Random projection: a new approach to VLSI layout

We show that random projection, the technique of projecting a set of points to a randomly chosen low-dimensional subspace, can be used to solve problems in VLSI layout. Specifically, for the problem of laying out a graph on a 2-dimensional grid so as to minimize the maximum edge length, we obtain an O(log/sup 3.5/ n) approximation algorithm (this is the first o(n) approximation), and for the bicriteria problem of minimizing the total edge length while keeping the maximum length bounded, we obtain an O(log/sup 3/ n, log/sup 3.5/ n) approximation. Our algorithms also work for d-dimensional versions of these problems (for any fixed d) with polylog approximation guarantees. Besides random projection, the main components of the algorithms are a linear programming relaxation, and volume-respecting Euclidean embeddings.