Linear-time algorithm

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed upper bound is Π P 2 -complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter three chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n β ) time, where β = 2ω/(ω + 1) ∼ = 1.41 and ω ∼ = 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

A famous theorem by Cauchy states that a convex polyhedron is determined by its incidence structure and face-polygons alone. In this paper, we prove the same for orthogonal polyhedra of genus 0 as long as no face has a hole. Our proof yields a linear-time algorithm to find the dihedral angles.

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding if a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and that deciding if the pebbling number has a prescribed upper bound is Π P 2 -complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter three chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n β ) time, where β = 2ω/(ω + 1) ∼ = 1.41 and ω ∼ = 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

Graph pebbling models the transportation of consumable resources. As two pebbles move across an edge, one reaches its destination while the other is consumed. The t-pebbling number is the smallest integer m so that any initially distributed supply of m pebbles can place t pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter two graphs is well-studied. Here we investigate the t-pebbling number of diameter two graphs under the lense of connectivity.

We generalize the k-means algorithm presented by the authors and show that the resulting algorithm can solve a larger class of clustering problems that satisfy certain properties (existence of a random sampling procedure and tightness). We prove these properties for the k-median and the discrete k-means clustering problems, resulting in O(2 (k/ε) O(1) dn) time (1 + ε)-approximation algorithms for these problems. These are the first algorithms for these problems linear in the size of the input (nd for n points in d dimensions), independent of dimensions in the exponent, assuming k and ε to be fixed. A key ingredient of the k-median result is a (1 + ε)-approximation algorithm for the 1-median problem which has running time O(2 (1/ε) O(1) d). The previous best known algorithm for this problem had linear running time.

Let S = {p1, p2, . . . , pn} be a set of pairwise disjoint geometric objects of some type in a 2D plane and let C = {c1, c2, . . . , cn} be a set of closed objects of some type in the same plane with the property that each element in C covers exactly one element in S and any two elements in C are interior-disjoint. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) has a vertex for each element of C and an edge between two vertices whenever the corresponding cover elements touch. It is known how to construct, for any given point seed set, a disk or triangle cover whose contact graph is 1or 2-connected but the problem of deciding whether a k-connected CCG can be constructed or not for k > 2 is still unsolved. A triangle cover contact graph (T CCG) is a cover contact graph whose cover elements are triangles. In this paper, we give algorithms to construct a 3-connected T CCG and a 4-connected T CCG for a given set of point seeds. We also show that any connected outerplanar graph has a realization as a T CCG on a given set of collinear point seeds. Note that, under this restriction, only trees and cycles are known to be realizable as CCG.

Let G = (V, E) be a planar triangulated graph (PTG) having every face triangular. A rectilinear dual or an orthogonal floor plan (OFP) of G is obtained by partitioning a rectangle into \mid V \mid rectilinear regions (modules) where two modules are adjacent if and only if there is an edge between the corresponding vertices in G. In this paper, a linear-time algorithm is presented for constructing an OFP for a given G such that the obtained OFP has B_{min} bends, where a bend in a concave corner in an OFP. Further, it has been proved that at least B_{min} bends are required to construct an OFP for G, where \rho - 2 \leq B_{min} \leq \rho + 1 and \rho is the sum of the number of leaves of the containment tree of G and the number of K_4 (4-vertex complete graph) in G.

Let G be a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d,(S) of S is the smallest number of edges in a connected subgraph of G that contains S. Let k, I, s and m be nonnegative integers with m > s > 2 and k and I not both 0. Then a connected graph G is said to be k-vertex I-edge (s,m)-Steiner distance stable, if for every set S of s vertices of G with d,(S) = m, and every set A consisting of at most k vertices of G-S and at most I edges of G, dG_A(S) =d,(S). It is shown that if G is k-vertex I-edges (s, m)-Steiner distance stable, then G is k-vertex I-edge (s, m + 1)-Steiner distance stable. Further, a k-vertex I-edge (s, m)-Steiner distance stable graph is shown to be a k-vertex I-edge (s -1, m)-Steiner distance stable graph for s > 3. It is then shown that the converse of neither of these two results holds. If G is a connected graph and S an independent set of s vertices of G such that d,(S) = m, then S is called an I(s, m)-set. A connected graph is k-vertex I-edge I(s, m)-Steiner distance stable if for every I(s, m)-set S and every set A of at most k vertices of G-S and I-edges of G, dc_A(S) = m. It is shown that a k-vertex I-edge 1(3, m)-Steiner distance stable graph, m>4, is k-vertex I-edge 1(3, m + 1)-Steiner distance stable.

The inference of consensus from a set of evolutionary trees is a fundamental problem in a number of fields such as biology and historical linguistics, and many models for inferring this consensus have been proposed. In this paper we present a model for deriving what we call a local consensus tree T from a set of trees T . The model we propose presumes a function f , called a total local consensus function, which determines for every triple A of species, the form that the local consensus tree should take on A. We show that all local consensus trees, when they exist, can be constructed in polynomial time and that many fundamental problems can be solved in linear time. We also consider partial local consensus functions and study optimization problems under this model. We present linear time algorithms for several variations. Finally we point out that the local consensus approach ties together many previous approaches to constructing consensus trees.

While graph embedding aims at learning low-dimensional representations of nodes encompassing the graph topology, word embedding focus on learning word vectors that encode semantic properties of the vocabulary. The first finds applications on tasks such as link prediction and node classification while the latter is systematically considered in natural language processing. Most of the time, graph and word embeddings are considered on their own as distinct tasks. However, word co-occurrence matrices, widely used to extract word embeddings, can be seen as graphs. Furthermore, most network embedding techniques rely either on a word embedding methodology (Word2vec) or on matrix factorization, also widely used for word embedding. These methods are usually computationally expensive, parameter dependant and the dimensions of the embedding space are not interpretable. To circumvent these issues, we introduce the Lower Dimension Bipartite Graphs Framework (LDBGF) which takes advantage of the fact that all graphs can be described as bipartite graphs, even in the case of textual data. This underlying bipartite structure may be explicit, like in coauthor networks. However, with LDBGF, we focus on uncovering latent bipartite structures, lying for instance in social or word co-occurrence networks, and especially such structures providing conciser and interpretable representations of the graph at hand. We further propose SINr, an efficient implementation of the LDBGF approach that extracts Sparse Interpretable Node Representations using community structure to approximate the underlying bipartite structure. In the case of graph embedding, our near-linear time method is the fastest of our benchmark, parameter-free and provides state-ofthe-art results on the classical link prediction task. We also show that low-dimensional vectors can be derived from SINr using singular value decomposition. In the case of word embedding, our approach proves to be very efficient considering the classical similarity evaluation.

Weighted finite-state transducers are used in many applications such as text, speech and image processing. This chapter gives an overview of several recent weighted transducer algorithms, including composition of weighted transducers, determinization of weighted automata, a weight pushing algorithm, and minimization of weighted automata. It briefly describes these algorithms, discusses their running time complexity and conditions of application, and shows examples illustrating their application.

Timing convergence problem arises when the estimations made during logic synthesis can not be met during physical design. In this paper, an efficient rewiring engine is proposed to explore maximal freedom after placement. The most important feature of this approach is that the existing placement solution is left intact throughout the optimization. A linear time algorithm is proposed to detect functional symmetries in the Boolean network and is used as the basis for rewiring. Integration with an existing gate sizing algorithm further proves the effectiveness of our technique. Experimental results are very promising.

The first polynomial time algorithm (O(n 4)) for modular decomposition appeared in 1972 [8] and since then there have been incremental improvements, eventually resulting in linear time algorithms [22, 7, 23, 9]. Although an optimal time complexity these algorithms are quite complicated and difficult to implement. In this paper we present an easily implementable linear time algorithm for modular decomposition. This algorithm use the notion of factorizing permutation and a new datastructure, the Ordered Chain Partitions.

In this paper, we consider the recognition problem on three classes of perfect graphs, namely, the HH-free, the HHDfree, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min{m α(n, n), m + n log n}) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min{m α(n, n), m + n log n})-time and O(n + m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement G of the graph G is HH-free can be efficiently resolved in O(nm) time using O(n 2) space, which leads to an O(n m)-time and O(n 2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n 3) time and O(n 2) space.

We present an algorithm for out-of-core simplification of large polygonal datasets that are too complex to fit in main memory. The algorithm extends the vertex clustering scheme of Rossignac and Borrel [13] by using error quadric information for the placement of each cluster's representative vertex, which better preserves fine details and results in a low mean geometric error. The use of quadrics instead of the vertex grading approach in [13] has the additional benefits of requiring less disk space and only a single pass over the model rather than two. The resulting linear time algorithm allows simplification of datasets of arbitrary complexity. In order to handle degenerate quadrics associated with (near) flat regions and regions with zero Gaussian curvature, we present a robust method for solving the corresponding underconstrained leastsquares problem. The algorithm is able to detect these degeneracies and handle them gracefully. Key features of the simplification method include a bounded Hausdorff error, low mean geometric error, high simplification speed (up to 100,000 triangles/second reduction), output (but not input) sensitive memory requirements, no disk space overhead, and a running time that is independent of the order in which vertices and triangles occur in the mesh.

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree ∆ with max{∆, 9} colors. Thus the coloring is optimal for graphs with maximum degree ∆ ≥ 9. Moreover for ∆ = 4, 5, 6 we give linear-time algorithms that use ∆ + 2 colors. These results improve over the algorithms of Chrobak and Yung [1] and of Chrobak and Nishizeki [2] which color planar graphs using max{∆, 19} colors in linear time or using max{∆, 9} colors in O(n log n) time.

We present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs. Partially supported by the German Science Foundation (JU204/11-1).

We present new linear time algorithms using the SPQR-tree data structure for computing planar embeddings of planar graphs optimizing certain distance measures. Experience with orthogonal drawings generated by the topology-shape-metrics approach shows that planar embeddings following these distance measures lead to improved quality of the final drawing in terms of bends, edge length, and drawing area. Given a planar graph, the algorithms compute the planar embedding with 1. the minimum depth among the set of all planar embeddings of G, 2. the external face of maximum size among the set of all planar embeddings of G, 3. the external face of maximum size among the set of all embeddings of G with minimum depth.

A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch-and-Cut algorithms which require multiple such subdivisions to generate cut constraints. The algorithm is not only described theoretically, but we also present an experimental study of its implementation.

A classic theorem by Steinitz states that a graph G is realizable by a convex polyhedron if and only if G is 3-connected planar. Zonohedra are an important subclass of convex polyhedra having the property that the faces of a zonohedron are parallelograms and are in parallel pairs. In this paper we give characterization of graphs of zonohedra. We also give a linear time algorithm to recognize such a graph. In our quest for finding the algorithm, we prove that in a zonohedron P both the number of zones and the number of faces in each zone is O(√ n), where n is the number of vertices of P .

We propose bipartite analogues of comparability and cocomparability graphs. Surprizingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids; this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.

Code clones are similar code fragments that occur at multiple locations in a software system. Detection of code clones provides useful information for maintenance, reengineering, program understanding and reuse. Several techniques have been proposed to detect code clones. These techniques differ in the code representation used for analysis of clones, ranging from plain text to parse trees and program dependence graphs. Clone detection based on lexical tokens involves minimal code transformation and gives good results, but is computationally expensive because of the large number of tokens that need to be compared. We explored string algorithms to find suitable data structures and algorithms for efficient token based clone detection and implemented them in our tool Repeated Tokens Finder (RTF). Instead of using suffix tree for string matching, we use more memory efficient suffix array. RTF incorporates a suffix array based linear time algorithm to detect string matches. It also provides a simple and customizable tokenization mechanism. Initial analysis and experiments show that our clone detection is simple, scalable, and performs better than the previous wellknown tools.

The distinguishability language of a regular language L is the set of words distinguishing between pairs of words under the Myhill-Nerode equivalence induced by L, i.e., between pairs of distinct left quotients of L. The similarity relation induced by a language L is a similarity relation inspired by the Myhill-Nerode equivalence and it was used to obtain compact representation of automata for a finite language L, i.e., deterministic finite cover automata, which are deterministic finite automata accepting all the words of L and possibly some other words that are longer than any word of L. The dissimilarity language of a finite language L is defined as the set of words that separate a pair of words which are not similar w.r.t. to a (finite) language L. In this paper we extend the study of distinguishability operation on regular languages to l-dissimilarity, for l ∈ N, and the dissimilarity operation on finite languages. We examine their properties, the state complexity, and relations...

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.

Let P be a graph property. A graph G is said to be locally P (closed locally P, respectively) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. A graph G of order n is said to satisfy Dirac’s condition if �(G) ≥ n/2 and it satisfies Ore’s condition if for every pair u,v of non-adjacent vertices in G, deg(u) + deg(v) ≥ n. A graph is locally Dirac (locally Ore, respectively) if the subgraph induced by the open neighbourhood of every vertex satisfies Dirac’s condition (Ore’s condition, respectively). In this paper we establish global properties for graphs that are locally Dirac and locally Ore. In particular we show that these graphs, of sufficiently large order, are 3-connected. For locally Dirac graphs it is shown that the edge connectivity equals the minimum degree and it is illustrated that this results does not extend to locally Ore graphs. We show that ⌊n/3⌋ − 1 is a sharp upper bound on the diameter of eve...

In this paper we show that the following problem, the even simple path (ESP) problem for directed planar graphs, is solvable in polynomial time: Given: a directed planar graph G = (V, E) and two nodes s (startingnode), t (targetnode) ∈ V ; Find: a simple path (i.e., without repeated nodes) from s to t of even length. (The length of the path is the number of edges it contains.

The ability to "define" propositions using default assumptions about the same propositions is identified as a significant source of computational complexity in nonmonotonic reasoning. If such constructs are not allowed, i.e. the knowledge base is stratified, a notable computational advantage is obtained. This is demonstrated by developing an iterative algorithm for propositional stratified autoepistemic theories the complexity of which is dominated by required classical reasoning. Thus efficient subclasses of stratified nonmonotonic reasoning can be obtained by further restricting the form of sentences in the knowledge base. As an example quadratic and linear time algorithms are derived for specific subclasses of stratified autoepistemic theories. The results are shown to imply efficient reasoning methods for stratified cases of default logic, logic programs, truth maintenance systems, and nonmonotonic modal logics. * This paper was first published as (Niemel~ and Rintanen, 1992). ** Ilkka Niemel~ gratefully acknowledges the support from the following foundations: Foundation of Technology, Jenny and Antti Wihuri Foundation, Emil Aaltonen Foundation, Alfred Kordelin Foundation, Heikki and Hilma Honkanen Foundation.

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

Given a graph G = (V, E), k natural numbers n 1 , n 2 , ..., n k such that k i=1 n i = |V |, we wish to find a partition V 1 , V 2 , ..., V k of the vertex set V such that |V i | = n i and V i induces a connected subgraph of G for each i, 1 ≤ i ≤ k. Such a partition is called a k-partition of G. The problem of finding a k-partition of a graph G is NP-hard in general. It is known that every k-connected graph has a k-partition. But there is no polynomial time algorithm for finding a k-partition of a k-connected graph. In this paper we give a simple linear-time algorithm for finding a k-partition of a "doughnut graph" G.

Given a graph G = (V, E), k natural numbers n 1 , n 2 , ..., n k such that k i=1 n i = |V |, we wish to find a partition V 1 , V 2 , ..., V k of the vertex set V such that |V i | = n i and V i induces a connected subgraph of G for each i, 1 ≤ i ≤ k. Such a partition is called a k-partition of G. The problem of finding a k-partition of a graph G is NP-hard in general. It is known that every k-connected graph has a k-partition. But there is no polynomial time algorithm for finding a k-partition of a k-connected graph. In this paper we give a simple linear-time algorithm for finding a k-partition of a "doughnut graph" G.

This paper studies linear-time algorithms on a hierarchical memory model called Block Move (BM), which extends the Block Transfer (BT) model of Aggarwal, Chandra, and Snir, and which is more stringent than a pipelining model studied recently by Luccio and Pagli. Upper and lower bounds are shown for various data-processing primitives, and some interesting open problems are given.

on the caterpillar arboricity of planar graphs. We prove that for every planar graph G = (V , E), the edge set E can be partitioned into four subsets (E i) 1 i 4 in such a way that G[E i ], for 1 i 4, is a forest of caterpillars. We also provide a linear-time algorithm which constructs for a given planar graph G, four forests of caterpillars covering the edges of G.

Vizing conjectured that γ (G H) ≥ γ (G)γ (H) for every pair G, H of graphs, where " " is the Cartesian product, and γ (G) is the domination number of the graph G. Denote by γ i (G) the maximum, over all independent sets I in G, of the minimal number of vertices needed to dominate I. We prove that γ (G H) ≥ γ i (G)γ (H). Since for chordal graphs γ i = γ , this proves Vizing's conjecture when G is chordal.

Vizing conjectured that γ (G H) ≥ γ (G)γ (H) for every pair G, H of graphs, where " " is the Cartesian product, and γ (G) is the domination number of the graph G. Denote by γ i (G) the maximum, over all independent sets I in G, of the minimal number of vertices needed to dominate I. We prove that γ (G H) ≥ γ i (G)γ (H). Since for chordal graphs γ i = γ , this proves Vizing's conjecture when G is chordal.

Given a point set S and a polygonal curve P in R d , we study the problem of finding a polygonal curve through S, which has a minimum Fréchet distance to P. We present an efficient algorithm to solve the decision version of this problem in O(nk 2) time, where n and k represent the sizes of P and S, respectively. A curve minimizing the Fréchet distance can be computed in O(nk 2 log(nk)) time. As a by-product, we improve the map matching algorithm of Alt et al. by a log k factor for the case when the map is a complete graph.

A set is called recurrent if its minimal automaton is strongly connected and birecurrent if it is recurrent as well as reversal. We prove a series of results concerning birecurrent sets. It is already known that any birecurrent set is completely reducible (that is, such that the minimal representation of its characteristic series is completely reducible). The main result of this paper characterizes completely reducible sets as linear combinations of birecurrent sets

We present a new model for OLAP, called the nested data cube (NDC) model. Nested data cubes are a generalization of other OLAP models such as f-tables [3], and hypercubes [2], but also of classical structures such as sets, bags, and relations. The model we propose adds to the previous models mainly flexibility in viewing the data, in that it allows for the assignment of priorities to the different dimensions of the multidimensional OLAP data. We also present an algebra in which all typical OLAP analysis and navigation operations can be formulated. We present a number of algebraic operators that work on nested data cubes and that preserve the functional dependency between the dimensional coordinates of the data cube and the factual data in it. These operations include nesting, unnesting, summary, roll-up, and aggregation operations. We show how these operations can be applied to sub-NDC's at any depth, and also show that the NDC algebra can express the SPJR algebra [1] of the relational model. A major motivation for defining an algebra rather than a calculus, is that an algebra naturally leads to an implementation strategy. Importantly, we show that the NDC algebra primitives can be implemented by linear time algorithms.

The principle of symmetry has beneficial applications in architecture. Symmetry mainly creates order and equilibrium in complex designs. This study presents a graph theoretic approach for the automatic generation of rectangular floorplans with block symmetry. Existing graph theoretical approaches focus on the floorplan's outer boundary design, different room shapes, and spatial arrangements. This paper introduces block symmetry as a new concept in floorplan generation. Based on this concept, an algorithm is proposed for generating a rectangular floorplan with rectangular blocks for a given adjacency graph if one exists. Further, suppose two blocks are required to be symmetric, i.e., of equal size. In that case, an optimisation framework is used to equate the widths and heights of the blocks, resulting in the generation of a rectangular floorplan with block symmetry. A GUI is provided for users to perform the automatic generation of floorplans.

Wc show how membership in classes of graphs definable m monwhc second-order ]oglc and of bounded treewldth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that wc describe an algorlthm that wdl produce, from J formula in monxhc second-order Ioglc and an mleger k such that the class dcfmed by the formul~IS of treewidth s k, a set of rewrite rules that rcducxs any member of the elms to one of' firrltely many graphs, in a number of steps bounded by the size c~f the graph. This rcductmn syjtem ymlds an algorlthm that runs m time linear m the size of the graph. We illustrate our results with rcductlon systems that recognux some families of outerplanar and planar graphs. Categories .ind Subject Descriptors F.2 2 [Analysis of Algorithms and Problem Complexity], Nonnurnerical Algorithms and Pr[>blems-cotyz/]~trczt~o/zs O;Zdzscrctc strut tztrcs; F.-1 3 [Mathematical Logic and Formal Languages]; Formal Languages-ckmscs defu~ed by~runzn~ars c~r uutomafa.

Under consideration is some optimization problem of data transmission in a hierarchical acyclic network. This problem is a special case of the makespan minimization problem with multiprocessor jobs on dedicated machines. We study computational complexity of the subproblems with a specific set of job types, where the type of a job is a subset of the machines required by the job.

Under consideration is some optimization problem of data transmission in a hierarchical acyclic network. This problem is a special case of the makespan minimization problem with multiprocessor jobs on dedicated machines. We study computational complexity of the subproblems with a specific set of job types, where the type of a job is a subset of the machines required by the job.

We define a new measure of complexity for finite strings using nondeterministic finite automata, called nondeterministic automatic complexity and denoted A N (x). In this paper we prove some basic results for A N (x), give upper and lower bounds, estimate it for some specific strings, begin to classify types of strings with small complexities, and provide A N (x) for |x| ≤ 8. iii TABLE OF CONTENTS

In a tournament, n players enter the competition. In each round, they are paired-up to compete against each other. Losers are thrown, while winners proceed to the next round, until only one player (the winner) is left. Given a prediction of the outcome, for every pair of players, of a match between them (modeled by a digraph D), the competitive nature of a tournament makes it attractive for manipulators. In the Tournament Fixing (TF) problem, the goal is to decide if we can conduct the competition (by controlling how players are paired-up) so that our favorite player w wins. A common form of manipulation is to bribe players to alter the outcome of matches. Kim and Williams [IJCAI 2015] integrated such deceit into TF, and showed that the resulting problem is NP-hard when < (1 −) log n alterations are possible (for any fixed > 0). For this problem, our contribution is fourfold. First, we present two operations that "obfuscate deceit": given one solution, they produce another solution. Second, we present a combinatorial result, stating that there is always a solution with all reversals incident to w and "elite players". Third, we give a closed formula for the case where D is a DAG. Finally, we present exact exponential-time and parameterized algorithms for the general case.

In a tournament, $n$ players enter the competition. In each round, they are paired-up to compete against each other. Losers are thrown, while winners proceed to the next round, until only one player (the winner) is left. Given a prediction of the outcome, for every pair of players, of a match between them (modeled by a digraph $D$), the competitive nature of a tournament makes it attractive for manipulators. In the Tournament Fixing (TF) problem, the goal is to decide if we can conduct the competition (by controlling how players are paired-up) so that our favorite player $w$ wins. A common form of manipulation is to bribe players to alter the outcome of matches. Kim and Williams [IJCAI 2015] integrated such deceit into TF, and showed that the resulting problem is NP-hard when $\ell&lt;(1-\epsilon)\log n$ alterations are possible (for any fixed $\epsilon&gt;0$). For this problem, our contribution is fourfold. First, we present two operations that ``obfuscate deceit&#39;&#39;: given o...

In a tournament, n players enter the competition. In each round, they are paired-up to compete against each other. Losers are thrown, while winners proceed to the next round, until only one player (the winner) is left. Given a prediction of the outcome, for every pair of players, of a match between them (modeled by a digraph D), the competitive nature of a tournament makes it attractive for manipulators. In the Tournament Fixing (TF) problem, the goal is to decide if we can conduct the competition (by controlling how players are paired-up) so that our favorite player w wins. A common form of manipulation is to bribe players to alter the outcome of matches. Kim and Williams [IJCAI 2015] integrated such deceit into TF, and showed that the resulting problem is NP-hard when < (1 −) log n alterations are possible (for any fixed > 0). For this problem, our contribution is fourfold. First, we present two operations that "obfuscate deceit": given one solution, they produce another solution. Second, we present a combinatorial result, stating that there is always a solution with all reversals incident to w and "elite players". Third, we give a closed formula for the case where D is a DAG. Finally, we present exact exponential-time and parameterized algorithms for the general case.