By Ivan Rival (auth.), Ivan Rival (eds.)

This quantity includes the texts of the valuable survey papers provided at ALGORITHMS -and ORDER, held· at Ottawa, Canada from June 1 to June 12, 1987. The convention was once supported by means of promises from the N.A.T.O. complicated examine Institute programme, the college of Ottawa, and the common Sciences and Engineering study Council of Canada. we're thankful for this enormous help. Over fifty years in the past, the Symposium on Lattice idea, in Charlottesville, usa, proclaimed the power of ordered units. in simple terms two decades later the Symposium on partly Ordered units and Lattice idea, held at Monterey, united states, had solved a number of the difficulties that were initially posed. In 1981, the Symposium on Ordered units held at Banff, Canada, endured this practice. It was once marked through a landmark quantity containing twenty-three articles on just about all present themes within the idea of ordered units and its functions. 3 years after, Graphs and Orders, additionally held at Banff, Canada, aimed to rfile the function of graphs within the idea of ordered units and its functions. due to its distinctive position within the panorama of the mathematical sciences order is principally delicate to new developments and advancements. this day, an important present within the idea and alertness of order springs from theoretical computing device seience. subject matters of machine technology prepared the ground. the 1st is facts constitution. Order is usual to facts structures.

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The diagrams should not only reflect the structure of the concept lattices but also unfold views for interpreting the data. Therefore different methods of decomposing and representing concept lattices are discussed to clarify how computer programs can be designed for drawing such lattices. Examples are given too. I. CONCEPT LATTICES Lattices in data analysis are more than just mathematical structures: they carry some meaning. Therefore, drawings of such lattices should not only reflect the mathematical structure but also give a meaningful presentation for the data.

Afinite graph is the covering graph of an ordered set if and only if it has as a graph retract the covering graph of an ordered set. Indeed, this led Duffus and Rival (1983) to characterize the covering graphs of distributive 22 M Figure 27 lattices as particular graph retracts of hypercubes. Subsequently, Bandelt (1984) extended this by showing that a finite, undirected graph is a median graph if and only if it is a graph retract of a Cartesian product of a family of graphs each graph isomorphic to the two-element path.

In an important sense the answer to the first question is m= 2. THEOREM [Rival and Urrutia (1988)]. Every ordered set has a subdivision which, in turn, has two-directional representation. This contrasts sharply with the striking answer to the second question. THEOREM [Rival and Urrutia (1988)]. There is an ordered set which has no m-directional representation at all,for any positive integer m. 28 The separability problem has received extensive attention in recent years. A typical result is that the order of separation of n disjoint rectangles in the plane in a given direction can be determined in O(n log n) time [Guibas and Yao (1980)].