Abstract Of the two major approaches to line drawing, run‐length slice algorithms are seldom used because of the division operation deemed necessary in these algorithms. The biggest advantage of these algorithms, the reduction of additions used, is considered outweighed by the division used. In this paper, a new run‐length slice algorithm that does not require a division operation is presented. Furthermore, it uses the double‐stepping paradigm in incremental line drawing algorithms to reduce the number of additions used by at least half. For sufficiently long lines, this algorithm uses at least 50% fewer arithmetic operations than Wu et al.'s bi‐directional double‐step incremental algorithm. But because of its high initialization cost, for short lines, it is less efficient. For a line with endpoints (0,0) and (δx, δy), the strategy is then to use the bi‐directional Bresenham algorithm for very short lines (δx < 20), the bi‐directional double‐step algorithm for moderate long lines (20 ≤δx ≤ 110), and the new algorithmfor the longer lines (δx > 110).

Let K ( m ) denote the smallest number with the property that every m -state finite automaton can be built as a neural net using K ( m ) or fewer neurons. A counting argument shows that K ( m ) is at least Ω(( m log m ) 1/3 ), and a construction shows that K ( m ) is at most O ( m 3/4 ). The counting argument and the construction allow neural nets with arbitrarily complex local structure and thus may require neurons that themselves amount to complicated networks. Mild, and in practical situations almost necessary, constraints on the local structure of the network give, again by a counting argument and a construction, lower and upper bounds for K ( m ) that are both linear in m .

A tour in a machine is a shortest input sequence taking the machine from some initial state, through all of its remaining states and back again into its initial state. A best upper bound for tour length is found for two types of machines: n-state sequential machines with unrestricted input alphabet and n-state sequential machines with a two-letter input alphabet. The problem of finding a best upper bound for length of tours in machines is restated and solved using the language of the theory of directed graphs. The solutions to the above special cases restated in this language seem obvious but require a nontrivial proof of their status as solutions.