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I need to make an application for the Quickest Route to the nearby hospitals from the incident place. I have read many articles and papers but I am confused whether I need polyline shapefile or network dataset or any other format for storing data in PostgreSQL

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It all depends of what you need to achieve... but If you need only data for routing to be done in PostgreSQL (so assuming postGIS and PGRouting) you'll need a topology (network).

You can use OSM2PO to convert OSM data into topology and write it as sql script, than import into database using PSQL.

Here is nice instruction how to do this

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