More on random walks, electrical networks, and the harmonic \(k\)-server algorithm. (Q1853151)

Techniques from electrical network theory have been used to establish various properties of random walks. We explore this connection further, by showing how the classical formulas for the determinant and cofactors of the admittance matrix, due to Maxwell and Kirchoff, yield upper bounds on the edge stretch factor of the harmonic random walk. For any complete, \(n\)-vertex graph with distances assigned to its edges, we show the upper bound of \((n-1)^{2}\). If the distance function satisfies the triangle inequality, we give the upper bound of \(\frac12n(n-1)\). Both bounds are tight. As a consequence, we obtain that the harmonic algorithm for the \(k\) server problem is \(\frac12k(k+1)\)-competitive against the lazy adversary.