I have configured a web based GIS Map using Geoserver & Openlayers controls.Now I want to add 'Shortest Route' analysis in the same...
Thanks In advance Regads Sandipan
Geoserver serves maps while OpenLayers displays them. None of those two can do routing maps as of this writing. What you need is something to do the shortest path computation.
I think it would be best to study PostGIS and pgrouting as @simplexio has suggested. PostGIS is a spatial database while pgrouting extends PostGIS by adding geospatial routing capabilities. Here's a workshop that combines PostGIS, OpenLayers and pgRouting. It should be enough to get you started. Good luck!
Your question is quite open ended, but I assume what you are after would be some sort of routing engine.
GeoServer and OpenLayers do not have routing capabilities built in (someone can correct me if I am wrong though), so you would need some sort of routing engine on your server such as RouteWare, or you could use a web routing service such as the Google Directions API.
I hope this helps get you started.