A "based" plane triangulation is a plane triangulation with one designated edge on the outer face. In this paper we give a simple algorithm to generate all biconnected based plane triangulations having exactly n vertices and with the maximum degree exactly D. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications.

A "rooted" plane triangulation is a plane triangulation with one designated vertex r and one designated edge incident to r on the outer face. In this paper we give a simple algorithm to generate all connected rooted plane triangulations with at most m edges. The algorithm uses O(m) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm we can generate all connected (non-rooted) plane triangulations with at most m edges in O(m3) time per triangulation.

Given a set of n disjoint intervals on a line and an integer k, we want to find k points in the intervals so that the minimum pairwise distance of the k points is maximized. Intuitively, given a set of n disjoint time intervals on a timeline, each of which is a time span we are allowed to check something, and an integer k, which is the number of times we will check something, we plan k checking times so that the checks occur at equal time intervals as much as possible, that is, we want to maximize the minimum time interval between the k checking times. We call the problem the k-dispersion problem on intervals. If we need to choose exactly one point in each interval, so k=n, and the disjoint intervals are given in the sorted order on the line, then two O(n) time algorithms to solve the problem are known. In this paper we give the first O(n) time algorithm to solve the problem for any constant k. Our algorithm works even if the disjoint intervals are given in any (not sorted) order. If the disjoint intervals are given in the sorted order on the line, then, by slightly modifying the algorithm, one can solve the problem in O(log n) time. This is the first sublinear time algorithm to solve the problem. Also we show some results on the k-dispersion problem on disks, including an FPTAS.

A rectangular drawing is a plane drawing in which every face is a rectangle. In this paper we give a simple encoding scheme for rectangular drawings. Given a rectangular drawing R with maximum degree 3, our scheme encodes R with m + o(n) bits where n is the number of vertices of R and m is the number of edges of R. Also we give an algorithm to supports a rich set of queries, including adjacency and degree queries on the faces, in constant time.

This paper presents an efficient algorithm to generate all (unordered) rooted trees with exactly n vertices including exactly k leaves. There are known results on efficient enumerations of some classes of graphs embedded on a plane, for instance, biconnected and triconnected triangulations [3],[6], and floorplans [4]. On the other hand, it is difficult to enumerate a class of graphs without a fixed embedding. The paper is on enumeration of rooted trees without a fixed embedding. We already proposed an algorithm to generate all “ordered” trees with n vertices including k leaves [11], while the algorithm cannot seem to efficiently generate all (unordered) rooted trees with n vertices including k leaves. We design a simple tree structure among such trees, then by traversing the tree structure we generate all such trees in constant time per tree in the worst case. By repeatedly applying the algorithm for each k=1,2, ...,n-1, we can also generate all rooted trees with exactly n vertices.

Given a graph G=(V,E), five distinct vertices u1,u2,u3,u4,u5 V and five natural numbers n1,n2,n3,n4,n5 such that Σ5i=1ni=|V|, we wish to find a partition V1,V2,V3,V4,V5 of the vertex set V such that ui Vi, |Vi|=ni and Vi induces a connected subgraph of G for each i, 1i5. In this paper we give a simple linear-time algorithm to find such a partition if G is a 5-connected internally triangulated plane graph and u1,u2,u3,u4,u5 are located on the outer face of G. Our algorithm is based on a "5-canonical decomposition" of G, which is a generalization of an st-numbering and a "canonical ordering" known in the area of graph drawings.

The dispersion problem is a variant of the facility location problem. Given a set P of n points and an integer k, we intend to find a subset S of P with |S|=k such that the cost minp∈S{cost(p)} is maximized, where cost(p) is the sum of the distances from p to the nearest c points in S. We call the problem the dispersion problem with partial c sum cost, or the PcS-dispersion problem. In this paper we present two algorithms to solve the P2S-dispersion problem(c=2) if all points of P are on a line. The running times of the algorithms are O(kn2 log n) and O(n log n), respectively. We also present an algorithm to solve the PcS-dispersion problem if all points of P are on a line. The running time of the algorithm is O(knc+1).

In this paper we give a simple algorithm to generate all partitions of {1,2,,n} into k non-empty subsets. The number of such partitions is known as the Stirling number of the second kind. The algorithm generates each partition in constant time without repetition. By choosing k = 1,2,,n we can also generate all partitions of {1,2,,n} into subsets. The number of such partitions is known as the Bell number.

Recently a compact code of mosaic floorplans with ƒ inner face was proposed by He. The length of the code is 3ƒ-3 bits and asymptotically optimal. In this paper, we propose a new code of mosaic floorplans with ƒ inner faces including k boundary faces. The length of our code is at most $3f - rac{k}{2} - 1$ bits. Hence our code is shorter than or equal to the code by He, except for few small floorplans with k=ƒ≤3. Coding and decoding can be done in O(ƒ) time.

In this paper we give a simple algorithm to compute a canonical code for fullerene graphs. Our algorithm runs in O(n) time, while the best known algorithm runs in O(n2) time. Our algorithm is simple. One can generalize the algorithm to compute a canonical code for the skeleton of a convex polyhedron with n vertices. The algorithm runs in O(n2) time.

Given a set P of n points and an integer k, we wish to place k facilities on points in P so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem, and the set of such k points is called a k-dispersion of P. Note that the 2-dispersion problem corresponds to the computation of the diameter of P. Thus, the k-dispersion problem is a natural generalization of the diameter problem. In this paper, we consider the case of k=3, which is the 3-dispersion problem, when P is in convex position. We present an O(n2)-time algorithm to compute a 3-dispersion of P.

In this paper we give a simple algorithm to generate all partitions of a positive integer n. The problem is one of the basic problems in combinatorics, and has been extensively studied for a long time. Our algorithm generates each partition of a given integer in constant time for each without repetition, while best known algorithm generates each partition in constant time on "average." Also, we propose some algorithms to generate all partitions of an integer with some additional property in constant time.

In this paper, we consider the problem of enumerating spanning subgraphs with high edge-connectivity of an input graph. Such subgraphs ensure multiple routes between two vertices. We first present an algorithm that enumerates all the 2-edge-connected spanning subgraphs of a given plane graph with n vertices. The algorithm generates each 2-edge-connected spanning subgraph of the input graph in O(n) time. We next present an algorithm that enumerates all the k-edge-connected spanning subgraphs of a given general graph with m edges. The algorithm generates each k-edge-connected spanning subgraph of the input graph in O(mT) time, where T is the running time to check the k-edge-connectivity of a graph.

An r-gathering of customers C to facilities F is an assignment A of C to open facilities F' ⊂ F such that r (≥ 2) or more customers are assigned to each open facility. (Each facility needs enough number of customers for its opening.) Then the r-gathering problem finds an r-gathering minimizing a designated cost. Armon gave a simple 3-approximation algorithm for the r-gathering problem and proved that with assumption P ≠ NP the problem cannot be approximated within a factor of less than 3 for any r ≥ 3. The running time of the 3-approximation algorithm is O(|C||F|+r|C|+|C|log|C|)). In this paper we improve the running time of the algorithm by (1) removing the sort in the algorithm and (2) designing a simple but efficient data structure.

A plane quadrangulation is a plane graph such that each inner face has exactly four edges on its contour. This is a planar dual of a plane graph such that all inner vertices have degree exactly four. A based plane quadrangulation is a plane quadrangulation with one designated edge on the outer face. In this paper we give a simple algorithm to generate all biconnected based plane quadrangulations with at most f faces. The algorithm uses O(f) space and generates such quadrangulations in O(1) time per quadrangulation without duplications. By modifying the algorithm we can generate all biconnected (non-based) plane quadrangulations with at most f faces in O(f3) time per quadrangulation.

Let P be a set of points on the plane, and d(p, q) be the distance between a pair of points p, q in P. For a point p∈P and a subset S ⊂ P with |S|≥3, the 2-dispersion cost, denoted by cost2(p, S), of p with respect to S is the sum of (1) the distance from p to the nearest point in Ssetminus{p} and (2) the distance from p to the second nearest point in Ssetminus{p}. The 2-dispersion cost cost2(S) of S ⊂ P with |S|≥3 is minp∈S{cost2(p, S)}. Given a set P of n points and an integer k we wish to compute k point subset S of P with maximum cost2(S). In this paper we give a simple 1/({4sqrt{3}}) approximation algorithm for the problem.

Given a plane graph G, we wish to find a drawing of G in the plane such that the vertices of G are represented as grid points, and the edges are represented as straight-line segments between their endpoints without any edge-intersection. Such drawings are called planar straight-line drawings of G. An additional objective is to minimize the area of the rectangular grid in which G is drawn. In this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.

A rectangular drawing is a plane drawing of a graph in which every face is a rectangle. Rectangular drawings have an application for floorplans, which may have a huge number of faces, so compact code to store the drawings is desired. The most compact code for rectangular drawings needs at most 4f-4 bits, where f is the number of inner faces of the drawing. The code stores only the graph structure of rectangular drawings, so the length of each edge is not encoded. A grid rectangular drawing is a rectangular drawing in which each vertex has integer coordinates. To store grid rectangular drawings, we need to store some information for lengths or coordinates. One can store a grid rectangular drawing by the code for rectangular drawings and the width and height of each inner face. Such a code needs 4f-4 + f⌈log W⌉ + f⌈log H⌉ + o(f) + o(W) + o(H) bits*, where W and H are the maximum width and the maximum height of inner faces, respectively. In this paper we design a simple and compact code for grid rectangular drawings. The code needs 4f-4 + (f+1)⌈log L⌉ + o(f) + o(L) bits for each grid rectangular drawing, where L is the maximum length of edges in the drawing. Note that L ≤ max{W,H} holds. Our encoding and decoding algorithms run in O(f) time.