In terminological/arear: Geospatial Convenience Functions and a Supply Demand Catchment Area Generator
{=latex}
\begin{algorithm}[H]
\setstretch{1.0}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwComment{Comment}{\--- }{}
\Input{$V_N$ - the $N$ regions of demand as a set of geographical polygons}
\Input{$stats::D(V_n)$ - the density of demand in any given region as a function of the region $V_n$}
\Input{$P_M$ - a set of $M$ labelled suppliers as a set of geographical points}
\Input{$S(P_m)$ - the capacity of supply at any given supply point as a function of the supplier $P_m$}
\Input{$C_{growth}$ - a rate constant defining rate of label propagation}
\Output{$G_M$ - $M$ labelled subgraphs of graph $G$, relating to the catchment areas of suppliers $P_M$}
\BlankLine
\Comment{define $G$ as the graph consisting of geographical regions $V_N$, connected by edges, $E_N$, given by their geographical neighbours $\nu(V_N)$:}
$E_N \gets \nu(V_N)$\;
$G \gets (V_N, E_N)$\;
\Comment{define $V_M$ and $V^{new}_{M,0}$ as the geographic regions of $G$ serviced by points $P_M$, and $G_{M,0}$ as a set of labelled sub-graphs (also initially consisting solely of the vertices $V_M$):}
$V_M \gets G \cap P_M$\;
$V^{new}_{M,0} \gets V_M$;
$G_{M,0} \gets V_M$\;
\Comment{define the initial unlabelled set of vertices:}
$U_0 \gets \neg V_{M}$\;
\Comment{define the initial un-labelled neighbours of labelled sub-graphs, $G_M$:}
$U_{M,0} \gets \nu(V_M)$\;
\Comment{define an accumulated growth score for each un-labelled neighbour $U_{M,0}$ of each $G_{M,0}$:}
$A_{U_{M,0}} \gets 0$\;
\BlankLine
$k \gets 0$\;
\Comment{execute the loop while there are still unlabelled vertices and there exist some unlabelled neighbours of labelled vertices}
\While{$|U_k| > 0$ and $|U_{M,k}| > 0$} {
$k \gets k+1$\;
\BlankLine
\Comment{define the un-labelled vertices as the set of $V$ not contained in any of $G_{M,k-1}$:}
$U_k \gets \neg G_{M,k-1}$\;
\Comment{define the un-labelled neighbours of $G_{M,k-1}$ as $U_{M,k}$ as the previously unlabelled neighbours and the neighbours of the most recently labelled neighbours $V^{new}_{M,k-1}$:}
$U_{M,k} \gets U_{M,k-1} \cup (U_k \cap \nu(V^{new}_{M,k-1}))$\;
\Comment{define the reserve capacity, $R_M$, to supply existing labelled, $G_{M,k-1}$, and un-labelled neighbours $U_{M,k}$, as:}
$R_M \gets \frac{S(P_M)}{stats::D(U_{M,k} \cup G_{M,k-1})}$\;
\Comment{for unlabelled areas only, update the accumulated growth score, $A_{U_{M,k}}$, with the normalised rank of the reserve capacity and multiplied by a constant $C_{growth} > 1$ representing the speed at which the accumulated growth score increases in all areas:}
$R_{M,k} \gets R_m \{m \in U_{M,k}\}$\;
$A_{U_{M,k}} \gets A_{U_{M,k-1}} + C_{growth} \times \text{rank}(R_{M,k})/|R_{M,k}|$\;
\Comment{for all the un-labelled vertices, select the label $M$, with the highest score, and if the accumulated score has reached the threshold of 1, incorporate it into the labelled sub-graph, $G_{M,k-1}$:}
$A^{max}_{U_k} = \text{max}(A_{U_{m,k}},m \in M)$\;
$V^{new}_{M,k} \gets U_{M,k} \in \{A^{max}_{U_k} > 1\}$\;
$G_{M,k} \gets G_{M,k-1} \cup V^{new}_{M,k}$\;
$U_{M,k+1} \gets U_{M,k} \cap \neg V^{new}_{M,k}$\;
}
\Return{$G_{M,k}$}
\caption{A weighted label propagation algorithm for matching geographical supply to demand}
\end{algorithm}
terminological/arear documentation built on April 27, 2024, 9:02 a.m.
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{=latex}
\begin{algorithm}[H]
\setstretch{1.0}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwComment{Comment}{\--- }{}
\Input{$V_N$ - the $N$ regions of demand as a set of geographical polygons}
\Input{$stats::D(V_n)$ - the density of demand in any given region as a function of the region $V_n$}
\Input{$P_M$ - a set of $M$ labelled suppliers as a set of geographical points}
\Input{$S(P_m)$ - the capacity of supply at any given supply point as a function of the supplier $P_m$}
\Input{$C_{growth}$ - a rate constant defining rate of label propagation}
\Output{$G_M$ - $M$ labelled subgraphs of graph $G$, relating to the catchment areas of suppliers $P_M$}
\BlankLine
\Comment{define $G$ as the graph consisting of geographical regions $V_N$, connected by edges, $E_N$, given by their geographical neighbours $\nu(V_N)$:}
$E_N \gets \nu(V_N)$\;
$G \gets (V_N, E_N)$\;
\Comment{define $V_M$ and $V^{new}_{M,0}$ as the geographic regions of $G$ serviced by points $P_M$, and $G_{M,0}$ as a set of labelled sub-graphs (also initially consisting solely of the vertices $V_M$):}
$V_M \gets G \cap P_M$\;
$V^{new}_{M,0} \gets V_M$;
$G_{M,0} \gets V_M$\;
\Comment{define the initial unlabelled set of vertices:}
$U_0 \gets \neg V_{M}$\;
\Comment{define the initial un-labelled neighbours of labelled sub-graphs, $G_M$:}
$U_{M,0} \gets \nu(V_M)$\;
\Comment{define an accumulated growth score for each un-labelled neighbour $U_{M,0}$ of each $G_{M,0}$:}
$A_{U_{M,0}} \gets 0$\;
\BlankLine
$k \gets 0$\;
\Comment{execute the loop while there are still unlabelled vertices and there exist some unlabelled neighbours of labelled vertices}
\While{$|U_k| > 0$ and $|U_{M,k}| > 0$} {
$k \gets k+1$\;
\BlankLine
\Comment{define the un-labelled vertices as the set of $V$ not contained in any of $G_{M,k-1}$:}
$U_k \gets \neg G_{M,k-1}$\;
\Comment{define the un-labelled neighbours of $G_{M,k-1}$ as $U_{M,k}$ as the previously unlabelled neighbours and the neighbours of the most recently labelled neighbours $V^{new}_{M,k-1}$:}
$U_{M,k} \gets U_{M,k-1} \cup (U_k \cap \nu(V^{new}_{M,k-1}))$\;
\Comment{define the reserve capacity, $R_M$, to supply existing labelled, $G_{M,k-1}$, and un-labelled neighbours $U_{M,k}$, as:}
$R_M \gets \frac{S(P_M)}{stats::D(U_{M,k} \cup G_{M,k-1})}$\;
\Comment{for unlabelled areas only, update the accumulated growth score, $A_{U_{M,k}}$, with the normalised rank of the reserve capacity and multiplied by a constant $C_{growth} > 1$ representing the speed at which the accumulated growth score increases in all areas:}
$R_{M,k} \gets R_m \{m \in U_{M,k}\}$\;
$A_{U_{M,k}} \gets A_{U_{M,k-1}} + C_{growth} \times \text{rank}(R_{M,k})/|R_{M,k}|$\;
\Comment{for all the un-labelled vertices, select the label $M$, with the highest score, and if the accumulated score has reached the threshold of 1, incorporate it into the labelled sub-graph, $G_{M,k-1}$:}
$A^{max}_{U_k} = \text{max}(A_{U_{m,k}},m \in M)$\;
$V^{new}_{M,k} \gets U_{M,k} \in \{A^{max}_{U_k} > 1\}$\;
$G_{M,k} \gets G_{M,k-1} \cup V^{new}_{M,k}$\;
$U_{M,k+1} \gets U_{M,k} \cap \neg V^{new}_{M,k}$\;
}
\Return{$G_{M,k}$}
\caption{A weighted label propagation algorithm for matching geographical supply to demand}
\end{algorithm}
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