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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.
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]
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.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices ; the length of a path may either be measured by its number of edges, or (in weighted graphs ) by the sum of the weights of its ...
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.
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 ...
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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 ]