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See Also [[Further Maths - Graphs]]S [[Further Maths - Prim's Algorithm]]S Flashcards What is a spanning tree?? A tree that includes all the vertices of a graph. What is a minimum spanning tree?? A tree that includes all the vertices of a graph at the minimum possible cost. What is the first step of Kruskal’s algorithm?? List the edges in the order of weight, smallest first. What step comes after listing out the edges of a graph in weight order for Kruskal’s algorithm?

See Also Flashcards What is a Eulerian trail?? A trail that goes along every edge exactly once. Why isn’t a trail that visits every edge multiple times a Eulerian trail?? Because it visits an edge more than once. What is a Eulerian trail that joins up to the beginning called?? A Eulerian cycle. How many edges does entering a vertex and then exiting a vertex “use up” when constructing a Eulerian trail?? 2 Why must all vertices be even for a Eulerian trail to be possible?

See Also Flashcards What is a Hamiltonian cycle?? A cycle that visits all the vertices of a graph. How many times should a Hamiltonian cycle visit a vertex?? Just one, apart from at the end. What must be true about the start and end vertex of a Hamiltonian cycle?? It must be the same. What is the formula for the number of Hamiltonian cycles for a complete graph $K_n$?? $$ \frac{(n - 1)!}{2} $$ Backlinks [[Further Maths - Syllabus]]S [[Further Maths - Graphs]]S Metadata date: 2021-11-04 12:14 tags: - '@?

See Also [[Further Maths - Eulerian Trails]]S [[Further Maths - Hamiltonian Cycles]]S [[Further Maths - Kruskals Algorithm]]?? Flashcards What is a graph with weighted edges called?? A network. What is the degree of a vertex?? The number of edges connected to it. What is the order of a vertex?? The number of edges connected to it. What is the valency of a vertex?? The number of edges connected to it.

Flashcards What is the bin-packing problem?? Minimising the amount of bins used to store things of a fixed sizes. What is the full-bin combinations algorithm for bin-packing?? Fill as many bins as possible using full-bin combinations and then pack the remaining items into the first available bin. What is it called when you solve a problem by listing all possibilities?? Complete enumeration. What is the first-fit algorithm for bin-packing?? Work through the list of items and place each item in the first bin with sufficient space.

Flashcards After 3 passes with shuttlesort, what must be true?? The first three numbers are in the correct position. What’s the most amount of passes for $n$ numbers that will be needed in shuttlesort?? $$ n - 1 $$ What is the worst case time complexity of shuttlesort?? $$ O(n^2) $$ When can you prematurely stop doing passes for shuttlesort?? You can’t, you have to do $n-1$ passes. When can you move to the next pass for shuttlesort?

Flashcards After 3 passes with bubblesort, what must be true?? The last three numbers are in the correct position. What’s the most amount of passes for $n$ numbers that will be needed in bubblesort?? $$ n - 1 $$ What is the worst case time complexity of bubblesort?? $$ n^2 $$ Backlinks [[Further Maths - Syllabus]]S Metadata date: 2021-10-10 13:31 tags: - '@?public' - '@?school' - '@?decision' - '@?further-maths' title: Further Maths - Bubblesort

Flashcards 2021-10-09 What is the $t$-substitution ($t = …$)?? $$ t = \tan^{-1}\left(\frac{\theta}{2}\right) $$ What is $\sin\theta$ in terms of $t$?? $$ \sin\theta = \frac{2t}{1 + t^2} $$ What is $\cos\theta$ in terms of $t$?? $$ \cos\theta = \frac{1 + t^2}{1 - t^2} $$ What is $\tan\theta$ in terms of $t$?? $$ \tan\theta = \frac{2t}{1 - t^2} $$ What triangle can you imagine for deriving the $t$-formulae?? 2021-10-12 What $t$-substitution could you make other than $t = \tan\left(\frac{\theta}{2}\right)$ in order to rewrite $\sin 2\theta$?

Flashcards Parabolas How can you form a parabola from a cone?? Slice it parallel to its slope. Why must you slice a cone PARALLEL to the slope to form a parabola?? Otherwise you’d either get an ellipse or intersect the cone twice and get a hyperbola. What is the parametric equation that defines a parabola?? $$ x = at^2 $$ $$ y = 2at $$ When thinking about parabolas as a conic section, is it better to think of them symmetrical around the $x$-axis or $y$-axis?

Flashcards Backlinks [[Further Maths - Syllabus]]S Metadata date: 2021-10-05 16:31 tags: - '@?public' - '@?school' - '@?further-maths' - '@?inequalities' - '@?further-pure-1' - '@?safe-to-post-online' title: Further Maths - Inequalities

See Also Flashcards 2021-09-10 What is the general process for solving a coupled first-order differential equation?? Eliminating one of the variables to form a single second-order differential equation. $$\frac{\text{d}x}{\text{d}t} = x + y \\ \frac{\text{d}y}{\text{d}t} = x - y$$ What are the 3 first steps?? Rewriting it as $y = …$, differentiating and then substituting. $$\frac{\text{d}x}{\text{d}t} = x + y \\ \frac{\text{d}y}{\text{d}t} = x - y$$ You’ve discovered $$x = Ae^{\sqrt{2}t} + Be^{-\sqrt{2}t}$$ What do you now do to find $y$ in terms of $t$?

See Also [[Maths - Integration]]S [[Maths - Differentiation]]S Flashcards What type of answer do you normally get for differential equation questions?? A family of curves. What’s the general process for solving a differential equation question like $$\frac{dy}{dx} = \frac{y + 1}{x}$$?? Seperate the two variables and put them on either side of the equation. Integrate both sides with respect to $x$. Rearrange. $$\int \frac{1}{y + 1} \frac{dy}{dx} dx$$ What does this simplify down to?