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A time–distance diagram is a chart with two axes: one for time, the other for location. The units on either axis depend on the type of project: time can be expressed in minutes (for overnight construction of railroad modification projects such as the installation of switches) or years (for large construction projects); the location can be (kilo)meters, or other distinct units (such as ...
In this decision problem, the input is a graph G and a number k; the desired output is yes if G contains a path of k or more edges, and no otherwise. [1] If the longest path problem could be solved in polynomial time, it could be used to solve this decision problem, by finding a longest path and then comparing its length to the number k ...
Distance-matrix methods may produce either rooted or unrooted trees, depending on the algorithm used to calculate them. [4] Given n species, the input is an n × n distance matrix M where M ij is the mutation distance between species i and j. The aim is to output a tree of degree 3 which is consistent with the distance matrix.
For example, the minimum spanning tree of the graph associated with an instance of the Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O(n log n) time for n points (considerably less than the number of edges). This enables the simple 2-approximation algorithm for TSP with triangle inequality above to ...
An alternative lower bound in the form of a ten-vertex four-chromatic unit distance graph, the Golomb graph, was discovered at around the same time by Solomon W. Golomb. [ 4 ] The lower bound was raised to five in 2018, when computer scientist and biogerontologist Aubrey de Grey found a 1581-vertex, non-4-colourable unit-distance graph.
Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
They then suggest a new DO that, for every k, requires space (+ /), such that any subsequent distance query can be approximately answered in time (). The approximate distance returned is of stretch at most 2 k − 1 {\displaystyle 2k-1} , that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and ...
Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space units of measurement are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.