Given two positions, and a list of all positions, find quickest path

Question

I have two points (x1, y1) & (x2, y2), and a list of all possible positions that can be traversed across with format (x, y), how would I find/estimate the quickest path between the two values.

I am implementing this in Java, but the language doesn't matter much.

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Some limitations & additional info around the question:

• The values of x1 & x2 will be never the same, unless y1 = y2, this is because these positions are always located on the edge of a box.
• There may not always be a path between (x1, y1) & (x2, y2) (for when checking for continuity between the paths
• Not all possible points will be connected to either (x1, y1) or (x2, y2)
• Shortest path isn't required but would be handy to know
• Would be happy with simply some hints about what direction to take or algorithms to look into
• Travel can happen via a diagonal ((x, y) & (x + 1, y + 1) are considered neighbours)

Answer 1

The answer I was looking for is to use A* as a search algorithm, this finds the shortest path between two points given a map of all the nodes.