Design of On-Line Algorithms Using Hitting Times

Prasad Tetali
1999 SIAM journal on computing (Print)  
Random walks are well known for playing a crucial role in the design of randomized off-line as well as on-line algorithms. In this work we prove some basic identities for ergodic Markov chains (e.g., an interesting characterization of reversibility in Markov chains is obtained in terms of first passage times). Besides providing new insight into random walks on weighted graphs, we show how these identities give us a way of designing competitive randomized on-line algorithms for certain
more » ... problems. Hence we have the claim. Remark 3. Any quantitative sharpening of the trace inequality (Theorem 2.6) immediately gives an improved upper bound on stretch for cost matrices of Type I. Definition 3.3. A cost matrix is ergodic if it is either the hitting time matrix of an ergodic chain (Type I) or the commute time matrix of a reversible chain (Type II). Note that Theorem 2.2 guarantees that we can test if a matrix is ergodic or not with essentially a single matrix inversion. Theorem 3.4. Every graph with an ergodic cost matrix has a random walk with stretch ≤ n − 1. Moreover, this walk (termed ergodic walk) can be designed with a single matrix inversion. Proof. From Claims 1 and 2, the first part of the theorem follows. We now show how Theorem 2.2 can be used to design the desired random walk. Let G be a graph with the ergodic cost matrix C. In view of (**) it is easy to see that C in + C nj − C ij = H in +H nj −H ij , regardless of whether C is of Type I or II. DefineH ij = C in +C nj −C ij , for 1 ≤ i, j ≤ n − 1. The rest should be obvious: We construct P , the transition probability matrix of the desired walk, by first computingP using Theorem 2.2. Remark 4. Recall that a cost matrix is resistive if the C ij can be interpreted as effective resistance R ij ∀i, j. Note that any resistive cost matrix is of Type II (modulo a constant factor), since effective resistance R ij is essentially the commute time H ij +H ji (modulo the same constant factor) of a reversible chain. This, together with Fact 1, shows that our results imply those of [8] . The lower and upper bounds are obviously tight under symmetry, since Ψ(C) = 1. The following example shows that the lower bound is, in general, tight.
doi:10.1137/s0097539798335511 fatcat:ke2weubvivdqfdyb54qpvzn6eu