Answers.Body: <p>Consider each cell as a point on a 2D lattice and let <span class="math-container">$u(x,y,t)$</span> represent a time-dependent “flow density” at cell <span class="math-container">$(x,y)$</span>. Let <span class="math-container">$\Omega\subset\mathbb{Z}^2$</span> be the set of all valid grid cells and let <span class="math-container">$\Gamma\subset\Omega$</span> be the target zone. Define a potential function <span class="math-container">$\phi(x,y)$</span> on <span class="math-container">$\Omega$</span> that decreases toward <span class="math-container">$\Gamma$</span>. One way is to solve an Eikonal-like equation
  <span class="math-container">$$
  \|\nabla \phi(x,y)\| = W(x,y)
  $$</span>
  where <span class="math-container">$W(x,y)$</span> is a nonnegative weight capturing travel cost or difficulty at each cell <span class="math-container">$(x,y)$</span>. Impose the boundary condition <span class="math-container">$\phi(x,y)=0$</span> for <span class="math-container">$(x,y)\in \Gamma$</span>. On a discrete grid, approximate gradients via forward/backward differences. For instance, in one dimension of the grid,
  <span class="math-container">$$
  \frac{\partial \phi}{\partial x}\bigg\rvert_{(x,y)} \approx \phi(x+1,y)-\phi(x,y),
  $$</span>
  and similarly in <span class="math-container">$y$</span>. After numerically solving for <span class="math-container">$\phi$</span>, define a velocity field
  <span class="math-container">$$
  \mathbf{v}(x,y)=-\nabla \phi(x,y),
  $$</span>
  so agents “flow” along negative gradients toward <span class="math-container">$\Gamma$</span>. Next, track the dynamics of the flow density <span class="math-container">$u(x,y,t)$</span>. Assume that cells can supply or lose objects at rate <span class="math-container">$S(x,y,t)$</span>, and the objects are advected by <span class="math-container">$\mathbf{v}$</span>. Model this via a continuum-like advection equation on the discrete grid:
  <span class="math-container">$$
  \frac{\partial u}{\partial t}(x,y,t) + \nabla \cdot \bigl(u(x,y,t)\,\mathbf{v}(x,y)\bigr) = S(x,y,t).
  $$</span>
  Discretize time in increments <span class="math-container">$\Delta t$</span> and use finite differences in space for the divergence term. Let
  <span class="math-container">$$
  u^{n}(x,y)\approx u(x,y,n\,\Delta t).
  $$</span>
  The discrete update can be written:
  <span class="math-container">$$
  u^{n+1}(x,y) = u^{n}(x,y) - \Delta t \,\nabla_{\text{disc}} \cdot \bigl(u^{n}(x,y)\,\mathbf{v}(x,y)\bigr) + \Delta t\,S^{n}(x,y).
  $$</span>
  Pick a scheme such as the upwind method to ensure stability. The quantity
  <span class="math-container">$$
  H(x,y,t)=\int_{0}^{t} \|u(x,y,\tau)\|\;d\tau
  $$</span>
  then measures cumulative flow through each cell. This <span class="math-container">$H$</span> acts as a dynamic heat map that grows faster where more agents have recently passed. As the environment changes (e.g., obstacles appear, objects move), update <span class="math-container">$\phi$</span> by re-solving the discrete Eikonal equation with new weights <span class="math-container">$W$</span>. Then update <span class="math-container">$\mathbf{v}$</span> and re-evolve <span class="math-container">$u$</span> according to the above advection model. Because the potential <span class="math-container">$\phi$</span> can be kept at low resolution away from changing regions, and because one can employ fast Eikonal solvers (e.g., fast marching), the method adapts to real-time changes while remaining computationally tractable.</p>
  <p>In more detail, for every time step, do: first, update <span class="math-container">$\phi$</span> by solving
  <span class="math-container">$$
  \|\nabla \phi\| = W, \quad \phi=0 \text{ on } \Gamma
  $$</span>
  using a fast-marching or fast-sweeping algorithm on the grid. Then define <span class="math-container">$\mathbf{v}=-\nabla \phi$</span>. Next, update <span class="math-container">$u$</span> in each cell by applying the discrete advection step:
  <span class="math-container">$$
  u^{n+1}(x,y)=u^{n}(x,y) - \Delta t \bigl[\text{flux in x-direction} + \text{flux in y-direction}\bigr] + \Delta t\,S^{n}(x,y).
  $$</span>
  Finally, update the heat map:
  <span class="math-container">$$
  H(x,y,t+\Delta t)=H(x,y,t) + \Delta t \,\bigl|u^{n+1}(x,y)\bigr|.
  $$</span>
  In this manner, the dynamically evolving heat map captures high-flow regions without explicitly storing or merging separate path solutions. Instead, it arises naturally from the continuous flow of agent densities. This resolves highlight areas of high agent or object traffic and does so in a way that neatly incorporates real-time changes in the environment.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-12-27T16:25:13.930Z
Answers.LastActivityDate: 2024-12-27T16:25:13.930Z
Answers.UserDisplayName: Raynard Bond
Body: <p>I’m working on a system where agents must move scattered objects (aggregates) from various locations (cells) to a target zone. The environment is divided into a grid of cells, where each cell contains objects and may have an optimal path to the target zone. I aim to create a dynamic heat map that highlights high-potential areas to guide agents effectively.</p>
  <p><strong>My Current Approach:</strong></p>
  <ol>
  <li><p><strong>Path Calculation</strong>:
  For each cell with objects, I calculate the best path to the target zone using graph
  search algorithms (e.g., A* or Dijkstra’s).
  The best path is defined as the sequence of cells an agent should traverse to maximize
  the number of objects collected and moved to the target zone. It considers directness,
  obstacles, and opportunities to collect additional objects along the way.</p>
  </li>
  <li><p><strong>Heat Map Creation</strong>:
  Convert these paths into smooth curves.
  Highlight regions where multiple paths are close to each other, similar to a
  topographic map, to identify high-potential areas (highways).</p>
  </li>
  </ol>
  <p><strong>Open Questions</strong>:</p>
  <ol>
  <li><p>Are there more efficient ways to aggregate paths and highlight high-potential areas, potentially using graph theory?</p>
  </li>
  <li><p>How can I represent flow density dynamically as the environment changes (e.g., objects move, agents act)?</p>
  </li>
  <li><p>What methods can ensure the heat map remains computationally efficient and adaptable for real-time updates?</p>
  </li>
  </ol>
  
Tags: <algorithm><multi-agent-systems><robotics><path-planning><shortest-path-problem>
Title: Creating a Dynamic Heat Map for Multi-Agent Path Planning
UserDisplayName: Nir Manor
ThreadText: # Creating a Dynamic Heat Map for Multi-Agent Path Planning
  
  (Asked by: username_0 on 2024-12-27)
  
  I’m working on a system where agents must move scattered objects (aggregates) from various locations (cells) to a target zone. The environment is divided into a grid of cells, where each cell contains objects and may have an optimal path to the target zone. I aim to create a dynamic heat map that highlights high-potential areas to guide agents effectively.
  
  **My Current Approach:**
  
  1. **Path Calculation**:
     For each cell with objects, I calculate the best path to the target zone using graph
     search algorithms (e.g., A\* or Dijkstra’s).
     The best path is defined as the sequence of cells an agent should traverse to maximize
     the number of objects collected and moved to the target zone. It considers directness,
     obstacles, and opportunities to collect additional objects along the way.
  2. **Heat Map Creation**:
     Convert these paths into smooth curves.
     Highlight regions where multiple paths are close to each other, similar to a
     topographic map, to identify high-potential areas (highways).
  
  **Open Questions**:
  
  1. Are there more efficient ways to aggregate paths and highlight high-potential areas, potentially using graph theory?
  2. How can I represent flow density dynamically as the environment changes (e.g., objects move, agents act)?
  3. What methods can ensure the heat map remains computationally efficient and adaptable for real-time updates?
  
  **Tags:** algorithm, multi-agent-systems, robotics, path-planning, shortest-path-problem
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2024-12-27. Score: 0
  
  Consider each cell as a point on a 2D lattice and let $u(x,y,t)$ represent a time-dependent “flow density” at cell $(x,y)$. Let $\Omega\subset\mathbb{Z}^2$ be the set of all valid grid cells and let $\Gamma\subset\Omega$ be the target zone. Define a potential function $\phi(x,y)$ on $\Omega$ that decreases toward $\Gamma$. One way is to solve an Eikonal-like equation
  $$
  \|\nabla \phi(x,y)\| = W(x,y)
  $$
  where $W(x,y)$ is a nonnegative weight capturing travel cost or difficulty at each cell $(x,y)$. Impose the boundary condition $\phi(x,y)=0$ for $(x,y)\in \Gamma$. On a discrete grid, approximate gradients via forward/backward differences. For instance, in one dimension of the grid,
  $$
  \frac{\partial \phi}{\partial x}\bigg\rvert\_{(x,y)} \approx \phi(x+1,y)-\phi(x,y),
  $$
  and similarly in $y$. After numerically solving for $\phi$, define a velocity field
  $$
  \mathbf{v}(x,y)=-\nabla \phi(x,y),
  $$
  so agents “flow” along negative gradients toward $\Gamma$. Next, track the dynamics of the flow density $u(x,y,t)$. Assume that cells can supply or lose objects at rate $S(x,y,t)$, and the objects are advected by $\mathbf{v}$. Model this via a continuum-like advection equation on the discrete grid:
  $$
  \frac{\partial u}{\partial t}(x,y,t) + \nabla \cdot \bigl(u(x,y,t)\,\mathbf{v}(x,y)\bigr) = S(x,y,t).
  $$
  Discretize time in increments $\Delta t$ and use finite differences in space for the divergence term. Let
  $$
  u^{n}(x,y)\approx u(x,y,n\,\Delta t).
  $$
  The discrete update can be written:
  $$
  u^{n+1}(x,y) = u^{n}(x,y) - \Delta t \,\nabla\_{\text{disc}} \cdot \bigl(u^{n}(x,y)\,\mathbf{v}(x,y)\bigr) + \Delta t\,S^{n}(x,y).
  $$
  Pick a scheme such as the upwind method to ensure stability. The quantity
  $$
  H(x,y,t)=\int\_{0}^{t} \|u(x,y,\tau)\|\;d\tau
  $$
  then measures cumulative flow through each cell. This $H$ acts as a dynamic heat map that grows faster where more agents have recently passed. As the environment changes (e.g., obstacles appear, objects move), update $\phi$ by re-solving the discrete Eikonal equation with new weights $W$. Then update $\mathbf{v}$ and re-evolve $u$ according to the above advection model. Because the potential $\phi$ can be kept at low resolution away from changing regions, and because one can employ fast Eikonal solvers (e.g., fast marching), the method adapts to real-time changes while remaining computationally tractable.
  
  In more detail, for every time step, do: first, update $\phi$ by solving
  $$
  \|\nabla \phi\| = W, \quad \phi=0 \text{ on } \Gamma
  $$
  using a fast-marching or fast-sweeping algorithm on the grid. Then define $\mathbf{v}=-\nabla \phi$. Next, update $u$ in each cell by applying the discrete advection step:
  $$
  u^{n+1}(x,y)=u^{n}(x,y) - \Delta t \bigl[\text{flux in x-direction} + \text{flux in y-direction}\bigr] + \Delta t\,S^{n}(x,y).
  $$
  Finally, update the heat map:
  $$
  H(x,y,t+\Delta t)=H(x,y,t) + \Delta t \,\bigl|u^{n+1}(x,y)\bigr|.
  $$
  In this manner, the dynamically evolving heat map captures high-flow regions without explicitly storing or merging separate path solutions. Instead, it arises naturally from the continuous flow of agent densities. This resolves highlight areas of high agent or object traffic and does so in a way that neatly incorporates real-time changes in the environment.
  
  
  ---
  
  