47,700 research outputs found

    Digraph Complexity Measures and Applications in Formal Language Theory

    Full text link
    We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.Comment: 19 pages, 1 figur

    Parametric shortest-path algorithms via tropical geometry

    Full text link
    We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

    Rectangular Layouts and Contact Graphs

    Get PDF
    Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct O(n2)O(n^2)-area rectangular layouts for general contact graphs and O(nlogn)O(n\log n)-area rectangular layouts for trees. (For trees, this is an O(logn)O(\log n)-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require Ω(n2)\Omega(n^2) (rsp., Ω(nlogn)\Omega(n\log n)) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

    Regular Cost Functions, Part I: Logic and Algebra over Words

    Full text link
    The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page

    Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

    Full text link
    We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter λ\lambda, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor bb, the hard-core model can be equivalently defined as a broadcasting process with a parameter ω\omega which is the positive solution to λ=ω(1+ω)b\lambda=\omega(1+\omega)^b, and vertices are occupied with probability ω/(1+ω)\omega/(1+\omega) when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at ωrlnb/b\omega_r\approx \ln{b}/b. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular bb-ary trees ThT_h of height hh and nn vertices, undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any ωlnb/b\omega \le \ln{b}/b, for ThT_h with any boundary condition, the relaxation time is Ω(n)\Omega(n) and O(n1+ob(1))O(n^{1+o_b(1)}). In contrast, above the reconstruction threshold we show that for every δ>0\delta>0, for ω=(1+δ)lnb/b\omega=(1+\delta)\ln{b}/b, the relaxation time on ThT_h with any boundary condition is O(n1+δ+ob(1))O(n^{1+\delta + o_b(1)}), and we construct a boundary condition where the relaxation time is Ω(n1+δ/2ob(1))\Omega(n^{1+\delta/2 - o_b(1)})
    corecore