317 research outputs found

    Harmonious Colourings of Temporal Matchings

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    Graph colouring is a fundamental problem in computer science, with a large body of research dedicated to both the general colouring problem and restricted cases. Harmonious colourings are one such restriction, where each edge must contain a globally unique pair of colours, i.e. if an edge connects a vertex coloured x with a vertex coloured y, then no other pair of connected vertices can be coloured x and y. Finding such a colouring in the traditional graph setting is known to be NP-hard, even in trees. This paper considers the generalisation of harmonious colourings to Temporal Graphs, specifically (k,t)-Temporal matchings, a class of temporal graphs where the underlying graph is a matching (a collection of disconnected components containing pairs of vertices), each edge can appear in at most t timesteps, and each timestep can contain at most k other edges. We provide a complete overview of the complexity landscape of finding temporal harmonious colourings for (k,t)-matchings. We show that finding a Temporal Harmonious Colouring, a colouring that is harmonious in each timestep, is NP-hard for (k,t)-Temporal Matchings when k ≥ 4, t ≥ 2, or when k ≥ 2 and t ≥ 3. We further show that this problem is inapproximable for t ≥ 2 and an unbounded value of k, and that the problem of determining the temporal harmonious chromatic number of a (2,3)-temporal matching can be determined in linear time. Finally, we strengthen this result by a set of upper and lower bounds of the temporal harmonious chromatic number both for individual temporal matchings and for the class of (k, t)-temporal matchings

    Ranking and Unranking k-subsequence universal words

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    A subsequence of a word ww is a word uu such that u=w[i1]w[i2],…w[i∣u∣]u = w[i_1] w[i_2] , \dots w[i_{|u|}], for some set of indices 1≤i1<i2<⋯<ik≤∣w∣1 \leq i_1 < i_2 < \dots < i_k \leq |w|. A word ww is kk-subsequence universal over an alphabet Σ\Sigma if every word in Σk\Sigma^k appears in ww as a subsequence. In this paper, we provide new algorithms for kk-subsequence universal words of fixed length nn over the alphabet Σ={1,2,…,σ}\Sigma = \{1,2,\dots, \sigma\}. Letting U(n,k,σ)\mathcal{U}(n,k,\sigma) denote the set of nn-length kk-subsequence universal words over Σ\Sigma, we provide: * an O(nkσ)O(n k \sigma) time algorithm for counting the size of U(n,k,σ)\mathcal{U}(n,k,\sigma); * an O(nkσ)O(n k \sigma) time algorithm for ranking words in the set U(n,k,σ)\mathcal{U}(n,k,\sigma); * an O(nkσ)O(n k \sigma) time algorithm for unranking words from the set U(n,k,σ)\mathcal{U}(n,k,\sigma); * an algorithm for enumerating the set U(n,k,σ)\mathcal{U}(n,k,\sigma) with O(nσ)O(n \sigma) delay after O(nkσ)O(n k \sigma) preprocessing

    Enumerating m-Length Walks in Directed Graphs with Constant Delay

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    In this paper, we provide a novel enumeration algorithm for the set of all walks of a given length within a directed graph. Our algorithm has worst-case constant delay between outputting succinct representations of such walks, after a preprocessing step requiring linear time relative to the size of the graph. We apply these results to the problem of enumerating succinct representations of the strings of a given length from a prefix-closed regular language (languages accepted by a finite automaton which has final states only)

    Ranking and Unranking k-Subsequence Universal Words

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    A subsequence of a word w is a word u such that u= w[ i1] w[ i2], ⋯ w[ i|u|], for some set of indices 1 ≤ i1&amp;lt; i2&amp;lt; ⋯ &amp;lt; ik≤ | w|. A word w is k-subsequence universal over an alphabet Σ if every word in Σk appears in w as a subsequence. In this paper, we provide new algorithms for k-subsequence universal words of fixed length n over the alphabet Σ= { 1, 2, ⋯, σ}. Letting U(n, k, σ) denote the set of n-length k-subsequence universal words over Σ, we provide: an O(nkσ) time algorithm for counting the size of U(n, k, σ) ;an O(nkσ) time algorithm for ranking words in the set U(n, k, σ) ;an O(nkσ) time algorithm for unranking words from the set U(n, k, σ) ;an algorithm for enumerating the set U(n, k, σ) with O(nσ) delay after O(nkσ) preprocessing.</p

    The k-center Problem for Classes of Cyclic Words

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    The problem of finding k uniformly spaced points (centres) within a metric space is well known as the k-centre selection problem. In this paper, we introduce the challenge of k-centre selection on a class of objects of exponential size and study it for the class of combinatorial necklaces, known as cyclic words. The interest in words under translational symmetry is motivated by various applications in algebra, coding theory, crystal structures and other physical models with periodic boundary conditions. We provide solutions for the centre selection problem for both one-dimensional necklaces and largely unexplored objects in combinatorics on words - multidimensional combinatorial necklaces. The problem is highly non-trivial as even verifying a solution to the k-centre problem for necklaces can not be done in polynomial time relative to the length of the cyclic words and the alphabet size unless P= NP. Despite this challenge, we develop a technique of centre selection for a class of necklaces based on de-Bruijn Sequences and provide the first polynomial O(k· n) time approximation algorithm for selecting k centres in the set of 1D necklaces of length n over an alphabet of size q with an approximation factor of O(1+logq(k·n)n-logq(k·n)). For the set of multidimensional necklaces of size n1× n2× … × nd we develop an O(k· N2) time algorithm with an approximation factor of O(1+logq(k·N)N-logq(k·N)) in O(k· N2) time, where N= n1· n2· … · nd by approximating de Bruijn hypertori technique

    Structural and combinatorial properties of 2-swap word permutation graphs

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    Funding: This work is supported by the Leverhulme Research Centre for Functional Materials Design and EPSRC grants EP/P02002X/1, EP/R018472/1.In this paper, we study the graph induced by the 2-swap permutation on words with a fixed Parikh vector. A 2-swap is defined as a pair of positions s=(i,j) where the word w induced by the swap s on v is v[1]v[2]⋯v[i-1]v[j]v[i+1]⋯v[j-1]v[i]v[j+1]⋯v[n]. With these permutations, we define the Configuration Graph, G(P) for a given Parikh vector. Each vertex in G(P) corresponds to a unique word with the Parikh vector P, with an edge between any pair of words v and w if there exists a swap s such that v∘s=w. We provide several key combinatorial properties of this graph, including the exact diameter of this graph, the clique number of the graph, and the relationships between subgraphs within this graph. Additionally, we show that for every vertex in the graph, there exists a Hamiltonian path starting at this vertex. Finally, we provide an algorithm enumerating these paths from a given input word of length n with a delay of at most O(log n) between outputting edges, requiring O(n log n) preprocessing

    Faster Exploration of Some Temporal Graphs

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    A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in ?(n log n) days. All these improve best known results in the literature
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