Advanced Math Vertex: Fill & Download for Free

I am assuming you know at least one programming language and the concept of object / pointers. I will mention algorithms and data structures in increasing order of difficulty.First start with Linear data structures and algorithms.ArraysLinked ListStackQueuesThen move to basic algorithms :Sorting - Merge Sort, Insertion Sort, Quick Sort, Number of inversionsMatrix Multiplication (just know the algo if not implement it)Prime SievingModular Math including multiplication and divisionEuclidean Algorithm for GCD, Modular Inverse, Fast ExponentiationFibonacci number with matrix multiplicationProbability distribution and expected valueStats - Mean, Median, Variance, Bayes theoremThen one can learn some popular algorithmic techniques:Divide and Conquer - Binary Search, Maximum SubarrayGreedy Algorithms - Activity Selection, Huffman encodingDynamic Programming - Matrix Chain Multiplication, Knapsack,Linear Programming - Variable Maximisation, Linear time sortingString Algorithms - Manacher, LCS, Edit DistanceThen comes some typical non-linear data structures:Trees - Binary Tree, General Tree, Lowest Common AncestorBinary Search Tree - Inorder Traversal, Level order traversal, finding kth largest element, diameter, depth, number of nodes, etc.Heaps - Array Implementation, Heapify, Heap SortUnion FindHash Table - Linear Probing, Open addressing, Collision avoidanceThen you can learn about Graphs:Adjacency List, Adjacency Matrix, Weighted Edge GraphsBasic Traversal algos - Breadth First Search, Depth First Search, etcShortest Path Finding Algorithm - Dijkstra, Floyd Warshal, Bellman FordMinimum Spanning Tree - Kruskal's Algo, Prim's AlgoBy this time you are already pretty good with programming. You will do better than most of undergrad CS students. If you want to learn more and delve deep read more.Advance Tree and Graph :Balanced Trees - AVL, Red-BlackHeavy Light Decomposition, B+ Trees, Quad TreeAdvance Graph - Min Cut, Max FlowMaximum Matching - Hall's MarriageHamiltonian CycleEdge Graphs / Line GraphsStrongly Connected ComponentsDominant Sub-Graph, Vertex Cover, Travelling Salesman - Approx algosAdvance String Algorithms :Knuth Morris Pratt AlgorithmRabin Karp AlgorithmTries and Compressed TriesPrefix Trees, Suffix Trees, Suffix Automation - Ukkonen AlgorithmAdvance Math:Fast Fourier TransformationPrimality TestingComputational Geometry - Closest point pair, Voronoi diagram, Convex HullGeneral Advance topics :Iterating through all combination / permutationBit manipulation

I will present a simple [math]\mathcal{O}(n^2(n [/math][math][/math][math]+ m))[/math] algorithm for testing if a graph is 3-connected (triconnected). Then, I will present a slightly faster and more advanced [math]\mathcal{O}(n(n+m))[/math] algorithm that reduces to testing for biconnectivity. The fastest algorithms for this problem run in linear [math]\mathcal{O}(n [/math][math][/math][math]+ m)[/math] time, but they are a bit complicated to describe, in my opinion. If you are interested in the fancier algorithms, let me suggest you read about the original 1973 algorithm of Hopcroft and Tarjan for computing the triconnected components of a graph[1][1][1][1]. There have been more developments since then, including the efficient calculation of a neat data structure called the SPQR tree[2][2][2][2].A graph is triconnected if deleting any two vertices results in a graph that is still connected. The naive algorithm I am about to describe is essentially a restatement of the definition. Here we go:For each pair of vertices, delete them, and check that the resultant graph is still connected via depth-first search (DFS) or breadth-first search (BFS) (as in the calculation of connected components).By the definition of triconnected, this algorithm is trivially correct. There are [math]\mathcal{O}(n^2)[/math] pairs of vertices, and DFS / BFS takes [math]\mathcal{O}(n [/math][math][/math][math]+ m)[/math] time. Hence, the above algorithm terminates after [math]\mathcal{O}(n^2(n [/math][math][/math][math]+ m))[/math] steps.The above algorithm likely leaves much to be desired. Without getting into the fastest [math]\mathcal{O}(n+m)[/math] algorithms, let’s look at one which runs in [math]\mathcal{O}(n(n+m))[/math] time.First, a biconnected graph is mostly the same as a triconnected graph except that you replace “any two vertices” with “any vertex” in the above definition of triconnectivity. With this, consider the following alternative (but equivalent) definition of triconnectivity: A graph is triconnected if deleting any vertex results in a graph that is biconnected.Cool! This means we can reduce testing triconnectivity to testing biconnectivity. Arguably, the well known [math]\mathcal{O}(n+m)[/math] algorithm using DFS for testing biconnectivity is much easier to understand than the ones for testing triconnectivity. It is interesting to note that this algorithm, like the one for triconnectivity, is due to Hopcroft and Tarjan (1973)[3][3][3][3]. The algorithm (for biconnectivity) takes advantage of a couple key observations. In a DFS tree for the graph [math]G[/math], the following are true:If the root [math]r[/math] has more than one child, then [math]G[/math] is not biconnected since removing [math]r[/math] disconnects [math]G[/math].If a non-root vertex [math]v[/math] has no descendant with a back edge to an ancestor of [math]v[/math], then [math]G[/math] is not biconnected since removing [math]v[/math] disconnects [math]G[/math]. Namely, the subtree of [math]v[/math] becomes disconnected from the rest of [math]G[/math].Check out the Wikipedia article[4][4][4][4] for a good description of the algorithm with pseudo code.Now, we can use this algorithm to get a slightly better result via the following algorithm for triconnectivity:For each vertex, remove it, and check that the resultant graph is biconnected via DFS.The algorithm is correct by the definition of triconnectivity. There are [math]n[/math] vertices and we have claimed that the DFS-based biconnectivity algorithm runs in [math]\mathcal{O}(n+m)[/math] time. Thus, this algorithm has an overall time complexity of [math]\mathcal{O}(n(n+m))[/math].Footnotes[1] Dividing a Graph into Triconnected Components[1] Dividing a Graph into Triconnected Components[1] Dividing a Graph into Triconnected Components[1] Dividing a Graph into Triconnected Components[2] SPQR tree - Wikipedia[2] SPQR tree - Wikipedia[2] SPQR tree - Wikipedia[2] SPQR tree - Wikipedia[3] Algorithm 447: efficient algorithms for graph manipulation[3] Algorithm 447: efficient algorithms for graph manipulation[3] Algorithm 447: efficient algorithms for graph manipulation[3] Algorithm 447: efficient algorithms for graph manipulation[4] Biconnected component - Wikipedia[4] Biconnected component - Wikipedia[4] Biconnected component - Wikipedia[4] Biconnected component - Wikipedia

I like fiendish math problems — ones that sound simple and don’t, in their text, suggest that any advanced math is need. My favourite one and one that I keep reading over the years is:Is there a point inside a unit square whose distances to all four corners is rational?This can be changed to:Is there a point inside a square whose sides are integers and whose distances foreach vertex is an integer?These belong to a set of problems call Rational Distance problems.Rational Distance Problem