Anna Lubiw

Doubly lexical orderings of matrices

Every matrix has a doubly lexical ordering an ordering of the rows and columns so that the row vectors are lexically (or “lexicographically”) increasing and the column vectors are lexically increasing. Every graph has a lexical ordering: a vertex ordering making the neighbourhood matrix doubly lexical. An almost linear time doubly lexical ordering algorithm is given. Doubly lexical orderings unify the orderings characterizing certain classes of matrices and graphs, including totally balanced matrices, subtree matrices and chordal graphs.

Pattern matching for permutations

Given a permutation T of 1 to n, and a permutation P of 1 to k, for k ≤ n, we wish to find a k-element subsequence of T whose elements are ordered according to the permutation P. For example, if P is (1,2,..., k), then we wish to find an increasing subsequence of length k in T; this special case can be done in time O(n log log n) [CW]. We prove that the general problem is NP-complete. We give a polynomial time algorithm for the decision problem, and the corresponding counting problem, in the case that P is separable—i.e. contains neither the subpattern (3,1,4,2) nor its reverse, the subpattern (2,4,1,3).

Some NP-Complete Problems Similar to Graph Isomorphism

The GRAPH ISOMORPHISM problem has so far resisted attempts at determining its complexity status—it has not been shown to be NP-complete nor in P. In this paper several altered or generalized versions of the ISOMORPHISM problem are presented and shown to be NP-complete. One of these is the problem of determining whether a given graph has a fixed-point-free automorphism. Some speculation is made on the possible implications of these results on deciding the complexity status of ISOMORPHISM. Various classes and hierarchies of problems in NP are discussed.

Efficient algorithms for Petersen's matching theorem

Petersens theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n3/2) time for 3-regular graphs. We have developed an O(nlog4n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

Efficient visibility queries in simple polygons

We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n3 log n) preprocessing time and O(n3) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(log n + k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(log n) time.The key notion behind the decomposition is the succinct representation of visibility regions, and tight bounds on the number of such regions. These techniques are extended to handle other types of queries, such as visibility of fixed points other than the polygon vertices, and for visibility from a line segment rather than a point. Some of these results have been obtained independently by Guibas, Motwani and Raghavan [18].

Upward Planar Drawing of Single-Source AcyclicDigraphs

An upward plane drawing of a directed acyclic graph is a plane drawing of the digraph in which each directed edge is represented as a curve monotone increasing in the vertical direction. Thomassen has given a nonalgorithmic, graph-theoretic characterization of those directed graphs with a single source that admit an upward plane drawing. This paper presents an efficient algorithm to test whether a given single-source acyclic digraph has an upward plane drawing and, if so, to find a representation of one such drawing. This result is made more significant in light of the recent proof by Garg and Tamassia that the problem is NP-complete for general digraphs. The algorithm decomposes the digraph into biconnected and triconnected components and defines conditions for merging the components into an upward plane drawing of the original digraph. To handle the triconnected components, we provide a linear algorithm to test whether a given plane drawing of a single-source digraph admits an upward plane drawing with the same faces and outer face, which also gives a simpler, algorithmic proof of Thomassens result. The entire testing algorithm (for general single-source directed acyclic graphs) operates in

Decomposing polygonal regions into convex quadrilaterals

O(n^2)

A Visibility Representation for Graphs in Three Dimensions

time and

Computing homotopic shortest paths efficiently

O(n)

Noncrossing subgraphs in topological layouts

space (