Chul E. Kim

Heuristic Algorithms for Scheduling Independent Tasks on Nonidentical Processors

The finishing time properties of several heuristic algorithms for scheduling <italic>n</italic> independent tasks on <italic>m</italic> nonidentical processors are studied. In particular, for <italic>m</italic> = 2 an <italic>n</italic> log <italic>n</italic> time-bounded algorithm is given which generates a schedule having a finishing time of at most (√5 + 1)/2 of the optimal finishing time. A simplified scheduling problem involving identical processors and restricted task sets is shown to be P-complete. However, the LPT algorithm applied to this problem yields schedules which are near optimal for large <italic>n</italic>.

Digital Straight Lines and Convexity of Digital Regions

It is shown that a digital region is convex if and only if every pair of points in the region is connected by a digital straight line segment contained in the region. The midpoint property is shown to be a necessary but not a sufficient condition for the convexity of digital regions. However, it is shown that a digital region is convex if and only if it has the median-point property.

On the Cellular Convexity of Complexes

In this paper we discuss cellular convexity of complexes. A new definition of cellular convexity is given in terms of a geometric property. Then it is proven that a regular complex is celiularly convex if and only if there is a convex plane figure of which it is the cellular image. Hence, the definition of cellular convexity by Sklansky [7] is equivalent to the new definition for the case of regular complexes. The definition of Minsky and Papert [4] is shown to be equivalent to our definition. Therefore, aU definitions are virtually equivalent. It is shown that a regular complex is cellularly convex if and only if its minimum-perimeter polygon does not meet the boundary of the complex. A 0(n) time algorithm is presented to determine the cellular convexity of a complex when it resides in n × m cells and is represented by the run length code.

Digital Convexity, Straightness, and Convex Polygons

New schemes for digitizing regions and arcs are introduced. It is then shown that under these schemes, Sklanskys definition of digital convexity is equivalent to other definitions. Digital convex polygons of n vertices are defined and characterized in terms of geometric properties of digital line segments. Also, a linear time algorithm is presented that, given a digital convex region, determines the smallest integer n such that the region is a digital convex n-gon.

Digital Disks

Geometric properties of digital disks are discussed. An algorithm is presented that determines whether or not a given digital region is a digital disk.

Approximation Algorithms for Certain Scheduling Problems

A subset of a set of tasks F is to be scheduled on a single processor. Associated with each task J time μJ and profit pJ. The tasks have precedence constraints in that a task cannot be scheduled until all of its predecessors have been scheduled. Approximation algorithms of polynomial time complexity are presented for the following problems: MAXPROFIT problem. Given T, the available processor time, find a subset L of F such that ΣL∈LμJ ≤ T and ΣJ∈LPJ is maximal. MINTIME problem. Given P, the minimum profit desired, find a subset L of F such that ΣJ∈LPJ ≥ p and ΣJ∈LμJ is minimal.

Three-Dimensional Digital Line Segments

Digital arcs in 3-D digital pictures are defined. The digital image of an arc is also defined. A digital arc is defined to be a digital line segment if it is the digital image of a line segment. It is shown that a digital line segment may be characterized by the chord property holding for its projections onto the coordinate planes. It is also shown that a digital line segment may not be characterized by its own chord property. A linear time algorithm is presented that determines whether or not a digital arc is a digital line segment.

Digital disks and a digital compactness measure

An O(n2) time algorithm is presented that determines whether or not a given convex digital region is a digital disk. A new compactness measure for digital regions is introduced, and an algorithm to evaluate the compactness measure of convex digital regions is also presented.

On cellular straight line segments

We discuss a scheme for digitizing curves that is consistent with a scheme for digitizing regions. It is shown that the cellular image of a region is determined by the cellular image of its boundary by the scheme. It is proved that the chord property is a necessary and sufficient condition for a cellular arc to be a cellular straight line segment. By showing that the chord property and the cellular convexity condition are equivalent, we prove thata cellular arc is a cellular straight line segment if and only if it is cellularly convex. This leads to an algorithm to determine whether or not a cellular complex is a cellular straight line segment in time linear in the number of rows of cells. Finally it is proven that a cellular complex is cellularly convex if and only if any pair of its cells is connected by a cellular straight line segment in the cellular complex.

Representation of digital line segments and their preimages

Abstract A period on digital line segments is defined. It is shown that each of the two nearest supports contains a preimage of the digital line segment and at least one of them contains two points of the digital line segment. Linear time algorithms are presented which find the nearest support of a 2-D digital line segment, four and eight parameters that represent 2-D and 3-D digital line segments, respectively, and construct the complete set of preimages of 2-D and 3-D digital line segments.