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We also introduce a function P(ω), the frequency density, which is related to the inverse Zipf function R(ω), and find a simple relationship between ζ and ψ, where ω( R)∼ R − ζ and P(ω)∼ ω − ψ. We study the relation between the exponent α characterizing long-range correlations and the exponent ζ characterizing power-law behavior in the Zipf plot. We analyze sequences with short-range Markovian correlations, as well as those with long-range correlations generated by three different methods: inverse Fourier transformation, Lévy walks, and the expansion-modification system. This time \(d\) will be the element at which a subsequence of length \(i\) terminates.We investigate correlated binary sequences using an n-tuple Zipf analysis, where we define ‘‘words’’ as strings of length n, and calculate the normalized frequency of occurrence ω( R) of ‘‘words’’ as a function of the word rank R. We will use the dynamic programming array \(d\). In order to obtain a faster solution for the problem, we construct a different dynamic programming solution that runs in \(O(n^2)\), and then later improve it to \(O(n \log n)\). Solution in \(O(n \log n)\) with dynamic programming and binary search This method leads to a slightly longer code, but in return we save some memory. We can simply recalculate the current value of \(d\) and also see how the maximum was reached. It is also possible to restore the subsequence without the auxiliary array \(p\). Int lis ( vector const & a ) Alternative way of restoring the subsequence If we fixate the index \(j\), then the longest increasing subsequence ending in the two elements \(a\) and \(a\) has the length \(d + 1\).Īll of these values \(d\) are already known, so we can directly compute \(d\) with: In this fashion we can compute \(d\) using the following formula: It can be any element \(a\) with \(j = 0 \dots i-1\) and \(a < a\). The Stern-Brocot Tree and Farey Sequences Optimal schedule of jobs given their deadlines and durationsġ5 Puzzle Game: Existence Of The Solution Search the subsegment with the maximum/minimum sum
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Smallest number of non-increasing subsequences covering a sequence Number of longest increasing subsequences Solution in O(n log n) with data structures Solution in O(n log n) with dynamic programming and binary search Solution in O(n^2) with dynamic programmingĪlternative way of restoring the subsequence RMQ task (Range Minimum Query - the smallest element in an interval)
![binary sequences binary sequences](http://www.itmeyer.at/umlet/imgdiag/sequencediagram.gif)
Kuhn's Algorithm - Maximum Bipartite Matching Maximum flow - Push-relabel algorithm improved Maximum flow - Ford-Fulkerson and Edmonds-Karp Lowest Common Ancestor - Tarjan's off-line algorithm
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Lowest Common Ancestor - Farach-Colton and Bender algorithm Second best Minimum Spanning Tree - Using Kruskal and Lowest Common AncestorĬhecking a graph for acyclicity and finding a cycle in O(M) Minimum Spanning Tree - Kruskal with Disjoint Set Union Number of paths of fixed length / Shortest paths of fixed length Strongly Connected Components and Condensation Graphĭijkstra - finding shortest paths from given vertexīellman-Ford - finding shortest paths with negative weightsįloyd-Warshall - finding all shortest paths Half-plane intersection - S&I Algorithm in O(N log N)Ĭonnected components, bridges, articulations points Search for a pair of intersecting segmentsĭelaunay triangulation and Voronoi diagram Pick's Theorem - area of lattice polygons Manacher's Algorithm - Finding all sub-palindromes in O(N)īurnside's lemma / Pólya enumeration theoremįinding the equation of a line for a segmentĬheck if points belong to the convex polygon in O(log N) Euclidean algorithm for computing the greatest common divisorĭeleting from a data structure in O(T(n) log n)ĭynamic Programming on Broken Profile.