Greedy routing steps S_n in the sequentially grown 1D graph equals the number of left-to-right and right-to-left minima in the insertion permutation minus 2, hence Theta(log n) whp.
Zhao, Ling Huang, Jeremy Stribling, Sean C
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Greedy Routing in a Sequentially Grown One-Dimensional Random Graph
Greedy routing steps S_n in the sequentially grown 1D graph equals the number of left-to-right and right-to-left minima in the insertion permutation minus 2, hence Theta(log n) whp.