Digital squares are defined and their geometric properties characterized. A linear time algorithm is presented that considers a convex digital region and determines whether or not it is a digital square. The algorithm also determines the range of the values of the parameter set of its preimages. The analysis involves transforming the boundary of a digital region into parameter space of slope and y-intercept.>

The slope of digital line segments is defined and an algorithm to evaluate it is presented. Parallelism and perpendicularity of two digital line segments are also defined. Finally, rectangular digital regions are defined and characterized, and an algorithm that determines whether or not a given digital region is a digital rectangle is presented.

We consider problems associated with the coordination of movement within a multiple robot system in which all motion is restricted to a single track. Our objective is to minimize the reconfiguration time, that is, the total time required to move a collection of robots from an initial to a goal configuration. We show that various models give rise to a wide range of problem complexities. For these problems we design and analyze optimization and approximation strategies.

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.

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

Definitions of 3-D digital surface and plane are introduced. Many geometric properties of these objects are examined. In particular, it is shown that digital convexity is neither a necessary nor a sufficient condition for a digital surface element to be a convex digital plane element, but it is both necessary and sufficient for a digital surface to be a digital plane. Also algorithms are presented to determine whether or not a finite set of digital points is a (convex) digital plane element.

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.

New schemes for digitizing regions and arcs are introduced. It is then shown that under these schemes, Sklansky's 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.

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.

A definition of convexity of digital solids is introduced. Then it is proved that a digital solid is convex if and only if it has the chordal triangle property. Other geometric properties which characterize convex digital regions are shown to be only necessary, but not sufficient, conditions for a digital solid to be convex. An efficient algorithm that determines whether or not a digital solid is convex is presented.

We define straightness and convexity of regions in a digital picture. Then it is shown that a few important properties satisfied by convex (Euclidean) regions are also satisfied by convex digital regions. We extend the definition of digital convexity of regions to digital solids. Efficient algorithms are presented that determine whether or not a digital region is convex, a digital arc is a digital straight line segment and a digital solid is convex.

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.

Several polynomial time approximation algorithms for some $NP$-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the stacker-crane problem (a modified traveling salesman problem). A tour-splitting heuristic is given for k-person variants of the traveling salesman problem, the Chinese postman problem, and the stacker-crane problem, for which a minimax solution is sought. This heuristic has a bound of $e + 1 - 1/k$, where e is the bound for the corresponding 1-person algorithm.

The finishing time properties of several heuristic algorithms for scheduling n independent tasks on m nonidentical processors are studied. In particular, for m = 2 an n log n 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 n .

Several polynomial time approximation algorithms for some NP-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the Stacker-Crane problem (a modified Traveling Salesman problem). A tour-splitting heuristic is given for k-person variants of the Traveling Salesman problem, the Chinese Postman problem, and the Stacker-Crane problem, for which a minimax solution is sought. This heuristic has a bound of e + 1 - 1/k, where e is the bound for the corresponding 1-person algorithm.

We look at a restricted model of a multihead pushdown automaton and use some of its properties to show the existence of algorithms for some decision problems concerning code sets and vector addition systems.

Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the sum~ ~p~ subject to the constramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* -P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n KEY WORDS AND PHRASES.knapsack problem, sum of subset problem, P-complete problem, polynomial time algorithm, approximation algorithm CR CATEGORIES. 5 25, 5.39, 5 42