bq. Find the shortest path between two nodes nodes in a large graph which you can’t keep in memory, for example the Facebook friendship graph.
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The breadth first search algorithm is a standard approach for this problem. If the distance between two nodes is $k$ and the average degree in the network is $p$ BFS explores $O(p^k^)$ nodes.
A better solution starts from both nodes and sees when the two search spaces meet.  This reduces the number of states explored to $O(p^k/2^)$.