paper fixing

Author: glucawork

paper/paper.md

@@ -75,15 +75,15 @@ This table provides a summary of the lengths in meters for each highway type.
 
 OpenStreetMap data consists of *nodes*, *ways*, and *tags*. Nodes are points with geographic coordinates. Ways are sequences of nodes that describe linear features like roads or paths. Tags are key-value pairs that describe attributes, such as `highway=residential` for a residential road.
 
-After selecting the input vector layer (`points_vector`) and specifying the two numerical parameters (`max_dist` and `min_loop_size`), the algorithm retrieves the OSM map section that includes the `points_vector`. It then constructs the road graph $G$ using the NetworkX library  [@networkx], labeling the edges of $G$ with tags extracted from OSM ways."
+After selecting the input vector layer (`points_vector`) and specifying the two numerical parameters (`max_dist` and `min_loop_size`), the algorithm retrieves the OSM map section that includes the `points_vector`. It then constructs the road graph $G$ using the NetworkX library  [@networkx], labeling the edges of $G$ with tags extracted from OSM ways.
 
 At each iteration, the algorithm attempts to match each point in `points_vector` with a node in $G$. Let $p$ be the last matched point and $v_p$ the corresponding node in $G$, and let $q$ be the next point to be matched. The algorithm computes the shortest path tree $T_p$ in $G$, rooted at $v_p$. It then searches for the edge $e_q$ in $T_p$ that is closest to $q$ and matches $q$ to the node $v_q$ of $e_q$ that is nearest to $q$. Finally, the nodes that belong to the shortest path from $v_p$ and $v_q$ are added to the solution and the algorithm continue from node $v_q$.
 
 ![How the algorithm works.\label{fig:shortestpath}](pictures/shortest_path.png){ width=60% }
 
 The picture illustrates an example of how the algorithm works. The green edges represent the shortest path tree from node $v_p$, while the black edge is another graph edge. The blue arrows highlight the shortest path from $v_p$ to $v_q$. Nodes $a$, $b$, and $v_q$ are added to the solution, and the algorithm proceeds by computing a shortest path tree from $v_q$.
 
-In the case where the distance between $q$ and $e_q$ is greater than `max_dist`, a dummy node is inserted into the solution instead of the shortest path between $p$ and $v_q$. Dummy nodes will be added in the same way until an edge is found whose distance from the current node is smaller than max_dist.
+In the case where the distance between $q$ and $e_q$ is greater than `max_dist`, a dummy node is inserted into the solution instead of the shortest path between $p$ and $v_q$. Dummy nodes will be added in the same way until an edge is found whose distance from the current node is smaller than `max_dist`.
 
 ## Optimizations