Alan P. Sprague

Performance of parallel branch-and-bound algorithms

Consideration is given to the performance of parallel best-bound-first branch-and-bound algorithms in which several nodes with least lower bounds are expanded simultaneously. It is well known that anomalies may occur in the execution of a parallel branch-and-bound algorithm. The authors show the conditions under which anomalies are guaranteed not to occur when the number of processors is doubled, or not even doubled.

Polar Spaces Having Some Line of Cardinality Two

Abstract The nonthick geometries of type C n and D n or equivalently all polar spaces having at least one line of cardinality 2 are classified. It turns out that there are two classes of such polar spaces. On the one hand, decomposable polar spaces or polar spaces which are direct sums of two or more polar spaces are obtained. On the other hand, polar spaces arising from the interval lattice of an irreducible projective geometry which can also be seen as being partitioned by a pair of disjoint maximal singular subspaces can be gotten.

Placement of the processors of a hypercube

The authors formalize the problem of minimizing the length of the longest interprocessor wire as the problem of embedding the processors of a hypercube onto a rectangular mesh, so as to minimize the length of longest wire. Where neighboring nodes of the mesh are taken as being at unit distance from one another, and where wires are constrained to be laid out as horizontal and vertical wires, the length of the wire joining nodes u and v of the mesh equals the graph-theoretic distance between u and v. The problem of minimizing delays due to interprocessor communication is then modeled as the problem of embedding the vertices of a hypercube onto the nodes of a mesh, so as to minimize dilation. Two embeddings which achieve dilations that (for large n) are within 26% of the lower bound for square meshes and within 12% for meshes with aspect ratio 2 are presented. >

Rank 3 incidence structures admitting dual-linear, linear diagram

Abstract The results, restricted to finite Buekenhout incidence structures, are the following. (1) Let Γ be a finite rank 3 Buekenhout incidence structure admitting diagram Then for some generalized projective geometry π and some integer i , Γ is isomorphic to the rank 3 Buekenhout incidence structure having all ( i − 2)-, ( i − 1)-, and i -dimensional subspaces of π as varieties, and comparability as the incidence relation. (2) Let Γ be a finite rank 3 Buekenhout incidence structure admitting diagram Then for some set X and integer i , Γ is isomorphic to the rank 3 Buekenhout incidence structure having all ( i − 1)-, i -, and ( i + 1)-subsets of X as varieties, and set inclusion as the incidence relation.

Characterization of projective incidence structures

Abstract : Consider a simple graph whose vertices are s-dimensional subspaces of a d-dimensional vector space V over (GF(q). Two vertices in this graph are adjacent if the corresponding s-dimensional subspaces intersect in an (s-1)-dimensional subspace. This graph will be called an (s,q,d)-projective graph. The Theorem 1 of this paper can be used to obtain a characterization of the (s,q,d)-projective graphs provided d is larger than some function of s and q. Characterization problems of Affine spaces and Polar spaces are also concerned in terms of flats of higher dimensions.

Pasch's axiom and projective spaces

Let @p be a generalized projective geometry and i @e Z^+ such that some i-dimensional subspace of @p contains finitely many (i - 1) dimensional subspaces. We give a characterization of the incidence structure formed by the (i - 1)-dimensional and i-dimensional subspaces of @p (where the incidence relation is set inclusion). The specialization of this characterization to projective geometries follows. Let @q be a connected incidence structure with more than one point and line, and let two lines have at most one point in common. If (i) both @q and the dual of @q satisfy Paschs Axiom, (ii) for all points x and lines m of @q not containing x, the number of points of m collinear with x is not 1 or 2, (iii) some line has finitely many points, then @q is the incidence structure having the (i - 1)-dimensional and i-dimensional subspaces (for some finite i) of some projective geometry of finite order as its points and lines respectively.

Incidence Structures whose Planes are Nets

A d-net is a connected semilinear incidence structure π such that (D1) every plane is a net, (D2) the intersection of two subspaces is connected, (D3) if two planes in a 3-space have a point in common then they have a second point in common, and (D4) the minimum number of points which generate π is d. Let V be a vector space over a skew field F, and W a subspace of finite codimension d. Let P, L be the set of d-, (d - 1)-dimensional subspaces respectively of V whose intersection with W is the zero vector. The incidence structure ( P , L ⊇ ) is called an attenuated space. We show every d-net for finite d ⩾ 3 is an attenuated space. We also characterize d-nets (together with AG(d, 2)) as those incidence structures belonging to the diagram where signifies a projective plane and signifies a net.

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhouts intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.

On the routability of a convex grid

Abstract We give necessary and sufficient conditions to lay out the nets of a convex grid, in the knockknee model. Convex grids, defined by Nishizeki, Saito, Suzuki [9], include the common rectilinear channel, as well as L- , T- , and X -shaped regions. Whether a convex grid can be laid out can be determined in time which is linear in the perimeter of the grid. The algorithm of [9] to accomplish this is incomplete.

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and pij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff dim(x ⌢ y) = s − 1. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p11(x, y) = λ whenever d(x, y) = 1, and p11(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where 2d3 ≤ s < d − 2 and (s, q, d) ≠ (2d3, 2, d), are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.