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What I'm trying to do is have two edit boxes for source and destination. Based on those two locations I need to draw the route on an ArcGIS map in Android.

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What area do you want your app to give results in? Do you want ArcGIS online related resources, Or can you host your own services? Do the Web services need to be from the ESRI stack or is OpenSource fine? – Devdatta Tengshe Mar 21 '13 at 14:50

The Basic algorithm is as follows:

  1. Geocode the two addresses to get the position for both of them.

  2. Pass these two points to a Network Analyst Service, and get the result from them.

If you want to stick to the ESRI stack, ArcGIS Online has Geocoding, as well as Network Analyst services available.

In the OpenSource Stack, you have several options like gisgraphy, OpenStreetMaps' Nominatim, MapQuest geocoder etc. You could even host your own. As for routing you could host your own pgRouting or use something like MapQuest's OpenDirection API

Please note that most of these online services have some kind of limits on them. You need to be aware of them, and make sure that you do not run afoul of their Terms & Conditions.

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