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As a programmer working on our online taxi service, one of the key components of our system is calculating the distances between clients and taxis to identify the nearest available taxi. To do this, we rely on graph algorithms. These algorithms allow us to find the shortest paths between nodes in a graph, taking into account weighted edges.

In our case, the nodes represent the locations of our clients and taxis, and the weighted edges represent the distances between these locations. By iteratively updating the distances from the source node to all other nodes, we can determine the most optimal path for our clients and taxis.

So, to answer your question, we find the shortest paths between the locations of our clients and taxis in order to identify the nearest available taxi for each client. This allows us to efficiently match clients with taxis and provide a seamless experience for our users.