Ewa Misiołek

Two flow network simplification algorithms

Flow network simplification can reduce the size of the flow network and hence the amount of computation performed by flow algorithms. We present the first linear time algorithm for the undirected network case. We also give an O(|E| × (|V| + |E|)) time algorithm for the directed case, an improvement over the previous best O(|V| + |E|2 log |V|) time solution. Both of our algorithms are quite simple.

Shape rectangularization problems in intensity-modulated radiation therapy

In this paper, we present a theoretical study of several geometric shape approximation problems, called shape rectangularization (SR), which arise in intensity-modulated radiation therapy (IMRT). Given a piecewise linear function f such that f(x) ≥0 for any x∈ℝ, the SR problems seek an optimal set of constant window functions to approximate f under a certain error criterion, such that the sum of the resulting constant window functions equals (or well approximates) f. A constant window function W is defined on an interval I such that W(x) is a fixed value h>0 for any x ∈I and is 0 otherwise. Geometrically, a constant window function can be viewed as a rectangle (or a block). The SR problems find applications in micro-MLC scheduling and dose calculation of the IMRT treatment planning process, and are closely related to some well studied geometric problems. The SR problems are NP-hard, and thus we aim to develop theoretically efficient and provably good quality approximation SR algorithms. Our main results include a polynomial time

Computing toolpaths for 5-axis NC machines

(\frac{3}{2}+\epsilon)

Efficient algorithms for simplifying flow networks

-approximation algorithm for a general key SR problem and an efficient dynamic programming algorithm for an important SR case that has been studied in medical literature. Our key ideas include the following. (1) We show that a crucial subproblem of the key SR problem can be reduced to the multicommodity demand flow (MDF) problem on a path graph (which has a known (2+e)-approximation algorithm); further, by extending the result of the known (2+e)-approximation MDF algorithm, we develop a polynomial time

Flattening topologically spherical surface

(\frac{3}{2}+\epsilon)

Optimal Surface Flattening

-approximation algorithm for our first target SR problem. (2) We show that the second target SR problem can be formulated as a k-MST problem on a certain geometric graph G; based on a set of interesting geometric observations and a non-trivial dynamic programming scheme, we are able to compute an optimal k-MST in G efficiently.

Finding many optimal paths without growing any optimal path trees

We present several algorithms for computing a feasible tool-path with desired features for sculpting a given surface using a 5-axis numerically controlled (NC) machine in computer-aided manufacturing. A toolpath specifies the orientation of a cutting tool at each point of a path taken by the tool. Previous algorithms are all heuristics with no quality guarantee of solutions and with no analysis of the running time. We present optimal quality solutions and provide time analysis for our algorithms. We model the problems using a directed, layered graph G such that a feasible toolpath corresponds to a certain path in G, and give efficient methods for solving several path problems in such graphs.

Stabbing Convex Polygons with a Segment or a Polygon

Computing flows in a network is a fundamental graph theory problem with numerous applications. In this paper, we present two algorithms for simplifying a flow network G=(V,E), i.e., detecting and removing from G all edges (and vertices) that have no impact on any source-to-sink flow in G. Such network simplification can reduce the size of the network and hence the amount of computation performed by maximum flow algorithms. For the undirected network case, we present the first linear time algorithm. For the directed network case, we present an O(|E|*(|V|+|E|)) time algorithm, an improvement over the previous best O(|V|+|E|2 log |V|) time solution. Both of our algorithms are quite simple.

Algorithms for interval structures with applications

The problem of optimal surface flattening in 3-D finds many applications in engineering and manufacturing. However, previous algorithms for this problem are all heuristics without any quality guarantee and the computational complexity of the problem was not well understood. In this paper, we prove that the optimal surface flattening problem is NP-hard. Further, we show that the problem of flattening a topologically spherical surface admits a PTAS and can be solved by a (1+ε)-approximation algorithm in O(nlog n) time for any constant ε>0, where n is the input size of the problem.

Free-Form Surface Partition in 3-D

The problem of optimal surface flattening in 3-D finds many applications in engineering and manufacturing. However, previous algorithms for this problem are all heuristics without any quality guarantee and the computational complexity of the problem was not well understood. In this paper, we prove that the optimal surface flattening problem is NP-hard. Further, we show that the problem admits a PTAS and can be solved by a (1 + ?)-approximation algorithm in O(nlogn) time for any constant ?> 0, where nis the input size of the problem.