Parallel algorithms for permutation graphs

In this paper, parallel algorithms are presented for solving some problems on permutation graphs. The coloring problem is solved inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM. The weighted clique problem, the weighted independent set problem, the cliques cover problem, and the maximal layers problem are all solved with the same complexities. We can also show that the longest common subsequence problem belongs to the class NC.     

Asymmetric directed graph coloring games

This note generalizes the (a,b)-coloring game and the (a,b)-marking game which were introduced by Kierstead [H.A. Kierstead, Asymmetric graph coloring games, J. Graph Theory 48 (2005) 169-185] for undirected graphs to directed graphs. We prove that the (a,b)-chromatic and (a,b)-coloring number for the class of orientations of forests is b+2 if b≤a, and infinity otherwise. From these results we deduce upper bounds for the (a,b)-coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.     

A Randomized Parallel Algorithm for Planar Graph Isomorphism

We present a parallel randomized algorithm running on aCRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms inO(log² n) time withn^{1 + ε}processors, for any ε > 0). Ifnis the number of vertices, our algorithm takesO(log(n)) time with processors and with a probability of failure of 1/nat most. The algorithm needs 2 · log(m) − log(n) + O(log(n)) random bits. The number of random bits can be decreased toO(log(n)) by increasing the number of processors ton^{3/2 + ε}, for any ε > 0. Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of Miller and Reif (Siam J. Comput.20(1991), 1128–1147), which requiresn⁴randomized processors orn⁵deterministic processors.     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

An algorithm for outerplanar graphs with parameter

For n-vertex outerplanar graphs, it is proven that O(n^2.87) is an upper bound on the number of breakpoints of the function which gives the maximum weight of an independent set, where the vertex weights vary as linear functions of a parameter. An O(n^2.87) algorithm for finding the solution is proposed.     

Rates for approximation of unbounded functions by positive linear operators

Let g_a(t) and g_b(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {L_n} be a sequence of linear positive operators, each with domain containing 1, t, g_a(t), and g_b(t). If L_n(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, g_a(t), g_b(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(g_a(t)) (t → a⁺) and ƒ(t) = O(g_b(t)) (t → b⁻). We estimate the convergence rate of L_nƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′.     

On consecutive edge magic total labeling of graphs

Let G=(V,E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection α from VE to the integers 1,2,…,n+e, with the property that for every xyE, α(x)+α(y)+α(xy)=k, for some constant k. Such a labeling is called an a-vertex consecutive edge magic total labeling if α(V)={a+1,…,a+n} and a b-edge consecutive edge magic total if α(E)={b+1,b+2,…,b+e}. In this paper we study the properties of a-vertex consecutive edge magic and b-edge consecutive edge magic graphs.     

Power optimization for connectivity problems

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-CSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log⁴ n)-approximation algorithm for MPb-EC. This gives an O(log⁴ n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an approximation algorithm for ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2^log1-ε n) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time. This paper was done when V. S. Mirrokni was at Computer Science and Artificial Intelligence Laboratory, MIT.     

Small Drawings of Outerplanar Graphs,Series-Parallel Graphs,and Other Planar Graphs

In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Θ(n ²) is the established upper and lower bound on the worst-case area. A long-standing open problem is to determine for what graphs a smaller area can be achieved. We show here that series-parallel graphs can be drawn in O(n ^3/2) area, and outerplanar graphs can be drawn in O(nlog n) area, but 2-outerplanar graphs and planar graphs of proper pathwidth 3 require Ω(n ²) area. Our drawings are visibility representations, which can be converted to polyline drawings of asymptotically the same area.     

Average‐case complexity of shortest‐paths problems in the vertex‐potential model

We study the average‐case complexity of shortest‐paths problems in the vertex‐potential model. The vertex‐potential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths, but without negative cycles. We show that on a graph with n vertices and with respect to this model, the single‐source shortest‐paths problem can be solved in O(n²) expected time, and the all‐pairs shortest‐paths problem can be solved in O(n² log n) expected time. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 33–46, 2000     

A Framework for Drawing Planar Graphs with Curves and Polylines

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n) × O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) × O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).     

Product graph representations

We study a hierarchy of canonical representations of grpahs as subgraphs of cartesian products of graphs. This hierarchy starts with the isometric representation, includes the 2-isometric represnetation, and ends with the cartesian prime factorization. We show that all three representations can be obtained in O(nm) time using O(m) space, for graphs with n vertices and m edges. The algorithms have immediate parallel versions that use n³ processors and run in O(log² n) time. © 1929 John Wiley & Sons, Inc.     

Convexity preserving and predicting by Bernstein polynomials

It is known that the Bernstein polynomials of a function f defined on [0, 1 ] preserve its convexity properties, i.e., if f⁽ⁿ⁾ 0 then for m n, (B_mf)⁽ⁿ⁾ 0. Moreover, if f is n-convex then (B_mf)⁽ⁿ⁾ 0. While the converse is not true, we show that if f is bounded on (a, b) and if for every subinterval [α, β] (a, b) the nth derivative of the mth Bernstein polynomial of f on [α, β] is nonnegative then f is n-convex.     

Space-optimal, backtracking algorithms to list the minimal vertex separators of a graph

For a graph G in read-only memory on n vertices and m edges and a write-only output buffer, we give two algorithms using only O(n) rewritable space. The first algorithm lists all minimal a−b separators of G with a polynomial delay of O(nm). The second lists all minimal vertex separators of G with a cumulative polynomial delay of O(n³m).One consequence is that the algorithms can list the minimal a−b separators (and minimal vertex separators) spending O(nm) time (respectively, O(n³m) time) per object output.     

Parallel solution of certain Toeplitz least-squares problems

We describe a systolic algorithm for solving a Toeplitz least-squares problem of special form. Such problems arise, for example, when Volterra convolution equations of the first kind are solved by regularization. The systolic algorithm is based on a sequential algorithm of Eldén, but we show how the storage requirements of Eldén's algorithm can be reduced from O(n²) to O(n). The sequential algorithm takes time O(n²); the systolic algorithm takes time O(n) using a linear systolic array of O(n) cells. We also show how large problems may be decomposed and solved on a small systolic array.     

Partitioning planar graphs: a?fast combinatorial approach for max-cut

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. (IEEE Trans. Comput. 39(5):694–697, 1990) and that of Berman et al. (WADS, Lecture Notes in Computer Science, vol. 1663, pp. 25–36, Springer, Berlin, 1999). The running time of the former can be bounded by O(|V|^\frac32log|V|)O(|V|^{\frac{3}{2}}\log|V|). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O(|V|^\frac32(log|V|)^\frac32)a(|V|)O(|V|^{\frac{3}{2}}(\log|V|)^{\frac{3}{2}})\alpha(|V|), where α(|V|) is the inverse Ackermann function, it can solve large instances in practice.     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.     

Ramsey-type results for oriented trees

For a graph G and a digraph (RIGHT ARROW ABOVE H), we write G → (RIGHT ARROW ABOVE H) (respectively, G (A ABOVE RIGHT ARROW)(RIGHT ARROW ABOVE H) if every orientation (respectively, acyclic orientation) of the edges of G results in an induced copy of (RIGHT ARROW ABOVE H). In this note we study how small the graphs G such that G → (RIGHT ARROW ABOVE H) or such that G (A ABOVE RIGHT ARROW) (RIGHT ARROW ABOVE H) may be, if (RIGHT ARROW ABOVE H) is a given oriented tree (RIGHT ARROW ABOVE T)ⁿ on n vertices. We show that there is a graph on O(n⁴ log n) vertices and O(n⁶(log n)²) edges such that G → Tⁿ for every n-vertex oriented tree (RIGHT ARROW ABOVE T)ⁿ. We also prove that there exists a graph G with O(n² log n) vertices and O(n³(log n)²) edges such that G → (RIGHT ARROW ABOVE T)ⁿ for any such tree (RIGHT ARROW ABOVE T)ⁿ. This last result turns out to be nearly best possible as it is shown that any graph G with G (A ABOVE RIGHT ARROW) (RIGHT ARROW ABOVE P)ⁿ, where (RIGHT ARROW ABOVE P)ⁿ is the directed path of order n, has more than n²/2 vertices and more than [n/3]³ edges if n ≥ 3. © 1996, John Wiley & Sons, Inc.     

Incremental algorithms for minimal length paths

We consider the problem of maintaining on-line a solution to the All Pairs Shortest Paths Problem in a directed graph G = (V,E) where edges may be dynamically inserted or have their cost decreased. For the case of integer edge costs in a given range [1…C], we introduce a new data structure which is able to answer queries concerning the length of the shortest path between any two vertices in constant time and to trace out the shortest path between any two vertices in time linear in the number of edges reported. The total time required to maintain the data structure under a sequence of at most O(n²) edge insertions and at most O(Cn²) edge cost decreases is O(Cn³ log(nC)) in the worst case, where n is the total number of vertices in G. For the case of unit edge costs, the total time required to maintain the data structure under a sequence of at most O(n²) insertions of edges becomes O(n³ logn) in the worst case. The same bounds can be achieved for the problem of maintaining on-line longest paths in directed acyclic graphs. All our algorithms improve previously known algorithms and are only a logarithmic factor away from the best possible bounds.