In my recent post I motivated the elimination tree and gave a loose derivation from the Cholesky task DAG. I thought it might be nice to visualize the actual tree-computation algorithm (without path compression) to help develop intuition for how it represents structural fill so compactly. I asked Codex to build this visualization, which I share below:
Open the elimination tree visualization
You can step forward or backward through the elimination-tree calculation. Structurally filled entries are colored differently from starting nonzeros. As the algorithm traverses the partially constructed parent structure, the visualization displays the new entries implied by the corresponding tree paths. The etree computation itself is constructing the dependency forest; the colored matrix pattern is the symbolic fill inferred from it, with numerical cancellation intentionally ignored.