The Expected Length of a Shortest Path

Davis, R. and A. Prieditis, The expected length of a shortest path, Information Processing Letters 46 (1993) 135-141. We derive an exact summation formula and a closed-form approximation for the expected length of a shortest path for a complete graph where the arc lengths are independent and exponentially distributed random variables. Experimental data validates both results. The property of completeness allows us to exploit certain symmetries to derive these results, which would otherwise require computing an exponential number of recursive equations. We have also found that this formula is a close approximation for the expected length of a shortest path in complete graphs with uniformly distributed arc lengths.