A 2-matching in an undirected graph G = (VG, EG) is a function x : EG -> {0, 1, 2} such that for each node v is an element of VG the sum of values x(e) for all edges e incident to v does not exceed 2. The size of x...
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A 2-matching in an undirected graph G = (VG, EG) is a function x : EG -> {0, 1, 2} such that for each node v is an element of VG the sum of values x(e) for all edges e incident to v does not exceed 2. The size of x is the sum Sigma(e) x(e). If {e is an element of EG | x(e) not equal 0} contains no triangles then x is called triangle-free. Cornuejols and Pulleyblank devised a combinatorial O(mn)-algorithm that finds a maximum triangle free 2-matching of size (hereinafter n := |VG|, m := |EG|) and also established a min-max theorem. We claim that this approach is, in fact, superfluous by demonstrating how these results may be obtained directly from the Edmonds-Gallai decomposition. Applying the algorithm of Micali and Vazirani we are able to find a maximum triangle-free 2-matching in O(m root n) time. Also we give a short self-contained algorithmic proof of the min-max theorem. Next, we consider the case of regular graphs. It is well-known that every regular graph admits a perfect 2-matching. One can easily strengthen this result and prove that every d-regular graph (for d >= 3) contains a perfect triangle-free 2-matching. We give the following algorithms for finding a perfect triangle-free 2-matching in a d-regular graph: an O(n)-algorithm for d = 3, an O(m + n(3/2))-algorithm for d = 2k (k >= 2), and an O(n(2))-algorithm for d = 2k + 1 (k >= 2). We also prove that there exists a constant c > 1 such that every 3-regular graph contains at least c(n) perfect triangle-free 2-matchings.
A hexagonal patch is a plane graph in which inner laces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and t...
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ISBN:
(纸本)9783642121999
A hexagonal patch is a plane graph in which inner laces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem.
graphs are canonical examples of high-dimensional non-Euclidean data sets, and are emerging as a common data structure in many fields. While there are many algorithms to analyze such data, a signal processing theory f...
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ISBN:
(纸本)9781424442966
graphs are canonical examples of high-dimensional non-Euclidean data sets, and are emerging as a common data structure in many fields. While there are many algorithms to analyze such data, a signal processing theory for evaluating these techniques akin to detection and estimation in the classical Euclidean setting remains to be developed. In this paper we show the conceptual advantages gained by formulating graph analysis problems in a signal processing framework by way of a practical example: detection of a subgraph embedded in a background graph. We describe an approach based on detection theory and provide empirical results indicating that the test statistic proposed has reasonable power to detect dense subgraphs in large random graphs.
The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J'GAA 99] gives the first linear...
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ISBN:
(纸本)9783939897163
The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J'GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order k, and arbitrary planar host graph, improving upon the O(n(root k))-time algorithm when using the "Color-coding" technique of Alon et al [J'ACM 95]. Eppstein's algorithm runs in time k(O(k)) n, that is, the dependency on k is superexponential. We improve the running time to 2(O(k)) n, that is, single exponential in k while keeping the term in n linear. Next to deciding subgraph isomorphism, we can construct a solution and count all solutions in the same asymptotic running time. We may enumerate omega subgraphs with an additive term O(omega k) in the running time of our algorithm. We introduce the technique of "embedded dynamic programming" on a suitably structured graph decomposition, which exploits the number and topology of the underlying drawings of the subgraph pattern (rather than of the host graph).
In this paper we explore a simple and general approach for developing parallel algorithms that lead to good cache complexity on parallel machines with private or shared caches The approach is to design nested-parallel...
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ISBN:
(纸本)9781450300797
In this paper we explore a simple and general approach for developing parallel algorithms that lead to good cache complexity on parallel machines with private or shared caches The approach is to design nested-parallel algorithms that have low depth (span. critical path length) and for which the natural sequential evaluation order has low cache complexity in the cache-oblivious model We describe several cache-oblivious algorithms with optimal work, polylogarithmic depth, and sequential cache complexities that match the best sequential algorithms, including the first such algorithms for sorting and for sparse-matrix vector multiply on matrices with good vertex separators Using known mappings. our results lead to low cache complexities on shared-memory multiprocessors with a single level of private caches or a single shared cache We generalize these mappings to multi-level cache hierarchies of private or shared caches, implying that our algorithms also have low cache complexities on such hierarchies The key factor in obtaining these low parallel cache complexities is the low depth of the algorithms we propose.
It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repe...
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ISBN:
(纸本)9783540771180
It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.
We present a polynomial-time approximation algorithm for legally coloring as many edges of a given simple graph as possible using two colors. It achieves an approximation ratio of roughly 0.842 and runs in O(n(3)m) ti...
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ISBN:
(纸本)9783642143540
We present a polynomial-time approximation algorithm for legally coloring as many edges of a given simple graph as possible using two colors. It achieves an approximation ratio of roughly 0.842 and runs in O(n(3)m) time, where n (respectively, m) is the number of vertices (respectively, edges) in the input graph. The previously best ratio achieved by a polynomial-time approximation algorithm was 5/6 approximate to 0.833.
Many algorithms for analyzing social networks assume that the structure of the network is known, but this is not always a reasonable assumption. We wish to reconstruct an underlying network given data about how some p...
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ISBN:
(数字)9783642135620
ISBN:
(纸本)9783642135613
Many algorithms for analyzing social networks assume that the structure of the network is known, but this is not always a reasonable assumption. We wish to reconstruct an underlying network given data about how some property, such as disease, has spread through the network. Properties may spread through a network in different ways: for instance, an individual may learn information as soon as one of his neighbors has learned that information, but political beliefs may follow a different type of model. We create algorithms for discovering underlying networks that would give rise to the diffusion in these models.
Rank correlation can be used to compare two linearly ordered rankings. If the rankings include noise values, the rank correlation coefficient will yield lower values than it actually should. In this paper, we propose ...
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ISBN:
(纸本)9783642140549
Rank correlation can be used to compare two linearly ordered rankings. If the rankings include noise values, the rank correlation coefficient will yield lower values than it actually should. In this paper, we propose an algorithm to remove pairs of values from rankings in order to increase Kendall's tau rank correlation coefficient. The problem itself is motivated from real data in bioinformatics context.
Finding shortest paths between two vertices in a weighted graph is a well explored problem and several efficient algorithms for solving it have been reported. We propose a new variation of this problem which we call t...
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Finding shortest paths between two vertices in a weighted graph is a well explored problem and several efficient algorithms for solving it have been reported. We propose a new variation of this problem which we call the Detour Admitting Shortest Path Problem (DASPP). We present an efficient algorithm for solving DASPP. This is the first algorithm that constructs a shortest path such that each edge of the shortest path admits a detour with no more than k-hops. This algorithm has important applications in transportation networks. We also present implementation issues for the detour admitting shortest path algorithm.
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