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--- title intersecting_segments --- # Search for a pair of intersecting segments Given $n$ line segments on the plane. It is required to check whether at least two of them intersect with each other. If the answer is yes, then print this pair of intersecting segments; it is enough to choose any of them among several ...	--- title intersecting_segments --- # Search for a pair of intersecting segments Given $n$ line segments on the plane. It is required to check whether at least two of them intersect with each other. If the answer is yes, then print this pair of intersecting segments; it is enough to choose any of them among several ...	Search for a pair of intersecting segments
--- title segments_intersection_checking --- # Check if two segments intersect You are given two segments $(a, b)$ and $(c, d)$. You have to check if they intersect. Of course, you may find their intersection and check if it isn't empty, but this can't be done in integers for segments with integer coordinates. The a...	--- title segments_intersection_checking --- # Check if two segments intersect You are given two segments $(a, b)$ and $(c, d)$. You have to check if they intersect. Of course, you may find their intersection and check if it isn't empty, but this can't be done in integers for segments with integer coordinates. The a...	Check if two segments intersect
--- title - Original --- # Convex hull trick and Li Chao tree Consider the following problem. There are $n$ cities. You want to travel from city $1$ to city $n$ by car. To do this you have to buy some gasoline. It is known that a liter of gasoline costs $cost_k$ in the $k^{th}$ city. Initially your fuel tank is emp...	--- title - Original --- # Convex hull trick and Li Chao tree Consider the following problem. There are $n$ cities. You want to travel from city $1$ to city $n$ by car. To do this you have to buy some gasoline. It is known that a liter of gasoline costs $cost_k$ in the $k^{th}$ city. Initially your fuel tank is emp...	Convex hull trick and Li Chao tree
--- title - Original --- # Basic Geometry In this article we will consider basic operations on points in Euclidean space which maintains the foundation of the whole analytical geometry. We will consider for each point $\mathbf r$ the vector $\vec{\mathbf r}$ directed from $\mathbf 0$ to $\mathbf r$. Later we will n...	--- title - Original --- # Basic Geometry In this article we will consider basic operations on points in Euclidean space which maintains the foundation of the whole analytical geometry. We will consider for each point $\mathbf r$ the vector $\vec{\mathbf r}$ directed from $\mathbf 0$ to $\mathbf r$. Later we will n...	Basic Geometry
--- title circles_intersection --- # Circle-Circle Intersection You are given two circles on a 2D plane, each one described as coordinates of its center and its radius. Find the points of their intersection (possible cases: one or two points, no intersection or circles coincide). ## Solution Let's reduce this prob...	--- title circles_intersection --- # Circle-Circle Intersection You are given two circles on a 2D plane, each one described as coordinates of its center and its radius. Find the points of their intersection (possible cases: one or two points, no intersection or circles coincide). ## Solution Let's reduce this prob...	Circle-Circle Intersection
--- title segments_intersection --- # Finding intersection of two segments You are given two segments AB and CD, described as pairs of their endpoints. Each segment can be a single point if its endpoints are the same. You have to find the intersection of these segments, which can be empty (if the segments don't int...	--- title segments_intersection --- # Finding intersection of two segments You are given two segments AB and CD, described as pairs of their endpoints. Each segment can be a single point if its endpoints are the same. You have to find the intersection of these segments, which can be empty (if the segments don't int...	Finding intersection of two segments
--- title convex_hull_graham --- # Convex Hull construction In this article we will discuss the problem of constructing a convex hull from a set of points. Consider $N$ points given on a plane, and the objective is to generate a convex hull, i.e. the smallest convex polygon that contains all the given points. We w...	--- title convex_hull_graham --- # Convex Hull construction In this article we will discuss the problem of constructing a convex hull from a set of points. Consider $N$ points given on a plane, and the objective is to generate a convex hull, i.e. the smallest convex polygon that contains all the given points. We w...	Convex Hull construction
--- title voronoi_diagram_2d_n4 --- # Delaunay triangulation and Voronoi diagram Consider a set $\{p_i\}$ of points on the plane. A Voronoi diagram $V(\{p_i\})$ of $\{p_i\}$ is a partition of the plane into $n$ regions $V_i$, where $V_i = \{p\in\mathbb{R}^2;\ \rho(p, p_i) = \min\ \rho(p, p_k)\}$. The cells of th...	--- title voronoi_diagram_2d_n4 --- # Delaunay triangulation and Voronoi diagram Consider a set $\{p_i\}$ of points on the plane. A Voronoi diagram $V(\{p_i\})$ of $\{p_i\}$ is a partition of the plane into $n$ regions $V_i$, where $V_i = \{p\in\mathbb{R}^2;\ \rho(p, p_i) = \min\ \rho(p, p_k)\}$. The cells of th...	Delaunay triangulation and Voronoi diagram
--- title nearest_points --- # Finding the nearest pair of points ## Problem statement Given $n$ points on the plane. Each point $p_i$ is defined by its coordinates $(x_i,y_i)$. It is required to find among them two such points, such that the distance between them is minimal: $$ \min_{\scriptstyle i, j=0 \ldots n-...	--- title nearest_points --- # Finding the nearest pair of points ## Problem statement Given $n$ points on the plane. Each point $p_i$ is defined by its coordinates $(x_i,y_i)$. It is required to find among them two such points, such that the distance between them is minimal: $$ \min_{\scriptstyle i, j=0 \ldots n-...	Finding the nearest pair of points
--- title: Point location in O(log n) title - Original --- # Point location in $O(log n)$ Consider the following problem: you are given a [planar subdivision](https://en.wikipedia.org/wiki/Planar_straight-line_graph) without no vertices of degree one and zero, and a lot of queries. Each query is a point, for which w...	--- title: Point location in O(log n) title - Original --- # Point location in $O(log n)$ Consider the following problem: you are given a [planar subdivision](https://en.wikipedia.org/wiki/Planar_straight-line_graph) without no vertices of degree one and zero, and a lot of queries. Each query is a point, for which w...	Point location in $O(log n)$
--- title: Finding faces of a planar graph title facets --- # Finding faces of a planar graph Consider a graph $G$ with $n$ vertices and $m$ edges, which can be drawn on a plane in such a way that two edges intersect only at a common vertex (if it exists). Such graphs are called planar. Now suppose that we are gi...	--- title: Finding faces of a planar graph title facets --- # Finding faces of a planar graph Consider a graph $G$ with $n$ vertices and $m$ edges, which can be drawn on a plane in such a way that two edges intersect only at a common vertex (if it exists). Such graphs are called planar. Now suppose that we are gi...	Finding faces of a planar graph
--- title pick_grid_theorem --- # Pick's Theorem A polygon without self-intersections is called lattice if all its vertices have integer coordinates in some 2D grid. Pick's theorem provides a way to compute the area of this polygon through the number of vertices that are lying on the boundary and the number of verti...	--- title pick_grid_theorem --- # Pick's Theorem A polygon without self-intersections is called lattice if all its vertices have integer coordinates in some 2D grid. Pick's theorem provides a way to compute the area of this polygon through the number of vertices that are lying on the boundary and the number of verti...	Pick's Theorem
--- title oriented_area --- # Oriented area of a triangle Given three points $p_1$, $p_2$ and $p_3$, calculate an oriented (signed) area of a triangle formed by them. The sign of the area is determined in the following way: imagine you are standing in the plane at point $p_1$ and are facing $p_2$. You go to $p_2$ an...	--- title oriented_area --- # Oriented area of a triangle Given three points $p_1$, $p_2$ and $p_3$, calculate an oriented (signed) area of a triangle formed by them. The sign of the area is determined in the following way: imagine you are standing in the plane at point $p_1$ and are facing $p_2$. You go to $p_2$ an...	Oriented area of a triangle
--- title: Check if point belongs to the convex polygon in O(log N) title pt_in_polygon --- # Check if point belongs to the convex polygon in $O(\log N)$ Consider the following problem: you are given a convex polygon with integer vertices and a lot of queries. Each query is a point, for which we should determine whet...	--- title: Check if point belongs to the convex polygon in O(log N) title pt_in_polygon --- # Check if point belongs to the convex polygon in $O(\log N)$ Consider the following problem: you are given a convex polygon with integer vertices and a lot of queries. Each query is a point, for which we should determine whet...	Check if point belongs to the convex polygon in $O(\log N)$
--- title lines_intersection --- # Intersection Point of Lines You are given two lines, described via the equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$. We have to find the intersection point of the lines, or determine that the lines are parallel. ## Solution If two lines are not parallel, the...	--- title lines_intersection --- # Intersection Point of Lines You are given two lines, described via the equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$. We have to find the intersection point of the lines, or determine that the lines are parallel. ## Solution If two lines are not parallel, the...	Intersection Point of Lines
--- title - Original --- # Half-plane intersection In this article we will discuss the problem of computing the intersection of a set of half-planes. Such an intersection can be conveniently represented as a convex region/polygon, where every point inside of it is also inside all of the half-planes, and it is this ...	--- title - Original --- # Half-plane intersection In this article we will discuss the problem of computing the intersection of a set of half-planes. Such an intersection can be conveniently represented as a convex region/polygon, where every point inside of it is also inside all of the half-planes, and it is this ...	Half-plane intersection
--- title circle_tangents --- # Finding common tangents to two circles Given two circles. It is required to find all their common tangents, i.e. all such lines that touch both circles simultaneously. The described algorithm will also work in the case when one (or both) circles degenerate into points. Thus, this alg...	--- title circle_tangents --- # Finding common tangents to two circles Given two circles. It is required to find all their common tangents, i.e. all such lines that touch both circles simultaneously. The described algorithm will also work in the case when one (or both) circles degenerate into points. Thus, this alg...	Finding common tangents to two circles
--- title: Finding area of simple polygon in O(N) title polygon_area --- # Finding area of simple polygon in $O(N)$ Let a simple polygon (i.e. without self intersection, not necessarily convex) be given. It is required to calculate its area given its vertices. ## Method 1 This is easy to do if we go through all edg...	--- title: Finding area of simple polygon in O(N) title polygon_area --- # Finding area of simple polygon in $O(N)$ Let a simple polygon (i.e. without self intersection, not necessarily convex) be given. It is required to calculate its area given its vertices. ## Method 1 This is easy to do if we go through all edg...	Finding area of simple polygon in $O(N)$
--- title circle_line_intersection --- # Circle-Line Intersection Given the coordinates of the center of a circle and its radius, and the equation of a line, you're required to find the points of intersection. ## Solution Instead of solving the system of two equations, we will approach the problem geometrically. T...	--- title circle_line_intersection --- # Circle-Line Intersection Given the coordinates of the center of a circle and its radius, and the equation of a line, you're required to find the points of intersection. ## Solution Instead of solving the system of two equations, we will approach the problem geometrically. T...	Circle-Line Intersection
--- title - Original --- # Lattice points inside non-lattice polygon For lattice polygons there is Pick's formula to enumerate the lattice points inside the polygon. What about polygons with arbitrary vertices? Let's process each of the polygon's edges individually, and after that we may sum up the amounts of latt...	--- title - Original --- # Lattice points inside non-lattice polygon For lattice polygons there is Pick's formula to enumerate the lattice points inside the polygon. What about polygons with arbitrary vertices? Let's process each of the polygon's edges individually, and after that we may sum up the amounts of latt...	Lattice points inside non-lattice polygon
--- title - Original --- # Minkowski sum of convex polygons ## Definition Consider two sets $A$ and $B$ of points on a plane. Minkowski sum $A + B$ is defined as $\{a + b\| a \in A, b \in B\}$. Here we will consider the case when $A$ and $B$ consist of convex polygons $P$ and $Q$ with their interiors. Throughout thi...	--- title - Original --- # Minkowski sum of convex polygons ## Definition Consider two sets $A$ and $B$ of points on a plane. Minkowski sum $A + B$ is defined as $\{a + b\| a \in A, b \in B\}$. Here we will consider the case when $A$ and $B$ consist of convex polygons $P$ and $Q$ with their interiors. Throughout thi...	Minkowski sum of convex polygons
--- title length_of_segments_union --- # Length of the union of segments Given $n$ segments on a line, each described by a pair of coordinates $(a_{i1}, a_{i2})$. We have to find the length of their union. The following algorithm was proposed by Klee in 1977. It works in $O(n\log n)$ and has been proven to be the a...	--- title length_of_segments_union --- # Length of the union of segments Given $n$ segments on a line, each described by a pair of coordinates $(a_{i1}, a_{i2})$. We have to find the length of their union. The following algorithm was proposed by Klee in 1977. It works in $O(n\log n)$ and has been proven to be the a...	Length of the union of segments
--- title segment_to_line --- # Finding the equation of a line for a segment The task is: given the coordinates of the ends of a segment, construct a line passing through it. We assume that the segment is non-degenerate, i.e. has a length greater than zero (otherwise, of course, infinitely many different lines pass...	--- title segment_to_line --- # Finding the equation of a line for a segment The task is: given the coordinates of the ends of a segment, construct a line passing through it. We assume that the segment is non-degenerate, i.e. has a length greater than zero (otherwise, of course, infinitely many different lines pass...	Finding the equation of a line for a segment
--- title triangles_union --- # Vertical decomposition ## Overview Vertical decomposition is a powerful technique used in various geometry problems. The general idea is to cut the plane into several vertical stripes with some "good" properties and solve the problem for these stripes independently. We will illustrate...	--- title triangles_union --- # Vertical decomposition ## Overview Vertical decomposition is a powerful technique used in various geometry problems. The general idea is to cut the plane into several vertical stripes with some "good" properties and solve the problem for these stripes independently. We will illustrate...	Vertical decomposition
--- title - Original --- # Knuth's Optimization Knuth's optimization, also known as the Knuth-Yao Speedup, is a special case of dynamic programming on ranges, that can optimize the time complexity of solutions by a linear factor, from $O(n^3)$ for standard range DP to $O(n^2)$. ## Conditions The Speedup is applie...	--- title - Original --- # Knuth's Optimization Knuth's optimization, also known as the Knuth-Yao Speedup, is a special case of dynamic programming on ranges, that can optimize the time complexity of solutions by a linear factor, from $O(n^3)$ for standard range DP to $O(n^2)$. ## Conditions The Speedup is applie...	Knuth's Optimization
--- title maximum_zero_submatrix --- # Finding the largest zero submatrix You are given a matrix with `n` rows and `m` columns. Find the largest submatrix consisting of only zeros (a submatrix is a rectangular area of the matrix). ## Algorithm Elements of the matrix will be `a[i][j]`, where `i = 0...n - 1`, `j = 0...	--- title maximum_zero_submatrix --- # Finding the largest zero submatrix You are given a matrix with `n` rows and `m` columns. Find the largest submatrix consisting of only zeros (a submatrix is a rectangular area of the matrix). ## Algorithm Elements of the matrix will be `a[i][j]`, where `i = 0...n - 1`, `j = 0...	Finding the largest zero submatrix
--- title profile_dynamics --- # Dynamic Programming on Broken Profile. Problem "Parquet" Common problems solved using DP on broken profile include: - finding number of ways to fully fill an area (e.g. chessboard/grid) with some figures (e.g. dominoes) - finding a way to fill an area with minimum number of figures ...	--- title profile_dynamics --- # Dynamic Programming on Broken Profile. Problem "Parquet" Common problems solved using DP on broken profile include: - finding number of ways to fully fill an area (e.g. chessboard/grid) with some figures (e.g. dominoes) - finding a way to fill an area with minimum number of figures ...	Dynamic Programming on Broken Profile. Problem "Parquet"
--- title - Original --- # Divide and Conquer DP Divide and Conquer is a dynamic programming optimization. ### Preconditions Some dynamic programming problems have a recurrence of this form: $$ dp(i, j) = \min_{0 \leq k \leq j} \\{ dp(i - 1, k - 1) + C(k, j) \\} $$ where $C(k, j)$ is a cost function and $dp(i, ...	--- title - Original --- # Divide and Conquer DP Divide and Conquer is a dynamic programming optimization. ### Preconditions Some dynamic programming problems have a recurrence of this form: $$ dp(i, j) = \min_{0 \leq k \leq j} \\{ dp(i - 1, k - 1) + C(k, j) \\} $$ where $C(k, j)$ is a cost function and $dp(i, ...	Divide and Conquer DP
--- title maximum_average_segment --- # Search the subarray with the maximum/minimum sum Here, we consider the problem of finding a subarray with maximum sum, as well as some of its variations (including the algorithm for solving this problem online). ## Problem statement Given an array of numbers $a[1 \ldots n]$....	--- title maximum_average_segment --- # Search the subarray with the maximum/minimum sum Here, we consider the problem of finding a subarray with maximum sum, as well as some of its variations (including the algorithm for solving this problem online). ## Problem statement Given an array of numbers $a[1 \ldots n]$....	Search the subarray with the maximum/minimum sum
--- title joseph_problem --- # Josephus Problem ## Statement We are given the natural numbers $n$ and $k$. All natural numbers from $1$ to $n$ are written in a circle. First, count the $k$-th number starting from the first one and delete it. Then $k$ numbers are counted starting from the next one and the $k$-th on...	--- title joseph_problem --- # Josephus Problem ## Statement We are given the natural numbers $n$ and $k$. All natural numbers from $1$ to $n$ are written in a circle. First, count the $k$-th number starting from the first one and delete it. Then $k$ numbers are counted starting from the next one and the $k$-th on...	Josephus Problem
--- title stern_brocot_farey --- # The Stern-Brocot tree and Farey sequences ## Stern-Brocot tree The Stern-Brocot tree is an elegant construction to represent the set of all positive fractions. It was independently discovered by German mathematician Moritz Stern in 1858 and by French watchmaker Achille Brocot in 1...	--- title stern_brocot_farey --- # The Stern-Brocot tree and Farey sequences ## Stern-Brocot tree The Stern-Brocot tree is an elegant construction to represent the set of all positive fractions. It was independently discovered by German mathematician Moritz Stern in 1858 and by French watchmaker Achille Brocot in 1...	The Stern-Brocot tree and Farey sequences
--- title 15_puzzle --- # 15 Puzzle Game: Existence Of The Solution This game is played on a $4 \times 4$ board. On this board there are $15$ playing tiles numbered from 1 to 15. One cell is left empty (denoted by 0). You need to get the board to the position presented below by repeatedly moving one of the tiles to ...	--- title 15_puzzle --- # 15 Puzzle Game: Existence Of The Solution This game is played on a $4 \times 4$ board. On this board there are $15$ playing tiles numbered from 1 to 15. One cell is left empty (denoted by 0). You need to get the board to the position presented below by repeatedly moving one of the tiles to ...	15 Puzzle Game: Existence Of The Solution
--- title dijkstra --- # Dijkstra Algorithm You are given a directed or undirected weighted graph with $n$ vertices and $m$ edges. The weights of all edges are non-negative. You are also given a starting vertex $s$. This article discusses finding the lengths of the shortest paths from a starting vertex $s$ to all ot...	--- title dijkstra --- # Dijkstra Algorithm You are given a directed or undirected weighted graph with $n$ vertices and $m$ edges. The weights of all edges are non-negative. You are also given a starting vertex $s$. This article discusses finding the lengths of the shortest paths from a starting vertex $s$ to all ot...	Dijkstra Algorithm
--- title tree_painting --- # Paint the edges of the tree This is a fairly common task. Given a tree $G$ with $N$ vertices. There are two types of queries: the first one is to paint an edge, the second one is to query the number of colored edges between two vertices. Here we will describe a fairly simple solution (...	--- title tree_painting --- # Paint the edges of the tree This is a fairly common task. Given a tree $G$ with $N$ vertices. There are two types of queries: the first one is to paint an edge, the second one is to query the number of colored edges between two vertices. Here we will describe a fairly simple solution (...	Paint the edges of the tree
--- title flow_with_limits --- # Flows with demands In a normal flow network the flow of an edge is only limited by the capacity $c(e)$ from above and by 0 from below. In this article we will discuss flow networks, where we additionally require the flow of each edge to have a certain amount, i.e. we bound the flow f...	--- title flow_with_limits --- # Flows with demands In a normal flow network the flow of an edge is only limited by the capacity $c(e)$ from above and by 0 from below. In this article we will discuss flow networks, where we additionally require the flow of each edge to have a certain amount, i.e. we bound the flow f...	Flows with demands
--- title fixed_length_paths --- # Number of paths of fixed length / Shortest paths of fixed length The following article describes solutions to these two problems built on the same idea: reduce the problem to the construction of matrix and compute the solution with the usual matrix multiplication or with a modified...	--- title fixed_length_paths --- # Number of paths of fixed length / Shortest paths of fixed length The following article describes solutions to these two problems built on the same idea: reduce the problem to the construction of matrix and compute the solution with the usual matrix multiplication or with a modified...	Number of paths of fixed length / Shortest paths of fixed length
--- title kuhn_matching --- # Kuhn's Algorithm for Maximum Bipartite Matching ## Problem You are given a bipartite graph $G$ containing $n$ vertices and $m$ edges. Find the maximum matching, i.e., select as many edges as possible so that no selected edge shares a vertex with any other selected edge. ## Algorithm D...	--- title kuhn_matching --- # Kuhn's Algorithm for Maximum Bipartite Matching ## Problem You are given a bipartite graph $G$ containing $n$ vertices and $m$ edges. Find the maximum matching, i.e., select as many edges as possible so that no selected edge shares a vertex with any other selected edge. ## Algorithm D...	Kuhn's Algorithm for Maximum Bipartite Matching
--- title strong_connected_components --- # Finding strongly connected components / Building condensation graph ## Definitions You are given a directed graph $G$ with vertices $V$ and edges $E$. It is possible that there are loops and multiple edges. Let's denote $n$ as number of vertices and $m$ as number of edges ...	--- title strong_connected_components --- # Finding strongly connected components / Building condensation graph ## Definitions You are given a directed graph $G$ with vertices $V$ and edges $E$. It is possible that there are loops and multiple edges. Let's denote $n$ as number of vertices and $m$ as number of edges ...	Finding strongly connected components / Building condensation graph
--- title - Original --- # Maximum flow - MPM algorithm MPM (Malhotra, Pramodh-Kumar and Maheshwari) algorithm solves the maximum flow problem in $O(V^3)$. This algorithm is similar to [Dinic's algorithm](dinic.md). ## Algorithm Like Dinic's algorithm, MPM runs in phases, during each phase we find the blocking fl...	--- title - Original --- # Maximum flow - MPM algorithm MPM (Malhotra, Pramodh-Kumar and Maheshwari) algorithm solves the maximum flow problem in $O(V^3)$. This algorithm is similar to [Dinic's algorithm](dinic.md). ## Algorithm Like Dinic's algorithm, MPM runs in phases, during each phase we find the blocking fl...	Maximum flow - MPM algorithm
--- title topological_sort --- # Topological Sorting You are given a directed graph with $n$ vertices and $m$ edges. You have to find an order of the vertices, so that every edge leads from the vertex with a smaller index to a vertex with a larger one. In other words, you want to find a permutation of the verti...	--- title topological_sort --- # Topological Sorting You are given a directed graph with $n$ vertices and $m$ edges. You have to find an order of the vertices, so that every edge leads from the vertex with a smaller index to a vertex with a larger one. In other words, you want to find a permutation of the verti...	Topological Sorting