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Therefore, the Euclidean shortest path problem may be decomposed into two simpler subproblems: constructing the visibility graph, and applying a shortest path algorithm such as Dijkstra's algorithm to the graph.
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Therefore and Euclidean
Therefore, in general Euclidean spaces every similarity is an affine transformation, because the Euclidean group E ( n ) is a subgroup of the affine group.
Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate.
Therefore the symmetry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group.
" Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.
Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article.
Therefore a symmetry group of rotational symmetry is a subgroup of E < sup >+</ sup >( m ) ( see Euclidean group ).
Therefore, finding the Euclidean minimum spanning tree as a spanning tree of the complete graph or as a spanning tree of the Delaunay triangulation both take O ( dn < sup > 2 </ sup >) time.
Therefore and shortest
Therefore, the length of an accepting computation of the DTM is, in general, exponential in the length of the shortest accepting computation of the NTM.
Therefore the shortest possible path from the root of T to any leaf consists of B black nodes.
Therefore, the most common symbols use the shortest codes.
Therefore, after i cycles, is at most the length of the shortest path from source to u that uses at most i edges.
Therefore they have the shortest possible stirrup length.
Therefore, we want the model which generates the shortest ( two-part ) encoding of the data.
Therefore of those with clear and effective terms, the Prime Minister with the shortest single one was Charles Watson-Wentworth, 2nd Marquess of Rockingham, whose second term lasted 96 days from 27 March 1782 until his death on 1 July 1782.
Therefore and path
Therefore the longest possible path consists of 2B nodes, alternating black and red.
Therefore the four schools can be seen as a gradual path from a rather easy-to-grasp, " realistic " philosophical point of view, to more and more complex and subtle views on the ultimate nature of reality, that is on emptiness and dependent arising, culminating in the philosophy of the Mādhyamikas, which is widely believed to present the most sophisticated point of view.
Therefore, for the first part of its history, the torpedo was guided only in the sense that its course could be regulated so as to achieve an intended impact depth ( because of the sine wave running path of the Whitehead, this was a hit or miss proposition, even when everything worked correctly ) and, through gyroscopes, a straight course.
Therefore, in magnetic circuits with an air gap, the strength of the magnetic field B depends strongly on the length of the air gap, and the length of the flux path in the core doesn't matter much.
Assuming that the cost of traversing any network segment is 1, the least cost path from bridge 4 to the root bridge goes through network segment c. Therefore, the root port for bridge 4 is the one on network segment c.
Therefore, by trying this replacement for all edges incident to some chosen vertex of G, the Hamiltonian cycle problem can be solved by at most n Hamiltonian path computations, where n is the number of vertices in the graph.
Therefore, for a given p, what is the probability that an open path exists from the top to the bottom?
Therefore, an Eulerian path is now possible, but since it must begin on one island and end on the other, it is impractical for tourists.
Therefore, each religion can hold an authentic but ultimately inadequate concept of divine reality, producing a partial understanding of the universal truth, which requires syncretism to achieve a complete understanding as well as a path towards salvation or spiritual enlightenment.
Therefore, is at most the length of the path from s to u.
' Therefore, let justice be done to all mankind, realizing that if the strong oppresses the weak, confusion and discontent will ever mark the path of man but with love, faith and charity towards all the reign of peace and plenty will be heralded into the world and the generations of men shall be called Blessed.
Therefore, only the existence of at least one straight path is necessary to determine whether a set of points is an alignment.
Therefore there is full step-free access coverage in the station, although a considerably steep path to the station still exists if approaching from the north and a steep approach from the road to station level from the south also remains.
Therefore, the red arrow labeled with a " 3 " in Figure 6 shows the final conduction path of the current.
Therefore, the red arrow labeled with a " 3 " in Figure 6 shows the final conduction path of the current.
Therefore the path of the harmonograph figure is described by the parametric equations
Therefore does not have the path independence property discussed above, and is not conservative.
Therefore, one who is traveling " west " on I-90 / 94 is actually driving north on the Dan Ryan as it passes through Chicago ; the interstates continue on a westernly path outside of the city.
Therefore, the ball keeps " moving " in the path of least resistance, which constantly changes.
Therefore, Roark engineers another plan to demolish a 22-story condominium under construction by Jaye's husband to block the lava's path from entering the city.
Therefore the distance from A to B is no bigger than the length of the straight-line path from A to C plus the length of the straight-line path from C to B.
Therefore the BCS condensate flowing around a closed path in a multiply connected superconducting sample acquires a phase difference Δφ determined by the magnetic flux Φ < sub > B </ sub > through the area enclosed by the path ( via Stokes ' theorem and ), and given by:
Therefore and problem
Therefore to Helva, the problem that she couldn't open her mouth to sing, among other restrictions, did not bother her.
Therefore, vapor lock is almost never a problem in a vehicle with fuel injection.
Therefore God's present desire is that the problem of sin be solved and that all these things be restored, thus bringing about the earthly and heavenly kingdom of God.
Therefore, designing a smooth V6 engine is a much more complicated problem than the straight-6, flat-6, and V8 layouts.
Therefore functional analysis of the mind alone will always leave us with the mind-body problem which cannot be solved.
Therefore Molniya orbit satellite have been launched, mainly in Russia, to alleviate this problem.
Therefore, its use in the production should not be a problem for vegans.
Therefore, the assertion that an object R → R * is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R * is a most efficient solution to our problem.
Therefore, preventing hydrate formation appears to be the key to the problem.
Therefore, we have to reword the problem so that the sequential version is in P. That is why this problem required T to be written in unary.
Therefore, for the purposes of locating and fixing bugs, it is vitally important to be able to retrieve and run different versions of the software to determine in which version ( s ) the problem occurs.
Therefore, in any geometric problem we have an initial set of symbols ( points and lines ), an algorithm, and some results.
Therefore, wear is a major problem in that area.
Therefore, there is an important overflying problem.
Therefore, brute-force search is typically used when the problem size is limited, or when there are problem-specific heuristics that can be used to reduce the set of candidate solutions to a manageable size.
Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.
Therefore, if King Juan Carlos had decided to follow the Belgian example in 2005 or 2010, a declaration of incapacity would only transfer the problem to Prince Felipe, his heir apparent.
Therefore, most fictional and non-fictional media productions about crime begin by briefly ' sensationalizing ' a concerning event before presenting a non-political resolution, provided by agents of the criminal justice system, to the event and the isolated moral problem it represents.
Therefore ( besides temporary strong discomfort to the animals ) any outbreak in the rich world should not be much more as a localized, cyclical economic problem.
For instance, the problem statement and the solution statement are printed in bold font, the latter is always preceded by the " Therefore :" keyword.
Therefore God's present desire is that the problem of sin be solved and that all these things be restored, thus bringing about the earthly and heavenly kingdom of God.
Therefore, the missing mass problem is not solved by MACHOs.
Therefore, some authors consider that the name " divide and conquer " should be used only when each problem may generate two or more subproblems.
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