Could someone explain to me how the nodes should select the heaviest subtree, when they don't have that information? It requires information from all(!) other nodes. How can this even make sense, because one node is connected to say 8 peers and not the entire network?
Block headers could be broadcast for orphaned blocks. However, it is possible that 2 nodes will have different headers accumulated.
With a linear blockchain, proof of work is always exactly agreed by all miners.
Since all block headers are linked. All nodes can agree on the proof of work linked to a particular header. If a node was missing any blocks on that chain, then it wouldn't be able to form the chain at all.
With GHOST, 2 nodes could disagree about the proof of work for a particular chain. There is no way to be sure that you have all the orphans.
Our proposal includes keeping the hashes of off-chain blocks in each block (new ones not included in any of the ancestors of the block). The chain, when read sequentially therefore contains all off-chain block hashes, and thus any node can know the entire tree (if a node is missing some block it simply requests it or its header).