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Starting with a shapefile of linestrings I want to assign street names to these lines if there is a match with a road/street in OpenStreetMap.

There are no exact matches as the linestrings have a different source. It could be that some linestrings have no counterpart in the OSM data or that some linestrings are longer than the roads in OSM in which case I will probably have to cut my own linestrings to be able to name them properly.

I love to hear a complete solution, but tips and hints are also welcome.

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These people discuss the idea of matching each individual point on the linestring to the road it is on. – flup Jan 23 at 18:12
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Conflation tools are what you need -- gis.stackexchange.com/questions/8262/… wiki.openstreetmap.org/wiki/Conflation Colleagues have used Map Merger mapmerger.com with some success, but there is license cost. – awesomo Jan 23 at 20:13

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