rewirte it in more homogenous way and explain the methods more scientifically and add the explain briefly. use less bullet points as possible  \documentclass{article} \usepackage{amsmath, amssymb, geometry, fancyhdr} \usepackage{graphicx} \usepackage{tikz} \usepackage{hyperref} \geometry{margin=1in} \pagestyle{fancy} \fancyhf{} \chead{\textbf{Collision Avoidance in Grid-Based Multi-Agent Path Finding}} \cfoot{\thepage}  \begin{document}  \title{Collision Avoidance in Grid-Based Multi-Agent Path Finding: A Literature Review} \author{[Your Name]} \date{\today}  \maketitle  \section{Introduction}  Multi-Agent Path Finding (MAPF) is a fundamental problem in robotics and artificial intelligence, where multiple agents must navigate from their respective start locations to designated goal locations without colliding with each other. This problem is particularly challenging in grid-based environments due to the discrete nature of the space and the potential for conflicts at shared vertices or edges. Effective collision avoidance strategies are critical for applications such as robotics fleets, automated warehouse systems, and autonomous traffic management. This literature review examines various collision avoidance strategies in grid-based MAPF, explaining each method and providing illustrative examples using a simple grid graph.  \section{Collision Definitions in Grid-Based Multi-Agent Path Finding}  In grid-based MAPF, a \emph{collision} occurs when agents interfere with each other's movements, violating the constraints of the environment or the rules governing agent interactions. Collisions are generally categorized into the following types:  \begin{enumerate}     \item \textbf{Vertex Collision (Node Collision)}:     \begin{itemize}         \item \textbf{Definition}: Occurs when two or more agents occupy the same grid cell (vertex) at the same time step.     \end{itemize}     \item \textbf{Edge Collision}:     \begin{itemize}         \item \textbf{Definition}: Happens when two agents traverse the same edge between grid cells in the same direction at the same time step.     \end{itemize}     \item \textbf{Swap Collision (Head-on Collision)}:     \begin{itemize}         \item \textbf{Definition}: Occurs when two agents move into each other's current positions along the same edge at the same time step.     \end{itemize} \end{enumerate}  Understanding these collision types is essential for designing algorithms that ensure agents navigate the grid efficiently and safely.  \section{Example Grid Graph}  To illustrate the collision avoidance methods discussed in this review, we define a simple grid graph that will be used consistently in all examples.  \subsection{Grid Graph Definition}  Consider a $5 \times 5$ grid graph where each cell is identified by its coordinates $(x, y)$, with $1 \leq x, y \leq 5$. Agents can move to adjacent cells in the four cardinal directions (up, down, left, right) unless obstructed.  \subsection{Agent Start and Goal Positions}  \begin{itemize}     \item \textbf{Agent A}: Start at $(1,1)$, Goal at $(5,5)$.     \item \textbf{Agent B}: Start at $(5,1)$, Goal at $(1,5)$. \end{itemize}  This setup creates a scenario where the agents' optimal paths intersect, necessitating collision avoidance strategies.  \section{Collision Avoidance Strategies in the Literature}  The literature provides a variety of strategies for avoiding collisions in grid-based MAPF. The following sections describe these methods and illustrate them using the example grid graph.  \subsection{Centralized Planning Methods}  Centralized methods involve a single planning process that considers all agents simultaneously.  \subsubsection{Conflict-Based Search (CBS)}  Conflict-Based Search (CBS) is an optimal algorithm that resolves conflicts through constraint propagation.  \paragraph{Method Explanation}  CBS operates on two levels: \begin{enumerate}     \item \textbf{High-Level Search}: Maintains a constraint tree (CT), where each node represents a set of constraints and a solution satisfying them.     \item \textbf{Low-Level Search}: For each node in the CT, individual shortest paths for agents are computed under the given constraints. \end{enumerate}  When a collision is detected in a solution: \begin{itemize}     \item The high-level search generates two child nodes, each adding a constraint to one of the conflicting agents to avoid the collision.     \item The search continues recursively until a collision-free set of paths is found. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Initial Paths}: Agents plan their shortest paths independently.         \begin{itemize}             \item Agent A moves right along row $y=1$ to $(5,1)$, then up to $(5,5)$.             \item Agent B moves left along row $y=1$ to $(1,1)$, then up to $(1,5)$.         \end{itemize}     \item \textbf{Collision Detection}: A vertex collision at $(3,1)$ at time step $t=3$ is detected.     \item \textbf{Constraint Addition}:         \begin{itemize}             \item Create two CT nodes:                 \begin{enumerate}                     \item Node 1: Add constraint forbidding Agent A from being at $(3,1)$ at $t=3$.                     \item Node 2: Add constraint forbidding Agent B from being at $(3,1)$ at $t=3$.                 \end{enumerate}         \end{itemize}     \item \textbf{Path Replanning}:         \begin{itemize}             \item In Node 1, Agent A replans to avoid the constraint, possibly moving up to $(3,2)$.             \item In Node 2, Agent B replans an alternative route.         \end{itemize}     \item \textbf{Solution}: A collision-free set of paths is found in one of the CT nodes. \end{enumerate}  \subsubsection{Increasing Cost Tree Search (ICTS)}  Increasing Cost Tree Search (ICTS) explores combinations of individual agent costs to find a collision-free solution.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Cost Tree Construction}: Builds a tree where each node represents a total cost (path length) for agents.     \item \textbf{Candidate Paths}: For each cost combination, generates all possible paths of that cost for each agent.     \item \textbf{Collisions Checking}: Checks combinations of paths for collisions.     \item \textbf{Search Expansion}: Increases path costs incrementally if collisions persist. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Initial Cost}: Start with minimum path costs (e.g., 8 steps for both agents).     \item \textbf{Path Generation}: Generate all possible paths of cost 8.     \item \textbf{Collision Detection}: Identify collisions among path combinations.     \item \textbf{Cost Increment}: Increase the cost for one agent to 9 and repeat the process.     \item \textbf{Collision-Free Paths}: Continue until a set of paths without collisions is found. \end{enumerate}  \subsubsection{Enhanced Partial Expansion A* (EPEA*)}  EPEA* is an optimization of the A* algorithm that reduces node expansions by using operator pruning.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Heuristic Guidance}: Uses a heuristic to estimate the cost to the goal.     \item \textbf{Operator Pruning}: Precomputes operator selections to expand only nodes that can lead to a solution.     \item \textbf{Partial Expansion}: Expands nodes partially based on the most promising operators. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Path Planning}: Agents plan their paths using EPEA*, considering potential collisions.     \item \textbf{Selective Expansion}: Nodes leading to immediate collisions are pruned.     \item \textbf{Collision Avoidance}: By expanding only viable nodes, agents find efficient, collision-free paths. \end{enumerate}  \subsubsection{M\textsuperscript{$\ast$} (Modified A\textsuperscript{$\ast$})}  M$^\ast$ is an extension of A$^\ast$ that handles multiple agents by merging their searches upon interaction.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Independent Planning}: Agents plan paths independently until a potential conflict arises.     \item \textbf{Meta-Agent Formation}: When agents' paths intersect, they form a meta-agent and plan jointly.     \item \textbf{Conflict Resolution}: The meta-agent searches for a combined path that avoids collisions. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Initial Planning}: Agents plan independently using A$^\ast$.     \item \textbf{Collision Detection}: Paths intersect at cell $(3,3)$ at time $t=3$.     \item \textbf{Meta-Agent Planning}:         \begin{itemize}             \item Agents A and B form a meta-agent.             \item Jointly plan paths to avoid the collision at $(3,3)$.         \end{itemize}     \item \textbf{Path Adjustment}: Agents find new routes, possibly adjusting their timelines or taking alternative paths. \end{enumerate}  \subsection{Decentralized Planning Methods}  Decentralized methods involve agents planning their paths independently with limited or no coordination.  \subsubsection{Priority-Based Planning}  Agents are assigned priorities, and lower-priority agents adapt their paths to avoid higher-priority agents.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Priority Assignment}: Assign static or dynamic priorities to agents.     \item \textbf{Sequential Planning}: Higher-priority agents plan first; lower-priority agents plan subsequently, avoiding conflicts. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Priority Assignment}: Agent A has higher priority than Agent B.     \item \textbf{Agent A's Path}: Plans the shortest path to its goal.     \item \textbf{Agent B's Planning}:         \begin{itemize}             \item Observes Agent A's path.             \item Plans a path avoiding conflicts, possibly taking a longer route.         \end{itemize} \end{enumerate}  \subsubsection{Local Repair Strategies}  Agents detect collisions during execution and locally adjust their paths.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Independent Planning}: Agents plan without considering others.     \item \textbf{Collision Detection}: Monitor for potential collisions during movement.     \item \textbf{Local Adjustment}: Agents wait, slow down, or reroute to resolve conflicts. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Collision Prediction}: Agents anticipate a collision at cell $(2,2)$.     \item \textbf{Agent B's Adjustment}:         \begin{itemize}             \item Chooses to wait at its current position for one time step.         \end{itemize}     \item \textbf{Collision Avoidance}: By adjusting timing, the collision is averted. \end{enumerate}  \subsubsection{Reciprocal Collision Avoidance (e.g., ORCA)}  Agents adjust their velocities and paths in response to the predicted movements of others.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Velocity Obstacles}: Agents compute their own velocity obstacles based on others' predicted paths.     \item \textbf{Reciprocity}: Each agent is responsible for half of the required collision avoidance effort.     \item \textbf{Dynamic Adjustment}: Agents continuously update their paths in real-time. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Velocity Calculation}: Agents determine potential collisions at future time steps.     \item \textbf{Path Adjustment}:         \begin{itemize}             \item Agent A slightly alters its path to the right.             \item Agent B adjusts its path to the left.         \end{itemize}     \item \textbf{Collision Avoidance}: The combined adjustments prevent the collision. \end{enumerate}  \subsection{Reservation-Based Methods}  Agents reserve nodes or edges in a time-space domain to prevent conflicts.  \subsubsection{Time-Expanded Networks}  Constructs a graph where nodes represent positions at specific time steps.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Graph Expansion}: Extends the spatial grid over time to create a time-space graph.     \item \textbf{Reservation of Nodes}: Agents plan paths by reserving nodes at specific times.     \item \textbf{Conflict Prevention}: Ensures no two agents occupy the same node or edge simultaneously. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Graph Construction}: Create time-stamped copies of the grid for each time step.     \item \textbf{Path Planning}: Agents plan paths through this expanded graph.     \item \textbf{Node Reservation}: Agent A reserves $(2,1)$ at $t=2$; Agent B must plan accordingly to avoid this node at the same time. \end{enumerate}  \subsubsection{Reservation Tables}  Agents maintain a schedule of their planned movements.  \paragraph{Method Explanation}  \begin{itemize}     \item \textbf{Scheduling}: Agents store their planned occupancy of cells over time.     \item \textbf{Consultation}: Before moving, agents check for reservations by others.     \item \textbf{Conflict Resolution}: Agents may wait or choose alternative paths if conflicts are detected. \end{itemize}  \paragraph{Example Using the Grid Graph}  \begin{enumerate}     \item \textbf{Agent A's Schedule}: Plans and records its path from $(1,1)$ to $(5,5)$.     \item \textbf{Agent B's Planning}: Checks Agent A's reservations and avoids conflicts.     \item \textbf{Path Adjustment}: Agent B may delay movement or select a different route. \end{enumerate}  \section{Conclusion}  Collision avoidance in grid-based Multi-Agent Path Finding is a critical area of research with significant practical implications. The methods discussed provide a range of solutions, from centralized algorithms that guarantee optimality to decentralized approaches that offer scalability. By applying these strategies to a common grid graph, we have illustrated their practical functioning and highlighted the importance of choosing the appropriate method based on specific application requirements.  \section*{References}  \begin{itemize}     \item Silver, D. (2005). \emph{Cooperative Pathfinding}. In \emph{Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment} (pp. 117--122).     \item Sharon, G., Stern, R., Felner, A., \& Sturtevant, N. R. (2015). \emph{Conflict-Based Search for Optimal Multi-Agent Pathfinding}. \emph{Artificial Intelligence}, 219, 40--66.     \item Wagner, G., \& Choset, H. (2015). \emph{Subdimensional Expansion for Multirobot Path Planning}. \emph{Artificial Intelligence}, 219, 1--24.     \item Van den Berg, J., Guy, S. J., Lin, M., \& Manocha, D. (2011). \emph{Reciprocal n-Body Collision Avoidance}. In \emph{Robotics Research} (pp. 3--19). Springer. \end{itemize}  \end{document}