Algorithms for Sparse Layered Graphs and Transition Graphs
E.2 Retrieval algorithms
In the worst case for shortest distance queries, the nodes are contracted already on the smallest scale. Then query will be performed on the entire graph. However, in all other cases the computation of the shortest path using Dijkstra’s algorithm on a TG is at least as fast a applying the algorithm to the SLG. This is due to the construction of the TG which ideally sparsifies SLG. This depends on the implementation of the union operator, though. The operator is required to merge cells of the SLG and TG as mentioned in the description of the algorithm.
Retrieval of a shortest path requires the expansion of contracted as well as all-pair-shortest-path edges. Denote the aforementioned union of cells of the SLGS with the TGT by the temporary graphQ, and the shortest path inQ asPQ. Expanding any all-pair-shortest-path on layer kyields the shortest path PSk in Sk. Furthermore, PQ crosses ζQ many cells in Q, with 0≤ζQ≤Ck−1 andCk the number of cells on layerk. Then, the run-time complexity of shortest path retrieval is given by
O(ζQESk Ck +VkS
Ck logVkS Ck
). (E.17)
Simulations suggest that thatζQ≈ C2k. In this case, Equation E.17 can be assumed to follow according to
O(EkS+VkSlogVkS
Ck)⊆ O(EkS+VkSlogVkS). (E.18) Recursively expanding all remaining contracted edges requires at most 2k−1·(PSk−1) computations in lower-level cells.
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