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navigation app with a different route finding algorithm

I wish to make an android app that leverages OSM services and runs an algorithm of my choice to find the route, say A* or some heuristic on Dijkstra.
The Routing wiki page of OSM has some good pointers but it does not give any insight about how to go about implementing your own algorithm in the app.

This response is overview which might help you to orient better in the landscape and find a way to approach the problem. Your question is too broad to give an exact simple answer but I do understand why are you asking such question.
You can use Osmdroid library to display the route you have on Android.
Keep in mind that the core feature of Osmdroid is to display the map tiles. Map tiles are just bitmaps so you need either an online source of tiles or an offline one. How to do that is covered in the documentation and partly in other questions on SO. It's a too big topic to explain it here in detail. Just to make it clear: .osm file with raw OpenstreetMap data cannot be directly rendered by Osmdroid. Map tiles are usually rendered from that data in quite complex way and not on the fly.
Another very important feature in Osmdroid library is the ability to display so called Overlays. E.g. Markers, polygons etc. You may use those to draw your route on the map and other features you may wish to display. This is again covered in the documentation and too big to go in details here.
There is a support for routing for Osmdroid (AFAIK in Osmbonuspack) but that's about using some 3rd party service to get your routing data and displaying that. According to your comment you want to implement your own algorithm and therefore this would be no use to you.
So the last part is how to integrate your algorithm into that. That's not a simple question with a simple answer. You'll need to do almost everything on your own, starting with getting the data you need to run the algorithm from the OpenstreetMap database (.osm file). I don't think there's an API which would provide you reasonable way to access the data in a form suitable for route computation. The approach I would choose is to develop some kind of preprocessing which would take either .osm file or a database with the data an produce my own structure representing the graph for navigation. I would than include this graph in my application and run my algorithm on that data. The output of my algorithm would be the route: a list of coordinates which I would use to draw the polyline.
If the area you need to cover is small enough, you may start with including the .osm file in your project and running the algorithm in the same was as you do now. Maybe it's going to be good enough for your situation.
One interesting project to look at to find inspiration is Graphopper - they have build exactly what I said above and they use it to provide a library for Android/iOs and a web API. But the core is the same - pre-process OSM data into graph data and run the algorithm on that. Since you wish to implement the algorithm on your own this library will not help you (unless you fork the project and modify it) but it might be a valuable learning resource.

Android indoor map using image generates different path from point a to b depending on destination's input (No use of indoor navigation technologies)

I'm working on an android app for my final year project and I'm unsure how to solve this problem. I would like some advices to get me started in the right direction.
The project is to develop an android app to output a direction path from point a to point b on an image representing a two dimensional indoor map. The direction displayed will be depending on the user's input (destination). The starting point will always be the same location on the image. Once the user select a specific object, the app should find its location on the map using a database and then draw the direction's path from Point "a" to point "b".
I do not want to use any indoor navigation technology for this app. The location of each object on the map should be predefined and stored in a database.
The part which I am really unsure about is how to do the following for and android app. --> How to predefine points on an image and how to use these predefined data to display the path to take in the image representing the indoor map. I have been advised to use SVG (Scalable Vector Graphics), I have found androidVG to use SVG with android but I haven't found much information on it.
I am currently clueless regarding what language and techniques to use in order to perform the above feature on android.
Questions:
1) What general advice would you give me on how to effectively tackle this problem from the information I've provided?
2) Was I correctly advised when I have been given SVG as one of the language to use for the development of this app? If I was, would anybody have any more information on using SVG for android to provide me with?
3) IF there would be a better way to solve this problem which language should be used?
I really appreciate all the help provided in this community. Hopefully I have been clear enough, hoping to have my questions answered. Thank you!

For this use case I think it would be much simpler to just use the standard Canvas API to draw your lines rather than using an SVG renderer.
I would extend the Android ImageView class. In the onDraw() method call super.onDraw() to let it plot the image. Then add your lines using (for example) a set of canvas.drawLine() calls.

Downloading specific maptiles to cache in OSMDroid

I am currently working on a school project and I am trying to learn the techniques on downloading OSM map tiles into the phone's local memory/cache, so that users are able to view the maps when offline.
This is to be done by giving a specific location, and the map tiles that are within specific radius/distance from the location will be downloaded into the phone memory (when there is online connection) for offline display.
However, I am having some slight trouble in understanding OSMDroid's APIs. From my understanding, it seems that the classes involved are:
MapTileBase
MapDownloader
IFilesystemCache
Please do correct me if I'm mistaken.
Also, in MapTileBase (which I believe is the provider), the getMapTile returns android.graphics.drawable.Drawable.
Is this Drawable class even related to the concept of offline map display?
Hope someone can enlighten me on where exactly maps are being called to be downloaded into the phone's memory, and how we can edit the maps to display interface items such as routes, pins, etc.

I don't have an exactly answer as you're looking for a method to cache offline outside of an area the MapView hasn't already navigated to. The built in caching support is there, but not exactly how you want to do it.
You're going to need online download capabilities if you plan to cache and not just supply your own offline map tiles. I would take a look at the MapTileDownloader and maybe try to implement your own version of it. The MapTileDownloader.TileLoader.loadTile() method takes a MapTileRequestState which I believe the MapTileProviderArray calls when a map tile is needed.
If you can replicate the existing MapTileDownloader's TileLoader functionality, you can probably inject your own relative MapTileRequestState or straight up MapTile's to the one requested. e.g. Take the TileLoader called MapTileRequestState, inspect it, figure out how to construct new MapTileRequestState/MapTile representing tiles outside the current view. It won't be easy though as you'll have to deal with the different zoom levels, and depending on the zoom level, the radius should be smaller, etc... A lower zoom level of 18 would have a larger tile radius than a higher 10.
Personally I was recently working on trying to work 100% offline so I don't have any example code to help. Good luck even though its been a couple of months.

First, go right the source
https://github.com/osmdroid/osmdroid
There's a few wiki articles that will help, especially the offline map tiles article. In short, use MOBAC or the tool osmdroid provides. OSMBonuspack also has a tool to perform the same task on device.
Map tiles are actually downloaded using this class
https://github.com/osmdroid/osmdroid/blob/master/osmdroid-android/src/main/java/org/osmdroid/tileprovider/modules/MapTileDownloader.java

Alternative(To Google Maps) Open Source Map API that provides a database of nodes and Arcs on roads

I need a Map API for Android that can provide me with indexed nodes and indices that make up the road network. The main idea is to determine if two GPS devices are on the same road. Thank you in advance

A Map API by itself will not have that information. Anyway, you can get it from OpenStreetMap freely. You can download it from here.
I don't understand from your question if you intend on displaying the results on a map. If so, and you want a nice and free map API, I would suggest Leaflet. It's not as mature as the likes of OpenLayers but, as you've tagged this post with "android", Leaflet just kicks ass in the mobile department.

OpenStreetMap is definitely a good source of data for this kind of project. Unlike google maps, it gives developers access to the underlying vector data of a map (fully open). This allows interesting new use cases which simply are not possible with google maps, and something involving geometric calculations like this would definitely fit into that category. You either need OpenStreetMap or some other source of "vector" map data, and beyond OpenStreetMap this can be expensive.
Unfortunately that's not the full answer to your question. You still have a lot of work to do to use the data in the way you intend. You need to calculate the proximity of two points (GPS readings from two devices?) to nearby roads, and figure out which road the point lies closest to. It's the kind of powerful geo calculation you might do using a GIS package such as QGIS or a functions of a geo-aware database system PostGIS.
But that's not the answer to your question, because you need to do these calculations on device. I'm not aware of an off-the-shelf library to do this on android. I think you would have to roll your own.
But another challenge is to get the vector data onto the device in a suitable format in the first place, and this is the first thing to solve. You'd want the vector data either as a large download for a whole country, or perhaps a smaller area, perhaps with an on-the-fly download feature within this app. Whole countries are not infeasible when working with maps in vector form (ever tried the awesome MayDroyd app?), but require some compact formatting. Happily some of these problems are starting to be solved in open source off-the-shelf libraries. You could try to build on top of MapsForge for example.
So then you're back to the challenge of writing on-device code to poke around in this data and do the calculations you want to do. I suppose it could be rather good if projects like MapsForge included generic PostGIS style geo-functions to make this easier. Something to ask the mapsforge developers about perhaps.

Can we use our own map in android?

Can we use our own map(Like our house map) and then use GPS to show where we are in the map?
If it is possible then how to do it?

You can do anything you want, but if you are asking if there is an easy way to do this, no.
You would have to produce a scale drawing, then map the pixel space to the scale of the image, know what geopoints represented the corners of your map, then project that onto your scaled display image.
Some open source products use tiled images similar to google maps, but they use OSM data, I suppose if you were dedicated you could use that and turn your map into tiles but I would think for your purposes it would be easier to do this your self.
You might check the google code repository etc. and other open source venues and see if anyone has done anything similar to this.
Also the best accuracy you are going to get is about 2m, and certainly not indoors (if you got a signal at all)