Graph embeddings and simplicial maps

An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhattet al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of thek-dimensional torus are determined. As corollaries to the main lower-bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genusg graphs, andk-dimensional meshes in a butterfly host graph. This research was supported by National Science Foundation Grant CCR-9009953. A preliminary version of some of this research appears in “Lower Bounds for Graph Embeddings via Algebraic Topology (Extended Abstract),”Proceedings of the 5th Annual ACM Symposium on Parallel Algorithms and Architectures, 1993, pp. 311–317.     

An approach to emulating separable graphs

A technique foremulating multicomputer interconnection networks that are based onseparable graphs (graphs having bounded degree and sublinear multicolor recursive bisectors) is presented. Efficient emulations among interconnection networks are necessary for porting programs designed for one network to another.Emulations are formalized asgraph embeddings, where the nodes (processors) of theguest graph (emulated network) are assigned to nodes of thehost graph (emulator), while the edges (communication links) of the guest are routed via paths in the host. The communication slowdown in an emulation depens on thedilation (length of the longest routing path) and thecongestion (number of paths that contend for a host edge) of the embedding. Theexpansion of the embedding (the ratio of the sizes of the host to guest) determines the inefficiency of processor utilization. Cell trees are introduced as interconnection networks whose special communication properties enable them to serve as intermediate devices in these emulations. Nodes in cell trees are organized into equinumerous parts calledcells; the cells are labeled by nodes of a complete binary tree. Communication in cell trees is restricted to two specific and distinct primitives:cell communication is confined within cells, whiletransfer communication occurs between adjacent cells. Rather than solved directly, the emulation problem for the original guest-host pair is decomposed into two independent parts: emulating the guest by the cell tree, and emulating the cell tree by the host.In emulations of separable graphs by cell trees, the node assignment that ensures small dilation is derived from the separator-based decomposition of guest graphs. The congestion-free edge routing is achieved by coordinatingglobal andlocal phases, which are based on two characteristic cell-tree communication primitives.The technique is instantiated by emulating cell trees on specific host graphs. Withshuffle-like hypercube-derivative networks as hosts new constant-expansion emulations are obtained that have both dilation and congestion logarithmic in the size of the multicolor bisector of guest graphs. These emulations are the first such to have optimal (up to constants)congestion; they provide the firstoptimal algorithm for emulating arbitrary separable graphs on shuffle-like networks. The application of the technique tohypercubes as hosts also produces optimal emulations that differ from those previously known by having smaller expansion constants.This research was supported in part by NSF Grants CCR-88-12567 and CCR-90-13184, and by the University of Massachusetts Graduate School Fellowship for the academic year 1991-92. A preliminary version of this paper was presented at the 3rd ACM Symposium on Parallel Algorithms and Architectures, July 22–24, 1991, in Hilton Head, South Carolina, USA.     

Multilayer grid embeddings for VLSI

In this paper we propose two new multilayer grid models for VLSI layout, both of which take into account the number of contact cuts used. For the first model in which nodes “exist” only on one layer, we prove a tight area × (number of contact cuts) = Θ(n ²) tradeoff for embeddingn-node planar graphs of bounded degree in two layers. For the second model in which nodes “exist” simultaneously on all layers, we give a number of upper bounds on the area needed to embed groups using no contact cuts. We show that anyn-node graph of thickness 2 can be embedded on two layers inO(n ²) area. This bound is tight even if more layers and any number of contact cuts are allowed. We also show that planar graphs of bounded degree can be embedded on two layers inO(n ^3/2(logn)²) area. Some of our embedding algorithms have the additional property that they can respect prespecified grid placements of the nodes of the graph to be embedded. We give an algorithm for embeddingn-node graphs of thicknessk ink layers usingO(n ³) area, using no contact cuts, and respecting prespecified node placements. This area is asymptotically optimal for placement-respecting algorithms, even if more layers are allowed, as long as a fixed fraction of the edges do not use contact cuts. Our results use a new result on embedding graphs in a single-layer grid, namely an embedding ofn-node planar graphs such that each edge makes at most four turns, and all nodes are embedded on the same line.     

A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.     

Tight bounds for the cover time of multiple random walks

We study the cover time of multiple random walks on undirected graphs G=(V,E). We consider k parallel, independent random walks that start from the same vertex. The speed-up is defined as the ratio of the cover time of a single random walk to the cover time of these k random walks. Recently, Alon et al. (2008) [5] derived several upper bounds on the cover time, which imply a speed-up of Ω(k) for several graphs; however, for many of them, k has to be bounded by O(logn). They also conjectured that, for any 1?k?n, the speed-up is at most O(k) on any graph. We prove the following main results:

•

We present a new lower bound on the speed-up that depends on the mixing time. It gives a speed-up of Ω(k) on many graphs, even if k is as large as n.

•

We prove that the speed-up is O(klogn) on any graph. For a large class of graphs we can also improve this bound to O(k), matching the conjecture of Alon et al.

•

We determine the order of the speed-up for any value of 1?k?n on hypercubes, random graphs and degree restricted expanders. For d-dimensional tori with d>2, our bounds are tight up to logarithmic factors.

•

Our findings also reveal a surprisingly sharp threshold behaviour for certain graphs, e.g., the d-dimensional torus with d>2 and hypercubes: there is a value T such that the speed-up is approximately min{T,k} for any 1?k?n.

     

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n−18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.     

Uniform data encodings

Given two finite, directed, edge-labeled graphs, G (the guest) and H (the host) an encoding of G into H is an injection of guest-graph edges into host-graph paths that induces an injection of G vertices into H vertices. An encoding is uniform if like-labeled edges map to like-labeled paths. Encodings and uniform encodings are motivated by data structure representation issues. Although uniform encodings are economically expressed, three types of analysis indicate diseconomies introduced by the uniformity condition. First, space usage (i.e., the size of the host as a function of the size of the guest) can be exponential for ‘natural’ encodings such as uniform encodings of arrays into trees. By contrast, strongly connected hosts of size equal to the guests are adequate for nonuniform encodings. Secondly, the edge dilation under uniform encodings can be nonpolynomial in the size of the guest. By contrast, the same graphs can be nonuniformly encoded with edges mapping to paths of unit length. Thirdly, finding uniform encodings, even when the guest is a line, is PSPACE-complete. By contrast, there is a linear time algorithm for nonuniform encoding of lines in graphs. Additional upper and lower bounds amplify the limitations of uniform data encodings.     

A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs

We study a problem of lower bounds on straight line drawings of planar graphs. We show that at least 1.235·n points in the plane are required to draw each n-vertex planar graph with edges drawn as straight line segments (for sufficiently large n). This improves the previous best bound of 1.206·n (for sufficiently large n) due to Chrobak and Karloff [Sigact News 20 (4) (1989) 83-86]. Our contribution is twofold: we improve the lower bound itself and we give a significantly simpler and more straightforward proof.     

On Spectral Bounds for the k-Partitioning of Graphs

When executing processes on parallel computer systems a major bottle-neck is interprocessor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the k-partitioning problem of the corresponding process graph, where k is the number of processors. The classical spectral lower bound of (|V|/2k)\sum ^k _i=1λ _i for the k-section width of a graph is well known. We show new relations between the structure and the eigenvalues of a graph and present a new method to get tighter lower bounds on the k-section width. This method makes use of the level structure defined by the k-section. We define a global expansion property and prove that for graphs with the same k-section width the spectral lower bound increases with this global expansion. We also present examples of graphs for which our new bounds are tight up to a constant factor.     

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2^k, we also exhibit a highly parallel ?L\oplus {\bf L} algorithm for finding the value of a permanent modulo 2^k. Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2^k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.     

On Triangulating Planar Graphs under the Four-Connectivity Constraint

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included. Received August 1, 1995; revised July 8, 1996, and August 23, 1996.     

Linear-time algorithms for problems on planar graphs with fixed disk dimension

The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most 2k−2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem.     

Treewidth Lower Bounds with Brambles

In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms often give excellent lower bounds, in particular when applied to (close to) planar graphs. This work was partially supported by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation) and partially by the DFG research group “Algorithms, Structure, Randomness” (Grant number GR 883/9-3, GR 883/9-4).     

Embeddings of Hypercubes and Grids into de Bruijn Graphs

An embedding of a graph G into a graph H is an injective mapping f from the vertices of G into the vertices of H together with a mapping P_f of edges of G into paths in H. The dilation of the embedding is tile maximum taken over all the lengths of the paths P_f(xy) associated with the edges xy of G. We show that it is possible to embed a D-dimensional hypercube into the binary de Bruijn graph of the same order and diameter with dilation at most [D/2]. Similarly a majority of planar grids can be embedded into a binary de Bruijn graph of the same or nearly the same order with dilation at most [D/2] where D is the diameter of the de Bruijn graph.     

On simultaneous straight-line grid embedding of a planar graph and its dual

Simultaneous representations of planar graphs and their duals normally require that the dual vertices to be placed inside their corresponding primal faces, and the edges of the dual graph to cross only their corresponding primal edges. Erten and Kobourov [C. Erten, S.G. Kobourov, Simultaneous embedding of a planar graph and its dual on the grid, Theory Computer Systems 38 (2005) 313-327] provided a linear time algorithm on simultaneous straight-line grid embedding of a 3-connected planar graph and its dual such that all the vertices are placed on grid points and each edge is drawn as one straight-line segment except for one which is drawn using two segments. Their drawing size is (2n−2)×(2n−2), where n is the total number of vertices in the graph and its dual. They raised an open question on whether there is a large class of planar graphs that allows this simultaneous straight-line grid embedding on a smaller grid. We answer this open question by giving a linear time simultaneous straight-line grid embedding algorithm for a 3-connected planar graph and its dual on a grid of size (n−1)×n.     

Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

Algebraic specification of interconnection network relationships by permutation voltage graph mappings

Symmetries of large networks are used to simplify the specification of a guest-host network relationship. The relevant kinds of symmetries occur not only in Cayley graphs and in group-action graphs, but elsewhere as well. In brief, the critical topological symmetry property of a guest or host is that it is algebraically specifiable as a covering space of a smaller graph. A first objective is to understand the circumstances under which a mapping (a.k.a. “embedding”) between two base graphs can be lifted topologically to a mapping between their respective coverings. A suitable assignment of algebraic elements called “permutation voltages” to a base graph for the intended host network facilitates the construction not only of the intended host, but also of the intended guest and of the intended mapping of the guest into the host. Explicit formulas are derived for measurement of the load, of the congestion, and of the dilation of the lifted mapping. A concluding example suggests how these new formulas open the opportunity to develop optimization methods for algebraically specified guest-host mappings. The first author was partially supported by ONR Contract N00014-85-0768, and the second author was partially supported by NSF Grant CCR-9110824.     

Lower and upper bounds on time for multiprocessor optimal schedules

The lower and upper bounds on the minimum time needed to process a given directed acyclic task graph for a given number of processors are derived. It is proved that the proposed lower bound on time is not only sharper than the previously known values but also easier to calculate. The upper bound on time, which is useful in determining the worst case behavior of a given task graph, is presented. The lower and upper bounds on the minimum number of processors required to process a given task graph in the minimum possible time are also derived. It is seen with a number of randomly generated dense task graphs that the lower and upper bounds we derive are equal, thus giving the optimal time for scheduling directed acyclic task graphs on a given set of processors     

Succinct Representation of Labeled Graphs

In many applications, the properties of an object being modeled are stored as labels on vertices or edges of a graph. In this paper, we consider succinct representation of labeled graphs. Our main results are the succinct representations of labeled and multi-labeled graphs (we consider planar triangulations, planar graphs and k-page graphs) to support various label queries efficiently. The additional space cost to store the labels is essentially the information-theoretic minimum. As far as we know, our representations are the first succinct representations of labeled graphs. We also have two preliminary results to achieve the main contribution. First, we design a succinct representation of unlabeled planar triangulations to support the rank/select of edges in ccw (counter clockwise) order in addition to the other operations supported in previous work. Second, we design a succinct representation for a k-page graph when k is large to support various navigational operations more efficiently. In particular, we can test the adjacency of two vertices in O(lg?k) time, while previous work uses O(k) time.     

Colorings with few Colors: Counting, Enumeration and Combinatorial Bounds

Edge coloring, total coloring and L(2,1)-labeling are well-studied NP-hard graph problems. Even the versions asking whether a graph has a coloring with few colors or a labeling with few labels remain NP-hard on graphs of small maximum degree. This paper studies enumeration and counting problems on edge colorings, total colorings and L(2,1)-labelings of graphs. One part deals with the enumeration of all edge 3-colorings, all total 4-colorings and all L(2,1)-labelings of span 5 of a given connected cubic graph. Branching algorithms to solve these enumeration problems are established. They imply upper bounds on the maximum number of edge 3-colorings, total 4-colorings and L(2,1)-labelings of span 5 in any n-vertex connected cubic graphs. Corresponding combinatorial lower bounds are also provided. The other part of the paper studies dynamic programming algorithms solving counting problems. On one hand, algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth are given. On the other hand, an algorithm to count the number of L(2,1)-labelings of span 4 for graphs of maximum degree three are given.