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When solving using a recursive assembly plan, elements we're now solving for may occur on one or more existing cluster's frontiers. The previous code correctly solves for those cluster's poses such that all clusters agree on those elements' global positions. However, the previous code does not take into account that changing one of those clusters' poses may move an element that we're not directly solving for, but which may itself occur on some other cluster's frontier. That cluster will then also have to be moved, etc. Therefore, we need to solve for the transitive closure of all elements affected. #97 passes once rebased on top of this.
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tomcur
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The old routine does a depth-first search, which doesn't actually find *minimum* dense subgraphs, which a a breadth-first search does find. Finding small dense subgraphs is important for solving performance. Note both the DFS and the BFS implementations have exponential time-complexity, but BFS will in general lead to constructing more efficient decomposition plans. #97 introduced a polynomial-time *minimal* dense subgraph finder (not *minimum*). Its correctness is harder to check, so I'm thinking to keep the exhaustive search too, for now, mainly for ease of debugging. In addition to changing to BFS, this now makes use of the sparsity of the graph (normally, most elements only have a few incident edges) to reduce the amount of scanning performed, making this more useful in practice.
tomcur
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The old routine does a depth-first search, which doesn't actually find *minimum* dense subgraphs, which a a breadth-first search does find. Finding small dense subgraphs is important for solving performance. Note both the DFS and the BFS implementations have exponential time-complexity, but BFS will in general lead to constructing more efficient decomposition plans. #97 introduced a polynomial-time *minimal* dense subgraph finder (not *minimum*). Its correctness is harder to check, so I'm thinking to keep the exhaustive search too, for now, mainly for ease of debugging. In addition to changing to BFS, this now makes use of the sparsity of the graph (normally, most elements only have a few incident edges) to reduce the amount of scanning performed, making this more useful in practice.
tomcur
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The old routine does a depth-first search, which doesn't actually find *minimum* dense subgraphs, which a a breadth-first search does find. Finding small dense subgraphs is important for solving performance. Note both the DFS and the BFS implementations have exponential time-complexity, but BFS will in general lead to constructing more efficient decomposition plans. #97 introduced a polynomial-time *minimal* dense subgraph finder (not *minimum*). Its correctness is harder to check, so I'm thinking to keep the exhaustive search too, for now, mainly for ease of debugging. In addition to changing to BFS, this now makes use of the sparsity of the graph (normally, most elements only have a few incident edges) to reduce the amount of scanning performed, making this more useful in practice.
tomcur
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Oct 28, 2025
The old routine does a depth-first search, which doesn't actually find *minimum* dense subgraphs, which a a breadth-first search does find. Finding small dense subgraphs is important for solving performance. Note both the DFS and the BFS implementations have exponential time-complexity, but BFS will in general lead to constructing more efficient decomposition plans. #97 introduced a polynomial-time *minimal* dense subgraph finder (not *minimum*). Its correctness is harder to check, so I'm thinking to keep the exhaustive search too, for now, mainly for ease of debugging. In addition to changing to BFS, this now makes use of the sparsity of the graph (normally, most elements only have a few incident edges) to reduce the amount of scanning performed, making this more useful in practice.
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This replaces the exhaustively searching
densefunction with a max-flow based one. The exhaustive search finds minimum dense subgeometry (which is known to be NP-hard), whereas we only need minimal dense subgeometry (in that there isn't a proper subset of that subgeometry that itself is also dense).Draft, as this needs more explanation in the code, and more validation. I believe this uncovers an issue in the solving: sometimes the final configuration of points is not correct. I've had a quick look, and I think the issue is that child clusters sometimes get wrong poses. When there are two child clusters that share more than one element, those clusters should all be posed such that they agree on the elements' positions. I believe currently, only the elements of the subgeometry now solved for are considered.