An exact algorithm for the statistical shortest path problem

Graph algorithms are widely used in VLSI CAD. Traditional graph algorithms can handle graphs with deterministic edge weights. As VLSI technology continues to scale into nanometer designs, we need to use probability distributions for edge weights in order to model uncertainty due to parameter variations. In this paper, we consider the statistical shortest path (SSP) problem. Given a graph G, the edge weights of G are random variables. For each path P in G, let L _P be its length, which is the sum of all edge weights on P. Clearly L_P is a random variable and we let μ_P and σ_P² be its mean and variance, respectively. In the SSP problem, our goal is to find a path P connecting two given vertices to minimize the cost function μ_P + Φ(σ_P ²) where Φ is an arbitrary function. (For example, if Φ (x) = 3√x, the cost function is μ_P + 3σ_P.) To minimize uncertainty in the final result, it is meaningful to look for paths with bounded variance, i.e., σ_P² ≤ B for a given fixed bound B. In this paper, we present an exact algorithm to solve the SSP problem in O(B(V + E)) time where V and E are the numbers of vertices and edges, respectively, in G. Our algorithm is superior to previous algorithms for SSP problem because we can handle: 1) general graphs (unlike previous works applicable only to directed acyclic graphs), 2) arbitrary edge-weight distributions (unlike previous algorithms designed only for specific distributions such as Gaussian), and 3) general cost function (none of the previous algorithms can even handle the cost function μ_P + 3σ_P. Finally, we discuss applications of the SSP problem to maze routing, buffer insertions, and timing analysis under parameter variations.