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Applications of the Quantum Algorithm for St-Connectivity
Applications of the quantum algorithm for st-connectivity KAI DELORENZO1 , SHELBY KIMMEL1 , AND R. TEAL WITTER1 1Middlebury College, Computer Science Department, Middlebury, VT Abstract We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algo- rithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity. 1 Introduction Quantum query algorithms are remarkably described by span programs [15, 16], a linear algebraic object originally created to study classical logspace complexity [12]. However, finding optimal span program algorithms can be challenging; while they can be obtained using a semidefinite program, the size of the program grows exponentially with the size of the input to the algorithm. Moreover, span programs are designed to characterize the query complexity of an algorithm, while in practice we also care about the time and space complexity. One of the nicest span programs is for the problem of undirected st-connectivity, in which one must decide whether two vertices s and t are connected in a given graph. It is “nice” for several reasons: It is easy to describe and understand why it is correct. • It corresponds to a quantum algorithm that uses logarithmic (in the number of vertices and edges of • the graph) space [4, 11]. -
The Normalized Laplacian Spectrum of Subdivisions of a Graph
The normalized Laplacian spectrum of subdivisions of a graph Pinchen Xiea,b, Zhongzhi Zhanga,c, Francesc Comellasd aShanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China bDepartment of Physics, Fudan University, Shanghai 200433, China cSchool of Computer Science, Fudan University, Shanghai 200433, China dDepartment of Applied Mathematics IV, Universitat Polit`ecnica de Catalunya, 08034 Barcelona Catalonia, Spain Abstract Determining and analyzing the spectra of graphs is an important and excit- ing research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spec- tra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact val- ues of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees. Keywords: Normalized Laplacian spectrum, Subdivision graph, Degree-Kirchhoff index, Kemeny's constant, Spanning trees 1. Introduction Spectral analysis of graphs has been the subject of considerable research effort in mathematics and computer science [1, 2, 3], due to its wide appli- cations in this area and in general [4, 5]. In the last few decades a large body of scientific literature has established that important structural and dynamical properties -
Coincidence Theorems Involving Composites of Acyclic Mappings in Contractible Spaces
Appl. Math. Lett. Vol. 11, No. 2, pp. 85-89, 1998 Pergamon Copyright(~)1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0893-9659/98 $19.00 + 0.00 PII: S0893-9659(98)00016.0 Coincidence Theorems Involving Composites of Acyclic Mappings in Contractible Spaces XIE PING DING Department of Mathematics, Sichuan Normal University Chengdu, Sichuan 610066, P.R. China (Received March 1997; accepted April 1997) Abstract--Some coincidence theorems involving a new class of set-valued mappings containing compact composites of acyclic mappings defined on a contractible space is proved. Keywords--Coincidence theorem, Contractible space, Acyclic mapping, Local intersection prop- erty. 1. INTRODUCTION AND PRELIMINARIES Let X and Y be two nonempty sets and let T : X --* 2 Y and S : Y --* 2 X be two set-valued mappings, where 2 X denotes the family of all subsets of X. Following Browder [1], a point (x0, Y0) E X × Y is said to be a coincidence point if Y0 E T(xo) and x0 E S(yo). Let An be the standard n-dimensional simplex with vertices e0, el,..., e~. If J is a nonempty subset of {0, 1,..., n}, we denote by Aj the convex hull of the vertices (ej : j E J}. A topological space X is said to be contractible if the identity mapping Ix of X is homotopic to a constant function. A topological space X is said to be an acyclic space if all of its reduced Cech homology groups over the rationals vanish. In particular, any contractible space is acyclic, and hence any convex or star-shaped set in a topological vector space is acyclic. -
Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 973920, 27 pages doi:10.1155/2012/973920 Research Article Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory Jianguo Tang,1, 2 Kun She,1 and William Zhu3 1 School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China 2 School of Computer Science and Engineering, XinJiang University of Finance and Economics, Urumqi 830012, China 3 Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, China Correspondence should be addressed to William Zhu, [email protected] Received 4 February 2012; Revised 30 April 2012; Accepted 18 May 2012 Academic Editor: Mehmet Sezer Copyright q 2012 Jianguo Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. -
On Generalized 3-Manifolds Which Are Not Homologically Locally Connected ∗ Umed H
Topology and its Applications 160 (2013) 445–449 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On generalized 3-manifolds which are not homologically locally connected ∗ Umed H. Karimov a,DušanRepovšb,c, a Institute of Mathematics, Academy of Sciences of Tajikistan, Ul. Ainy 299A , Dushanbe 734063, Tajikistan b Faculty of Education, University of Ljubljana, Kardeljeva pl. 16, Ljubljana 1000, Slovenia c Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia article info abstract Article history: We show that the classical example X of a 3-dimensional generalized manifold constructed Received 10 September 2011 by van Kampen is not homologically locally connected (i.e. not HLC) space. This space X Accepted 14 October 2012 is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either. MSC: © 2012 Elsevier B.V. All rights reserved. primary 54F15, 55N15 secondary 54G20, 57M05 Keywords: Singular quotient n-manifold (Co)homology manifold (Co)homological local connectedness van Kampen generalized 3-manifold 1. Introduction In our earlier paper [10] we constructed for every natural number n > 2, examples of n-dimensional compact metrizable cohomology n-manifolds which are not homologically locally connected with respect to the singular homology (i.e. they are not HLC spaces). In the present paper we shall call them singular quotient n-manifolds. Subsequently, we have discovered that van Kampen constructed a compact metrizable generalized 3-manifold which “is not locally connected in dimension 1 in the homotopy sense” [2, p. -
Helly Numbers of Acyclic Families
Helly numbers of acyclic families Eric´ Colin de Verdi`ere∗ Gr´egoryGinoty Xavier Goaocz October 16, 2012 1 Abstract 2 The Helly number of a family of sets with empty intersection is the size of its largest inclusion- 3 wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of 4 an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family 5 G ⊆ F the intersection of the elements of G has at most r connected components, each of which 6 is a Q-homology cell. We show that the Helly number of F is at most r(dΓ + 1), where dΓ is 7 the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and 8 higher. (In particular d d = d.) This bound is best possible. We prove, in fact, a stronger R 9 theorem where small sub-families may have more than r connected components, each possibly 10 with nontrivial homology in low dimension. As an application, we obtain several explicit bounds 11 on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were 12 previously known; in certain cases, the bound we obtain is better than what was previously 13 known. 14 1 Introduction d 15 Helly's theorem [32] asserts that if, in a finite family of convex sets in R , any d + 1 sets have 16 non-empty intersection, then the whole family has non-empty intersection. Equivalently, any finite d 17 family of convex sets in R with empty intersection must contain a subfamily of at most d + 1 sets 18 whose intersection is already empty. -
Permutation of Elements in Double Semigroups
PERMUTATION OF ELEMENTS IN DOUBLE SEMIGROUPS MURRAY BREMNER AND SARA MADARIAGA Abstract. Double semigroups have two associative operations ◦, • related by the interchange relation: (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). Kock [13] (2007) discovered a commutativity property in degree 16 for double semigroups: as- sociativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer alge- bra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9. 1. Introduction Definition 1.1. A double semigroup is a set S with two associative binary operations •, ◦ satisfying the interchange relation for all a,b,c,d ∈ S: (⊞) (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). The symbol ≡ indicates that the equation holds for all values of the variables. We interpret ◦ and • as horizontal and vertical compositions, so that (⊞) ex- presses the equivalence of two decompositions of a square array: a b a b a b (a ◦ b) • (c ◦ d) ≡ ≡ ≡ ≡ (a • c) ◦ (b • d). c d c d c d This interpretation of the operations extends to any double semigroup monomial, producing what we call the geometric realization of the monomial. The interchange relation originated in homotopy theory and higher categories; see Mac Lane [15, (2.3)] and [16, §XII.3]. It is also called the Godement relation by arXiv:1405.2889v2 [math.RA] 26 Mar 2015 some authors; see Simpson [21, §2.1]. -
G-Parking Functions and Tree Inversions
G-PARKING FUNCTIONS AND TREE INVERSIONS DAVID PERKINSON, QIAOYU YANG, AND KUAI YU Abstract. A depth-first search version of Dhar’s burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph solves a problem posed by R. Stanley. 1. Introduction Let G = (V, E) be a connected simple graph with vertex set V = {0,...,n} and edge set E. Fix a root vertex r ∈ V and let SPT(G) denote the set of spanning trees of G rooted at r. We think of each element of SPT(G) as a directed graph in which all paths lead away from the root. If i, j ∈ V and i lies on the unique path from r to j in the rooted spanning tree T , then i is an ancestor of j and j is a descendant of i in T . If, in addition, there are no vertices between i and j on the path from the root, then i is the parent of its child j, and (i, j) is a directed edge of T . Definition 1. An inversion of T ∈ SPT(G) is a pair of vertices (i, j), such that i is an ancestor of j in T and i > j. It is a κ-inversion if, in addition, i is not the root and i’s parent is adjacent to j in G. The number of κ-inversions of T is the tree’s κ-number, denoted κ(G, T ). -
The Normalized Laplacian Spectrum of Subdivisions of a Graph
The normalized Laplacian spectrum of subdivisions of a graph Pinchen Xiea,c, Zhongzhi Zhangb,c, Francesc Comellasd aDepartment of Physics, Fudan University, Shanghai 200433, China bSchool of Computer Science, Fudan University, Shanghai 200433, China cShanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China dDepartment of Applied Mathematics IV, Universitat Polit`ecnica de Catalunya, 08034 Barcelona Catalonia, Spain Abstract Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the nor- malized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny’s constant and number of spanning trees. Keywords: Normalized Laplacian spectrum, Subdivision graph, Degree-Kirchhoff index, Kemeny’s constant, Spanning trees 1. Introduction arXiv:1510.02394v1 [math.CO] 7 Oct 2015 Spectral analysis of graphs has been the subject of considerable research effort in theoretical computer science [1, 2, 3], due to its wide applications in this area and in general [4, 5]. In the last few decades a large body of scientific literature has established that important structural and dynamical properties -
Classes of Graphs Embeddable in Order-Dependent Surfaces
Classes of graphs embeddable in order-dependent surfaces Colin McDiarmid Sophia Saller Department of Statistics Department of Mathematics University of Oxford University of Oxford [email protected] and DFKI [email protected] 12 June 2021 Abstract Given a function g = g(n) we let Eg be the class of all graphs G such that if G has order n (that is, has n vertices) then it is embeddable in some surface of Euler genus at most g(n), and let eEg be the corresponding class of unlabelled graphs. We give estimates of the sizes of these classes. For example we 3 g show that if g(n) = o(n= log n) then the class E has growth constant γP, the (labelled) planar graph growth constant; and when g(n) = O(n) we estimate the number of n-vertex graphs in Eg and eEg up g to a factor exponential in n. From these estimates we see that, if E has growth constant γP then we must have g(n) = o(n= log n), and the generating functions for Eg and eEg have strictly positive radius of convergence if and only if g(n) = O(n= log n). Such results also hold when we consider orientable and non-orientable surfaces separately. We also investigate related classes of graphs where we insist that, as well as the graph itself, each subgraph is appropriately embeddable (according to its number of vertices); and classes of graphs where we insist that each minor is appropriately embeddable. In a companion paper [43], these results are used to investigate random n-vertex graphs sampled uniformly from Eg or from similar classes. -
Tverberg's Theorem for Cell Complexes
TVERBERG’S THEOREM FOR CELL COMPLEXES SHO HASUI, DAISUKE KISHIMOTO, MASAHIRO TAKEDA, AND MITSUNOBU TSUTAYA Abstract. The topological Tverberg theorem states that given any continuous map (d+1)(r−1) d (d+1)(r−1) f : ∆ → R , there are pairwise disjoint faces σ1,...,σr of ∆ such that f(σ1) ∩···∩ f(σr) =6 ∅ whenever r is a prime power. We generalize this theorem to a continuous map from a certain CW complex into a Euclidean space. 1. Introduction Let d ≥ 1 and r ≥ 2 be integers. Tverberg’s theorem states that for any given (d+1)(r−1)+1 points in Rd, there is a partition of points into r subsets whose convex hulls intersect. This theorem has been of great interest in combinatorics for more than 50 years. Clearly, Tverberg’s theorem can be restated in terms of an affine map from a (d + 1)(r − 1)-simplex into Rd. The topological Tverberg theorem replaces an affine map in Tverberg’s theorem by a continuous maps: for any continuous map f : ∆(d+1)(r−1) → Rd, there are pairwise (d+1)(r−1) disjoint faces σ1,...,σr of the simplex ∆ such that f(σ1) ∩···∩ f(σr) 6= ∅ whenever r is a prime power. This was first proved by B´ar´any, Shlosman and Sz˝ucs [4] when r is a prime, and later by Ozaydin¨ [16] and Volovikov [17], independently, when r is a prime power. Remark that Frick [8] proved that the result does not hold unless we assume r is a prime power. See the surveys [3, 6] for more on the topological Tverberg theorem. -
Homology Equivalences Inducing an Epimorphism on the Fundamental Group and Quillen’S Plus Construction
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 3, Pages 891{898 S 0002-9939(03)07221-6 Article electronically published on October 21, 2003 HOMOLOGY EQUIVALENCES INDUCING AN EPIMORPHISM ON THE FUNDAMENTAL GROUP AND QUILLEN'S PLUS CONSTRUCTION JOSEL.RODR´ ´IGUEZ AND DIRK SCEVENELS (Communicated by Paul Goerss) Abstract. Quillen's plus construction is a topological construction that kills the maximal perfect subgroup of the fundamental group of a space without changing the integral homology of the space. In this paper we show that there is a topological construction that, while leaving the integral homology of a space unaltered, kills even the intersection of the transfinite lower central series of its fundamental group. Moreover, we show that this is the maximal subgroup that can be factored out of the fundamental group without changing the integral homology of a space. 0. Introduction As explained in [8], [9], Bousfield's HZ-localization XHZ of a space X ([2]) is homotopy equivalent to its localization with respect to a map of classifying spaces Bf : BF1 ! BF2 induced by a certain homomorphism f : F1 !F2 between free groups. This means that a space X is HZ-local if and only if the induced ∗ map Bf :map(BF2;X) ! map(BF1;X) is a weak homotopy equivalence. More- over, the effect of Bf-localization on the fundamental group produces precisely the group-theoretical HZ-localization (i.e., f-localization) of the fundamental group, ∼ ∼ i.e., π1LBf X = Lf (π1X) = (π1X)HZ for all spaces X. A universal acyclic space for HZ-localization (i.e., Bf-localization), in the sense of Bousfield ([4]), was studied by Berrick and Casacuberta in [1].