Abstract. We consider the problem of drawing Venn diagrams for which each region's area is proportional to some weight (e.g., population or percentage) assigned to that region. These area-proportional Venn diagrams have an enhanced ability over traditional Venn diagrams to visually convey information about data sets with interacting characteristics. We develop algorithms for drawing area-proportional Venn diagrams for any population distribution over two characteristics using circles and over three characteristics using rectangles and near-rectangular polygons; modifications of these algorithms are then presented for drawing the more general Euler diagrams. We present results concerning which population distributions can be drawn using specific shapes. A program to aid further investigation of area-proportional Venn diagrams is also described.

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One of the most important sets associated with a poset P is its set of linear extensions, E(P). In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)), where e(P) = jE(P)j. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)), where n is the number of elements of the poset. Our algorithm is the rst constant amortized time algorithm for generating a \naturally de ned" class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension di ers from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modi ed to e ciently count linear extensions, and to compute P (x < y), for all pairs x; y, in time O(n 2 + e(P)).

A ranking function for the permutations on n symbols assigns a unique integer in the range [0, n! − 1] to each of the n! permutations. The corresponding unranking function is the inverse: given an integer between 0 and n! − 1, the value of the function is the permutation having this rank. We present simple ranking and unranking algorithms for permutations that can be computed using O(n) arithmetic operations.  2001 Elsevier Science B.V. All rights reserved.Keywords: Permutation; Ranking; Unranking; Algorithms; Combinatorial problems A permutation of order n is an arrangement of n symbols. For convenience when applying modular arithmetic, this paper considers permutations of {0, 1, 2, . . ., n − 1}. The set of all permutations over {0, 1, 2, . . ., n − 1} is denoted by S n .There are many applications that call for an array indexed by the permutations in S n [2]. One example is the development of programs that search for Hamilton cycles in particular types of Cayley graphs [10,11]. To do such indexing, what is desired is a bijective ranking function r that takes as input a permutation π and produces r(π), a number in the range 0, 1, . . ., n! − 1. The inverse of r is also often useful, and is called the unranking function.The traditional approach to this problem is to first define an ordering of permutations and then find ranking and unranking functions relative to that ordering.

Many applications call for exhaustive lists of strings subject to various constraints, such as inequivalence under group actions. A k-ary necklace is an Ž . equivalence class of k-ary strings under rotation the cyclic group . A k-ary unlabeled necklace is an equivalence class of k-ary strings under rotation and permutation of alphabet symbols. We present new, fast, simple, recursive algo-Ž . rithms for generating i.e., listing all necklaces and binary unlabeled necklaces. These algorithms have optimal running times in the sense that their running times are proportional to the number of necklaces produced. The algorithm for generating necklaces can be used as the basis for efficiently generating many other equivalence classes of strings under rotation and has been applied to generating bracelets, fixed density necklaces, and chord diagrams. As another application, we describe the implementation of a fast algorithm for listing all degree n irreducible Ž . and primitive polynomials over GF 2 . ᮊ