Showing papers on "Disjoint sets published in 1995"

Disjoint paths in densely embedded graphs

Abstract: We consider the following maximum disjoint paths problem (MDPP). We are given a large network, and pairs of nodes that wish to communicate over paths through the network-the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share an edge in the network. This classical problem has been brought into focus recently in papers discussing applications to routing in high-speed networks, where the current lack of understanding of the MDPP is an obstacle to the design of practical heuristics. We consider the class of densely embedded, nearly-Eulerian graphs, which includes the two-dimensional mesh and other planar and locally planar interconnection networks. We obtain a constant-factor approximation algorithm for the maximum disjoint paths problem for this class of graphs; this improves on an O(log n)-approximation for the special case of the two-dimensional mesh due to Aumann-Rabani and the authors. For networks that are not explicitly required to be "high-capacity," this is the first constant-factor approximation for the MDPP in any class of graphs other than trees. We also consider the MDPP in the on-line setting, relevant to applications in which connection requests arrive over time and must be processed immediately. Here we obtain an asymptptically optimal O(log n)competitive on-line algorithm for the same class of graphs; this improves on an O(log n log log n) competitive algorithm for the special case of the mesh due to B. Awerbuch et al (1994).

A projection technique for partitioning the nodes of a graph

Abstract: LetG=(N,E) be an undirected graph. We present several new techniques for partitioning the node setN intok disjoint subsets of specified sizes. These techniques involve eigenvalue bounds and tools from continuous optimization. Comparisons with examples taken from the literature show these techniques to be very successful.

A survey of efficient reliability computation using disjoint products approach

Abstract: Several algorithms have been developed to solve the reliability problem for nonseries-parallel networks using the sum of disjoint products (SDP) approach This paper provides a general framework for most of these techniques It reviews methods that help improve computer time and memory requirements in reliability computation These parameters are generally used to compare SDP algorithms We also overview three multiple variable inversion algorithms that result in sum of disjoint products expressions with fewer terms than that of algorithms that use only a single-variable inversion One common network is solved for two-terminal network reliability using each of these algorithms Finally, we have provided a comparison among these techniques

Approximations for the disjoint paths problem in high-diameter planar networks

Abstract: We consider the problem of connecting distinguished terminal pairs in a graph via edgedisjoint paths. This is a classical NP-complete problem for which no general approximation techniques are known; it has recently been brought into focus in papers discussing applications to admission control in high-speed networks and to routing in all-optical networks. In this paper we provide O(logn)-approximation algorithms for two natural optimization versions of this problem for the class of nearly-Eulerian, uniformly high-diameter planar graphs, which includes two-dimensional meshes and other common planar interconnection networks. We give an O(logn)-approximation to the maximum number of terminal pairs that can be simultaneously connected via edge-disjoint paths, and an O(logn)-approximation to the minimum number of wavelengths needed to route a collection of terminal pairs in the “optical routing” model considered by Raghavan and Upfal, and others. The latter result improves on an O(log n)-approximation for the special case of the mesh obtained independently by Aumann and Rabani. For both problems the O(logn)-approximation is a consequence of an O(1)-approximation for the special case when all terminal pairs are roughly the same distance apart. Our algorithms make use of a number of new techniques, including the construction of a “crossbar” structure in any nearly-Eulerian planar graph, and develops some connections with classical matroid algorithms.

Computing the visibility graph via pseudo-triangulations

Abstract: We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k + n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudo-triangulations, whose combinatorial properties are crucial for our method.

Preserving and Increasing Local Edge-Connectivity in Mixed Graphs

Abstract: Generalizing and unifying earlier results of W. Mader, and A. Frank and B. Jackson, we prove two splitting theorems concerning mixed graphs. By invoking these theorems we obtain min-max formulae for the minimum number of new edges to be added to a mixed graph so that the resulting graph satisfies local edge-connectivity prescriptions. An extension of Edmonds's theorem on disjoint arborescences is also deduced along with a new sufficient condition for the solvability of the edge-disjoint paths problem in digraphs. The approach gives rise to strongly polynomial algorithms for the corresponding optimization problems.

Restricted planar location problems and applications

Abstract: Facility location problems in the plane are among the most widely used tools of Mathematical Programming in modeling real-world problems. In many of these problems restrictions have to be considered which correspond to regions in which a placement of new locations is forbidden. We consider center and median problems where the forbidden set is a union of pairwise disjoint convex sets. As applications we discuss the assembly of printed circuit boards, obnoxious facility location and the location of emergency facilities.

Method for deriving optimal code schedule sequences from synchronous dataflow graphs

Abstract: A method is disclosed for deriving code schedule sequences for a target code generator from an input ordering of nodes and prime factors of their respective ordered invocation rates from an SDF graph representative of a system. The method involves first creating a loop set for each prime factor wherein the elements of each loop set are the actors, the invocation frequency from which are factorable by that prime factor and are ordered. The redundant created loop sets are merged so as to eliminate those sets with identical nodes. The merged loop sets are then sorted in decreasing order by the total number of node elements in each set. A determination is then made as to whether each loop set is a proper subset of its sorted ordered predecessor loop set with which it intersects and, if not, then breaking the non-disjoint sets into sublists of sets which are proper subsets of their predecessor sets and then determining whether the parent sets of the broken sublists are then disjoint from one another. If they are not then repeating these two substeps. The next step of the present method then involves extracting a loop schedule for each sublist and combining the extracted loop schedules in accordance with the sorted precedence ordering of the nodes in each of said loop sets to produce the code schedule sequences. In such a manner, the code schedule sequences generated corresponds to every potential type of actor or block in the system and can then be used to minimize both program and data memory requirements of the scheduled systems.

Taxonomies of logically defined qualitative spatial relations

Abstract: This paper develops a taxonomy of qualitative spatial relations for pairs of regions, which are all logically defined from two primitive (but axiomatized) notions. The first primitive is the notion of two regions being connected, which allows eight jointly exhaustive and pairwise disjoint relations to be defined. The second primitive is the convex hull of a region which allows many more relations to be defined. We also consider the development of the useful notions of composition tables for the defined relations and networks specifying continuous transitions between pairs of regions. We conclude by discussing what kind of criteria to apply when deciding how fine grained a taxonomy to create.

Asymptotic packing via a branching process

Abstract: It is shown that under certain side conditions the natural random greedy algorithm almost always provides an asymptotically optimal packing of disjoint hyperedges from a hypergraph H.

Transversals of 2-intervals, a topological approach

Abstract: Fix two distinct parallel linese andf. A 2-interval is the union of an interval one and an interval onf. We study thetransversal number τ (ℋ) of families of 2-intervals ℋ. This is the cardinality of the smallest set which intersects every 2-interval in ℋ. A Gyarfas and J. Lehel [6] proved that τ(ℋ)=O(υ(ℋ)2) where ν(ℋ) is the maximum number of disjoint 2-intervals in ℋ. In the present paper we prove the tight bond τ(ℋ)≤2v(ℋ). Our result has applications in the estimation of the transversal number of other types of set systems. The method we use is topological. We associate a simplicial complexK with our system of 2-intervals and prove that a given subcomplex is contractible inK unless the required transversal exists. Then we construct a cocycle of (another subcomplex of)K to prove that the subcomplex is not contractible inK. We hope that this approach will be applicable to a wider variety of combinatorial optimization problems.

Signal comprising binary spreading-code sequences

Abstract: A method and apparatus for electrically generating sets of binary spreading-code sequences for use in a multi-node communication network. Also, a method for assigning disjoint sets of binary spreading-code sequences to different nodes of such a network. Each set of binary spreading-code sequences consists of multiple sequences, which are generated using two binary shift registers. The sequences can be generated simultaneously, or sequence segments can be generated sequentially. To generate sequences simultaneously, the contents of multiple pairs of stages of two linear-feedback binary shift registers are combined by modulo-2 addition, where each pair of stages consists of one stage from each of the two binary shift registers. To generate sequence segments sequentially, the contents of a single stage of a first binary shift register are combined by modulo-2 addition with the contents of a single stage of a second binary shift register, where new fills are switched into each of the registers at the beginning of each period. To assign disjoint sets of binary spreading-code sequences to different nodes of the network, the initial fill of the first binary shift register is fixed and different initial fills are specified for the second binary shift register.

Supports of doubly stochastic measures

Abstract: Recent work has shown that extreme doubly stochastic measures are supported on sets that have no axial cycles. We give a new proof of this result and examine the supporting set structure more closely. It is shown that the property of no axial cycles leads to a tree like structure which naturally partitions the support into a collection of disjoint graphs of functions from the x-axis to the y-axis and from the y-axis to the x-axis. These functions are called a limb numbering system. It is shown that if the disjoint graphs in the limb numbering system are measurable, then the supporting set supports a unique doubly stochastic measure. Further, the limb structure can be used to develop a general method for constructing sets which support a unique doubly stochastic measure.

Computing Common Tangents Without a Separating Line

Abstract: Given two disjoint convex polygons in standard representations, one can compute outer common tangents in logarithmic time without first obtaining a separating line. If the polygons are not disjoint, there is an additional factor of the logarithm of the intersection or convex hull, whichever is smaller.

Digraphs with the path-merging property

Abstract: In this paper we study those digraphs D for which every pair of internally disjoint (X, Y)-paths P1, P2 can be merged into one (X, Y)-path P* such that V(P1) ∪ V(P2), for every choice of vertices X, Y ϵ V(D). We call this property the path-merging property and we call a graph path-mergeable if it has the path-merging property. We show that each such digraph has a directed hamiltonian cycle whenever it can possibly have one, i.e., it is strong and the underlying graph has no cutvertex. We show that path-mergeable digraphs can be recognized in polynomial time and we give examples of large classes of such digraphs which are not contained in any previously studied class of digraphs. We also discuss which undirected graphs have path-mergeable digraph orientations. © 1995, John Wiley & Sons, Inc.

On Families of Sets of Integral Vectors Whose RepresentativesForm Sum-Distinct Sets

Abstract: We study the size of a collection $\{{\cal P}_1,\ldots,{\cal P}_T\}$ whose members ${\cal P}_i$, $i=1,\ldots,T$, are disjoint sets of integral vectors such that $\sum_{i=1}^T \bar x_i$ are all distinct and each $n$-tuple $\bar x_i$ comes from a different set ${\cal P}_i$. In particular, if ${\cal P}_i=\{\bar 0_n,\bar x_i\}$, we have a well-known problem on maximum cardinality of sum-distinct sets of integral vectors. We state bounds on $n^{-1}\sum_{i=1}^T \log_2\vert{\cal P}_i\vert$ and give a construction that meets the lower bound.

Combinatorics of inductively factored arrangements

Abstract: A real arrangement of hyperplanes is a finite family A of hyperplanes through the origin in a finite-dimensional real vector space V = R1. A real arrangement A of hyperplanes is said to be factored if there exists a partition Π = (Π1, …, Π1) of A into l disjoint subsets such that the Orlik-Solomon algebra of A factors according to this partition. A real arrangement A of hyperplanes is called inductively factored if it is factored and there exists a hyperplane H ϵ A such that the arrangement obtained by removing H from A and the arrangement on H consisting of all intersections of elements of A — H with H are both inductively factored. A chamber of A is a connected component of the complement of A . For a fixed base chamber, we may define a partial order on the set of chambers according to their combinatorial distances from the base chamber. Given an inductive factorization Π = (Π1,…(Π1 and a base chamber C0, we define the counting map of A with respect to C0 as a morphism from the poset of chambers to the poset 0, 1,…,|Π1| x·.x (0, 1,…,Π1). We prove that, for a suitable base chamber, the counting map is a bijection, the poset of chambers is a lattice, and its rank-generating function has a nice factorization. We consider the dual decomposition of the sphere of V induced by A . We prove that, if A is inductively factored, then this cellular decomposition can be viewed as a decomposition of the boundary of the l-cube [0,|Π1|]x·.x]0, |Π1|[ by cubic cells.

On the resolvability of normally distributed vector parameter estimates

Abstract: This paper presents a metric useful in analyzing the resolution of multidimensional (vector) parameter estimators that produce normal estimates. This metric is based on the distance between ellipsoidal confidence regions about each of the parameters. When the ellipsoids are disjoint the parameters are deemed resolvable. This paper presents a method for finding the noise level at which all of the ellipsoids are disjoint, but at least two are tangent. This is deemed the resolution threshold noise level. The underlying mathematical result treats the problem of finding a dilation (noise level) at which two coupled quadratic equations have a unique solution.

Span-disjoint paths for physical diversity in networks

Abstract: Diversity has been widely recognized as a critical factor for robustness in networks. In order to resist isolations due to link and node failures, we need to route along paths which are both span-disjoint (physically disjoint) and node-disjoint. In this way we can design the network for fault tolerance or graceful degradation. We demonstrate the procedure to obtain span-disjoint paths. This procedure may be applied to an arbitrary network topology, without restrictions on the span sharing configurations. In those cases where two completely disjoint paths do not exist, the procedure still tries to find a pair of paths which are as diverse as possible. The solution path pair we obtain is not guaranteed to be optimal, but gives us an optimal or near-optimal design, relatively quickly. The algorithm also accepts parameters which can be used to specify the cost-robustness tradeoffs.

Geometric Pattern Matching in d-Dimensional Space

Abstract: We show that, using the L∞ metric, the minimum Hausdorff distance under translation between two point sets of cardinality n in d-dimensional space can be computed in time O(n(4d−2)/3 log2n) for d>3. Thus we improve the previous time bound of O(n2d−2 log2n) due to Chew and Kedem. For d=3 we obtain a better result of O(n3 log2n) time by exploiting the fact that the union of n axis-parallel unit cubes can be decomposed into O(n) disjoint axis-parallel boxes. We prove that the number of different translations that achieve the minimum Hausdorff distance in d-space is Θ(n⌊3d/2⌋). Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L2 metric in d-space in time O(n⌊3d/2⌋+1 log3n).

Finding common ancestors and disjoint paths in DAGs

Abstract: Author(s): Eppstein, David | Abstract: We consider the problem of finding pairs of vertex-disjoint paths in a DAG, either connecting two given nodes to a common ancestor, or connecting two given pairs of terminals. It was known how to find a single pair of paths, for either type of input, in polynomial time. We show how to find the k pairs with shortest combined length, in time O(mn + k). We also show how to count all such pairs of paths in O(mn) arithmetic operations. These results can be extended to finding or counting tuples of d disjoint paths, in time O(mn^(d-1 + k). We give further results on finding the subset of the DAG involved in pairs of disjoint paths, and on finding disjoint paths in linear space.