tried to use Dijkstra algorithm with spatial data offline on the browser, unfortunately I didn't found anything about it, so why not make my own ... I tried it for now and it's working ( at least on my setup)

1- check if your road network is perfectly connected

2- make sure that the vertices of your network aren't complicated otherwise you may have false routes

How to use it :

<script src="http://openlayers.org/en/v3.18.2/build/ol.js"></script>
<script src="required/underscore.js"></script>
<script src="required/graph.js"></script>
<script src="OR-OL3.js"></script>

make sure to get all the js files above

create your network from a featureCollection (linestring geometries):

var network = createNetwork(yourFeatureCollection);

then you get the geometry of the shortest path between two set of coordinates following your network

var shortestPath = getShortestPath(network, coordsSource, coordsDestination);

there is a working example on Example.html for the old Medina of Tangier

NOTE: credits for the graph.js goes to https://github.com/andrewhayward/dijkstra