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This class of problem is associated with Rank revealing QR factorizations and D optimal experimental design. [39] Minimal addition chains for sequences. [40] The complexity of minimal addition chains for individual numbers is unknown. [41] Modal logic S5-Satisfiability; Pancake sorting distance problem for strings [42]
The Erdős Distance Problem consists of twelve chapters and three appendices. [5]After an introductory chapter describing the formulation of the problem by Paul Erdős and Erdős's proof that the number of distances is always at least proportional to , the next six chapters cover the two-dimensional version of the problem.
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 [ 1 ] [ 2 ] and almost proven by Larry Guth and Nets Katz in 2015.
Shortest path (A, C, E, D, F) between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The best path is taken to be the one that minimizes the worst-case distance to travel before reaching the edge of the forest. Other variations of the problem have been studied. Although non-contrived real-world applications are not apparent, the problem falls into a class of geometric optimization problems, including search strategies that are ...
If the distance measure is a metric (and thus symmetric), the problem becomes APX-complete, [53] and the algorithm of Christofides and Serdyukov approximates it within 1.5. [ 54 ] [ 55 ] [ 10 ] If the distances are restricted to 1 and 2 (but still are a metric), then the approximation ratio becomes 8/7. [ 56 ]
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In 2023, Haeupler, Rozhoň, Tětek, Hladík, and Tarjan (one of the inventors of the 1984 heap), proved that, for this sorting problem on a positively-weighted directed graph, a version of Dijkstra's algorithm with a special heap data structure has a runtime and number of comparisons that is within a constant factor of optimal among comparison ...