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Home » Articles » Letter to the editor
Micronutrients lead the way Discusses options and strategies for improving micronutrient health in deficient populations. Provides details on USAID's efforts to reduce micronutrient deficiency through supplementation, fortification, and food-based programs. Updated in 2004 to include zinc deficiency as a significant public health issue. Enquête sur la carence en vitamine A / Madagascar This document presents a survey done in Madagascar to assess the extent of the vitamin A deficiency and anemia problems on a national scale. The survey was based on a national sample of 600 children aged 6 to 59 months and a sample of 300 women, ranging from 15 to 49 years of age. In addition, an investigation of anemia in the primary schools was conducted to assess the extent of the anemia problem in schoolchildren aged 6 to 14 years. The survey revealed that Vitamin A deficiency is a serious public health problem for children aged 6–59 months, as well as for women. It also shows that almost two of every five school children from the ages of 6 to 14 years is affected by anemia, making it an important public health problem among school age children. [This publication is in French only.] National Micronutrient Survey translation of the 2000 Summary Report – Nicaragua The Government of Nicaragua has paid special attention to micronutrient deficiencies, particularly those of vitamin A, iron, and iodine, since the 1993 National Micronutrient Survey. This summary report of the subsequent 2000 survey provides information about the evolution of these deficiencies, which are considered to be significant public health problems in Nicaragua.
DOI https://doi.org/10.52439/SJCB4017
DOI Number 10.52439/SJCB4017


LONDON, Sept 27 (Thomson Reuters Foundation) - A zero waste shop won an ethical business award in Wales on Wednesday night, highlighting a growing trend among retailers to cut plastic use as worry over its environmental impact rises.
Judges praised Natural Weigh in south Wales for educating people about the dangers of single-use plastic and for providing a solution to minimise its use - by asking customers to bring their own containers to buy food by weight.
U.N. figures show 8 million tonnes of plastic - bottles, packaging and other waste - enter the ocean each year, degrading precious habitats, killing marine life and entering the human food chain.


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Home // OARDC Report // May-June 2015 // OARDC's Linda Saif is first woman to win prestigious Wolf Prize in Agriculture
OARDC's Linda Saif is first woman to win prestigious Wolf Prize in Agriculture
OARDC scientist Linda Saif (pictured, right) has received the 2015 Wolf Prize in Agriculture for her work on viral diseases of critical importance to farm animals, food safety and human health.
A Distinguished University Professor in the Food Animal Health Research Program, Saif is the first woman and the first Ohio State scientist to receive the recognition, which the Wolf Foundation of Israel has given since 1978.
In addition to agriculture, the Wolf Prize is given for achievement in the fields of chemistry, mathematics, medicine, physics and the arts. Past recipients include Stephen Hawking (in physics) and Jared Diamond (agriculture).
"Dr. Linda Saif's work exemplifies our college's commitment to conducting meaningful research that benefits agriculture as well as human health and well-being," said Bruce McPheron, vice president for agricultural administration and dean of the College of Food, Agricultural, and Environmental Sciences. "This prestigious award is a testament to the world-class research we do at Ohio State."
"Linda Saif's work exemplifies our college's commitment to conducting meaningful research that benefits agriculture as well as human health and well-being."—Bruce McPheron
'Extensive contributions' to health, food security
In selecting Saif for the award, the Wolf Foundation noted how "her discoveries of novel enteric and respiratory viruses of food animals and humans have led to her extensive contributions of fundamental knowledge of the gut-mammary immunologic axis and has provided new ways to design vaccines and vaccination strategies."
Her discoveries, the foundation said, "have contributed immensely to the improvement of global food safety, food production as well as animal and human health."
To contact the scientist: Linda Saif at saif.2@osu.edu.

redpajama_set_name: RedPajamaCommonCrawl


In September 2017 Florencius International School opened its doors in Amstelveen. Located in the same building and based on the same principles as Florencius Primary School, Florencius International School aims to give all children with an international background (expat or repat) a broad educational foundation. The school works in small groups with each child following their own development plan. There are plenty opportunities for individual support and guidance. As we follow a holistic approach, our students are educated on physical, academic, emotional as well as spiritual levels.
Florencius International School accepts students between the ages of 5 and 12 and follows the International Primary Curriculum (IPC), giving our students the opportunity to become independent world citizens.
All lessons are conducted in English, but the students have exposure to the customs and habits of their host country. Dutch is offered as a second language if requested.


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\title{
Optimizing Space Utilization for More Effective Multi-Robot Path Planning
}
\author{Shuai D. Han$^{1}$
\quad Jingjin Yu$^{1}
\thanks{
$^{1}$S. D. Han and J. Yu are with the Department of Computer Science,
Rutgers University.
Emails: {\tt\small \{shuai.han, jingjin.yu\}@rutgers.edu}.
}%
}

\begin{document}

\maketitle
\thispagestyle{empty}
\pagestyle{empty}

\begin{abstract}
We perform a systematic exploration of the principle of Space Utilization
Optimization (\textsc{SUO}\xspace) as a heuristic for planning better individual paths
in a decoupled multi-robot path planner, with applications to both one-shot
and life-long multi-robot path planning problems. We show that the decentralized
heuristic set, \textsc{SU-I}\xspace, preserves single path optimality and significantly
reduces congestion that naturally happens when many paths are planned without
coordination. Integration of \textsc{SU-I}\xspace into complete planners brings
dramatic reductions in computation time due to the significantly reduced number
of conflicts and leads to sizable solution optimality gains in
diverse evaluation scenarios with medium and large maps, for both one-shot
and life-long problem settings.
\end{abstract}

\section{Introduction}
Recent years have witnessed a dramatic acceleration in the deployment of
multi-robot systems for general logistic tasks~\cite{dekhne2019automation},
especially in the domain of shipping and warehousing~\cite{WurDanMou08}.
Fast-paced expansion is predicted across the board, with the warehouse
domain alone expecting a $14\%$ year-over-year growth in the next five
years~\cite{covid-auto}.
This in turn demands the push for enhancing the scalability of multi-agent
and multi-robot systems, which in the end boils down to achieving the
maximum possible output attainable. Holding other variables constant,
maximizing system throughput is most readily achieved by increasing robot
density and plan optimality, which calls for faster and better computational
methods for Multi-Robot Path Planning (\texttt{MPP}\xspace) and Life-long Multi-Robot Path
Planning (\texttt{LMPP}\xspace) problems.

Toward the development of more efficient and higher performance multi-robot systems
catering to the rapidly growing need of automation, in this work, following the
decoupled planning paradigm~\cite{ErdLoz86}, we perform a systematic study of an
intuitive principle for the design of better performing heuristics for \texttt{MPP}\xspace. The
decoupled setting generally involves two planning phases, where the first phase
plans individual robot paths ignoring other robots and the second one resolves
robot-robot conflicts within some spatio-temporal window. Traditionally, this
phase is executed by running single robot path plannings ignoring other robots.
The \emph{Space Utilization Optimization} (\textsc{SUO}\xspace) principle tackles the first
planning phase, seeking to make robots use the free space ``evenly''.

Based on vertex, edge, and temporal usage information, \textsc{SU-I}\xspace, as our
implementation of the \textsc{SUO}\xspace principle, builds a global heuristic that tracks
how the free space is being used among all participating robots.
We then exploit applying \textsc{SU-I}\xspace as both an \emph{estimated cost-to-go},
for reducing congestion, and as part of the \emph{cost-to-come}, for
reducing conflicts along the entire robot path. Theoretically, we prove
that \textsc{SU-I}\xspace does not compromise individual path optimality while simultaneously
achieves its design goal of providing better space utilization. In practice,
\textsc{SU-I}\xspace leads to significant improvement in initial path quality, resulting in
over $40\%$ reduction of path conflicts.

The introduction of \textsc{SU-I}\xspace brings notable improvement to a multitude of \texttt{MPP}\xspace
and \texttt{LMPP}\xspace benchmarks. \textsc{SU-I}\xspace, which may be applied as an orthogonal heuristic to
many \texttt{MPP}\xspace algorithms, leads to sizable gains in both computation time and
solution optimality when combined with efficient methods for the second phase
of a decouple planner for path scheduling, like DDM~\cite{han2020ddm} and
ECBS~\cite{barer2014suboptimal}. For example, using \textsc{SU-I}\xspace with the database-driven
collision resolution from \cite{han2020ddm} leads to $15\%+$ reduction in
computation time and improves solution optimality by roughly $25\%$.
For \texttt{LMPP}\xspace, \textsc{SU-I}\xspace, with additional planning horizon management, could reduce the
computation time of a state-of-the-art methods \cite{li2020lifelong} by more than
$65\%$, while keeping the same level of optimality.

\begin{comment}
\begin{enumerate}
    \item MRPP background.
    \item Existing algorithms often use individual paths but do not care about single robot path finding:
    the path are planned randomly and independently, which causes unbalanced graph resource utilization.
    \item Noticing the above fact, we study the \textsc{SUO}\xspace principle: the robot paths should have a more
    balanced usage of graph avoid congestion from a higher level.
    \item We propose \textsc{SU-I}\xspace, which is a global heuristic that can help generate single robot paths
    with fewer conflicts.
    \item (maybe move Fig.~\ref{fig:example-suo} here)
    \item \textsc{SU-I}\xspace is based on vertex, edge, temporal information. Each component has its own justification
    to be used for \textsc{SUO}\xspace.
    \item We use \textsc{SU-I}\xspace based on two objectives: reducing the conflicts at the most congested graph areas
    (use as part of the estimated cost-to-go), or reducing the conflicts along an entire path
    (use as part of cost-to-come).
    \item Theoretical results are provided: our \textsc{SU-I}\xspace method can find the shortest path that fulfills
    the objectives.
    \item Both concepts was implemented and evaluated. The evaluation result shows significant
    improvement in initial path quality (e.g., $40\%$ (number may change) less vertex conflict).
    \item We propose a horizon-cut technique to an existing bounded-horizon lifelong algorithm.
    We further apply \textsc{SU-I}\xspace to it to improve both its efficiency and throughput.
    \item We propose an adaptive horizon method to automatically adjust the planning horizon to
    optimize for computation time.
    \item \textsc{SU-I}\xspace is them combined with DDM and a priority based solver. We see improvement in
    computation time, solution optimality, and success rate.
\end{enumerate}
\end{comment}

\textbf{Related Work. }
Multi-Robot Path Planning (\texttt{MPP}\xspace), or equivalently, Multi-Agent Path Finding
(\texttt{MAPF}\xspace)~\cite{stern2019multi}, has been actively studied for decades from many
angles including computational complexity and effective algorithm
design~\cite{Gol84, ErdLoz86, GuoPar02}.
Until a few years back, studies on \texttt{MPP}\xspace focus mainly on \emph{one-shot} or
\emph{static} problems, where $n$ robots are to reach $n$ specific goals.
Many algorithms for computing high-quality or optimal solutions have been proposed.
Decoupled solutions~\cite{ErdLoz86} dominate the algorithmic attack, with methods
using techniques including independence detection~\cite{StaKor11}, sub-dimensional
expansion~\cite{wagner2015subdimensional},
conflict-based search~\cite{boyarski2015icbs, cohen2016improved}, among others.
Methods have also been proposed through the reduction to other problems including
satisfiability (SAT)~\cite{Sur12},
Answer Set Programming (ASP)~\cite{erdem2013general},
and multi-commodity flow~\cite{YuLav16TRO}.
There also exists prioritized
methods~\cite{BerOve05, bennewitz2002finding, saha2006multi,BerSnoLinMan09}
and a divide-and-conquer approach~\cite{yu2018constant}
which achieve good scalability but at the cost of either completeness or optimality.
In \cite{vedder2019x}, an any time algorithm is proposed to quickly find a feasible
solution, which is subsequently improved.
A learning-assisted approach~\cite{sigurdson2019automatic} has been developed to
automatically select the algorithm for solving \texttt{MPP}\xspace challenges.

With the rise of multi-robot applications in the logistics domain~\cite{WurDanMou08},
\emph{dynamic} or \emph{life-long} \texttt{MPP}\xspace variants, or \texttt{LMPP}\xspace, have attracted
attentions in the past few years. Recent work has focused on dynamic warehouse
setups, pursuing both better planning algorithms~\cite{ma2017lifelong} and robust
execution schedules~\cite{hoenig2019persistent}. Prioritized planning method
with a flexible priority sequence has also been developed~\cite{okumura2019priority}.

The general idea behind the \textsc{SUO}\xspace principle, better usage of the shared free space,
has been explored under both single and multi-robot settings. For single robot
exploring a obstacle-laden domain, a path ensemble can increase the chance of
succeeding in finding a longer horizon
plan~\cite{branicky2008path, erickson2009survivability}.
Similar to what we explore in the current study, path diversity is just one of the
relevant factors in \texttt{MPP}\xspace resolution~\cite{knepper2009path}.
Survivability is also examined under a probabilistic framework for multi-robot
systems \cite{lyu2016k}. Under a similar context, a heuristic based on path conflicts
expedites the solution process of an \texttt{MPP}\xspace algorithm~\cite{barer2014suboptimal}.

Despite the fact that the \textsc{SUO}\xspace principle is intuitive, to our knowledge, our
exploitation of the principle, building on our initial work \cite{han2020ddm},
makes novel contributions to the field. In contrast to \cite{han2020ddm}, this study
{\em (i)} thoroughly exploits the \textsc{SUO}\xspace principle with the introduction of the \textsc{SU-I}\xspace
heuristics with a number of variations;
{\em (ii)} proves \textsc{SU-I}\xspace's collision avoidance property:
it finds the shortest paths while minimizes certain collision-based metrics;
{\em (iii)} integrates \textsc{SUO}\xspace to \texttt{MPP}\xspace and \texttt{LMPP}\xspace algorithms and empirically shows that
using \textsc{SU-I}\xspace  benefits both computation time and solution optimality.

\section{Preliminaries}
\subsection{Problem Statement}
Multi-Robot Path Planning (\texttt{MPP}\xspace) tasks to find collision-free paths
that efficiently route robots.
Consider an undirected graph $\mathcal G(V, E)$ and $n$ robots
with start configuration $S = \{s_1, \dots, s_n\} \subseteq V$ and goal configuration
$G = \{g_1, \dots, g_n\} \subseteq V$. Each robot has start and goal vertices $s_i$, $g_i$.
We define a {\em path} for robot $i$ as a map
$P_i: \mathbb {N} \to V$ where $\mathbb N$ is the set of non-negative integers.
A feasible $P_i$ must be a sequence of vertices that connects $s_i$ and $g_i$:
1) $P_i(0) = s_i$;
2) $\exists T_i \in \mathbb N$, s.t. $\forall t \geq T_i, P_i(t) = g_i$;
3) $\forall t > 0$, $P_i(t) = P_i(t - 1)$ or $(P_i(t), P_i(t - 1)) \in E$.

Here, we first define the single robot version of the problem.

\begin{problem}{\bf Single-Robot Path Planning (\texttt{SPP}\xspace).}\label{prob:spp}
    Given $\mathcal G, s, g$, find a feasible path $P$.
\end{problem}

Following the feasibility definition of $P_i$, we denote $T_i$ as the
{\em length} of $P_i$.
We call $P_i$ the {\em shortest path} for robot $i$ if and only if it minimizes $T_i$.
Given a set of paths $\{P_1, \dots, P_n\}$ for all robots,
we call them {\em collision-free} if and only if there are no simultaneous
occupancy of the same vertex or edge.
That is, $\forall 1 \leq i < j \leq n$, $P_i, P_j$ must satisfy:
1) $\forall t \geq 0$, $P_i(t) \neq P_j(t)$;
2) $\forall t > 0$, $(P_i(t - 1), P_i(t)) \neq (P_j(t), P_j(t - 1))$.

We say paths are {\em independent} if they are
feasible single robot paths but not necessarily collision-free.

The traditional one-shot \texttt{MPP}\xspace problem is defined as

\begin{problem}{\bf Multi-Robot Path Planning (\texttt{MPP}\xspace).}\label{prob:mpp}
    Given $\mathcal G, S, G$, find a collision-free path set $\{P_1, \dots, P_n\}$.
\end{problem}

An {\em optimal} solution for \texttt{MPP}\xspace may minimize the {\em makespan}
$\max_{1 \leq i \leq n} |T_i|$
or {\em sum-of-cost} $\sum_{1 \leq i \leq n} T_i$.

Apart from \texttt{MPP}\xspace, we also study the {\em life-long}
variation \texttt{LMPP}\xspace where each robot has {\em a list of goal vertices}.
We denote the goal configuration as $\mathbf G = \{\mathbf g_1, \dots, \mathbf g_n\}$
where $\mathbf g_i = (g_i^1, g_i^2, \dots)$.
Here, $g_i^k$ is the $k$-th goal in robot $i$'s goal list $\mathbf g_i$.
Note that  for an actual \texttt{LMPP}\xspace instance, $\mathbf g_i$ is often a list that
is constantly updated.
In \texttt{LMPP}\xspace, the second condition for a feasible path $P_i$ becomes:
2) $\exists T_i^1 < T_i^2 < \dots \in \mathbb N$, s.t. $P_i(T_i^k) = g_i^k$.

With all other conditions inherited from \texttt{MPP}\xspace, we have

\begin{problem}{\bf Life-long Multi-Robot Path Planning (\texttt{LMPP}\xspace).}\label{prob:mppl}
    Given $\mathcal G, S, \mathbf G$, find a collision-free path set
    $\{P_1, \dots, P_n\}$.
\end{problem}

\texttt{LMPP}\xspace algorithms often optimizes {\em throughput}, i.e., the average
number of goal reaches in a unit time step.
Given a large $T \in \mathbb N$, the throughput can be expressed by
$(\sum_{1 \leq i \leq n}\argmax_{k}  (T_i^k | T_i^k \leq T)) / T$.

For practicality and simplifying explanation, we assume $\mathcal G$ is a
$4$-connected grid. All algorithms and heuristics proposed in this paper
apply to arbitrary graphs.

\subsection{Importance of \texttt{SPP}\xspace in Solving \texttt{MPP}\xspace and \texttt{LMPP}\xspace}\label{subsec:spp-mpp}

Solving \texttt{SPP}\xspace is a stepping stone toward \texttt{MPP}\xspace and \texttt{LMPP}\xspace.
Many \texttt{MPP}\xspace solvers incorporate \texttt{SPP}\xspace planners as sub-routines, e.g.,
decoupled planners generally use a two-phase approach to first plan
independent paths and then resolve collisions.
Such methodologies are popular for two reasons.
First, unlike multi-robot planning which is hard to
optimize \cite{Yu2015IntractabilityPlanar},
\texttt{SPP}\xspace has been thoroughly studied and can be solved efficiently and optimally
using A* search with simple heuristics.
Second, in a practical setting with relatively low robot density,
usually only small sub-groups of robots have local interactions
at any given time, which can be quickly resolved.

\begin{wrapfigure}[6]{r}{0.25\linewidth}
    \hspace*{-10pt}
    \includegraphics[width = \linewidth]{figures/histo.png}
\end{wrapfigure}
However, such a two-phase approach tends to overuse parts of the free space while leaving
other parts underutilized. To demonstrate the effect, we plan shortest paths
from the top left corner to the bottom right corner on a $7 \times 7$
grid with randomized node exploration and randomized tie-breaking.
The shades of cells in the figure visualizes the probability a
cell is to be used. The paths concentrate along the diagonal connecting
the two corners, leaving the top right and bottom left corners underutilized.
This effect gets more pronounced when obstacles exist and the available
path choices are limited.

\texttt{LMPP}\xspace, more general and practical than \texttt{MPP}\xspace, is often solved iteratively
using existing \texttt{MPP}\xspace algorithms. Thus, unbalanced graph utilization issue also
negatively impacts \texttt{LMPP}\xspace solvers.

\section{\textsc{SUO}\xspace: Principles and Implementation}\label{sec:suo}
\subsection{Optimizing Space Utilization: Principles}
From the discussion in Section~\ref{subsec:spp-mpp}, robots' individual paths
should be spread ``evenly'' across the free space to reduce congestion.
We call this idea the {\em Space Utilization Optimization (\textsc{SUO}\xspace)
principle}. In this work, we develop {\em a first \textsc{SUO}\xspace implementation},
\textsc{SU-I}\xspace (SU-First), which
serves as a global heuristic to help generate independent paths for
\texttt{MPP}\xspace/\texttt{LMPP}\xspace algorithms to better utilize graph resources.
The example in Fig.~\ref{fig:example-suo} demonstrates that \textsc{SU-I}\xspace
could potentially
enhance both computational efficiency and optimality.

\begin{figure}[h!]
    \centering
    \subfloat[Without \textsc{SU-I}\xspace]{\includegraphics[width=0.32\linewidth]{figures/example-wo-suo.pdf}\label{fig:example-wo-suo}}
    \hspace{20pt}
    \subfloat[With \textsc{SU-I}\xspace]{\includegraphics[width=0.32\linewidth]{figures/example-w-suo.pdf}\label{example-w-suo}}
    \caption{Two sets of individual paths in a $5 \times 3$ grid.
    The robots' start configuration is shown as colored disks.
    The planned individual paths are drawn as colored lines.
    (a) Randomly generated paths may result in congestion.
    (b) When using \textsc{SU-I}\xspace, which ``spreads out'' the robots' paths,
    the number of conflicts is reduced.
    }
    \label{fig:example-suo}
\end{figure}

In \textsc{SU-I}\xspace, each path uses vertex, edge, and temporal information from other paths
to avoid more congested areas or to reduce the sum of conflicts
along the path.
\textsc{SU-I}\xspace improves upon its predecessor~\cite{han2020ddm} which only considered vertex
information.
In Appendix~\ref{app:examples}, we use several examples to highlight the essence of
vertex, edge, and temporal information when applying \textsc{SUO}\xspace.

\subsection{Path Planning with \textsc{SU-I}\xspace: High Level Procedure}

We now describe the implementation of \textsc{SU-I}\xspace, including
theoretical guarantees and evaluation results to show
that \textsc{SU-I}\xspace with the proposed path planning procedure indeed optimizes
space utilization. Since \texttt{LMPP}\xspace algorithms are often based on \texttt{MPP}\xspace solvers,
we only describe \textsc{SU-I}\xspace in the context of \texttt{MPP}\xspace; \textsc{SU-I}\xspace is directly
applicable to \texttt{LMPP}\xspace.

\begin{algorithm}
\begin{small}
\DontPrintSemicolon
\caption{\label{alg:search} Generate independent paths with \textsc{SU-I}\xspace}
\KwIn{Graph $\mathcal G$, $n$ robots with starts $S$ and goals $G$}
$\pi \gets \textsc{GetOrderByDistanceToGoal}(\mathcal G, S, G)$\;\label{alg:search:order}
$S \gets \pi(S), G \gets \pi(G)$\;\label{alg:search:sort}
\lFor{$1 \leq i \leq n$}{$P_i \gets {\text{None}}$}
\For{number of planning iterations $r$}{\label{alg:search:multiple}
\For{$1 \leq i \leq n$}{\label{alg:search:for}
    $\mathcal T \gets \textsc{BuildSuo}(P_1, \dots, P_{i-1}, P_{i+1},\dots, P_n)$\;\label{alg:search:suo}
    $P_i \gets \textsc{FindPath}(\mathcal G, s_i, g_i, \mathcal T)$\;\label{alg:search:search}
}
}
\Return{$\pi^{-1}(P_1, \dots, P_n)$}
\end{small}
\end{algorithm}

Alg.~\ref{alg:search} shows the high level procedure that uses \textsc{SU-I}\xspace
to generate individual paths.
In lines~\ref{alg:search:order}-\ref{alg:search:sort}, robots are sorted in
{\em descending} distance-to-goal order.
Then, paths for the robots are planned sequentially
(lines~\ref{alg:search:for}-\ref{alg:search:search}) while using \textsc{SU-I}\xspace to
avoid previously planned paths.
The rationale for the descending ordering is that planning longer paths first helps
quickly collect graph utilization information.
As indicated in line~\ref{alg:search:multiple}, the path planning procedure
is repeated for $r$ iterations as more iterations capture the space
utilization more accurately, improving the collision avoidance.
Both robot ordering and number of iterations are tested in Section~\ref{sec:evaluation}.

\subsection{\textsc{SU-I}\xspace Heuristic Construction and Computation}
At line~\ref{alg:search:suo}, before each time
we call \texttt{SPP}\xspace planner, \textsc{BuildSuo} first builds an \textsc{SU-I}\xspace lookup table $\mathcal T$
to memorize the space usage of existing paths.
Denoting $\mathcal T(v, t)$ (resp. $\mathcal T(v_1, v_2, t)$) as the expected
usage of vertex $v$ (resp. directed edge $v_1, v_2$) at time step $t$,
$\mathcal T$ aggregates vertex,
edge, and temporal information over all \textsc{BuildSuo} input paths:
\begin{align*}
    \mathcal T (v, t) & = \textstyle \sum_{P} [(v) \preceq P_i],
\end{align*}
\begin{align*}
    \mathcal T (v_1, v_2, t) & = \textstyle \sum_{P} [(v_1, v_2) \preceq P_i].
\end{align*}
Here, $[\cdot]$ on the right hand side of the equations is an \emph{indicator} variable:
it is $1$ if the expression inside is true, otherwise it takes $0$.
We use $\preceq$ to denote a sub-sequence relationship.

\textsc{SU-I}\xspace is mainly used to generate initial paths for \texttt{MPP}\xspace algorithms.
Since the initial paths can be modified (e.g. delayed or diverted
for collision avoidance) in a full \texttt{MPP}\xspace algorithm,
we add integer parameters $\alpha_L \geq 0$ and $\alpha_H \geq 0$
to reason about graph usage in adjacent time steps
and handle robot movement uncertainty:
\begin{align*}
    \mathcal T (v, t) & = \textstyle \sum_{P} [(v) \preceq P(t - \alpha_H: t + \alpha_L)],
\end{align*}
\begin{align*}
    \mathcal T (v_1, v_2, t) & = \textstyle \sum_{P} [(v_1, v_2) \preceq P(t - \alpha_H: t + \alpha_L)].
\end{align*}
For each path, the occupancy of graph utility at time $t$
can now expand its influence to time steps in $(t - \alpha_L, t + \alpha_H)$.
The values of $\alpha_L, \alpha_H$ are empirically determined based on
the \texttt{MPP}\xspace algorithm itself; in Section~\ref{sec:evaluation}, we observe that such an
temporal reasoning feature delivers better collision avoidance results when
the movement of robots is uncertain.

We use $\mathcal T$ to calculate \textsc{SU-I}\xspace heuristic and then use it during
\texttt{SPP}\xspace path planning (line~\ref{alg:search:search}).
Given a state transition $(v_1, v_2, t)$ which means the robot moves
from vertex $v_1$ to vertex $v_2$ at time $t$, the \textsc{SU-I}\xspace heuristic value is
\begin{equation*}\label{equation:suo}
    H_\text{\textsc{SU-I}\xspace}(v_1, v_2, t) = \beta_v \frac{\mathcal T(v_2, t)}{n} +
    \beta_e \frac{\mathcal T(v_2, v_1, t)}{n}.
\end{equation*}
Here, the first term indicates the amount of vertex conflicts the robot
may encounter for the inferred state transition, while the second term
is associated with head-to-head edge collisions.
Parameters $\beta_v, \beta_e>0$ are used to balance between vertex and
edge information; $\beta_v + \beta_e = 1$.
When each vertex/edge in the graph is visited for at most once by each single robot path, we have
\begin{lemma}\label{lemma:occupancy-bound}
When $\beta_v + \beta_e = 1$, $0 \leq H_\text{\textsc{SU-I}\xspace}(\cdot) < 1$.
\end{lemma}

The condition $H_\text{\textsc{SU-I}\xspace}(\cdot) < 1$ is essential for
\textsc{SU-I}\xspace to behave as a tie-breaker,
which facilitates a good balance between single path optimality and congestion avoidance.

\begin{remark}\label{remark:no-temp}
A special case for constructing $\mathcal T$ is to ignore temporal information,
i.e., instead of using $(v, t)$ and $(v_1, v_2, t)$ as lookup table keys, we
use $(v)$ and $(v_1, v_2)$. Thus, $\mathcal T(v)$ (resp., $\mathcal T(v_1, v_2)$) simply
records the total number of times the vertex $v$ (resp., the edge $(v_1, v_2)$)
is used by the existing paths. This leads to smaller lookup tables
but potentially worse collision avoidance as a result.
\end{remark}

\begin{remark}
Note that for the actual implementation, in Alg.~\ref{alg:search}, line~\ref{alg:search:suo},
$\mathcal T$ is not re-constructed but updated based on the previous iteration which makes
the computational complexity for the construction step $O(|P|(\alpha_L + \alpha_H))$.
\end{remark}

Line~\ref{alg:search:search} uses standard A* to find a path from $s_i$ to $g_i$.
In the next two subsections, we discuss two ways to integrate \textsc{SU-I}\xspace into A*.
For simplicity, we now assume \textsc{SU-I}\xspace only uses vertex information
(i.e., $\beta_v = 1, \beta_e = 0$) without temporal information
(see Remark~\ref{remark:no-temp}), unless otherwise specified.

\subsection{\textsc{SU-I}\xspace as Part of Estimated Cost-To-Go}\label{sec:suo-cost-to-go}
We use $H_{\text{short}}(v, g_i)$ to denote the shortest path
distance between $v$ and $g_i$ in $\mathcal G$.
$H_{\text{short}}(v, g_i)$ is graph-dependent and can be calculated before
path planning. For a grid without obstacles, $H_{\text{short}}(v, g_i)$ is
the Manhattan distance heuristic. The heuristic we use in A* search is
\[\mathcal H(v) = H_{\text{short}}(v, g_i) + H_{\textsc{SU-I}\xspace}(v).\]

\begin{lemma}\label{lemma:shortest}
A path planned using A* search with $\mathcal H$ as heuristic
is a shortest path from $s_i$ to $g_i$.
\end{lemma}
\vspace*{-6pt}
\begin{proof}
See Appendix~\ref{app:proof}.
\end{proof}
\vspace*{-3pt}

$\mathcal H$ not only ensures a shortest path is found; the path also
\emph{minimizes the maximum single step conflict}.
Given path $P_i$, its maximum single step
conflict can be represented as
\[\textstyle C_{\text{single}}(P_i) = \max_{0 < t < |P_i|} \mathcal T[P_i(t)].\]

\begin{lemma}\label{lemma:minimum}
A path $P_i$ planned using A* search with $\mathcal H$ heuristic
is a shortest path minimizing {\normalfont $C_{\text{single}}(P_i)$}.
\end{lemma}
\vspace*{-6pt}
\begin{proof}
See Appendix~\ref{app:proof}.
\end{proof}
\vspace*{-3pt}

Now, given paths $P_1, \dots, P_n$ returned from Alg.~\ref{alg:search},
denoting the maximum conflict on a single vertex as
\[\textstyle \mathcal C_{\text{single}} =
\max_{v \in \mathcal G} \sum_{1 \leq i \leq n} [v \in Im(P_i)],\]
we reach the following property.

\begin{lemma}\label{lemma:cost-to-go-converge}
$\mathcal C_{\text{single}}$ cannot increase after the first \textsc{SU-I}\xspace planning iteration.
\end{lemma}
\vspace*{-6pt}
\begin{proof}
See Appendix~\ref{app:proof}.
\end{proof}
\vspace*{-3pt}

Lemma~\ref{lemma:cost-to-go-converge} directly leads to the following theorem.
\begin{theorem}
$\mathcal C_{\text{single}}$ will converge as the number of planning iterations increases.
\end{theorem}

\subsection{\textsc{SU-I}\xspace as Part of Cost-To-Come}

With the default transition cost as $1$ for all states during A* search,
when aggregating \textsc{SU-I}\xspace it into cost-to-come.
we define the new transition cost leading to vertex $v$ as
\begin{equation*}\label{equation:cost}
    \textstyle
    C(v) = 1 + H_{\textsc{SU-I}\xspace}(v) / (\max_{1 \leq i \leq n} H_\text{short}(s_i, g_i) + 1) .
\end{equation*}
The collision avoidance property of using \textsc{SU-I}\xspace with cost-to-come is different from
that of using \textsc{SU-I}\xspace with cost-to-go,
We hereby define the number of vertex collisions on $P_i$ as
\begin{align*}
    C_\text{path}(P_i)
    &= \textstyle \sum_{1 \leq j \leq n, j \neq i} |\text{Im}(P_i) \cap \text{Im}(P_j)|.
\end{align*}

\begin{lemma}\label{lemma:cost-to-come-shortest}
A path $P_i$ planned using A* search with $C$ as transition cost
and an admissible heuristic
is the shortest path which minimizes {\normalfont $C_{\text{path}}(P_i)$}.
\end{lemma}
\vspace*{-6pt}
\begin{proof}
See Appendix~\ref{app:proof}.
\end{proof}
\vspace*{-3pt}

Given resulting paths as $P_1, \dots, P_n$ when using \textsc{SU-I}\xspace as cost-to-come,
we denote the total number of collisions as
\[\textstyle \mathcal C_{\text{path}} = \sum_{1 \leq i \leq n} C_{\text{path}}(P_i), \]
we find the following property.

\begin{lemma}\label{lemma:cost-to-come-converge}
$\mathcal C_{\text{path}}$ cannot increase after the first \textsc{SU-I}\xspace iteration.
\end{lemma}
\vspace*{-6pt}
\begin{proof}
See Appendix~\ref{app:proof}.
\end{proof}
\vspace*{-3pt}

Lemma~\ref{lemma:cost-to-come-converge} directly leads to the following theorem.
\begin{theorem}
$\mathcal C_{\text{path}}$ will converge as the number of planning iterations increases.
\end{theorem}

\begin{remark}
Regardless of whether \textsc{SU-I}\xspace is used as part of the cost-to-come or the cost-to-go,
even though the properties mentioned above are only proved for \textsc{SU-I}\xspace without
temporal information, similar properties exist for \textsc{SU-I}\xspace with temporal information
when using state-time A*.
Instead of just considering vertex conflicts,
all lemmas in this section remain true when edge conflicts are considered (i.e. $\beta_e > 0$),
except for Lemma~\ref{lemma:cost-to-go-converge} and Lemma~\ref{lemma:cost-to-come-converge}.
However, in evaluation, we empirically observe that running multiple planning iterations
considering edge conflicts is still beneficial.
\end{remark}

\section{Space Utilization Optimization Application}

The paths generated by \textsc{SU-I}\xspace can be directly used as input to some \texttt{MPP}\xspace/\texttt{LMPP}\xspace
algorithms.
The more balanced graph utilization and reduced conflicts facilitate collision
resolution, improving both computation time and solution optimality.

\textsc{SU-I}\xspace can also be combined with time-based divide-and-conquer to provide better intermediate goals.
With a baseline structure adapted from \cite{li2020lifelong},
we first propose a {\em horizon cut} technique to reduce unnecessary
node explorations and then use \textsc{SU-I}\xspace to further enhance the performance.

\subsection{Baseline Bounded-Horizon Search for \texttt{LMPP}\xspace}\label{sec:algorithm:lifelong}

It is well known that solving an entire \texttt{LMPP}\xspace instance in one-shot is impractical,
not only because the long lists of goals makes the problem computationally demanding,
but also because the goal lists could be dynamically updated in real world scenarios,
which invalidates the current solution and brings the need for online re-planning.

Given the above factors, \texttt{LMPP}\xspace is usually solved by a bounded-horizon approach,
The basic idea is to plan the paths for the $h$ time steps,
execute the paths, and then iteratively re-plan and execute.
Here, $h$ is called the {\em planning horizon}.
The pseudocode of such a baseline horizon-based structure~\cite{li2020lifelong} is
provided in Alg.~\ref{alg:horizon}; readers may ignore
lines~\ref{alg:horizon:cut-start}-\ref{alg:horizon:cut} for now as we will
discuss later.
In the beginning, the goal list $\mathbf{g}_i$ for each robot $i$ is shortened
until there is at most one unreachable goal for horizon $h$
(lines~\ref{alg:horizon:truncate:1}-\ref{alg:horizon:truncate:2});
we use $(-1)$ to index the last element in a sequence, and $+$ to indicate sequence
concatenation.
Then, in line~\ref{alg:horizon:search}, the current state (denoted as $S$)
and the modified goal list are sent to a windowed \texttt{MPP}\xspace solver.
The behavior of the windowed solver is to output an $h$-step collision-free path set,
where each robot $i$ aims to traverse the shortened goal list $\mathbf{\hat{g}}_i$ in order.
The selection of the windowed \texttt{MPP}\xspace solver is flexible.
In this work, we use Bounded-Horizon Enhanced Collision Based Search~\cite{li2020lifelong}
with weight parameter $w = 1.5$ and treat it as a black box.

\begin{algorithm}
\small
\DontPrintSemicolon
\caption{\label{alg:horizon} \textsc{BoundedHorizonSearch}}
\KwIn{Graph $\mathcal G$, current state $S$, goal lists $\mathbf G$, horizon $h$.}
\KwOut{A $h$-step solution}
$\forall 1 \leq i \leq n$, $\mathbf{\hat{g}}_i \gets (s_i)$, $d_i \gets 0$\;\label{alg:horizon:truncate:1}
\For{$1 \leq i \leq n$}{
    \For{$g \in \mathbf{g}_i$}{
        $\mathbf{\hat{g}}_i \gets \mathbf{\hat{g}}_i + (g)$\;
        $d_i \gets d_i + \textsc{ShortestDistance}(\mathbf{\hat{g}}_i(-1), g)$\;
        \lIf{$d_i \geq h$}{break\label{alg:horizon:truncate:2}}
    }
}
\If{using horizon cut}{\label{alg:horizon:cut-start}
$\forall 1 \leq i \leq n$, $P_i \gets ()$\;
\For{$1 \leq i \leq n$}{
    \If{using \textsc{SU-I}\xspace}{
        $H_\textsc{SU-I}\xspace \gets \textsc{BuildSuo}(P_1, \dots, P_n)$\;\label{alg:horizon:suo2}
        $P_i \gets \textsc{FindPath}(\mathcal G, \mathbf{\hat{g}}_i(-2), \mathbf{\hat{g}}_i(-1), H_\textsc{SU-I}\xspace)$\label{alg:horizon:suo1}
    }
    \lElse{
        $P_i \gets \textsc{FindPath}(\mathcal G, \mathbf{\hat{g}}_i(-2), \mathbf{\hat{g}}_i(-1))$\label{alg:horizon:simple-cut}
    }
    $\mathbf{\hat{g}}_i(-1) \gets P_i(- (|P_i| - d_i + h - 1))$\;\label{alg:horizon:cut}
}
}
\Return{$\textsc{WindowedSolver}(S, \mathbf{\hat{g}}_1(1:), \dots, \mathbf{\hat{g}}_n(1:), h)$}\label{alg:horizon:search}
\end{algorithm}

\subsection{Horizon Cut and \textsc{SU-I}\xspace Integration}

Due to using \texttt{SPP}\xspace algorithm as subroutine, the baseline bounded-horizon
method wastes computation power due to unnecessary reasoning about
paths outside horizon $h$: although these paths are not collision-checked,
they are planned by the windowed solver and are discarded afterwards.
To overcome this weakness, we propose {\em horizon cut},
which reduces the number of search nodes generated outside of $h$
while still ensures that the robots move toward their future goals.
Horizon cut further truncate the goal list by changing the last goal to a vertex that is
in-between the second last goal (i.e., $\mathbf{\hat{g}}_i(-2)$) and the last goal
(see Alg.~\ref{alg:horizon}, lines~\ref{alg:horizon:simple-cut}-\ref{alg:horizon:cut}).
In our implementation, we find a shortest path between $\mathbf{\hat{g}}_i(-2)$ and
$\mathbf{\hat{g}}_i(-1)$ and select the vertex at $t = h + 1$.
The dashed paths in
in Fig.~\ref{fig:horizon-comparison:baseline} and Fig.~\ref{fig:horizon-comparison:cut}
demonstrate that we can avoid planning redundant paths when using horizon cut.

\begin{figure}[h!]
    \centering
    \subfloat[Baseline]{\includegraphics[width=0.32\linewidth]{figures/example-horizon-baseline.pdf}\label{fig:horizon-comparison:baseline}}
    \hspace*{1pt}
    \subfloat[Horizon cut]{\includegraphics[width=0.32\linewidth]{figures/example-horizon-cut.pdf}\label{fig:horizon-comparison:cut}}
    \hspace*{1pt}
    \subfloat[Adding \textsc{SU-I}\xspace]{\includegraphics[width=0.32\linewidth]{figures/example-horizon-suo.pdf}\label{fig:horizon-comparison:suo}}
    \caption{An example comparing bounded-horizon baseline, horizon cut,
    and horizon cut with \textsc{SU-I}\xspace.
    The robots each has $2$ goals to reach, marked as colored numbers.
    The planned configuration after the current horizon ($h = 4$) is visualized as transparent disks.
    (a) The paths planned by the baseline bounded-horizon method.
    The dashed part is planned but not executed.
    (b) Using horizon cut, we avoid planning unnecessary steps by setting the last goals as $2^*$.
    (c) With \textsc{SU-I}\xspace, we have a better selection of $2^*$ so that the conflicts in
    the next planning horizon may be avoided beforehand.
    }
    \label{fig:horizon-comparison}
\end{figure}

We then integrate \textsc{SU-I}\xspace to help select better target vertices
(see Alg.~\ref{alg:horizon}, lines~\ref{alg:horizon:suo2}-\ref{alg:horizon:suo1},
which is a similar procedure as Alg.~\ref{alg:search}).
By selecting target vertices on paths which better utilize
graph resources, we can avoid conflicts in the future planning iterations.
We visualize the effect in Fig.~\ref{fig:horizon-comparison:suo},
where we anticipate less conflicts between the two robots in
the next planning and execution iteration when using \textsc{SU-I}\xspace.

\section{Evaluation}\label{sec:evaluation}

We performed comprehensive evaluation of \textsc{SU-I}\xspace and associated algorithms on
randomly generated graphs, warehouse-style graphs, and large DAO
maps (Fig.~\ref{fig:graphs} shows a subset of graphs). All experiments are performed on an
Intel\textsuperscript{\textregistered} Core\textsuperscript{TM} i7-6900k CPU.
Data points are averaged over $30$ to $100$ runs on randomly generated problem
instances.

\begin{figure}[h!]
    \centering
    \subfloat[$30 \times 20$ random graph]{\includegraphics[height=0.22\linewidth]{figures/30x20.pdf}\label{fig:graphs:random}}
    \hspace*{1pt}
    \subfloat[$37 \times 20$ warehouse]{\includegraphics[height=0.22\linewidth]{figures/warehouse.pdf}\label{fig:graphs:warehouse}}
    \hspace*{1pt}
    \subfloat[den520d]{\includegraphics[height=0.22\linewidth]{figures/den520d.pdf}\label{fig:graphs:den520d}}
    \caption{Example of graphs used for evaluation.
    The bright cells visualize vertices. The black and green cells visualize obstacles.
    }
    \label{fig:graphs}
    \vspace{-2mm}
\end{figure}

Our evaluation focuses on testing \textsc{SU-I}\xspace with existing \texttt{MPP}\xspace algorithms.
Before that, we also conducted analytical experiments, showing that
the graph utilization conflicts indeed converges to some minimal value with \textsc{SU-I}\xspace,
especially when considering both vertex and edge information with
a descending ordering of robots. For details, see Appendix~\ref{app:evaluation}.

\subsection{\textsc{SU-I}\xspace in a Full \texttt{MPP}\xspace Algorithm}\label{sec:evaluation:a}

Combining \textsc{SU-I}\xspace with an existing collision resolution method makes \texttt{MPP}\xspace solver
more efficient and optimal. For this evaluation, we used the database-driven
collision avoidance routine from~\cite{han2020ddm}. The test cases are in $30
\times 20$ grids with $10\%$ obstacles (see an random example in
Fig.~\ref{fig:graphs:random}). We report computation time and solution
optimality (based on sum-of-cost) with varied number of robots. For both
metrics, lower is better. All data points are normalized in terms of the
baseline algorithm's performance where \textsc{SU-I}\xspace is not used.

Starting with $\beta_v = \beta_e = 0.5$, $r = 1$, and no temporal information,
we first compare using \textsc{SU-I}\xspace as a part of estimated cost-to-go and cost-to-come.
As shown in Fig.~\ref{fig:ddm-ctcg}, using \textsc{SU-I}\xspace as cost-to-go can significantly
reduce the computation time (by $15\%+$) when robots have interactions.
As a comparison, using $\textsc{SU-I}\xspace$ as cost-to-come only makes the algorithm slightly
more efficient when the number of robots is large.
Both \textsc{SU-I}\xspace variations significantly improve the optimality, by up to $25\%$.
The efficiency difference was due to \textsc{SU-I}\xspace as cost-to-come minimizes collisions
along single paths and thus has a larger search space (state-time).
Hence forth, we use \textsc{SU-I}\xspace as part of estimated cost-to-go by default.

\begin{figure}[h!]
    \centering
    \includegraphics{figures/ddm-ctcg-time.pdf}
    \includegraphics{figures/ddm-ctcg-opt.pdf}
    \includegraphics{figures/ddm-ctcg-legend.pdf}
    \caption{Comparison between using \textsc{SU-I}\xspace as a part of estimated cost-to-come
    and cost-to-go in a full \texttt{MPP}\xspace algorithm.}
    \label{fig:ddm-ctcg}
\end{figure}

We then try different $\beta_v$, $\beta_e$ to demonstrate that using vertex and
edge information together is beneficial.
Shown in Fig.~\ref{fig:ddm-ve}, the performance improvement of using vertex
information alone over the baseline is already significant.
While using edge information alone is weaker than vertex,
combining the two elements pushes both computation time and solution
optimality even lower.
From now on, we set $\beta_v = \beta_e = 0.5$.

\begin{figure}[h!]
    \centering
    \includegraphics{figures/ddm-ve-time.pdf}
    \includegraphics{figures/ddm-ve-opt.pdf}
    \includegraphics{figures/ddm-ve-legend.pdf}
    \caption{\textsc{SU-I}\xspace \texttt{MPP}\xspace solver evaluation with different $\beta_v$, $\beta_e$ values.}
    \label{fig:ddm-ve}
\end{figure}

In Fig.~\ref{fig:ddm-t}, temporal information is added with different $\alpha_L$
and $\alpha_H$ values. As stated earlier, due to the modifications to the
initial paths in the second planning phase, using ``soft'' temporal information
via extended temporal reasoning is beneficial. This effect is shown here as the
green line ($\alpha_L = \alpha_H = 0$) under-performs \textsc{SU-I}\xspace without temporal
information, while the brown line ($\alpha_L = 2$, $\alpha_H = 15$) improves
solution quality. Computation time wise, using temporal information adds overhead
since it takes a longer time to construct \textsc{SU-I}\xspace. So whether to use temporal
information is a choice to be made by practitioners.

\begin{figure}[h!]
    \centering
    \includegraphics{figures/ddm-t-time.pdf}
    \includegraphics{figures/ddm-t-opt.pdf}
    \includegraphics{figures/ddm-t-legend.pdf}
    \caption{Comparison between different $\alpha_L$, $\alpha_H$ values.}
    \label{fig:ddm-t}
\end{figure}

Fig.~\ref{fig:ddm-mr} shows that using multiple planning iterations generates better solutions
when $r \leq 4$.

\begin{figure}[h!]
    \centering
    \includegraphics{figures/ddm-mr-time.pdf}
    \includegraphics{figures/ddm-mr-opt.pdf}
    \includegraphics{figures/ddm-mr-legend.pdf}
    \caption{Comparison between different planning iteration $r$.}
    \label{fig:ddm-mr}
\end{figure}

\subsection{\texttt{LMPP}\xspace Bounded-Horizon Search with \textsc{SU-I}\xspace}

We evaluate \texttt{LMPP}\xspace algorithms discussed in Sec.~\ref{sec:algorithm:lifelong} in a
warehouse-style environment (see Fig.~\ref{fig:graphs:warehouse}).
Each robot starts from a random vertex and is given a random list of goal vertices.
The goal lists are continuously extended to make sure each robot always has
future goals.
The horizon-based planners with $h = 5$ and internal ECBS~\cite{barer2014suboptimal}
parameter $w = 1.5$ are called iteratively until a total of $10000$ goals are reached.
The evaluated methods are the bounded-horizon search~\cite{li2020lifelong}
(the baseline), with horizon cut, and with \textsc{SU-I}\xspace. We report the total computation
time in Fig.~\ref{fig:ecbs-time}, and the system throughput in Table~\ref{table:ecbs-opt}.
The throughput is calculated as the average number of goal reached in a single time step;
the higher, the better.
The data shows that using horizon cut can effectively decrease the computation time
by more than $50\%$. At the same time, using \textsc{SU-I}\xspace not only makes the computation time
even lower (by about $65\%$), while keeping a same level of throughput as the baseline.
\begin{figure}[h!]
    \centering

    \includegraphics{figures/ecbs-time.pdf}
    \caption{Computation time of bounded-horizon methods.}
    \label{fig:ecbs-time}
\end{figure}

\begin{table}[h!]
    \scriptsize
    \setlength\tabcolsep{1mm}
    \centering
    \caption{\label{table:ecbs-opt} Throughput of bounded-horizon methods}
    \begin{tabular}{c|c|c|c|c}
        $n$   & Baseline &Horizon cut &\textsc{SU-I}\xspace w/o temporal&\textsc{SU-I}\xspace w/ temporal\\ \hline

        $40 $ & 1.981   &1.945        &1.968  &1.974\\ \hline

        $80 $ & 3.737   &3.647        &3.710  &3.719\\ \hline

        $120$ & 5.421   &5.273        &5.370  &5.387\\ \hline

        $160$ & 6.873   &6.714        &6.888  &6.896\\ \hline
    \end{tabular}
\end{table}

\subsection{Bounded-Horizon Search with \textsc{SU-I}\xspace on \texttt{MPP}\xspace}\label{sec:evaluation:c}

As the last evaluation, we directly apply bounded-horizon search to \texttt{MPP}\xspace
by considering \texttt{MPP}\xspace as a special case for \texttt{LMPP}\xspace where all goal lists have length $1$.
The tested graph is DAO den520d (see Fig.~\ref{fig:graphs:den520d}), a public
\texttt{MPP}\xspace benchmark~\cite{sturtevant2012benchmarks}. The map size is $257 \times 256$
with $28178$ vertices. We set $h = 50$.
Fig.~\ref{fig:dao-time} shows that \textsc{SU-I}\xspace \begin{figure}[h!]
    \centering

    \includegraphics{figures/dao-time.pdf}
    \caption{Bounded-horizon search's computation time on a den520d.}
    \label{fig:dao-time}
\end{figure}
remains effective
in large environments with a sparse robot setup. Note that when not using horizon
cut and \textsc{SU-I}\xspace, the baseline bounded-horizon search is significantly slower due
to its unnecessary exploration. When we divide the solution quality metric over
under-estimated lower-bounds (see Table~\ref{table:dao-opt}), we find that our
method is able to generate solutions very close to the optimal.

\begin{table}[h!]
\scriptsize
    \setlength\tabcolsep{1mm}
    \centering
    \caption{\label{table:dao-opt} Optimality ratio of \texttt{MPP}\xspace solutions in den520d}
    \begin{tabular}{c|c|c|c|c|c|c|c}
        $n$   & 50      & 100     & 150     & 200     & 250     & 300     & 350       \\ \hline
   Makespan   & 1.0002  & 1.0011  & 1.0020  & 1.0031  & 1.0040  & 1.0052  & 1.0064    \\ \hline
Sum-of-time   & 1.0019  & 1.0023  & 1.0034  & 1.0051  & 1.0057  & 1.0076  & 1.0090    \\ \hline
    \end{tabular}
\end{table}

Experiments in Section~\ref{sec:evaluation:a} and Section~\ref{sec:evaluation:c}
were repeated with different randomly generate graphs,
more DAO maps (e.g., brc200d, lak201d), and different number of robots.
The results show similar trends as provided in this section.

\section{Conclusion}
In this work, with \textsc{SU-I}\xspace, we performed an in-depth exploration optimizing
the space utilization for planning better individual robot paths in the first
phase of a modern decoupled \texttt{MPP}\xspace and \texttt{LMPP}\xspace pipeline. In addition to proving
that \textsc{SU-I}\xspace's desirable properties as a heuristic, thorough simulation study
validates the effectiveness of \textsc{SU-I}\xspace in significantly reducing the
computation load while maintaining or improving the optimality of the resulting
solution, for both one-shot and life-long multi-robot path planning problems.

Together with \cite{han2020ddm}, this research opens up a new direction in
multi-robot path planning. In a sense, \textsc{SUO}\xspace, and its implementation, \textsc{SU-I}\xspace, are
taking the decoupled multi-robot path planning paradigm a step further by
reducing possible
robot-robot interactions, making the process more like planning single robot
paths with loose interactions. In future research, it would be interesting to
exploit the \textsc{SUO}\xspace principle further to observe how far we can further minimize
robot-robot interaction to boost the performance of the system. One immediate
direction is to add weights to \textsc{SU-I}\xspace so that non-optimal single robot paths
will be generated and gauge the trade-off between optimality loss at the first
phase and the gain (in computation time and optimality) in the second phase of
a decouple multi-robot path planner.
Apart from handling uncertainty in time, we also plan to add mechanisms to treat space uncertainty.

\textbf{Acknowledgement.} We sincerely thank Joseph W. Durham for many insightful discussions of the work.

\clearpage
\bibliographystyle{IEEETran}

redpajama_set_name: RedPajamaArXiv


For 2016 MINI's taken their Cooper model to a whole new level, and we're here to tell you the top 5 things you need to know about the 2016 MINI Cooper.
Ah, the MINI Cooper. We dare you not to smile when you see it tottering down the road. Its iconic grin and shape are known the world over. For 2016 MINI's taken their Cooper model to a whole new level, and we're here to tell you the top 5 things you need to know about the 2016 MINI Cooper.
Now, don't get your knickers in a twist, just because it's a MINI doesn't mean it has to be, well, mini. And it's still as compact as ever, but the all-new MINI now provides more interior space for occupants and cargo in the rear. But don't worry, MINI's kept the four-corner wheel placement or that dynamic and oh-so-fun go-kart like feeling on the road despite the size increase.
While we did kind of like the centre-stack speedometer, it made it awfully tricky to weasel our way out of a pinch with the rozzers… but we digress. For the 2016 MINI Cooper, the speedometer is now behind the steering wheel (in both analogue and digital form), and in the centre stack you'll now find a nifty circle of light that indicates gear changes and in the middle of that a colour display screen for all the goodies MINI has to offer you in terms of onboard entertainment and navigation.
It seems the choices are limitless when it comes to the MIN Cooper these days! Most reently added to the stable is the brand spanking new MINI Cooper Clubman and Clubman S. Wider, longer, taller and offering all-wheel drive, this MINI is a superstar. Of course, there's also the brand new MINI Cooper Convertible, almost as new MINI Cooper 5-Door, the MINI Cooper Countryman ALL4, and the hallowed and revered MINI Cooper John Cooper Works. Essentially, you just have to buy one for every member of the family. Seriously.
No, we're not pulling your leg; all MINI Cooper models are equipped with a turbo, even the base Cooper. So, no matter which model you choose, you're going to be a-spoolin' and having a blast behind the wheel. In base trim, the MINI Cooper offers a 1.5L 3-cylinder turbo-pot, while the Cooper S steps it up with a 2.0L 4-cylinder turbocharged mill. Horsepower for each is 124 and 189 respectively. Oh, and did we mention you can order all 3-Door models with a 6-speed manual transmission? Um, yes please.
We might be biased, but we think the MINI Cooper might be the most fun you can have (legally) for the price of ownership. Built like a go-kart with all the right go-fast bits, handling paraphernalia, and a distinctive look that's as unique as the personality of the driver behind the wheel, there really is nothing else like the MINI Cooper on the road today. To get yourself behind the wheel of a 2016 MINI Cooper (where we guarantee the smile will never leave your face), head down to Mini Ottawa today to speak to one of our experts. And do not forget to book yourself a test drive!


NASA has entered the (virtual) travel business with the opening of its new Exoplanet Travel Bureau. Part of the space agency's Exoplanet Exploration website, the new "bureau" is really an interactive educational site that provides internet visitors with 360-degree artist's impressions of what the surface of a number of exoplanets might look like, as well as a collection of downloadable travel posters for those who can't wait for someone to invent a hyperdrive.
For over a quarter of a century, space scientists have been finding and confirming the existence of exoplanets – planets that orbit stars other than our Sun or even float free in interstellar space. Thanks to space telescopes like Kepler, the number of confirmed exoplanets is now in the thousands, with many more candidates awaiting confirmation.
But one frustrating aspect of all this success is that it doesn't provide much for the public imagination to latch onto. The conquest of the Moon and the solar system has been aided by the return to Earth of moon rocks and spectacular images from as far off as Pluto, but exoplanet hunting only gives us spectrograms and light intensity curves – hardly the stuff of romance and space opera.
The communications team and program chief scientists at NASA's Exoplanet Exploration Program based at the Jet Propulsion Laboratory (JPL) in Pasadena, California hope to change that with the Exoplanet Travel Bureau. By taking dull astronomical data, mixing it with a bit of imagination and a dash of poetic license, they've created explorable landscapes of what the surface of an exoplanet might look like.
NASA stresses that the alien panoramas are largely hypothetical because very little is actually known about individual exoplanets, but the goal is to give the public some idea of what these worlds might look like under given conditions. The interactive visualizations include the planet TRAPPIST-1d with a blood-red sky, an imaginary moon of the superplanet Kepler-16b that circles twin suns, and Kepler-186f, which circles a cool red dwarf star.
The views not only show the surface of the exoplanets, but are also annotated with click-on areas that tell more about the landscape and sky features. In addition, Kepler-186f allows the visitor to switch between a view of the planet with and without an atmosphere to show the effects on the sky and the surface.
For those of a more touristy bent who are annoyed that there aren't faster-than-light cruises laid on to these new worlds, NASA has also uploaded posters extolling the attractions of vacation spots beyond the solar system, including PSO J318.5-22, an extrasolar planet that orbits no star, but floats in interstellar space, making it a place where the "nightlife never ends."
"Because Kepler-186f and the majority of Kepler-discovered planets are so distant, it is currently impossible to detect their atmospheres – if they exist at all – or characterize their atmospheric properties," says Martin Still, program scientist for NASA's Transiting Exoplanet Survey Satellite (TESS). "Consequently, we have limited knowledge about what these distant worlds are really like, but these surface visualizations allow us to imagine some of the possibilities. Current and future NASA missions, including TESS and the James Webb Space Telescope, will find the nearest exoplanets to our solar system and characterize their atmospheres, bridging the gap between speculation and what's really out there."
Take a trip via NASA's Exoplanet Travel Bureau.


text:
Sharing the limelight
One problem with mergers is that the number of top executives doubles—temporarily. How can newlyweds avoid trouble in the executive suite?
TWO big mergers; two different approaches to management. The announcement on April 13th that NationsBank and BankAmerica would join forces (see article), creating America's biggest commercial bank, came fresh on the heels of the previous week's stunner by Citicorp and Travelers Group, which have agreed to the biggest merger in corporate history. The different strategies behind the two financial-services deals—Citigroup hopes to broaden its product range, the new BankAmerica to cut costs—have given analysts and investors plenty to think about. But the two new firms are also noteworthy because of the way they have dealt with their bosses.
At Citigroup, the two chief executives are going to share the top job. John Reed and Sandy Weill have each been credited with doing wonders for the companies they run (Citicorp and Travelers, respectively). But most analysts reckon that job-sharing is more suited to part-time secretaries than to the co-CEOs of a $700 billion financial conglomerate. They expect Mr Reed to be gone within a year or so. BankAmerica and Nationsbank, by contrast, have gone for a more traditional approach. Although the new bank will keep BankAmerica's name, it will be run by Hugh McColl, the 62-year-old head of NationsBank. David Coulter, BankAmerica's 50-year-old boss, must wait.
Financial-services firms are not the only ones trying to merge two top management teams into one. From drugs to computers to telecoms, companies are combining to cut costs and increase their clout. But even pundits who are inclined to see synergies behind every deal rarely expect rival management teams to mesh neatly together. While the markets are merrily popping champagne corks, senior managers are usually busy behind the scenes sharpening their knives.
It is not hard to see why. "The exact place where redundancy is most apparent is in the executive suite," says Jeffrey Sonnenfeld, a former Harvard Business School professor who now heads the Chief Executive Leadership Institute in Atlanta, Georgia. To solve this problem, newly merged companies must overcome "shrieking self-interest".
The proposed £100 billion ($165 billion) merger between Glaxo Wellcome and SmithKline Beecham, which was applauded by the markets when it was announced in February, fell apart over precisely such conflicts. And that was despite the fact that the two firms' bosses, Sir Richard Sykes and Jan Leschly, were considered to be close friends. The Glaxo-SmithKline example may seem extreme, but in many ways was typical. Because of the secrecy and speed with which the deal was negotiated, many of the important issues—such as who would occupy the top slots—had yet to be worked out when the merger was announced. Not all deals end as embarrassingly as that one, but often there are just as many loose ends.
Nor is haste the only reason. Even when companies have ample time to plan the new management team, they often put it off deliberately, reasoning that it is wise to wait. Otherwise, given the potential for conflict, many deals would never get done.
This begs the question whether such deals are worth doing in the first place. The evidence on mergers and acquisitions suggests that most of them actually destroy value. The reasons vary, but empire-building by power-hungry bosses is implicated in many failed deals. Most companies pay too much for the firms they acquire.
Because mergers are not exposed to those risks, some see them as a better bet than acquisitions. In the past, there has been little academic support for this claim. In one recent study*, however, J.P. Morgan, an investment bank (which makes some of its money from promoting mergers), examined 22 "mergers of equals" among European companies. It claims that two-thirds of these created value for shareholders. Perhaps some mergers of equals do indeed promote a more co-operative—and lucrative—approach. But that is unlikely to happen where the partners' bosses are unwilling or unable to work out the power-sharing arrangements in advance.
In some cases, therefore, the potential for management conflict is reason enough to avoid merging. But when companies do decide to join forces, how should they divide up the top jobs? Unfortunately, companies which take this question seriously will find little useful research to guide them—despite the mountain of books on leadership and teamwork published by management gurus every year. Most of these books will tell you the same thing: that co-operation is good.
But why are some managers so good at it, while others fail? Often, the support of the board of directors is crucial. If those at the top have visible and reliable backing from the board, they will find it much easier to work together. A classic example of this was the merger of ABC with Capital Cities in the 1980s. Although Capital Cities was puny compared with the giant network, its two top officers, Dan Burke and Tom Murphy, ended up running the new company—and turning it into a raging success. Not only were the two bosses themselves a model of co-operation (in 1990, they were presented with alumni achievement awards by Harvard Business School in tandem, to highlight their symbiotic relationship); but they were also able to soothe many of those at ABC who had been passed over. Some of the best people were given new roles; the rest were given time to leave gracefully.
Since ABC had acquired its smaller rival mainly to lay its hands on a better management team, the potential for conflict was huge. But the combination of a strong board and a diplomatic duo at the top defused the tension. Nor did it hurt that Messrs Burke and Murphy knew what they were doing strategically. A few mis-steps would no doubt have weakened the board's resolve.
This combination, of course, is a rare one. The problems of weak boards and poor strategies are perennial. Finding humility at the top may be even trickier.
* "European Merger of Equals". Published by J.P. Morgan, November 1997.
In search of mastery
Can China create a world-beating AI industry?
Why Microsoft is splashing $69bn on video games
Going green-ish
What is ExxonMobil's new climate strategy worth?
Where next for air travel?
Less tea, more toothpaste
Unilever's £50bn health cheque
Don't hold your breath
The tech giant's acquisition of Activision Blizzard is its biggest-ever deal
Big oil's most reluctant decarboniser lays out its green plan
New covid variants stall aviation's upward flight path
Purchasing GSK's consumer-health division may be hard to swallow


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Going to the market or store is fun when you get to pick your favourite fruits! This cute fruit basket provides hours and hours of play and you can even peel the banana! The set includes a fruit basket, a slice of melon, a kiwi, two apples, two oranges, a banana and a bunch of grapes.


I download the torrent from this url and installed a Mac using Oracle Virtual Box 4.0.4.
Installation done but didn't ask me about the user account info and now when I power on the virtual machine it stops on the user log in screen. What username and password should I try this??? Please advise?


text:
<!DOCTYPE html>

  <meta charset="utf-8">

  <title>Bus to Market</title>

  <meta name="description" content="">
  <meta name="author" content="">

  <meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
  <link href='http://fonts.googleapis.com/css?family=Happy+Monkey' rel='stylesheet' type='text/css'>

  <link rel="stylesheet" href="http://libs.cartocdn.com/cartodb.js/v3/3.14/themes/css/cartodb.css" />

  <link rel="stylesheet" href="css/theme.css">
  <link rel="stylesheet" href="css/makeitresponsive.css">

maps created by @medmond78 on cartodb

The Census

          The colored regions on this map are the 2010 Census tracts for Wake County, NC, as pulled from the Census.gov data website. On average, a tract has about 1500 households in this area. To define whether a particular area is served by the transit system
          and potentially the farmers market(s), we'll simply intersect the transit system map and the census data.

I say simply, because CartoDB's built in functionality for merging and joing tables with SQL make this drop dead simple, even for a non-programmer like me.

      back

      next

redpajama_set_name: RedPajamaGithub


Here at CoverLayout we offer tons of high quality Relationships - Husband and Wife facebook covers such as Married To My Soulmate for your Facebook profile. Installing Relationships - Husband and Wife facebook profile covers with something like the Wedding Rings cover is simple! Simply choose the Relationships - Husband and Wife facebook profile cover you would like to use, for example the Yes I still Love You cover then just click the button to update the cover to facebook, then follow the easy instructions on the next page.


We're counting down the five most popular songs in the Billboard Hot 100 Pop Singles Chart, for the week ending June 6, 2015.
The Canadian singer was one of several headliners at the eighth annual Roots Picnic in Philadelphia last weekend. Hosted by the famed rap group The Roots, this year's event also featured DJ Mustard, A.$.A.P. Rocky, and Erykah Badu.
FILE - Eli Maiman, from left, Nicholas Petricca, Kevin Ray and Sean Waugaman of the band Walk the Moon pose for photographers.
Holding in fourth place, we have Walk The Moon, whose members say they have an advantage over other bands. I'll have details after "Shut Up And Dance." This Cincinnati, Ohio band started five years ago, when music student Nick Petricca met the other three members…who were also music majors. Nick says their academic background enables them to write on a "different level" than competing bands.
Fetty Wap, left, and Pete Wentz perform at the MTV Movie Awards at the Nokia Theater, on April 12, 2015, in Los Angeles.
Wiz Khalifa arrives at the Billboard Music Awards at the MGM Grand Garden Arena, May 17, 2015, in Las Vegas.
Here's your tip-off that something big happens this week: Wiz Khalifa and Charlie Puth are no longer your champs. They end their six-week stint at the top, as "See You Again" falls to number two. Wiz hopes to keep the momentum alive as he tours North America with Fall Out Boy…they're on the road together through August 10.
Singer Taylor Swift poses at the 2015 iHeartRadio Music Awards.
When Taylor Swift does something, she does it big. Her fourth career Hot 100 championship happens this week, as "Bad Blood" rockets from 53rd to first place. Kendrick Lamar appears on the single mix, and this gives Kendrick his very first Hot 100 victory. More than ever, it pays to be a friend of Taylor!
And so we come to the end of another countdown. Join us next week for an all-new lineup.


text:
AJR's Debut Album 'Living Room' Out Now
By Acoustic Nation 04 March 2015
Breakout indie pop trio AJR have released their debut full-length album Living Room, through AJR Productions/Warner Bros. Records.
The entire album was completely self-penned, -mixed, and -produced by the brothers in the living room of their NYC apartment, with that true artistry and DIY-feel reflected in the 13 tracks on the record.
The success of Certified Gold breakout single "I'm Ready" in 2014 saw AJR perform everywhere from The Today Show and Live! With Kelly and Michael to The X-Factor in Australia (where "I'm Ready" is also certified double platinum).
New single "Infinity" is already following suit, recently added at both SiriusXM Hits1 and MTV Hits, and AJRwill perform the track on VH1's "Big Morning Buzz Live" on March 11. The band will also play their Album Release Show, presented by SiriusXM Hits1, at The Studio at Webster this Thursday, March 5, with more upcoming spring tour dates to be announced soon.
Check out the "Infinity" Official Video:
AJR got their musical start ten years ago busking on the streets of New York City. Through their earnings, they were able to buy their first $90 microphone and eventually a keyboard, ukulele, and computer, the basic tools brothers Adam, Jack and Ryan needed to begin writing their own brand of pop music.
Raised on the vinyl of the '50s and '60s,AJR was initially inspired by the likes of The Beach Boys and Simon & Garfunkel, but the sound they've created has now been favorably compared to a modern mix of those classic sounds, with flavors of Imagine Dragons, Fun., and even hip-hop production elements reflecting Kanye West.
"Pop has almost become the 'Anti-Pop' genre, where it's ashamed to be itself," says middle brother, Ryan. "There was a great time in the '60s where artists would lift the pop genre up to a new standard instead of avoiding it completely." And the debut album does exactly that - combining electronic and sample-filled production with honest lyrics drawn from experience, to appeal to the music lover and pop culture lover alike.
The band is drawing from the 'Living Room' concept to bring their acoustic setup right to their fans in their own living rooms, in a series of free concerts across the U.S. More information on the campaign is available at wedemand.com/ajr.
Acoustic Nation


Joy Developments is a licensed, bonded and insured (Washington license #JOYDEDL872NA) general contracting company specializing in residential and commercial construction. We serve the Seattle area on custom building projects ranging from new construction to remodeling/renovation jobs.
We've been listed #1 recently on Yelp.com for Best Residential General Contractor in Seattle and Joy Developments takes pride in our reviews on social media! We take pride in meeting and exceeding the needs of our clients, making sure that the project is completed on time, on or below budget and each client is truly satisfied with the results.
Snohomish Counties! The MBAKS strives to keep the Puget Sound region among the best places in the world to live.
Our West Seattle based team will assist you at every stage, from pre-construction to completion. If you need direction and plans, Joy Developments can also refer you to our team of architects and designers!
We care about Mother Earth! Our crew doesn't demolish everything to a large dumpster and send it to the landfill. Joy Developments salvages as much material as possible and delivers it to Second Use, Earth Wise, and Ballard Reuse so that it can be used again. Also, we try to recycle any materials that cannot be resold so that it helps keep our environment clean.
Now offering Solar! Joy Developments is teaming up with Seattle's expert in Solar products, Artisan Electric. Not only can we help you add on, reconfigure and/or update your roof but we can install a new solar panel system as well! Artisan installed the solar panels on our own home 7 years ago and we benefit immensely by decreasing our carbon footprint and our electric bill as well.
Joy Developments abides with the EPA's Lead Renovation, Repair and Painting Program (RRP Rule) which requires that firms performing renovation, repair, and painting projects that disturb lead-based paint in homes, child care facilities and pre-schools built before 1978 be certified.
In order to continue our effort to be as enviromentally conscious as possible, Joy Developments has teamed up with Green Latrine for our job site toilet facilities. Green Latrine not only uses 100% recycled paper products, has a rain water collection system to drastically reduce their water consumption. Green Latrine also uses natural enzymes to treat the waste water.


Super excited that @fashionofpure nominated me for the Liebster Award! Thank you!! Everyone should go check out her fashion blog for outfit and style inspiration.
How did you come up with the idea of starting your blog?
I had a ton of photos in my phone from recent travels, new outfits, and things that inspire me daily, so I decided it was time to share with everyone!
It's my creative outlet. Most importantly, i genuinely enjoy doing it.
The one moment that motivated you to start your blog?
I was flying home from Atlanta (on a business trip) and I was scrolling through my photos from the trip (amazing restaurants, fashion, drinks, scenery, etc.) and I said to myself If I shared this information for others I'm sure they would find it helpful if they are traveling to Atlanta. Who doesn't want some great restaurant tips or fun attractions to visit? And then my mind just filled with all of the other possibilities.
How is your blog different from the other blogs?
I read very few other blogs. I try to keep RTW unique to me and my inspirations. I think keeping your blog organic and true to yourself is so important. I hope that when others read RTW they are connecting directly with me, my visions and my life.
What is the one post made by you that you like the most, and why?
This is a tough one, but I'd have to say 'Boyfriend Jeans' because it was the most fun shooting the looks. My sister shoots all of my content and we laughed the entire day.
The one novel/song/movie that you relate to the most, and why?
Sunday Morning by Maroon 5. It's my all time favorite. It reminds me of weekends spent laying with the one you love. What's better than that?
If you could have one magical power, what would you want it to be?
To see the future. I am a planner. I like to know what's happening, when it's happening and how it's happening. I guess I will just have to continue to wait and see what's in store for me!
Which is your favorite clothing brand/designer and why?
Hmm.. There's a lot! But i'd have to say Free People.
The best piece of advice you've ever recieved?
To be happy – in all aspects of your life. Do what you have to do to make sure you are happy with your job, relationships, friendships, and most importantly yourself. There is no greater feeling than the balance of feeling content with all parts of your life.
A blogging trend you like the most and why?
Product reviews because they are so informative!
Fellow bloggers, please answer the same questions as i did above!
With Spring just around the corner it felt necessary to treat myself to some affordable, fun shoes. Black is just my thing. No matter what season – black is my go-to color, so don't be surprised that my spring shoe picks are mostly black. Some things never change! Shop the links below.
First, I went onto snap chat and Kylie posted that 46K active users were on her Kylie Cosmetics site, so i went into a panic. The first two times her lip kits were available I completely missed it and I was not going to let that happen again! Koko K was my first choice and that was sold out (I plan on ordering it whenever it becomes available again) and 22 was my second choice. I added 22 to my cart and typed in my credit card number as quickly as possible!!! Order confirmed. Order shipped. Happy human.
Positive Note: The kit comes with a matching lip pencil and matte lip gloss. I applied the lip pencil, then the lip gloss and let it set for a few minutes. The matte finish is amazing. For $29 I was SO impressed.
Side Note: I applied it at 7pm on friday night. I told myself I would get out of bed before i fell asleep to wipe it off and of course, I never did. I woke up in the morning and it was still on perfectly. (And it wasn't on my pillows!!) Long lasting is an understatement.
Negative Note: It smells like vanilla. It looks so good that I can overlook the sweet smell.
Another Side Note: It comes off easily with a make up wipe or remover.
A friend recommended Glossier because she loved a few of their products, so i decided to see for myself. I ordered their generation g matte lipstick (color: jam), boy brow (color: blonde) and milky jelly cleanser. All of them exceeded my expectations! Yes, i am in love.
The generation g matte lipstick has the finish like a stain, but can be applied like a balm. I ordered the color Jam and it's definitely something I can throw in my bag and apply at anytime. I'm so glad that I ordered it – i will definitely be purchasing the other shades. Below are a few pictures of the jam shade on me.
The boy brow gives me life. I've been looking for a good brow filler because my eye brows are blonde and barely exist. The boy brow is basically like mascara for your eyebrows. It works as a filler & shaper at the same time. I absolutely love it. Below is a photo with boy brow applied to my right eye brow and nothing on my left. Holy difference.
The milky jelly cleanser is amazing. It's both gentle and strong. Leaving your face feeling soft and clean with the ability to dissolve your make up off of your face too. I used it after I showered and it made my face feel like new. I am super excited to use it again tomorrow!
I am looking forward to trying more of Glossier's products. Stay tuned for more reviews!
4Love's mission is to empower indigenous women through stable employment, profit sharing and education.
Similar to 4Love Clothing, Run The World was originally created to empower and inspire women. 4Love's mission hits close to home for me and I am honored that I've had the opportunity to team up with them!
Lindsey's journey started when she moved to The Republic of Panama in 2011 to create Sowing Seeds of Love, a Christian non-profit organization focused on empowering indigenous women and children through education, creative arts, reading programs, and team sports. Soon she would tie her love for fashion with her passion to help others. The idea was to create a fashionable clothing line that empowered the women of Panama through stable employment and fair wages. Lindsey taught the women how to use electric sewing machines to bring her designs to life.
Fast foward a few years later and 4Love Clothing has exceeded all of their original expectations, continued to grow and most importantly, empowered women along the way.
Ive had the opportunity to support and wear 4Love's newest collection and I could not be happier. 4Love Clothing is currently doing a big Christmas & School Supply fundraiser for two orphanages they support and the children who live on the reservation. A portion of everything sold on 4Love Clothing will support this effort.
With the holidays approaching I hope we can all team up and support 4Love Clothing's mission. Use code runtheworld to receive 40% off your entire purchase.
Thank you 4Love Clothing & Lindsey Parry.
By supporting one another, we can run the world.


Page 708 - Gallery Design of Amazing Home Design | Alabamapbisnetwork White Tufted Headboard King. Small Kitchen Islands. Floating Media Cabinet.
collection of galleries from Amazing Home Design like White Tufted Headboard King. Small Kitchen Islands. Floating Media Cabinet. and other designs you might like Modern Bathroom Cabinets. Metal Shelf Brackets. Room Darkening Drapes. Twin Bed Frames With Storage. Drapery Window Treatment. Large Bean Bag Chairs. Queen Storage Bed Frame.