Fitting the \best" straight line to a scatter plot of data D f(x1; y1) : : :(xn; yn)g in which both variables xi; yi are subject to unknown error is undoubtedly the most common problem of inference faced by scientists, engineers, medical researchers, and economists. The problem is to estimate the parameters ; in the straight line equation y = + x , and assess the accuracy of the estimates. When...

It is known that for any set V of n ≥ 4 points in the plane, not in convex position, there is a 3-connected planar straight line graph G = (V, E) with at most 2n− 2 edges, and this bound is the best possible. We show that the upper bound |E| ≤ 2n continues to hold if G is constrained to contain a given graph G0 = (V,E0), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i...

Given a planar graph G on n vertices, let fix (G) denote the maximum k such that any straight line drawing of G with possible edge crossings can be made crossing-free by moving at most n − k vertices to new positions. Let v̄(G) denote the maximum possible number of collinear vertices in a crossingfree straight line drawing of G. In view of the relation fix (G) ≤ v̄(G), we are interested in exampl...

Consider a planar straight line graph (PSLG), G, with k connected components, k 2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k−1 edges so that we get a color conforming...

Straight line factoring algorithms include a variant Lenstra’s elliptic curve method. This note proves lower bounds on the length of straight line factoring algorithms.

Solving symbolically polynomial equation systems when intermediate and final polynomials are represented in the usual dense encoding turns out to be very inefficient: the sizes of the systems one can deal with do not respond to realistic needs. Evaluation representations appeared in this frame a decade ago as a new possibility to treat new families of problems. We present a survey of the most r...

We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed m...

We present a new algorithm for the computation of resultants associated with multihomogeneous (and, in particular, homogeneous) polynomial equation systems using straight-line programs. Its complexity is polynomial in the number of coefficients of the input system and the degree of the resultant computed.

The paper presents a design for the Denali-2 super-optimizer, which will generate minimum-instruction-length machine code for realistic machine architectures using automatic theorem-proving technology: specifically, using E-graph matching (a technique for pattern matching in the presence of equality information) and boolean satisfiability solving. The paper presents a precise definition of the ...

In this paper, we investigate planar upward straight-line grid drawing problems for bounded-degree rooted trees so that a drawing takes up as little area as possible. A planar upward straight-line grid tree drawing satisses the following four constraints: (1) all vertices are placed at distinct grid points (grid), (2) all edges are drawn as straight lines (straight-line), (3) no two edges in th...