0.48\linewidth]{cylindrical/cylindrical_cloud_blue.png}
\includegraphics[width=0.48\linewidth]{cylindrical/cylindrical_ASk_blue.png}

\includegraphics[width=0.48\linewidth]{branched/branched_cloud_blue.png}
\includegraphics[width=0.48\linewidth]{branched/branched_ASk_blue.png}
\caption{
\textbf{Left}: point clouds $C$ from real micelles.
\textbf{Right}: Approximate Skeletons $\mathrm{ASk}(C)$.}
\label{fig:cylindrical+branched}
\end{figure}

To automatically characterize branching shapes of micelles (clusters of molecules in colloids), an Approximate Skeleton $\mathrm{ASk}(C)$ allows us to compute
\smallskip

\noindent
$\bullet$
the topological type of any unorganized cloud $C$, e.g. count all {\em non-trivial} vertices of $\mathrm{ASk}(C)\subset\mathbb R^m$ whose degree is 1 (endpoints) or more than 2 (branching);
\smallskip

\noindent
$\bullet$
the geometric characteristics of $C$, e.g. edge-lengths of $\mathrm{ASk}(C)$;
\smallskip

\noindent
$\bullet$
the error of approximating a cloud $C$ by its skeleton $\mathrm{ASk}(C)$, see  Table~\ref{tab:micelles}.
\smallskip

Here is the pipeline of the Approximate Skeleton $\mathrm{ASk}(C)$.
\smallskip

\noindent
Stage 1 in section~\ref{sec:core}: for a cloud $C\subset\mathbb R^m$, we build an initial tree $\mathrm{core}(C)$, which has a small number of branching vertices within a Minimum Spanning Tree of $C$.
\smallskip

\noindent
Stage 2 in section~\ref{sec:ask}: replace polygonal paths of $\mathrm{core}(C)$ by approximate paths with much fewer vertices to get $\mathrm{ASk}(C)$ in a near linear time within a given error.

\begin{figure}[h]
\includegraphics[width=1.0\linewidth]{images/ASk_pipeline.png}
\caption{Pipeline to compute an Approximate Skeleton $\mathrm{ASk}(C)$ :
$\mathrm{MST}(C)$ is classical, the new subtree $\mathrm{core}(C)\subset\mathrm{MST}(C)$ is introduced in Definition~6 in section~3, final $\mathrm{ASk}(C)$ is built in section~4}.
\label{fig:pipeline}
\end{figure}

\noindent
The key novelty and contributions to the data skeletonization are the following.
\smallskip

\noindent
$\bullet$
Theorem~\ref{thm:size} guarantees a small number of vertices in the Approximate Skeleton $\mathrm{ASk}(C)$ close to the minimum by factor 2 in an optimal tree within a given error.
\smallskip

\noindent
$\bullet$
Corollary~\ref{cor:time} guarantees a near linear time  to compute $\mathrm{ASk}(C)$ within an error.

\section{Basic definitions and a review of the related past work}
\label{sec:review}

\begin{dfn}[a straight-line graph, $\varepsilon$-approximation]
\label{dfn:graph}
A {\em straight-line} graph $G\subset\mathbb R^m$ (briefly, a {\em graph}) consists of vertices at points $q_1,\dots,q_k\in\mathbb R^m$ and undirected straight-line edges connecting pairs $q_i,q_j$, $i\neq j$, in such a way that any edges meet only at their common vertex.
Let $d$ be the Euclidean distance.
For $\varepsilon>0$, a cloud $C\subset\mathbb R^m$ is $\varepsilon$-{\em approximated} by a graph $G$ if $C$ is within the $\varepsilon$-{\em offset} that is the union of $\varepsilon$-balls at all points of $G$, i.e.
$G^{\varepsilon}=\{p\in\mathbb R^m \;|\; d(p,q)\leq\varepsilon \mbox{ for some } q\in G\}$.
\end{dfn}

\noindent
\textbf{Past algorithms without guarantees}.
Singh et al. \cite{SCP00} approximated a cloud $C\subset\mathbb R^m$ by a subgraph of a Delaunay triangulation, which requires $O(n^{\lceil{m/2}\rceil})$ time for $n$ points of $C$
and the three thresholds: a minimum number $K$ of edges in a cycle and $\delta_{min},\delta_{max}$ for inserting/merging 2nd order Voronoi regions.
Similar parameters are need for {\em principal curves} \cite{KK02}, which were later extended to iteratively computed {\em elastic maps} \cite{GZ09}.
Since it is often hard to estimate a rate of convergence for iterative algorithms, we discuss below non-iterative methods with theoretical guarantees.
\medskip

\noindent
\textbf{The metric graph reconstruction} (MGR) takes as an input a large metric graph $Y$, which is an abstract graph with weighted edges  and outputs a smaller abstract metric graph $\hat X$.
The distance between any points of a metric graph is defined as the length of a shortest path these points.
If $Y$ is a good $\varepsilon$-approximation to an unknown graph $X$, then M.~Aanjaneya et al. \cite[Theorem~5]{ACCGGM12} proved the existence of a homeomorphism $X\to\hat X$ that distorts the metrics on $X$ and $\hat X$ with a multiplicative factor $1+c\varepsilon$ for $c>\frac{30}{b}$, where $b>14.5\varepsilon$ is the length of a shortest edge of $X$.
\smallskip

\noindent
The authors of the Reeb graph skeletonization \cite[page~3]{GSBW11} have checked that for the MGR algorithm from \cite{ACCGGM12} ``it is often hard to find suitable parameters in practice, and such local decisions tend to be less reliable when the input data are not as nice (such as a `fat' junction region)", see this junction in the 2nd picture of Fig.~\ref{fig:cylindrical+branched}.

\begin{dfn}[a Reeb graph]
\label{dfn:Reeb}
Given a topological space $K\subset\mathbb R^m$ (or ) with a function $f:K\to\mathbb R$, the {\em Reeb} graph $R_f(K)$ is obtained from $K$ by collapsing each connected components of every level set of $f$ to a single point, so the {\em Reeb} graph $R_f(K)$ is the quotient of $K$ by the equivalence relation $a\sim b$ if and only if $f(a)=t=f(b)$ and the points $a,b\in K$ are in the same connected component of $f^{-1}(t)\subset K$.
\end{dfn}

\noindent
\textbf{Skeletonization via Reeb-type graphs}.
The {\em Vietoris-Rips} complex $\mathrm{VR}(C;\alpha)$ on a cloud $C$ consists of all simplices spanned by points whose pairwise distances are at most $\alpha$.
Starting from a noisy sample $C$ of an unknown graph $G$ with a scale parameter, X.~Ge et al. \cite[Theorem~3.1]{GSBW11} proved that the Reeb graph of $\mathrm{VR}(C;\alpha)$ has a correct homotopy type if there is a triangulated space $K$ with a continuous deformation $h:K\to G$ that $\varepsilon$-approximates the metrics of $K,G$.
The {\em homotopy type} of a graph is the equivalence class of graphs under deformations when any edge (with distinct endpoints) can be collapsed to a point.
\medskip

\noindent
\textbf{The graph reconstruction by discrete Morse theory} (DMT).
T.~Dey et al. \cite{dey2018graph} substantially improved the discrete Morse-based framework \cite{delgado2015skeletonization} and proved new homotopy guarantees when an input is a density function $\rho:K\to\mathbb R$, which `concentrates' around a hidden geometric graph $G$.
The key advantage of this approach is the unbounded noise model that allows outliers far away from the underlying graph $G$, which has found practical applications to map reconstructions \cite{wang2015efficient, dey2017improved}.
\medskip

Since the molecules of a micelle form an unorganized cloud of points (with large bounded noise) around hidden tree structures, the Tree Reconstruction problem in section~\ref{sec:intro} essentially differs from the above approaches.
An initial unorganized cloud of points is not an abstract metric graph (as in the metric graph reconstruction problem) and not a simplicial complex with scalar values at vertices (as in the discrete Morse theory approach), so extra pre-processing was needed in section~\ref{sec:experiments}.
\medskip

\noindent
\textbf{The $\alpha$-Reeb graph $G$} by F.~Chazal et al. \cite{CHS15} solves the metric graph reconstruction problem, where the input is not an unorganized cloud, but a large metric graph $X$ that should be approximated by a smaller graph $\hat X$.
For a base point $p\in X$, the image of the distance function $d(p,*):X\to\mathbb R$ is covered by intervals $I_j$ having a length $\alpha$ and 50\% overlap.
Every connected component of $f^{-1}(I_j)\subset X$ defines a node in the $\alpha$-Reeb graph $G$.
Two nodes are linked if the corresponding components overlap.
\smallskip

\noindent
Informally, $\alpha$ controls the size of a subset of $X$ that maps to a single vertex of $G$.
Theorem~4.9 in \cite{CHS15} says that if $X$ is $\varepsilon$-close to an unknown graph with edges of minimum length $8\varepsilon$, the output $G$ is $34(\beta(G)+1)\varepsilon$-close to $X$ in the Gromov-Hausdorff distance between spaces, not within one space, where $\beta(G)$ is the first Betti number of $G$.
The algorithm has the fast time $O(n\log n)$ for $n$ points in $X$.
Similarly to Reeb graphs, $\alpha$-Reeb graphs are abstract without an intrinsic embedding into the space of the cloud $C$ and can have self-intersections even for $X\subset\mathbb R^2$.
\medskip

\noindent
\textbf{The Mapper} \cite{SMG07} extends any clustering algorithm and outputs a network of interlinked clusters and needs a user-defined function $f:C\to\mathbb R$, which helps to link different clusters of a cloud $C$.
Another parameter is a covering of the image of $f$ by a given number $k$ of intervals $I_j$ (often with 50\% overlap).
Each of $k$ subclouds $f^{-1}(I_j)\subset C$ is clustered.
Every cluster defines a node in the Mapper graph.
Two nodes are linked if the corresponding clusters overlap.
M.~Carri\'ere et al. \cite{CS17} have proved first theoretical guarantees for the Mapper output.
\medskip

More recent persistence-based algorithms for graph reconstruction \cite{kurlin2015homologically, kurlin2015one, kalisnik2019higher} and image segmentation \cite{forsythe2016resolution, forsythe2017convex, kurlin2017superpixels, kurlin2020persistence} essentially find most persistent cycles hidden in a cloud, hence go beyond the tree reconstruction problem in section~\ref{sec:intro}.
\medskip

\noindent
\textbf{Straightening polygonal curves} is a key ingredient in many skeletonization algorithms.
Douglas-Peucker's heuristic \cite{DP73} approximates a long zigzag line by a simpler line with fewer vertices, see section~\ref{sec:ask}.
The elegant algorithm by P.~Agarwal et al. \cite{AHMW05} guarantees a near linear time and a small number of vertices in a final polygonal approximation when used with the Frechet distance between curves in $\mathbb R^2$.
For the Hausdorff distance and  higher dimensions, there is no near linear time straightening with guarantees on the size of a skeleton to our best knowledge.

\begin{dfn}[$\mathrm{MST}(C)$]
\label{dfn:MST}
For a cloud $C\subset\mathbb R^m$, a {\em Minimum Spanning Tree} $\mathrm{MST}(C)$ is a connected graph that has (1) the vertex set $C$, (2) no cycles, and (3) a minimum total length, where lengths of edges are measured in the Euclidean distance.
\end{dfn}

If all distances between points of $C$ are distinct, then $\mathrm{MST}(C)$ is unique.
We write a Minimum Spanning Tree, similarly an Approximate Skeleton, to cover all cases.

\begin{thm}
\label{thm:MST_time}
\cite[Theorem~5.1]{EMST}
For any cloud $C\subset\mathbb R^m$ of $n$ points, a Minimum Spanning Tree $\mathrm{MST}(C)$ can be computed in time $O(\max\{c^6,c_p^2c^2_l) \}c^{10}n\log n\,\alpha(n))$, where
 $\alpha(n)$ is the inverse Ackermann function;
 $c,c_p,c_l$ are defined in Appendix~A.
\end{thm}

\section{A new tree $\mathrm{core}(C)$ defined for any point cloud $C\subset\mathbb R^m$}
\label{sec:core}

This section introduces an important subtree $\mathrm{core}(C)\subset\mathrm{MST}(C)$, which has many fewer non-trivial vertices than a usually `hairy' $\mathrm{MST}(C)$ from Definition~\ref{dfn:MST}.
\smallskip

A tree $\mathrm{core}(C)$ might still have too many zigzags and will be replaced by a better tree $\mathrm{ASk}(C)$ with fewer vertices in section~\ref{sec:ask}.
A vertex of a degree $k\neq 2$ is called (topologically) {\em non-trivial}, because any vertex of degree~2 can be potentially removed by straightening algorithms in section~\ref{sec:ask}.
Since $\mathrm{MST}(C)$ contains many non-trivial vertices, the next hard step is to identify those few vertices of $\mathrm{MST}(C)$ that represent `true' vertices of a tree $T$, which we try to reconstruct from $C$.
\medskip

\begin{figure}
\includegraphics[width=0.48\linewidth]{images/MST_core_paths.png}
\hspace*{2mm}
\includegraphics[width=0.48\linewidth]{images/MST_core_clouds.png}
\caption{\textbf{Left}. One vertex (large red dot at the bottom) of $\mathrm{MST}(C)$ has a high depth by Definition~\ref{dfn:depth} and is connected by longest paths to 3 vertices of degree~1.
The other vertices have at most 2 disjoint long paths within $\mathrm{MST}(C)$.
\textbf{Right}. The red monotone paths of $\mathrm{core}(C)$ and subclouds from Algorithm~\ref{alg:subclouds} are shown disjointly.}
\label{fig:depth}
\end{figure}

Definition~\ref{dfn:depth} introduces the depth characterizing how deep a vertex sits within $\mathrm{MST}(C)$.
At a deep vertex of a degree $k\geq 3$ at least 3 sufficiently long paths (without common edges) should meet, see the 3 red long paths in Fig.~\ref{fig:depth}.
The previous procedural approach by M.~Aanjaneya et al. \cite[Fig.~1b]{ACCGGM12} to detect branching points in a shape of $C$ used more parameters than a single branching factor $\beta$ below.

\begin{dfn}[deep vertices]
\label{dfn:depth}
For a cloud $C\subset\mathbb R^m$ and a vertex $v\in\mathrm{MST}(C)$ of a degree $k\geq 3$, let $B_1,\dots,B_k\subset\mathrm{MST}(C)$ be the branches (subtrees) joined at the vertex $v$.
Let $l_i$ be the length of a longest path within the branch $B_i$ from $v$ to another vertex, $i=1,\dots,k$.
Assuming that $l_1\geq l_2\geq\dots$, set $\mathrm{depth}(v)=\min\{l_1,l_2,l_3\}$.
Let $l(C)$ be the average edge-length of $\mathrm{MST}(C)$.
For a branching factor $\beta>0$, the vertices of $\mathrm{MST}(C)$ whose depths are larger than $\beta l(C)$ are called {\em deep}.
\end{dfn}

Taking the minimum $\mathrm{depth}(v)=\min\{l_1,l_2,l_3\}$ above guarantees that vertices in any short branches of $\mathrm{MST}(C)$ are not deep, hence deep vertices can not form small cliques.
The experiments on real micelles in section~\ref{sec:experiments} justify that Definition~\ref{dfn:depth} separates deep vertices from other shallow vertices for a long range of the factor $\beta$.
\smallskip

\begin{figure}[h]
\includegraphics[width=0.48\linewidth]{cylindrical/cylindrical_MST_black.png}
\includegraphics[width=0.48\linewidth]{cylindrical/cylindrical_core_red.png}

\includegraphics[width=0.48\linewidth]{branched/branched_MST_black.png}
\includegraphics[width=0.48\linewidth]{branched/branched_core_red.png}
\caption{\textbf{Left}: black $\mathrm{MST}(C)$ for the two clouds $C$ in Fig.~\ref{fig:cylindrical+branched}. \textbf{Right}: red $\mathrm{core}(C)$ in Definition~\ref{dfn:core}.}
\label{fig:MST+core}
\end{figure}

Definition~\ref{dfn:core} introduces a subtree $\mathrm{core}(C)$, which non-essentially depends on the branching factor $\beta$ and better approximates a cloud $C$ than $\mathrm{MST}(C)$, see Fig.~\ref{fig:depth}.

\begin{dfn}[$\mathrm{core}(C)$]
\label{dfn:core}
In Definition~\ref{dfn:depth}, if we remove all deep vertices $v_1,\dots,v_m$, $\mathrm{MST}(C)$ splits into several subtrees.
If the closure of such a subtree $S$ has two deep vertices $v_i,v_j$, they are joined by a unique path $P_{ij}\subset S$.
If $S$ has one deep vertex $v_i$, take a longest path $P_i\subset S$ from $v_i$ to another vertex $v'_i\in S$.
We ignore $P_i$ if its length is less than $\beta l(C)$, where $\beta$ is the branching factor from Definition~\ref{dfn:depth}.
All the vertices $v_i,v'_i$ and the paths $P_{ij},P_i$ between them form the subtree $\mathrm{core}(C)\subset\mathrm{MST}(C)$.
\end{dfn}

If the closure of a subtree $S$ above has $k\geq 3$ deep vertices $v_1,v_2,v_3$, then $S$ contains a vertex $v$ with at least 3 paths to $v_1,v_2,v_3$.
Then $\mathrm{depth}(v)>\mathrm{depth}(v_i)$, $i=1,2,3$, so $v$ is also deep and $S$ should be split by removing $v$.
Hence $k\leq 2$.
\smallskip

In Fig.~\ref{fig:depth} the two black edges at the red deep vertex $v$ of degree~5 are too short, hence ignored in Definition~\ref{dfn:core}.
The tree $\mathrm{core}(C)$ consists of only 3 red long paths meeting at $v$.
Here are the steps of Stage 1 for the Approximate Skeleton $\mathrm{ASk}(C)$.
\smallskip

\noindent
\textbf{Step 1a}.
If needed, split a cloud $C$ in clusters to approximate them below.
\smallskip

\noindent
\textbf{Step 1b}.
If $C$ is one cluster, find $\mathrm{MST}(C)$ by the fast algorithm from Theorem~\ref{thm:MST_time}.
\smallskip

\noindent
\textbf{Step 1c}.
Find the depths of vertices in $\mathrm{MST}(C)$ by Algorithm~\ref{alg:depths} in Appendix~A.
\smallskip

\noindent
\textbf{Step 1d}.
Identify all deep vertices of $\mathrm{MST}(C)$ by their $\mathrm{depth}(v)=\min\{l_1,l_2,l_3\}$.
\smallskip

\noindent
\textbf{Step 1e}.
The subtree $\mathrm{core}(C)\subset\mathrm{MST}(C)$ is formed by all the paths  $P_i$ and $P_{ij}$ from Definition~\ref{dfn:core} that have lengths more than $\beta l(C)$, where $\beta$ is a given branching factor.

\section{$\mathrm{ASk}(C)$: Approximate Skeleton of a cloud $C\subset\mathbb R^m$}
\label{sec:ask}

The tree $\mathrm{core}(C)$ from Definition~\ref{dfn:core} has only few non-trivial vertices, but contains noisy zigzags with too many {\em trivial} vertices of degree~2.
This section discusses how to straighten these zigzags and decrease the total number of vertices.
\smallskip

We have tried Douglas-Peucker's heuristic \cite{DP73}, which was rather unstable and produced large zigzags on curved micelles in Fig.~\ref{fig:cylindrical+branched}.
The worst complexity is $O(n^2)$ in the number $n$ of points for $d>2$.
A final approximation can have a size $\Omega(n)$ even in $\mathbb R^2$.
Another problem with \cite{DP73} are potential self-intersections even in $\mathbb R^2$, which are caused by large zigzags that approximate non-monotone curves \cite{WSM04}.
\smallskip

The problem of straightening polygonal paths in a tree $\mathrm{core}(C)$ is harder than the curve simplification, because the input is a cloud of unorganized points.
So a final approximation should take into account the points of a cloud $C$ outside $\mathrm{core}(C)$.
\medskip

\begin{dfn}
\label{dfn:monotone}
Let $L\subset\mathbb R^m$ be a straight line.
An ordered cloud $C=\{p_1,\dots,p_n\}\subset\mathbb R^m$ is called \emph{monotone} with respect to $L$ if the order of points is preserved by the orthogonal projection of $C$ to $L$.
\end{dfn}

Since there are many paths of $\mathrm{core}(C)$ to straighten, we split the cloud $C$ into monotone subclouds as formalized in Algorithm~\ref{alg:subclouds} in Appendix~A.
Since monotone subpaths can be quickly found only in $\mathbb R^2$ \cite{shin2004optimal}, Theorem~\ref{thm:size} below will assume that each subpath $P$ between non-trivial vertices of $\mathrm{core}(C)$ is monotone by Definition~\ref{dfn:monotone} with respect to the straight line connecting the endpoints of $P$.
\smallskip

All results in this section are proved in Appendix A.
Here are the Stage~2 steps.
\medskip

\noindent
\textbf{Step 2a}.
Split every polygonal path between non-trivial vertices (of degrees $k\neq 2$) in the subtree $\mathrm{core}(C)\subset\mathrm{MST}(C)$ into monotone subpaths by Algorithm~\ref{alg:subclouds}.
\smallskip

\noindent
\textbf{Step 2b}.
Each monotone subpath of $\mathrm{core}(C)$ with endpoints (say) $p_1,p_n$ has the subcloud $C'$ approximated by a polygonal path via points of $C'$ by Steps 2c--2f.
\smallskip

\noindent
\textbf{Step 2c}.
For each subcloud $C'=\langle p_1,\dots,p_n\rangle$ of points ordered by their orthogonal projections to $[p_1,p_n]$, start from $\mathrm{ind}(1)=1$ and find the next index $\mathrm{ind}(i)$ for $i=2,\dots,m$ by repeating Steps 2d--2e, which is  possible by Lemma~\ref{lem:algorithm} in Appendix~A.
\smallskip

\noindent
\textbf{Step 2d} (exponential).
Find the smallest index $j$ such that $d( [p_{\mathrm{ind}(i-1)} p_l], C' )>\varepsilon$ for $l=\mathrm{ind}(i-1)+2^{j+1}$, $j=0,1,2\dots$
For every index $l$, compute the distance $d( [p_{\mathrm{ind}(i-1)} p_l], C' )$ orthogonally to the line segment $[p_1 p_n]$ as in Definition~\ref{dfn:distances}.
\smallskip

\noindent
\textbf{Step 2e} (binary).
Search for the maximum $\mathrm{ind}(i)$ between $\mathrm{ind}(i-1)+2^j$ and $\mathrm{ind}(i-1)+2^{j+1}$ such that $d( [p_{\mathrm{ind}(i-1)} p_{\mathrm{ind}(i)}], C' )\leq\varepsilon$ by dividing the range in 2 halves.
\smallskip

\noindent
\textbf{Step 2f}.
The found indices $\mathrm{ind}(i)$ specify a polygonal path $\varepsilon$-approximating each monotone subcloud from Step~2b.
Combine all these paths into a full skeleton.
\smallskip

\noindent
\textbf{Step 2g}.
Any edges of a length more than $\beta l(C)$ from Definition~\ref{dfn:depth} are temporarily removed from the skeleton.
Each remaining connected component with only short edges is collapsed to its center of mass.
The resulting vertices are connected according to the temporarily removed edges to get the Approximate Skeleton $\mathrm{ASk}(C)$.
\medskip

For a cloud $C\subset\mathbb R^m$, mark the endpoints of all monotone subpaths in $\mathrm{core}(C)$ obtained by Algorithm~\ref{alg:subclouds}.
Consider all skeletons $S\subset\mathbb R^m$ that have fixed vertices at the marked points of $C$ such that any polygonal path between fixed vertices (say $u,v$) is monotone under the orthogonal projection to the line segment $[u,v]$.
\smallskip

\noindent
\textbf{The approximation problem} for an error $\varepsilon>0$ is to minimize the total number of vertices in a straight-line graph $S\subset\mathbb R^m$ whose each monotone path should $\varepsilon$-approximate the corresponding subcloud of $C$ by the distance in Definition~\ref{dfn:distances}.

\begin{thm}
\label{thm:size}
Let $k$ be the minimum number of vertices over all graphs $\varepsilon$-approximating a given cloud $C\subset\mathbb R^m$.
Then $\mathrm{ASk}(C)$ lies in $C^{2\varepsilon}$ and has at most $k$ vertices.
\end{thm}

Theorem~\ref{thm:size} estimates the number of vertices of $\mathrm{ASk}(C)$ when the geometric error is $2\varepsilon$.
In practice, the tree $\mathrm{core}(C)$ has an initial approximation error for a given cloud $C$, because many points of $C$ may not be vertices of $\mathrm{core}(C)\subset\mathrm{MST}(C)$.
\smallskip

We measure the initial error $d(\mathrm{core}(C),C)$ by Definition~\ref{dfn:distances} and take the maximum of $d([v_iv_j],C)$ over monotone paths of $\mathrm{core}(C)$ computed in Algorithm~\ref{alg:subclouds}.
Stage~2 approximates $C$ by a graph simpler than $\mathrm{core}(C)$, but keeps the approximation error small.
The error $\varepsilon$ in Corollary~\ref{cor:time} is $\gamma\times d(\mathrm{core}(C),C)$, where $\gamma$ is an error factor that takes values in the interval $[1.1,1.5]$ for the experiments in section~\ref{sec:experiments}.

\begin{cor}
\label{cor:time}
For any $n$ points $C\subset\mathbb R^m$ and any error factor $\gamma>1$, an Approximate Skeleton $\mathrm{ASk}(C)\subset\mathbb R^m$ within the $\gamma d(\mathrm{core}(C),C)$-offset of the cloud $C$ (as in Definition~\ref{dfn:graph}) can be computed in time $O(\max\{c^6,c_p^2c^2_l) \}c^{10}n\log n\,\alpha(n))$, where $\alpha(n)$ is the inverse Ackermann function,
 the constants $c,c_p,c_l$ are defined in Appendix~A.
\end{cor}

\section{Comparisons of 5 algorithms on real and synthetic data}
\label{sec:experiments}

This section experimentally compares the Approximate Skeleton $\mathrm{ASk}(C)$ with those four skeletonization algorithms from section~\ref{sec:review} that have theoretical guarantees and accept any cloud $C$ of points: Mapper \cite{SMG07}, Metric Graph Reconstruction MGR \cite{ACCGGM12}, $\alpha$-Reeb graphs \cite{CHS15} and most recent discrete Morse theory (DMT) algorithm \cite{dey2018graph}.
\smallskip

The Mapper \cite{SMG07} is very flexible in the sense that its parameters might be manually tuned for given data over numerous clustering algorithms.
Having tried several possibilities, we have settled on the following choices from the original work \cite{SMG07}.
\smallskip

\noindent
1) Convert a cloud $C$ into a connected neighborhood graph $N(C)$ with Euclidean edge-lengths by using a distance threshold.
The filter function is the distance function in $N(C)$ from a root that is the furthest point from a random point in $C$.
\smallskip

\noindent
2) The image of the filter function is covered by 10 intervals with the 50\% overlap so that $C$ splits into 10 subclouds when filter values are in one of the 10 intervals.
\smallskip

\noindent
3) Each subcloud $C'$ is clustered by the single-linkage clustering with the threshold $\tau\times$the average edge-length of $\mathrm{MST}(C')$, where values of the factor $\tau$ are given in Table~\ref{tab:micelles}.
The final Mapper graph has a single node representing each cluster.
\medskip

The authors of the DMT algorithm by T.~Dey et al. \cite{dey2018graph} have kindly made their code available at https://github.com/wangjiayuan007/graph\_recon\_DM.
Starting from an unorganized cloud of points, e.g. centers of molecules of a micelle, we generated scalar values at nodes of a regular grid required for the DMT algorithm.
\smallskip

\noindent
1) We subdivide the axis-aligned bounding box of a cloud $C\subset\mathbb R^3$ into small boxes: minimum 20 rectangular boxes (as close to cubic as possible) along each side.
\smallskip

\noindent
2) The scalar values are found from the Kernel Density Estimate $KDE(p)=\sum\limits_{q\in C}\exp(-d(p,q))$ at every grid node $p$.
The computed values are passed to the DMT with a parameter $\delta$ that regulates how small density values are replaced by 0.
\smallskip

The MGR algorithm has required much more efforts, because the original code was lost as confirmed by the main author of \cite{ACCGGM12}.
Since the algorithm was well-explained, we have implemented MGR ourselves and confirmed the earlier claim that ``it is often hard to find suitable parameters'' \cite[page~3]{GSBW11}.
Trying many values of the key parameter $r$ gave the zero success rate on the homeomorphism type.
\smallskip

Hence we have improved MGR by splitting this parameter into two: the first $r_1=15$ (values used in all experiments) was used for detecting vertex points, the second $r_2$ (three values $1,1.5,2$ in Table~\ref{tab:micelles} experiments) was used for clustering points of different types.
Only after using these different values, we have managed to push the success rates of MGR closer to $50\%$ on the homeomorphism type.
\smallskip

The $\alpha$-Reeb graph has the essential parameter $\alpha$ whose values $20,25,30$ were tried in all experiments.
$\mathrm{ASk}(C)$ has little dependence on the branching factor $\beta$, e.g. all values $[20,50]$ produced almost identical results in Table~\ref{tab:micelles} and Fig.~\ref{fig:synthetic}.
\smallskip

Since three output graphs (Mapper, $\alpha$-Reeb and MGR) are abstract, to compute any geometric error of approximation, we map them to $\mathbb R^3$ by sending every node $v$ of $G$ to the average point (center of mass) of the cluster (for Mapper and MGR) or subgraph (for $\alpha$-Reeb) corresponding to $v$.
Each link between nodes is mapped as a line segment between the corresponding points in $\mathbb R^3$.
\smallskip

Figures~\ref{fig:cylindrical_outputs}--\ref{fig:8star_outputs} show clouds and outputs of 5 algorithms.
Since real micelles have irregular shapes in $\mathbb R^3$, their 2D projections may contain intersections of edges.
\smallskip

Table~\ref{tab:micelles} shows the average results of the three algorithms on the dataset of more than 100 real micelles (clouds of about 300 molecules) whose endpoints and homeomorphism types were manually detected.
A \emph{homeomorphism} is a 1-1 continuous map with a continuous inverse, so a homeomorphism type is a stronger shape descriptor than a homotopy type, which counts only linearly independent cycles.
\smallskip

\smallskip

\begin{figure*}[t]
\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_cloud_blue.png}
\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_Mapper_green.png}
\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_alphaReeb_red.png}

\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_MGR_orange.png}
\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_DMT_olive.png}
\includegraphics[width=0.33\textwidth]{cylindrical/cylindrical_ASk_blue.png}
\caption{\textbf{1st}: a cylindrical micelle with no branching vertices.
\textbf{2nd}: Mapper,
\textbf{3rd}: $\alpha$-Reeb,
\textbf{4th}: MGR,
\textbf{5th}: DMT,
\textbf{6th}: new $\mathrm{ASk}(C)$.
}
\label{fig:cylindrical_outputs}
\end{figure*}

\begin{figure*}[h!]
\includegraphics[width=0.33\textwidth]{branched/branched_cloud_blue.png}
\includegraphics[width=0.33\textwidth]{branched/branched_Mapper_green.png}
\includegraphics[width=0.33\textwidth]{branched/branched_alphaReeb_red.png}

\includegraphics[width=0.33\textwidth]{branched/branched_MGR_orange.png}
\includegraphics[width=0.33\textwidth]{branched/branched_DMT_olive.png}
\includegraphics[width=0.33\textwidth]{branched/branched_ASk_blue.png}
\caption{\textbf{1st}: a branched micelle with exactly one degree 3 vertex,
\textbf{2nd}: Mapper,
\textbf{3rd}: $\alpha$-Reeb,
\textbf{4th}: MGR,
\textbf{5th}: DMT,
\textbf{6th}: new $\mathrm{ASk}(C)$.
}
\label{fig:branched_outputs}
\end{figure*}

\begin{figure*}[h!]
\includegraphics[width=0.33\textwidth]{Christmas/Christmas_cloud_blue.png}
\includegraphics[width=0.33\textwidth]{Christmas/Christmas_mapper_green.png}
\includegraphics[width=0.33\textwidth]{Christmas/Christmas_alphaReeb_red.png}

\includegraphics[width=0.33\textwidth]{Christmas/Christmas_MGR_orange.png}
\includegraphics[width=0.33\textwidth]{Christmas/Christmas_DMT_olive.png}
\includegraphics[width=0.33\textwidth]{Christmas/Christmas_ASk_blue.png}
\caption{\textbf{1st}: `Christmas tree' micelle with several degree 3 vertices).
\textbf{2nd}: Mapper,
\textbf{3rd}: $\alpha$-Reeb,
\textbf{4th}: MGR,
\textbf{5th}: DMT,
\textbf{6th}: new $\mathrm{ASk}(C)$.
All intersections come only from planar projections.
}
\label{fig:Christmas_outputs}
\end{figure*}

\begin{figure*}[t]
\includegraphics[width=0.33\textwidth]{8star/8star_cloud_crop.png}
\includegraphics[width=0.33\textwidth]{8star/8star_Mapper_crop.png}
\includegraphics[width=0.33\textwidth]{8star/8star_alpha-Reeb_crop.png}

\includegraphics[width=0.33\textwidth]{8star/8star_MGR_crop.png}
\includegraphics[width=0.33\textwidth]{8star/8star_DMT_crop.png}
\includegraphics[width=0.33\textwidth]{8star/8star_ASk_crop.png}
\caption{\textbf{1st}: a random point sample around an 8-star in $\mathbb R^3$,
\textbf{2nd}: Mapper,
\textbf{3rd}: $\alpha$-Reeb,
\textbf{4th}: MGR,
\textbf{5th}: DMT,
\textbf{6th}: new $\mathrm{ASk}(C)$.
}
\label{fig:8star_outputs}
\end{figure*}

The \emph{most important error} measure for the tree reconstruction problem in section~\ref{sec:intro} is the success rate for detecting a correct homeomorphism type.
Indeed, an incorrect graph can be perfect on other errors, e.g. $\mathrm{MST}(C)$ is extremely fast, has the zero geometric error (for many distances between a cloud and a reconstructed graph) and even has a correct homotopy type (no cycles) for any underlying tree $T$.
\smallskip

Hence the key results are in the middle column of Table~\ref{tab:micelles} and the top right picture of Fig.~\ref{fig:synthetic}.
We included the success rate on the number of endpoints (degree~1 vertices) as a weaker topological error.
As $\mathrm{MST}(C)$ shows above, only if an algorithm performs well on a topological reconstruction, it makes sense to evaluate the performance on other measures such as geometric distances and time.
\smallskip

\begin{table}[t]
\caption{
Columns 3-4 contain success rates for detecting the correct number of endpoints and a homeomorphism type of a graph over more than 100 real micelles.
Column 5 contains the maximum Euclidean distance from points of a given cloud $C$ to the reconstructed graph $G\subset\mathbb R^3$.}
\label{tab:micelles}
\begin{tabular}{c|c|c|c|c|c}
\toprule
algorithm & parameters & endpoints success & homeomorphism success & time, ms  & max distance from $C$
\\
\midrule
Mapper & $\tau=1.25$ & 54.21\% & 54.21\% & \textbf{18}  & 4.59 \\
Mapper & $\tau=1.75$ & 66.36\% & 66.36\%  & \textbf{18} & 4.62 \\
Mapper & $\tau=2.25$ & 68.20\% & 68.20\%  & 19 & 4.67 \\
\midrule
MGR & $r_2=1$ & 48.60\% & 45.79\%  & 25010 & 5.95 \\
MGR & $r_2=1.5$ & 40.19\% & 40.19\% & 17410 & 5.51 \\
MGR & $r_2=2$ & 29.91\% & 29.91\% & 25480 & \textbf{3.46} \\
\midrule
$\alpha$-Reeb & $\alpha=20$ & \textbf{98.13\%} & \textbf{98.13\%}  & 367 & 10.49 \\
$\alpha$-Reeb & $\alpha=25$ & 97.20\% & 97.20\%  & 375 & 12.50 \\
$\alpha$-Reeb & $\alpha=30$ & \textbf{98.13\%} & \textbf{98.13\%}  & 373 & 14.19 \\
\midrule
DMT & $\delta=0.1$ & 48.60\% & 45.79\% & 6290 & 5.95 \\
DMT & $\delta=0.2$ & 40.19\% & 40.19\% & 6192 & 5.51 \\
DMT & $\delta=0.3$ & 29.91\% & 29.91\% & 6410 & \textbf{3.46} \\
\midrule
$\mathrm{ASk}(C)$ & $\beta=20$ & \textbf{98.13\%} & \textbf{98.13\%} & 42 & 5.16 \\
$\mathrm{ASk}(C)$ & $\beta=30$ & \textbf{98.13\%} & \textbf{98.13\%} & 42 & 5.16 \\
$\mathrm{ASk}(C)$ & $\beta=40$ & 97.20\% & 97.20\% & 42 & 5.31 \\
\bottomrule
\end{tabular}
\end{table}

Table~\ref{tab:micelles} shows that the Mapper, MGR and DMT essentially depend on their parameters, because the success rates, run time and distance error significantly vary when the parameters are only slightly changed.
The $\alpha$-Reeb and $\mathrm{ASk}$ were stable, because Table~\ref{tab:micelles} contains almost identical success rates for different parameters.
\smallskip

Both algorithms achieved best results on the most important measure of the homeomorphism success rate, though the top right picture in Fig.~\ref{fig:synthetic} highlights $\mathrm{ASk}(C)$ and MGR as the best for homeomorphism.
In comparison with $\alpha$-Reeb and MGR, the Approximate Skeleton $\mathrm{ASk}(C)$ is much faster and achieves similar distance errors, see the relevant results in both Table~\ref{tab:micelles} and Fig.~\ref{fig:synthetic}.
\smallskip

\begin{figure*}[t]
\includegraphics[width=0.48\textwidth]{images/stars_endpoints.png}
\includegraphics[width=0.48\textwidth]{images/stars_homeo.png}
\medskip

\includegraphics[width=0.48\textwidth]{images/stars_time.png}
\includegraphics[width=0.48\textwidth]{images/stars_max_distance.png}
\caption{For $N=3,\dots,8$, each dot represents the average over 100 noisy samples around a random $N$-star graph in $\mathbb R^3$ and 3 parameters as in Table~\ref{tab:micelles}.
Mapper: green and long-dashed, MGR: orange and sparsely dotted, $\alpha$-Reeb: red and densely dotted, DMT: olive and short-dashed, $\mathrm{ASk}(C)$: blue and solid.
\textbf{Top left}: the success rate in percentages for detecting a correct number of endpoints.
\textbf{Top right}: the success rate in percentages for detecting the homeomorphism type.
\textbf{Bottom left} (logarithmic scale): average run times in milliseconds.
\textbf{Bottom right}: the max distance from a cloud $C$ to reconstructed graphs.
The exact numbers are in the txt files in the supplementary materials.
}
\label{fig:synthetic}
\end{figure*}

In addition to the comparison on more than 100 micelles, we have tested the algorithms on the much larger dataset of synthetic clouds generated as follows.
\smallskip

\noindent
1) An \emph{$N$-star} in $\mathbb R^3$ has one vertex at $0\in\mathbb R^3$ and straight edges of length 100 to $N$ {\em endpoints}in random directions with a minimum angle $\dfrac{\pi}{4}$ between edges.
\smallskip

\noindent
2) For $N=3,\dots,8$ and every of 100 random $N$-stars $T$, we found a minimum axis-aligned box containing $T$, enlarged this box by the noise bound of 10\%.
\smallskip

\noindent
3) We uniformly chose a random point $p$ in the resulting box and checked if $p$ is at a distance at most 10 (=10\% of edge-lengths) from $T$.
If successful, $500N$ such points form a noisy sample of the ground truth $N$-star $T\subset\mathbb R^3$.
\smallskip

Fig.~\ref{fig:synthetic} shows 4 plots for the 4 error measures of 5 algorithms, which were averaged over 3 values of essential parameters as in Table~\ref{tab:micelles}.
The Mapper threshold factor for single-edge clustering was $\tau\in\{1.25,1.75,2.25\}$.
The $\alpha$-Reeb scale was $\alpha\in\{20,25,30\}$.
The branching factor of $\mathrm{ASk}(C)$ was $\beta\in\{20,30,40\}$.
\smallskip

For the correct number of endpoints, the new skeleton $\mathrm{ASk}(C)$ achieves 100\% results on the synthetic clouds, because Definition~\ref{dfn:depth} provides a very stable concept of a deep vertex not critically depending on a branching factor $\beta$.
For the homeomorphism type, the minimum success of $\mathrm{ASk}(C)$ is 96\%, because all short branches of $\mathrm{MST}(C)$ are removed to get $\mathrm{core}(C)$ homeomorphic to an underlying tree.
\smallskip

For the random point sample of the 8-star graph in Fig.~\ref{fig:8star_outputs}, the 2nd, 3rd and 5th graphs have several branched vertices instead of one.
The 5th graph has several zigzags, which would be straightened in $\mathrm{core}(C)$.
The 4th graph has a triangular cycle because of incorrectly detected overlaps of clusters corresponding to vertices.

\section{Conclusions and a discussion of the Approximate Skeleton}
\label{sec:conclusions}

Though the current implementation was tested in $\mathbb R^3$, all steps and results work in any $\mathbb R^m$.
Here is the summary of the key contributions to data skeletonization.
\smallskip

\noindent
$\bullet$
The detection of deep (branched) vertices in Definition~\ref{dfn:depth} uses a global structure of longest paths within $\mathrm{MST}(C)$, hence is more stable under a change of parameters.
\medskip

\noindent
$\bullet$
To improve the Metric Graph Reconstruction by M.~Aanjaneya et al. \cite{ACCGGM12}, we have split one parameter $r$ (used for detecting vertex points and also for clustering later) into two separate parameters (with default values) $r_1=15$, $r_2\in[1,2]$, which led to more successful (20-40\% rates instead of 0\%) reconstructions in Table~\ref{tab:micelles}.
\medskip

\noindent
$\bullet$
Theorem~\ref{thm:size} proves the first size guarantees (on a small number of vertices) for the Approximate Skeleton $\mathrm{ASk}(C)$, while all past methods from section~\ref{sec:review} considered topological (mostly homotopy type) or metric properties of reconstructed graphs.
\medskip

\noindent
$\bullet$
Corollary~\ref{cor:time} says that the Approximate Skeleton $\mathrm{ASk}(C)$ can be quickly computed within a given error as required in the Tree Reconstruction Problem from section~\ref{sec:intro}.
\medskip

Because of the page limit the last author couldn't include one more result on $\mathrm{ASk}(C)$ with realistic conditions on an underlying tree $T\subset\mathbb R^m$ and its noisy sample $C$ to guarantee that $\mathrm{MST}(C)$ and $\mathrm{ASk}(C)$ are homeomorphic to $T$.
This is the first advance after Giesen's guarantees for shortest paths through sample points \cite{giesen1999curve} in 1999.
The C++ code of $\mathrm{ASk}(C)$ is at https://github.com/YuryUoL/AsKAlgorithm.
\smallskip

In comparison with the past methods in section~\ref{sec:review}, $\mathrm{ASk}(C)$ starts from the most challenging input (an unorganized cloud of points $C\subset\mathbb R^m$ without any extra structure such a metric graph or a regular grid or a mesh), outputs an embedded graph in $\mathbb R^m$ and provides two guarantees: combinatorial in Theorem~\ref{thm:size} and geometric in Corollary~\ref{cor:time} and topological.
Appendix~A has all missed proofs.
Appendix~B includes more experiments.
We thank all reviewers for their helpful suggestions.

\bibliographystyle{spmpsci}

redpajama_set_name: RedPajamaArXiv


Elgin is located between Rockford Illinois and Chicago along I-90, which draws dense trucking traffic into and through the city. Chicago is a major transportation hub, and the entire state is influenced by the extreme volume of semi-tractors that traverse Illinois's major roads. With an increased volume of trucks on the highways comes the added risk that other motorists will be involved in accidents due to negligent actions of reckless or fatigued drivers. The Elgin truck accident attorneys of Rosenfeld Injury Lawyers LLC are dedicated to representing the rights of those who have been seriously injured or killed in truck crashes.
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Since 1973, the East Baton Rouge Council on Aging (EBRCOA) has served the aging population of East Baton Rouge Parish through a spectrum of nutritional and social services. Additionally, the EBRCOA is charged with both meeting and discovering the growing needs of the ever-increasing aging population.
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Traditionally known in India as 'Champissage' this treatment combines both gentle and stimulating techniques on the upper back, shoulders, neck and head to improve blood flow, nourish the scalp and induce a deep sense of calm.
Used traditionally by Shamen healers to cleanse and harmonise the different energy fields to eliminate negative energies. Together with a sinus massage, a candle will be placed inside the ear canal to enhance and clear the sinus airways. The rising air gives the ear drum a gentle massage and helps to balance the ear pressure.
The Sinus Relief treatment also includes a relaxing herbal balm back, neck and shoulder massage followed by an ear candling treatment with a head and sinus massage.


Hill's West Denali is a stocky little fellow who came to us from our friend Tamara Marlin at Hills West Labradoodles in the Tri Cities of WA. He is cream colored, has a thick lovely coat and a large beaming nose; one of my favorite Australian Labradoodle traits. Denali lives with his guardian family of four in Sammamish WA. He is blessed to be part of a family of runners and hikers, and he keeps up with them every step of the way. He has a busy social schedule that every dog in the Northwest would envy.
I have been given glowing reviews about Denali and am grateful he is with us.
Size 18 inches at shoulder, 37lb.
Timber is a super fellow who sired his first litter late 2018. After not seeing him for quite some time I was stunned when his guardian owners met me at a Green Lake cafe for an outdoor lunch. Among the many many dogs who hang out there, Timber was simply a standout; thanks to his guardian parents diligent training. Sometimes a stud dog will not be known for his keen sense of calm with the scent of many dogs around. But Timber had the demeanor of an elder statesman, and I was so proud! Gentle, calm, confident. Timber also had the blocky head of an English Lab. A blocky head and a large nose makes my heart happy. His structure was superior. Timber lives with his young family north of Seattle. They are considering breeding in the future and have chosen their first gal from Shedless in Seattle Labradoodles, Sperry. For now, Timber is her "big brother" and excellent role model.
Now tell me, have you ever seen a more expensive face? Little Bear is a super wooly, super curly gal who reminds me of one of my favorite childhood book characters, Little Bear, by M. Sendak and E.H. Minarik. Raising her brought he warm fuzzies, and memories of reading on the lap of my Mom. But with her terrifically appealing temperament brought enthusiasm for every dog and human she met; an Australian Labradoodle trait that can simply go too far! But her guardian family of six slowly trained her to chill. She is now honestly one of the most easy going dogs I have had the pleasure of breeding. In my home with her first litter, she oozes calm. Running your fingers through her wooly coat is like going to yoga class. Little Bear would make a terrific visiting dog. And I am so grateful I kept her as a breeder. Her guardian family has two Shedless in Seattle Labradoodles.
Size 18 aches at shoulder, 38 lb.
Tallulah of Sundance Labradoodles in AZ is a little lump of love. I say lump because she is a compact build, who surprises you with her density when you pick her up. She is a highly affectionate lump of love, but also the most protective of her babes of all the Mama dogs I have loved. Day six since the birth of her little ones, and I am the only person who is allowed to touch her pups. Tallulah lives like a princess in her Seattle area home with her empty nester guardian family. She gets to live the Northwest RV life, going to the Oregon coast; a lovely life that we all aspire to, human and canine. One of my favorite "Tallulahisms" is the way she honks and grunts to speak to me. Tallulah has her very own language. In her first littler there were a few pups who shared her same language. I love to watch those genes passed down from one generation to the next. Tallula has a coat that does not mat as much as some Doodles, and is non shedding. It is fairly easy maintenance.
Dottie Bug is a beloved girl to us because she lived with us for most of her life before choosing her guardian owner. Dot was named after my aunt, Dorotha Grable, who I thought was the coolest Aunt ever because she lived in CA. That was a big deal for a KY girl to have a west coast Auntie. She was nicknamed "Bug" after my daughter, Ellie" Bug". Our canine Dottie has an intensely passionate exuberance. She does everything with gusto. She stares into my eyes like I made the sun and the moon just for her. She presses her big snout under my neck and holds it there for as long as I will be still. Her snout has magical calming powers, which is interesting because nothing about Dottie shouts calm. Like I said, it is magical. Ahhhhh! Dottie lives in Woodinville WA. with her guardian when she is not attending classes at Whitworth University. She is a hit visiting campus. Her first litter was summer 2018 and pups are ready October '18.
Size 21 inches at shoulder, 45 lb.
Introducing, with exuberance, Seattle's Handsome. Handsome is forty pounds of soft, sink your nose in, fabulous. Handsome is a medium, apricot, spiraling fleece Australian Labradoodle. His parents are Evergreen's Bluebells for Seattle and Pacific Rim's Rogue River. Handsome and his brother, My Best Buddy, had both been reserved as pets dogs but by the time they were ready to go home I knew that neutering would be a mistake! Both families were willing to work out arrangements with me for duplicates of little Handsome's and little Buddie's, for which I am very grateful. Handsome lives with his family in Redmond, Washington.
Sophie is one of our few stunning chalk colored dogs. I am really fond of this color and will be excited to have a litter in 2016 with Sophie both because of her color and the genetics brought by her parents. Her Mother, Creme de la Creme Dolly is an incredibly calm and tolerant dog; a wonderful trait to perpetuate in the Australian Labradoodle breed. Dolly, Sophie's Mom, was accepted into a service dog training program. Sophie's Sire, Johann Bach is one of the most perfect temperaments I have known in an Australian Labradoodle, and the most perfect conformation as well. He went to work with his guardian daily in her home decor store and i got a steady flow of inquiries from his fans. Sophie lives with her guardian family of five in Silverdale. She gets along well with other dogs when she visits here, and has a pleasant and calm temperament, except when she is chewing her stuffed toys! That is her moment of high enthusiasm! We hope to see Sophie's pups in summer 2016.
Dam Waltzing Matilda's Creme De La Creme "Dolly"
Carolina Rain is one of the best socialized and de-sensitized dogs I have had, as she is the only one of my breeding dogs who lives with a family including four children (indoors) and an ark load of other creatures (outdoors). Her family has pot bellied pigs, alpacas, llamas, and chickens. Her guardian family lives on a farm in Maple Valley. They are a crazy busy household. The last day I visited they had an alpaca due to give birth on that day, as well as the cross-country team expected that night for a pasta feed. The family has an F1 Labradoodle they had bred before I met them, named Dakota. The Zosel family cherished raising Dakota's one litter of pups, but did not enjoy the selling, administrative, and business end of being breeders. They approached me in 2011 about being guardians and helping to raise that the litter of that Mama for me. They did a fabulous job, and we have had a wonderful friendship since. Carolina Rain had her first litter in 2014. See photo. Carolina has a very curly thick non-shedding coat. She is a total dear since she stopped chasing the chickens! I love her temperament, and she will offer multiple colors and lovely calm to our breeding program.
Seattle's Missy was named by her Guardian Forever Family. But I was thrilled with the name, as Missy Baby was the name of my first dog ever as an infant. My mother told that I used Missy Baby as my canine pillow for my first years of life. Missy the Labradoodle lives in Mt. Vernon WA with her family of three generations. She is the beloved companion of all three. She is the canine daughter of my former breeder, Evergreen's Bluebells for Seattle and Evergreen's Epic. Her sister, who is a breeder for Legendary Labradoodles in Plano TX, recently had her first litter of eleven. Missy came from a litter of 11 and gave birth her first litter of eleven on October 11, 2014. Missy is a hilarious girl, expressive and highly affectionate. She has curly fleece, very soft, and excellent for those with allergies. Missy is terrific with children and gets along superbly with other dogs. She has an unusual smile, reminiscent of my childhood Missy Baby. We have very much enjoyed having the third generation of this line here at Shedless in Seattle.
Size Medium; 21 in. tall, 46 lbs.
Rogue is a jewel of a guy. He lives with his Forever Family just north of Seattle, where he has three little ones to watch over. He has a busy social schedule of baseball, birthday parties, and camping. Rogue has sired three incredibly soft Labradoodle litters for us. We're excited about the softness of the coats, as well as the deep color he produces. Rogue seems to float wherever he moves.
Size 19 in. to shoulder, 35 lbs.
A recent 2013 addition to our small family breeding program is Aussie L's Mr. Snuffleapagus. If you know where his name comes from, you may be dating yourself. I visited my friend Kari to choose a breeding dog, and was so enamored by the whole litter and the Mama that I came home with two; a male and female. Sister Bubblegum will be a Mama in 2015. Snuffleapagus sired his first litter in 2014 and we enjoyed a beautiful litter of eleven red and caramels. We expected medium 35 to 50 lb. pups around 18 to 20 inches, but we have one boy who is raising the bar with the possibility of maturing to a standard. Oops! Snuffleapagus sired both red and caramel pups. We are thrilled with his calm temperament. He is one of the calmest stud dogs I have ever met, or had the pleasure of having in our program. He lives with his guardian Forever family of seven in Bellevue. They call him Wrigley. He is very well stimulated and socialized.
Size Medium; 19 in. to shoulder, 32 lbs.
Seattle's Russel was named after…..in unison now…..Russell Wilson. In fact the year 2014/2015 brought us three new Russells. This new boy is as beloved as our Seahawks star. He is gentle natured, amicable, warm, confident, graceful, and runs like the wind. His first litter was sired in 2015, and we look forward to little Russel's for years to come. His favorite thing to do is visit the dog park since that is something a stud dog rarely gets to do. Russell lives in Lacey WA with his empty nester guardians, who are sports enthusiasts. I believe they are extremely proud of his athleticism. He has a deep red coat, which thankfully carried on in his first litter.
Size 19 inches to shoulder, 35 lb.
Amber Waves of Grain is a daughter of our sweet Wild Liberty, and Bocelli of Eden Valley Manor Labradoodles. Amber lives with her Forever Family of six on Whidbey Island, as well as her Mother, Liberty. Amber shares her Mother's zany love for life, high expressiveness, and non shedding coat (with an extra dose of relaxed silkiness). She enjoys jogging with her family, riding in the car, and digging in the back yard (I am brutally honest). Amber accompanies me on walks without a leash, and is very intuitive about my wishes. It is with great excitement that we welcome her first litter in Autumn of 2009.
Annie was so named because she lives on Bainbridge Island with her forever family, the Balls. The ferry in her name is a tribute to the first Labradoodle our family ever laid eyes on. We were traveling from Seattle to Bainbridge via ferry when a handsome and stately standard Doodle came aboard. We inquired about the breed of this creature and thus began our odyssey with Doodles.
Annie lives a spoiled life with her large family of school-age siblings. Since Annie's dad is a camp director, she gets to hang out with him at the "office". Annie will deliver her first litter in Summer '09. As the sire is yet a mystery, we cannot give details of this exciting event quite yet. Annie has a soft, curly fleece non-shedding coat. She is a smart and trainable companion with a warm temperament.
Size 19 in. tall, 32 lbs.
Jasper was raised right here, in our home; the son of Bainbridge Island Annie and Sunset Hills Red Chipotle. He was a hotshot from the beginning of the time when temperaments first begin to show (four to five weeks). Meanwhile, back on Bainbridge Island, some friends of Annie's Guardian family met Jasper, and began asking me to please place the little hotshot with them. We'll do whatever it takes to have him, including keep him for you as a future Australian Labradoodle Stud. They are empty-nesters, and treat this fellow like the heir to the family fortune. He is even allowed to join his human d	ad at work most days, as an apprentice in the financial advising industry. His briefcase, however, contains his well loved toys. Jasper spends weekends guarding the family property from neighboring deer, and jumping to catch his favorite toys with incredible accuracy. His first litter is due in the fall of 2012.
Size 17 inches tall, 22 lbs.
Our Bell of the Beach is a family member at the Johansen household, on Fox Island. She is the youngest of her human family of seven. In a family of blonds, she stands out with her shiny raven coat. In a family of virtuous folks, she stands out at slightly naughty. She gets this from her Mama, (canine, that is) Pretty Precocious Piper. Bella's first litter is in the Autumn of 2013.
Meet our recent sire, added in 2013. What a character! Fozzie has the temperament of a boy who just wants to have fun. He really really really loves everyone he meets. We plan to see his little ones in 2014. Fozzie lives on Whidbey Island with his human guardian parents, Gabby and Craig. They refer to him as Ozzy. Fozzie thinks the beach life suits him perfectly. He is their only little one, and has the life of a prince. No competition for this boy! The odds are in his favor with two people to throw a ball, and only one dog to fetch. Available for stud for knowledgeable responsible home breeders.
We are happy to have Tebow on our team! He is the grandson to Seattle's Wild Liberty and exhibits some of her same highly expressive traits. Tebow is exuberant about his life with another Labradoodle, Charlie, and empty nester guardian parents in Redmond, Washington. Tebow's enthusiasm is contagious. His first litter is expected Spring 2014.
Bluebells came to us from our dear friends Ann and Steve of Evergreen Manor Labradoodles. Bluebells is as gentle-natured and loving as the two of them. She lives with a guardian family of two school-age boys and their parents in Auburn, Washington. Her temperament would be perfect for a visiting or service dog if given the opportunity.
Size 20 in. tall, 43 lbs.
California Callie is a daughter of our beautiful Sunshine in Seattle. Callie lived with her forever family in Bellevue until they unexpectedly relocated to sunny Redondo Beach, California. Callie didn't give up her post as a breeding dog. She still blessed us with a litter of four sweet little ones despite her relocation. Callie has returned to the beach life but we are hopeful for a second long-distance litter in the fall of '09.
Callie has a curly fleece non-shedding coat. She is extremely affectionate and a joy to have around.
Size 19 in. to shoulder, 29 lbs.
There is nothing as exciting to a dog breeder than to witness new generations from a beloved dog. Seattle's Bainbridge Island Chelsea is one of these. Her, Mom, Pretty Precocious Piper, is a retired breeder and our family pet. Piper a a goofy thing, and we have been waiting impatiently for little Piper repeats to arrive in the form of grandbabies. Chelsea is a sweet, loyal, and mellow girl who got all the Piper sweet but without the naughtiness! Chelsea has a soft spiraling fleece coat, non -shedding, and a wise choice for those with dog allergies. Chelsea lives with her human Mom and cats on Whidbey Island. She has beautiful mothering instincts, birthing her first puppies in January 2013.
I toyed with they idea of having Waltzing Matilda's Creme de la Creme join our breeding program after seeing her eight-week-old pictures, and hearing her breeder, Michelle, describe her. "Gentle natured, intuitive, affectionate" she beamed about this little Australian Labradoodle girl. As a breeder, we are sometimes guilty of wanting every lovely pup we run across. "Self control" I said to myself. I waited … at sixteen weeks of age Creme de la Creme had her preliminary testing, and then I was sold. She joined our family briefly to get acquainted, and settled in with her doting Labradoodle Enthusiast family in Maple Valley. Her human parents are empty nesters, and this gal, who they call Dolly, and her sidekick Labradoodle, Tura are raised with incredible tender loving care. We are grateful to have them be part of our Forever Family (Guardian) Program. We hope to have Creme De La Creme's first litter available in 2012.
We are awaiting test results for the newest addition to our breeding program.
Creme De La Creme Dolly's Autumn 2013 litter have joined their new owners. Her next litter of medium, wavy- fleece cream colored puppies will be available in Autumn 2014.
Miss Effie came to us from our friend Maggie at Bedrock Labradoodles in Oregon. She is a second generation American Labradoodle (F2) with an intuitive, warm temperament. She is so forbearing with her human toddler sibling, Maggie. Her pups will be Multigenerational American Labradoodles. She is built exactly like a Lab, a trait I really seek for my Labradoodle lines. Her hip scores are outstanding—90th percentile! Her pups will have a very soft spiraling fleece, low/non-shedding coats. This coat is an excellent choice for families dealing with a dog allergy.
Size 21 inches to shoulder, 45 lbs.
Liberty Belle could be the poster dog for the ideal temperament of a Labradoodle. Clown-like, joyful, intuitive, fun-loving. She is the most expressive dog I have ever known. She "talks", spanning an entire octave. Belle lives in Issaquah with her Forever Family, pictured here. Belle has the type of non-shedding curly wool coat that is the easiest to groom; can be scissored at home. I love her block Lab boning. Belle is a sister to Amber Waves of Grain and daughter to Wild Liberty.
Size 17 inches to shoulder, 30 lbs.
Millie was recruited to join Shedless in Seattle Labradoodles at eight weeks of age. We chose her from Mountain View Labradoodles because we love her sire, Legend. Millie's name fits her perfectly, if you remember the cheerful Millie from the Andy Griffith Show. She is so chipper from dawn to dusk. Aside from her A+ temperament, she sports an A+ coat. She is one of those gals who always has "good hair days." Millie lives with this precious family in Auburn, WA. Millie's first litter is expected in Summer 2011.
Maya, as called by her forever family, is a regal lady. Maya sailed through her initial obedience class and was recommended for agility. Her obedience training continued at home, where her family cleverly taught her to play hide and seek and to sprint through the house on command. She loves to please and be given directions. Her trainer noted that Maya has the temperament of a service dog.
Maya lives with her forever family in Bonney Lake, WA. Her family homeschools, which gives them ample time to include Maya in daily routines. As part of the homeschooling science curriculum, her family requested the honor of birthing and raising her first litter in their home. This is an unusual arrangement, but given that both parents are physicians, I felt confidence that Maya and pups would be incredibly well cared for, even in an emergency setting. Maya's family has graciously served as nannies to our litters here in our home when we travel as well as attending several births. We are excited about their highly capable service.
Maya has plenty of experience as an auntie, as she regularly puppysits for our puppies and litters when we are out of town. She is patient and welcoming to her canine charges and companions.
Size 17 inches to shoulder, 25 lbs.
How to describe our pretty, precocious Piper? Maria from The Sound of Music? Mary Poppins? Tigger from Winnie the Pooh? Piper carries more joy in her tail than most dogs do in their entire body. Much to her delight, Piper lives in a Forever Family (guardians) who hike energetically and often. It was difficult for us to let her leave our nest, but this family is her dream come true. Almost two years old, Piper has retained her deep dark chocolate coloring and loose, loopy curls.
Size 19 in. to shoulder, 40 lbs.
Rainy is now retired from breeding and lives with the Benson family in Vancouver, WA.
Rainy is docile, cheerful, and desires nothing more than abundant companionship and maximum skin contact. She is not content just to sit on the sofa - she perches on the back so she can wrap her body across your shoulders and tuck her head under your chin. She will cause an owner to beam as she waddles with excitement each time you walk into the room. Rainy is a nature lover. She will bask in the sunshine watching birds and even deer without excitement. If she were in politics, she would be Laura Bush. If she were a book author, she would be Laura Ingles Wilder. If she were an athlete, she would be a hiker.
Rainy joined us from Mountain View Labradoodles in Sisters, Oregon. Her breeder, Terrie, is such a joy to work with. Rainy, who will whelp F1B puppies, brings fresh bloodlines and particularly strong hips to the Labradoodle gene pool.
Size 17 in. tall, 32 lbs.
Sunshine is now retired from breeding and reigns supreme in our own household.
Sunshine is not only beautiful; she is confident, graceful and poised. She most enjoys catching a tennis ball or playing fetch. If she were a politician she would be Hillary Clinton. If she were a movie star; Marilyn Monroe. If she were an athlete, she would probably be a figure skater. She is eager to please her human family and determined in leading her canine family. She will happily take commands from adults and children alike. She enjoys jumping on the trampoline with children and dogs as well as jogging with our family. We must confess that she also enjoys television. Don't be concerned...we carefully monitor her viewing and try to stick with Animal Planet and PBS. We have been amazed by her intelligence. Petting her is like eating chips - you just can't get enough.
Sunshine, a Multi-Generational Labradoodle, came to us from Spring Creek Labradoodles in Eugene, Oregon. Waiting for her was a learning experience as her breeder, Rochelle Sundholm, carefully chose her for our program. Rochelle has such a wealth of knowledge and is a backbone to the Labradoodle breeding community. We are so grateful to Rochelle for her dedication to the future of this breed. Sunshine's roots are both Tegan Park and Rutland Manor. She is non-shedding and has a thick, wavy fleece coat.
Size 19 in. tall, 28 lbs.
Tessie, although having a rather proper name, is actually a farm girl and adventure girl extraordinaire. She lives one mile from us on a gentleman's sheep farm. Her friends include sheep, doves, chickens, and a very naughty rooster. Her best chum is a squirrel, hand-raised on a bottle. In fact, when Tessie nursed her first litter of puppies, the squirrel was allowed to nurse right alongside. Tessie lives a much more interesting life than I; she has custom doggy goggles for four-wheeling trips in the mountains. On weekends she travels with her guardian parents, the Rainwaters, to their mountain retreat or their island getaway.
Tessie hails from our own Little Black Raincloud and Maestro Verdi of Tuscan Ridge Labradoodles. She stands 15 inches tall and 18 pounds of red, silky, springy, non-shedding fluff. Her first litter gave us two beautiful and rare phantom sable puppies. The male pup, My Best Buddy, is pictured here along with Tess.
Size 15 in. tall, 18 lbs.
Liberty is a rascal of a gal, who is bursting with enthusiasm for life. She lives with a family of four children in Oak Harbor and follows their every move. As a homeschooling family, they have had the flexibility to be very involved during her birthing of puppies. They enjoyed the forever family program and Liberty's addition to their family so much that they requested a second dog from her first litter. Amber, Liberty's daughter, will hopefully join us as a breeder in 2009 pending completion of health testing.
Liberty's expressive almond shaped eyes and clown-like nose are some of her heartwarming features. She has a dense non-shedding coat. The combination of wool and fleece keeps her from matting. Her springy, curly coat is addictive to the touch.
Liberty joined us from Mountain View Labradoodles in Bend, Oregon, where she was tenderly whelped and raised in a bedroom from birth to eight weeks. When she joined Seattle Labradoodles we enjoyed her enthusiasm and her ability to "play well" with other dogs. After becoming acquainted with our family and dogs, she joined her guardian family where she lives until time to be a mom. Currently three of Liberty's puppies are hopeful future breeders both with Shedless in Seattle Labradoodles and a new breeder in Phoenix. One of Liberty's puppies was recently recruited as a professional model in Miami. We are excited to see his print debut!
Size Medium, 21 in. to shoulder, 40 lbs.
Annie's Seaside Maimie is a calm, gentle soul, just the perfect companion for the rainy Pacific Northwest, "snuggle in front of the fire" type days. Maimie is a super companion for her guardians in Graham, Washington and their sweet family. Her lines go back four generations on her mother's side, to one of our original Labradoodles, Little Black Raincloud (Rainy). Her first litter is expected Spring 2014.
Size 19 inches tall, 32 lbs.
In the summer of 2011 I traveled to Atlanta to meet Mr. JJ. Bear. My co-owner for this stud, Tamara at Hills West Labradoodles, raved about his conformation and pedigree, but I needed to meet this fellow in person (or in canine). I reported back to her on the spot, "He is more incredible in the flesh (or in the fleece) than on paper". This boy was what I call an A+ Stud.
Soon he had joined us here in Sammamish, on the Eastside of Seattle, WA. His guardian family, the Nakatsus, had a Labradoodle from one of my first litters in 2006. They fell in love with JJ Bear while here visiting, and wanted to bring him into their family. He is impressive: he can walk off-leash better than any dog I have ever seen. JJ Bear goes to work with his guardian owner, Sandy Nakatsu, at her furniture store Diggs & Dwellings in Redmond, WA. If you are interested in meeting an Australian Labradoodle, you are invited to visit him at the store. Call ahead to be sure he is working that day. JJ Bear has sired multiple litters in 2012.
Size 20 inches to shoulder, 40 lbs.
Sunset Hills Red Chipotle arrived from Australia in Spring 2009. Chip is a medium/small standard, soft wool, non-shedding Australian Labradoodle. Chip carries red and chocolate. He lives with and is co-owned by the Schaffs of Mountain View Labradoodles, Bend, Oregon. Chip is shared with Mountainview Labradoodles, and Bedrock Labradoodles, both in Oregon. His services are available for trade and hire to approved breeders. Please contact the appropriate breeder for trade, or contact Maggie Palmblad to hire his services.
Size 20 in. tall, 35 lbs.
Seattle's Starbucks Stud has an enthusiastic work ethic.
Skill Set: Was the proud papa of three litters. Starbucks excelled in natural breeding and is also available via fresh AI or frozen. In select cases he may travel to you and your gal for courting and R&R.
Starbucks is a stocky sturdy fellow, whose Labrador-like build complements the slender build of so many Labradoodles. These pups are stunning. While visiting with an owner and one of these pups outside Starbucks recently, no less than three people stopped us in 30 minutes to remark on her pup. He is a walking billboard. Thank you Starbucks!
Size 17 in. tall, 29 lbs.
Epic Joined us in summer of 2011 from our dear friends Steve and Ann Green at Evergreen Manor Labradoodles south of Portland. He was scooped up immediately by a family who was waiting to be a Guardian for a male. They had no idea how their life would change. Epic, who they call Archie, goes everywhere they go including the Oregon Coast, the Cascade Mountains of Washington, and in their day to day life out and about the University District of Seattle. His parents feel like they are touring with a rock star with the number of requests they get about him. No more privacy for Judy and John. They say life with this dog is almost as much fun as being a grandparent!
Epic is very fun loving and gregarious, curious and energetic. He is very soft, non-shedding, and red, hence their calling him Archie. He sired his first litters for us in Summer of 2012, and we are delighted with his offspring. Below is a photo of his litter with Evergreen's Bluebells. His services as sire will be considered for conscientious breeders upon request.
Limited Edition Chester is so named because we will have only three litters with this fun-loving fellow. Chester was sold as a pet puppy; however, before he left us, I began to regret not having kept him as a future sire. I was crazy about his loyal and warm temperament. I decided to inquire of his family if they would allow him a brief breeding career before having him neutered. Thankfully they agreed and Chester passed his health testing. He was able to sire two litters due April 2009, as well as putting one litter of future Chesterlings into storage.
Size 19 in. tall, 30 lbs.


Being able to physical exercise is always very important to your health and therefore, something that you have to properly prioritize. Exercising on a regular basis is very important because it's going to help you to avoid a lot of diseases. However, the kind of exercise that you're going to do is going to give you different levels of benefits. Some of the things that people do in order to exercise include, going for yoga classes or even, going to the gym. One of the forms of exercise that is becoming quite popular is exercising rebounder. Exercising on a rebounder is recommended because it is perfect for a lot of things. One thing you realize is that, there are very many companies that are able to supply the rebounder to you today. Because of how, they have been and the competition in the industry, rebounders are also very affordable today. It would be possible for you to get a lot of perspectives when it comes to exercising with the rebounder and that's the information provided in this article.
Losing weight is going to be the first benefit that is that enjoy from exercising on the rebounder. Many people do not think about this as being very serious that, losing weight is able to help you to start breathing much better and also, to avoid a number of other conditions. Another benefit of exercising on the rebounder is that you get some stress reduction which is an important thing. One thing you realize is that, reducing your stress levels helps you to focus much better and also, to become more productive whatever you do. Another reason why, rebounder exercising is also good is simply the fact that, it's going to help you to strengthen your muscles. When you are able to strengthen your muscles, doing different kinds of activities becomes very easy for you.
If you are having some problems with your thyroid, it's possible that you're going to get a lot of treatment when you decide to exercise regularly on the rebounder. The reason why the exercising on the rebounder is going to be good is simply the fact that you will be going against gravity. Use the rebounder that you're going to buy to exercise regularly and you will be able to notice a lot of changes in how your body is going to perform in different things.


One vital issue to consider when it involves pre-foreclosure investment, nevertheless, is the issue of liens. A lien is a case that some party aside from the lender has on a portion of the residential properties value. For example, a professional that serviced the residential or commercial property however never ever obtained his pay could submit a lien against it. As a pre-foreclosure capitalist, if you purchase a residential or commercial property with a lien, you come to be in charge of paying that lien, which could cut into your earnings. Given that allowing the residential or commercial property go to repossession will erase all liens, a lot of lienholders want to bargain at this point. However, that is not an assurance, as well as finding every one of the details on residential property liens could be problematic. The absence of liens is something that makes acquiring foreclosed homes at auction preferable to acquiring pre-foreclosure buildings.
The new angle on i buy houses Paw Creek just released.
Consumers that file for bankruptcy protection could see their credit report decrease anywhere from 130 to 240 factors, according to credit scoring firm FICO. NC lending institutions are typically seeking a credit rating of at the very least 620, which type of decline knocks several debtors out of certifying variety.
Many of these foreclosures are brand-new property owners that had actually constructed little or no equity in their property; for that reason, the amount the lender is aiming to recover might be close to the complete market value of the house or at most only a 10 percent discount rate.
To be effective in realty investing, a strong collaboration needs to be developed with lots of company that have an intimate expertise of the neighborhoods we have selected. This listing of business calls consists of but is not restricted to property brokers/agents, chamber of business, regional financial investment clubs, financial institutions, utility company fixing employees, title insurance companies, local government personnel, as well as others.
Getting A Home In Spain These firms give you access to foreclosed home listings in your area, when they come into the market. Some of them supply info on pre-foreclosure listings and realty public auctions, also. There are a number of web-based services you could use for this purpose. RealtyTrac.com is one of one of the most widely valued (as well as commonly used) of these solutions. RealtyStore.com is additionally worth an appearance.
The city of Paw Creek has a city price of 0.0298%. The rate for the Paw Creek Unified College Area is 0.1756%. In the city of Bell, which component of the NC Unified College Area, there is a city price of 0.34%, and the overall rate is almost 1.57%, amongst the highest possible in the region.
Escrow Home Loan Requirements As a fast refresher course, an escrow account is an account that holds the funds you have to pay your property tax obligations and also home owners insurance policy. Its not an account that you handle straight. Its just a holding account which contains the funds you pay each month to guarantee your tax obligations and also insurance expenses are paid.
CalHFA has initially as well as junior car loan options for reduced to modest earnings families, including low to no rate of interest deposit support fundings. Although most of our programs are for new buyers, you do not need to be a first-time buyer to use CalHFA's FHA and also Standard first home loan. CalHFA does decline lending applications directly. A CalHFA approved Loan provider will qualify you for a mortgage, so you will certainly have to apply with one of our Preferred Loan Officers or authorized Lenders (See Step 2). You could find the information concerning CalHFA mortgage loans by clicking the Lending Programs tab over.
Alex Khalil- Dlc Home Mortgage Evolution Start by amounting to up all the monthly expenses related to a house purchase, including your forecasted mortgage settlement, insurance policy and also tax obligation quotes, HOA costs and also house maintenance costs. And don't worry if you don't have concrete numbersthe point is to see if you can pay for a ballpark quantity.
Buying A House At Auction Once you have actually made a list of your must-have's, remember to think about the type of area you desire, kinds of schools in the area, the size of your commute to and also from job, as well as the ease of neighborhood shopping. Take into consideration your security concerns in addition to exactly how great the price of house appreciation remains in the area.
There's a likelihood the foreclosure residential property will be valued below market price. Keep in mind the typical 27% discount rate we spoke about earlier? If it's valued above the equivalent sales, you should hesitate about making a deal-- unless the target home has particular unique functions that account for the higher sale price.
Word of mouth is a great method to utilize if you are simply beginning your quest for a great realtor. Why? When, you obtain a name and also a recommendation all at. The best point to do is speak to any individual you understand who could have recently purchased a residence. This resource of info can consist of buddies, family colleagues, acquaintances, or members, such as your stylist or your preferred salesperson at the nearest chain store. He or she could offer you the name and variety of the real estate professional they usedprovided they were pleased with the solution, that is.


My self-esteem began to plummet around the time I got married. Now I'm divorced I feel I've been taken back in time to when I was twenty-five and younger. Then I had a very strong sense of who I was and I was really proud of myself. I feel I'm almost that person again.
RMJ and her husband met in college and were married for nine years. They separated for two years while working on their marriage and were divorced two years after that. RMJ is now living openly as a single mother which is rare in the African country where she now lives.


Reading Wait for Me, by Susan May Warren, is very much like taking a raft down a river. It has a nice flow to it – a current – but every once in a while we get to some white water rapids, so hold on to your seat – you're in for a bumpy ride! I don't mean that in a negative way. In this book, just when you see things smoothing out, Warren throws in another bump. And then another. You get my point.
I found this story unique with the number of switchbacks in it. Not only does the plot take a number of twists, but even the established relationships between some of the characters get jumbled around with all the bumps.
Why Wait for Me is So Bumpy!
This is the sixth and final book in the Montana Rescue Series. I did not read any of the first five novels in the series. I started with this one. Thus, I don't know whether all the books are this bumpy, or if Warren saved the best bumps for last, so to speak.
It's quickly apparent that the characters have been together for a number of adventures. Relationships are already established, and the characters are well defined. Most of the characters are in the rescue business, so the abundance of story action was appreciated. However, Warren left a lot of headroom for romance.
There are a number of "backstory moments," where characters remember love and adventures gone by. I imagine this is the author's way of making this final book in the series a stand alone novel, meaning you don't have to read the previous five to understand and enjoy it. Susan May Warren was very successful in that attempt. I did enjoy the story, and it does stand on it's own.
HOWEVER – I do wish I could have read the other books in the series before reading this one. This story was very good, and it kept me fully engaged, but I think I could have enjoyed it even more if I read the stories before they became backstories. Does that make sense?
Some action stories have some romance, and some romance stories have some action, but this story has a lot of both! That's what I enjoyed the most about it. If you love romance, you'll be satisfied, and if you love action, you'll be content. But regardless of which element you like the most, you'll find enough bumps in the plot to keep you off balance.
I did come across a few SPAG problems. Thankfully, the problems weren't nearly as plentiful as the bumps. Thus, they didn't detract from my enjoyment of the story.
The pacing was sort of stop and go. This was not only necessary to include a sufficient amount of backstory, but it also helped to round out the characters as we read of their innermost thoughts and desires.
The character arcs were a little strange. They were there, but they certainly weren't smooth. Again with the bumps. I thought I knew where a character would be going, and then something would change with the character.
This leads us to "climax central" in the final chapter. I don't really want to say too much about it, because I'd have to include a spoiler or two. However, I will say that it was worth all the bumps and twists!
I hope you give this book a try. You'll enjoy it!
Susan May Warren is the USA Today bestselling author of over fifty novels with more than 1 million books sold, including Wild Montana Skies, Rescue Me, A Matter of Trust, Troubled Waters, and Storm Front. Winner of a RITA Award and multiple Christy and Carol Awards, as well as the HOLT Medallion and numerous Readers' Choice Awards, Susan has written contemporary and historical romances, romantic suspense, thrillers, romantic comedy, and novellas. She makes her home in Minnesota.
Posted: November 19, 2018 November 21, 2018 .	Posted in: Book Adverts, LSBBT. Tagged: Contemporary, LoneStarLit, LSBBT, MontanaRescue, Romance, Suspense, WaitForMe. Bookmark the permalink.
Great review (should I say, bumps and all? ha!) Cool that this book is still so enjoyable even as a stand alone and a series finale.


text:
Deja vu? Neocons tout Ahmed Chalabi as Iraq's next leader
Former Iraqi deputy prime minister Ahmed Chalabi in October 2010. (Manuel Balce Ceneta/Associated Press)
By Adam Taylor
As Iraq's crisis continues, there has been a growing consensus that much of the blame lies at the door of Prime Minister Nouri al-Maliki and his Shiite-led government. Many people believe that Maliki will soon be out of office, and they are probably right.
But who should replace him? There's a long list of possible candidates, but one seems to have won the backing of at least two former advisers to George W. Bush. It's quite a surprising choice.
This week the National Journal's Clara Ritger spoke to Richard Perle, the famously neoconservative adviser to Bush at the time of the Iraq war. Asked about who should next lead Iraq, Perle said 69-year-old Ahmed Chalabi was best suited for the job. "I think he's got the best chance," Perle said. "It would be foolish if we expressed a preference for somebody less competent, which we've done before."
Chalabi, of course, is a man Perle knows well. In the buildup to the Iraq war, the wealthy, politically connected Iraqi exile supplied the U.S. government a large amount of information that linked Saddam Hussein to al-Qaeda and weapons of mass destruction. For that, he was dubbed the "George Washington of Iraq." Of course, it was later discovered that the information was false, and, after that, a new nickname was found: "The Man Who Pushed America to War" (or, as the New York Times put it, "Neoconner").
Amazingly, it gets worse. In the years after the invasion, Chalabi's reputation for graft and deception expanded. In 2004, U.S. troops raided his Iraqi home and office on fraud and grand-theft charges, and allegations that he had embezzled $230 million from a Lebanese bank years earlier continued to dog him. Later that year, the United States accused Chalabi of providing information to Iran on U.S. troop positions in Iraq. In turn, Chalabi broke with the Bush administration, becoming vocally opposed to the occupation of Iraq.
As you can probably understand, Chalabi became a toxic figure for the United States. "He will use whatever vehicle or platform that presents itself to further his own agenda," Ryan C. Crocker, U.S. ambassador to Iraq from 2007 to 2009, told The Post as Chalabi campaigned in the 2010 elections — a move that caused significant concern among U.S. officials at the time.
So, Perle is championing a man who provided false information that led to a war now widely viewed as disastrous, is accused of stealing millions of dollars and is widely thought to have helped spy on the United States for Iran. And Perle isn't alone. Paul Wolfowitz, another leading neoconservative who was key to Bush's foreign policy, also has come out to say that for all his flaws, Chalabi is a viable candidate. "The man is a survivor," Wolfowitz said in an interview on Bloomberg Television. "That's impressive. I think he wants to succeed in what he does, he's smart; maybe he'll figure out a way to do it."
The reality is that Chalabi lacks popular support, though his political guile and status as a secular Shiite has helped him emerge as one of the few viable candidates for prime minister. Liz Sly, The Post's Beirut bureau chief, explained his appeal this month: He is "a perennial who presents himself as a compromise candidate after every election. He has no real constituency and won only one seat in parliament, but he has the support of the Sadrists and is being regarded by Sunnis as a chance to move away from [Maliki's] Shiite Islamism."
In many ways, the current chaos in Iraq has renewed focus on the logic of the 2003 invasion and its unintended consequence. It seems a sad indication of the absurdity of the past 11 years of Iraqi history that the man who helped dupe U.S. officials into that invasion should now be backed in his bid for leadership by those very same people.

redpajama_set_name: RedPajamaCommonCrawl


text:
Carr helps broaden Hallmark Channel's view
December 10, 2007 by Tonya Garcia
Two major announcements were keeping Nancy Carr, SVP of corporate communications for the Hallmark Channel and Hallmark Movie Channel, very busy of late. The Hallmark Channel just acquired three programs - I Love Lucy, Cheers, and 7th Heaven - adding to its list of popular "off-net" series.
Two major announcements were keeping Nancy Carr, SVP of corporate communications for the Hallmark Channel and Hallmark Movie Channel, very busy of late. The Hallmark Channel just acquired three programs - I Love Lucy, Cheers, and 7th Heaven - adding to its list of popular "off-net" series. And Comcast renewed the distribution agreement with both television stations, adding an HD version of the movie channel planned for March 2008.
Carr's role, handling press strategies, special events, and internal communications, was critical to getting the word out. And in all her efforts, she never strays from the company's core strengths.
"Hallmark is the premier channel for family programming," she says. "We're not about sexy. We don't have forensic crime. We're programming for the underserved."
Despite the proliferation of edgy viewing options, Hallmark is determined to stick to its family-friendly guns, since there seems to be an audience for it. The Hallmark Channel reaches 85 million households; the movie channel about 10 million. The 25- to 54-year-old age group is the overall target - with baby boomers at the core.
"We're the poster child for decency," explains Carr. "This is certainly a time when policy-makers and the FCC have been looking at content. That's not an issue for us."
While the Hallmark brand has been a staple for 50-plus years, the Hallmark Channel was not an immediate hit with the media when it debuted in August 2001.
But by the time Carr joined from CBS in 2005, things had changed remarkably.
"[At] Hallmark, the whole PR team has had to work hard over the past six years," she says. "Now the phone is ringing more. I had no idea what a success story I was walking into."
Unbeknownst to Carr, she was also walking into a period of great transition.
Within a few months of her start, Hallmark, which had been looking for a strategic partner or buyer, called the search off, laid off 20 people, including the CMO, and the CEO stepped down.
"It was [key] to provide continuity for not only our business activities, but the way we related to the investment community," says programming EVP David Kenin. "She has provided that link so that the company did not miss a beat in our public information servicing."
The key for Carr was crafting a single, unified message for all.
"Publicity is 'I've got a release. How do I get it in the [media]?'" notes Carr. "PR is establishing a rapport between an organization and its various audiences."
Enhanced online offerings, such as behind-the-scenes footage and a community section for blogging and chatting, are intended to further engage an already emotionally invested following. Carr says the Web site has seen a jump in traffic over the past year, including in boomer-audience participation. And the holiday season is a time when viewers are even more receptive to the Hallmark message.
"The Hallmark Channel is relationship-oriented... [and] we're coming to a period that's particularly tied to brand essence," she notes. "When we go online, we enhance our experience on-air."
Next year, the schedule will be packed with more than 30 original movies. Each film will have its own microsite and the network is adding more program streaming, which began with a series called Adoption that, as of press time, appeared on the homepage.
"We're doing the right thing," says Carr, "and we'll do more of what we've been doing."
Nancy Carr
September 2007 to present
Hallmark Channel and Hallmark Movie Channel, SVP of Corporate Communications
Hallmark Channel and Hallmark Movie Channel, VP of Corporate Communications
CBS Television Network, VP of communications
Various posts at CBS, ranging from associate director of publicity to VP of publicity
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