![]() ![]() But if at any step in the \(i\)-th column there are no rows with an non-empty entry among those that we didn't selected already, then we skip this step. We will perform the same operations as when solving the system or finding its determinant. You can search for the rank using Gaussian elimination. Generally, the rank of a matrix does not exceed \(\min (N, M)\). Note that if the matrix is square and its determinant is non-zero, then the rank is \(N\) ( \(=M\)) otherwise it will be less. Let the matrix be rectangular and have size \(N \times M\). The rank of a matrix can also be defined as the largest order of any non-zero minor in the matrix. The rank is not only defined for square matrices. The rank of a matrix is the largest number of linearly independent rows/columns of the matrix. The Stern-Brocot Tree and Farey Sequences ![]() Optimal schedule of jobs given their deadlines and durationsġ5 Puzzle Game: Existence Of The Solution MEX task (Minimal Excluded element in an array) Search the subsegment with the maximum/minimum sum RMQ task (Range Minimum Query - the smallest element in an interval) 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 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
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