Arc Diagrams Research Articles

Evaluation of visual display techniques for assembly sequence planning

AbstractThis study examined the effects of visual display formats for assembly planning tasks. A single assembly sequence was presented in: (a) a nonintegrated display format containing an arc and node diagram with an exploding graphic view of each step, and (b) an integrated display that incorporated pictures in the arc and node diagram and allowed enlargement of nodes. Both displays were tested with and without verbal and contextual information using five assembly rules. Overall, participants performed better and preferred the integrated display. Performance depended on the type of rule being examined. An integrated display format with verbal and contextual information is recommended for assembly sequence visualization. © 2001 John Wiley & Sons, Inc.

The permuted analogues of three Catalan sets

This paper considers three sets counted by the odd double factorials that are derived by permutation from sets counted by the Catalan numbers: sequences of paired events, complete binary trees (dendrograms), and arc diagrams. The distributions of certain parameters in both kinds of sets are compared and it is found that those with the same distributions in the Catalan sets do not always have the same distributions in the permuted sets. After reviewing bijections between the Catalan sets, the problems associated with transferring bijections to the permuted sets are discussed.

An algorithm for solving the jump number problem

First, Cogis and Habib (RAIRO Inform. Théor. 13 (1979), 3–18) solved the jump number problem for series-parallel partially ordered sets (posets) by applying the greedy algorithm and then Rival (Proc. Amer. Math. Soc. 89 (1983), 387–394) extended their result to N-free posets. The author (Order 1 (1984), 7–19) provided an interpretation of the latter result in the terms of arc diagrams of posets explaining partly tractability of this special case.In this paper, we present an algorithm for solving the jump number problem on arbitrary posets which extends the author's approach applied to N-free posets and makes use of two new types of greedy chains in posets introduced in companion papers [8,9]. Complexity analysis of the algorithm supports our expectation that, going from N-free to arbitrary posets, the complexity of the problem increases with the number of dummy arcs required in their arc diagrams. The algorithm works in time which is linear in the poset size but factorial in the number of dummies, therefore it is a polynomial-time algorithm for posets with bounded number of dummies in their arc diagrams.

Minimizing the jump number for partially-ordered sets: a graph-theoretic approach, II

This paper is a continuation of another author's work (Order 1 (1985) 7–19), where arc diagrams of posets have been successfully applied to solve the jump number problem for N-free posets. Here, we consider arbitrary posets and, again making use of arc diagrams of posets, we define two special types of greedy chains: strongly and semi-strongly greedy. Every strongly greedy chain may begin an optimal linear extension (Theorem 1 and Corollary 1). If a poset has no strongly greedy chains, then it has an optimal linear extension which starts with a semi-strongly greedy chain (Theorem 2). Therefore, every poset has an optimal linear extension which consists entirely of strongly and semi-strongly greedy chains. This fact leads to a polynomial-time algorithm for the jump number problem in the class of posets whose arc diagrams contain a bounded number of dummy arcs.