Example: at most K flips
Shortest path where at most K times you can flip a coin (change mode). Layers = flip count. Edge across layers = flip.
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Time-dependent edges
Layers = time. Edge (u, v, t) → cross-layer edge. Traffic-aware routing.
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State-space blowup
K layers → K·V vertices, K·E + cross-edges. Manageable if K small.
Alternative to state augmentation
Layered graph = concrete realization of state-augmented BFS/Dijkstra. Sometimes easier to reason about.