We show that the cost (length) of the shortest traveling salesman tour through n points in the unit square is, in the worst case, $\alpha _{{\text{opt}}}^{{\text{tsp}}} \sqrt n + o(\sqrt n )$, where $1.075 \leqq \alph...
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We show that the cost (length) of the shortest traveling salesman tour through n points in the unit square is, in the worst case, $\alpha _{{\text{opt}}}^{{\text{tsp}}} \sqrt n + o(\sqrt n )$, where $1.075 \leqq \alpha _{{\text{opt}}}^{{\text{tsp}}}\leqq 1.414$. The cost of the minimum matching of n points in the unit square is shown to be, in the worst case, $\alpha _{{\text{opt}}}^{{\text{mat}}} \sqrt n + o(\sqrt n )$, where $0.537 \leqq \alpha _{{\text{opt}}}^{{\text{mat}}} \leqq 0.707$. Furthermore, for each of these two problems there is an almost linear time heuristic algorithm whose Worst case cost is, neglecting lower order terms, as low as b Bible.
Two common problems in image analysis are described—the region-labeling problem and the clustering problem. Both are shown to be instances of a search-and-rename problem which can be solved in parallel by a system ar...
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Two common problems in image analysis are described—the region-labeling problem and the clustering problem. Both are shown to be instances of a search-and-rename problem which can be solved in parallel by a system architecture inherently suitable for VLSI implementation.
The technique of depth-first search is familiar in graph algorithms. Depth-first search is a technique which assumes that only one operation at a time is performed on a graph. Consideration is made of the possibilit...
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The technique of depth-first search is familiar in graph algorithms. Depth-first search is a technique which assumes that only one operation at a time is performed on a graph. Consideration is made of the possibility of parallel graph operations and a parallel traversal method which is analogous to depth-first search. A method is presented for the detection of properties of general graphs in an environment in which each node can be considered an autonomous processor, interacting with its neighbors by passing messages. The algorithms are decentralized in that they depend on no central controlling process nor on global storage. It is not required to have a node to know the configuration or extent of the graph, nor is a global clock required. These algorithms are inherently asynchronous, and they generally require execution time proportional to the diameter of the graph. Figures.
NP-completeness is established for the existence of a dominating vertex in a digraph lacking a k length path for a given k that is fixed at a value greater than or equal to 2. An implicit method of enumeration is use...
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NP-completeness is established for the existence of a dominating vertex in a digraph lacking a k length path for a given k that is fixed at a value greater than or equal to 2. An implicit method of enumeration is used in the proof to show that, for any k, there is a digraph with a path of k length for each dominating set. The study was motivated by a problem of multiobjective optimization. In dealing with a set of potential actions, including partial and nontransitive relations, optimization requires the selection of an optimal subset of actions. Previous solutions were found to be empty when the digraph is an odd cycle.
Recent technological development has drawn the attention of many researchers in computer science to problems of parallel computation. Three possible generalizations of the rule that no 2 different processors may writ...
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Recent technological development has drawn the attention of many researchers in computer science to problems of parallel computation. Three possible generalizations of the rule that no 2 different processors may write simultaneously into the same memory location are shown: 1. weak conflict-resolution rule, in which simultaneous writing is permitted only to selected memory locations which can contain the numbers 0 or 1 only; 2. equality conflict-resolution rule, in which simultaneous writing into the same memory location is permitted, provided all processors involved store the same value; and 3. priority conflict-resolution rule, where processors are linearly ordered according to their priorities. It is apparent that each of these rules leads, in turn, to the next one. It is demonstrated, at the expense of the number of processors, that the computational power of all of these rules is equal in a certain sense. Five theorems are presented.
Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter....
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Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter. The “greedy” heuristic of repeatedly matching the two closest unmatched points can be implemented in worst-case time $O(n^2 \log n)$, a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to $n^3 $ to find the minimum cost matching in a weighted graph. We show that, for an even number n of vertices whose distances satisfy the triangle inequality, the ratio of the cost of the matching produced by this greedy heuristic to the cost of the minimal matching is at most ${}_3^4 n^{\lg _2^3 } - 1$, $\lg _2^3 \approx 0.58496$, and there are examples that achieve this bound. We conclude that this greedy heuristic, although desirable because of its simplicity, would be a poor choice for this problem.
A jogger is confronted with a network of roads, each of which possesses some measure of undesirability. Beginning at a designated point, the jogger must plan a route that will return him to the starting point, have a...
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A jogger is confronted with a network of roads, each of which possesses some measure of undesirability. Beginning at a designated point, the jogger must plan a route that will return him to the starting point, have a minimum total length, include any single path no more than once, and achieve the minimum measure of undesirability for the entire route. Version B of the jogger's problem allows the undesirability of a road to be dependent upon the direction of travel. An algorithm has been developed for both the basic jogger's problem and for version B. In the algorithm, the cost in undesirability of the minimum jog can be calculated.
An algorithm is presented for determining whether a graph has a completely connected cut-set. If a cut-set exists, the algorithm can produce it in time proportional to the cube of the number of vertices of the graph....
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An algorithm is presented for determining whether a graph has a completely connected cut-set. If a cut-set exists, the algorithm can produce it in time proportional to the cube of the number of vertices of the graph. The algorithm can be used for determining the chromatic number and largest clique size for families of graphs defined inductively on primitive pieces. An initialization procedure produces a chordless odd cycle or a clique cut-set, or discovers that the function is a clique. Next a search is made for a pair of non-adjacent vertices. These vertices are then used for a chordless path. Intermediate vertices are then joined. The previous steps are repeated until a clique cut-set has been found.
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