Far-Play

Far-Play (stylized fAR-Play, from augmented reality) was a software platform developed at the University of Alberta, for creating location-based, scavenger-hunt style games which use the GPS and web-connectivity features of a player's smartphone. According to the development team, "our long-term objective is to develop a general framework that supports the implementation of AARGs that are fun to play and also educational". It utilizes Layar, an augmented reality smartphone application, QR codes located at particular real-world sites, or a phone's web browser, to facilitate games which require players to be in close physical proximity to predefined "nodes". A node, referred to by the developers as a Virtual Point of Interest (vPOI), is a point in space defined by a set of map coordinates; fAR-Play uses the GPS function of a player's smartphone — or, for indoor games, which are not easily tracked by GPS satellites, specially-created QR codes— to confirm that they are adequately near a given node. Once a player is within a node's proximity, Layar's various augmented reality features can be utilized to display a range of extra content overlaid upon the physical play-space or launch another application for extra functionality. == Development and features == fAR-Play began development in 2008, emerging from a collaborative project undertaken by a group of University of Alberta students from the Computer Science and Humanities Computing departments. fAR-Play is still under development, but a beta version is available for testing by request. fAR-Play's development is managed by a team of interdisciplinary professors and students at the University of Alberta. Currently, the developing team's roster includes Supervising Professors Geoffrey Rockwell and Eleni Stroulia, Developers Lucio Gutierrez and Matthew Delaney, and Website Developers Calen Henry and Garry Wong. === Technology === fAR-Play relies on a number of open- and closed-source web technologies as tools to create, and enhance the users' experience. Layar is the recommended client-side frontend for delivering game content to the player; it is available on Android and iOS, which covers over 91% of smartphones. While Layar is not a requirement to play fAR-Play games, the application does supply additional augmented reality functionality; Layar also includes a built-in QR scanner. Depending on the design of the particular game, the player may instead use a dedicated QR code scanner; the developers recommend BeeTagg, but any such application will do. Layar or a QR code scanner are the maximum software requirements to play a fAR-Play game, making implementation of games on a wide variety of platforms relatively straightforward. fAR-Play games can also be designed for play strictly within a mobile phone's web browser. On the server side, fAR-Play's engine is composed of an Apache server which manages the system's web interface, including the mobile and desktop versions of the fAR-Play website, and a Java-based REST framework for managing the database of nodes. === Features === As a platform for designing AR games, as opposed to an AR game itself, fAR-Play offers little in the way of explicit shapes or patterns for games to take; instead, these elements are left to the game designer or players to develop. However, the nonspecific nature of nodes, the many options they offer for content delivery, and the open design of the platform are such that these elements can be developed extensively. Functionally, fAR-Play is a tool for tracking arbitrary points in space and a given player's proximity to them; what it does beyond that is up to the developers' and players' discretion. However, the fAR-Play website contains a leaderboard which tracks registered user's total scores. Players are assigned levels based on their total score, ranging from Novice — Super Player. Player profiles will display nodes that the player has recently caught, and any achievements the player has gained. Additionally, players can share their adventure progress, achievements, and the capture of vPOIs on Facebook. == How to play == In order to participate in the locative aspects of fAR-Play games, users must have an Android or iOS mobile device and access to wireless internet. Players can participate in fAR-Play anonymously, or create and sign into a fAR-Play account. Those who choose to play anonymously will lose the ability to track their progress across multiple games. When signed in, the player is presented with a list of games that are currently available for play. Each game includes a brief description and the various "adventures" available to the player. Once the game has been started, the player has three different methods for capturing nodes: they may scan a QR in the physical space, discover a node through the Layar camera virtual view, or receive a link in their device's web browser. === QR codes and Layar === QR codes can only be used as a method for capturing nodes and initiating games when there is a physical code present. In order to scan a QR code, players are required to have an application which can capture and recognize QR codes. If the player is utilizing a QR scanning application that has a built in browser, they will be required to log into fAR-Play through the app. Layar is a free to download augmented reality app, containing a built in QR code scanner, which enables its users to participate in fAR-Play games. === Capturing nodes === Layar permits the player to see nodes on their mobile device, guiding the player to their goal. Using this application, the player is able to navigate to their objective with map provided by Google Maps' API or by using their camera — Layar overlays a virtual image onto the real-world scene presented by the camera. The representations on screen expand in size as the player approaches the node destination, simulating relative distance. If the player taps any of the nodes that are presented on the screen, they will be provided additional information about that node, including the node's name and a brief description. Nodes can be captured by tapping the "capture" button. === Playing on browsers === The player can also play fAR-Play games within their mobile device's browser. By visiting https://archive.today/20131123223038/http://farplay.ualberta.ca/far-play/ on a mobile device, players will be presented with a fully realized user interface, permitting full interaction with the games. The player can capture the in game vPOIs through their browser by tapping the "nodes" button. This will bring up a list of all the accessible nodes, complete with a brief description for each location. By clicking on one of the nodes, the player is shown to a screen with a mapped location of the vPOI, an in-depth description of it, and hints. At the top of the page, the player can tap "CAPTURE THIS NODE" and advance in the game. When attempting to capture a node, the developer may or may not associate a challenge with the node. For example, in the game "Zombies ate my Campus", when players are attempting to capture a node, they're presented with a multiple choice question associated with the current node. === Game types === Players complete an adventure when they have captured all of the nodes within it. fAR-Play provides two game modes: in a Virtual Scavenger Hunt, nodes must be captured in a specific order; in a Virtual Treasure Hunt, the order is unimportant. == Existing fAR-Play games == Games currently available through fAR-Play include: Giselle Ever After Thought Hub Comics Arts Capture Challenge Pioneering Edmonton The Intelliphone Challenge A Tour of Atwater Zombies ate my Campus == For developers == fAR-Play's ultimate goal is to provide a simple, effective platform for the creation of locative augmented reality games, but the developer tools are still under active development and not openly available to the public. Access can be granted on a case-by-case basis, however, and a developer's manual is available. Users with development privileges can create new games or edit their existing games, in addition to playing their own or others' games. === Adventures === Games that are developed with fAR-Play are segmented into components called "Adventures". To progress through each game adventure, the player must reach and capture virtual points of interest, referred to in the game as vPOIs. In order to capture a vPOI, the player must travel to a physical location that is set by the developer. It is the developer's choice to include a challenge question to capture the vPOI, though it is not mandatory. A deduction of points can be implemented if the player submits an incorrect answer to a challenge question. === Points and achievements === Each of the nodes will reward the player with a predetermined number of points once they have been captured by the player. These points are added to the player's total points. Each of the adventures that are created require a predetermined number of vPOIs

Floyd–Steinberg dithering

Floyd–Steinberg dithering is an image dithering algorithm first published in 1976 by Robert W. Floyd and Louis Steinberg. It is commonly used by image manipulation software, for example, when converting an image from a Truecolor 24-bit PNG format into a GIF format, which is restricted to a maximum of 256 colors. == Implementation == The algorithm achieves dithering using error diffusion, meaning it pushes (adds) the residual quantization error of a pixel onto its neighboring pixels, to be quantized after. It spreads the debt out according to the distribution (shown as a map of the neighboring pixels): [ ∗ 7 16 … … 3 16 5 16 1 16 … ] {\displaystyle {\begin{bmatrix}&&&{\frac {\displaystyle 7}{\displaystyle 16}}&\ldots \\\ldots &{\frac {\displaystyle 3}{\displaystyle 16}}&{\frac {\displaystyle 5}{\displaystyle 16}}&{\frac {\displaystyle 1}{\displaystyle 16}}&\ldots \\\end{bmatrix}}} The pixel indicated with a star () indicates the pixel currently being scanned, and the blank pixels are the previously scanned pixels. The specific values (7/16, 3/16, 5/16, 1/16) were originally found by trial-and-error, "guided by the desire to have a region of desired density 0.5 come out as a checkerboard pattern". The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one. Each time, the quantization error is transferred to the neighboring pixels, while not affecting the pixels that already have been quantized. Hence, if a number of pixels have been rounded downwards, it becomes more likely that the next pixel is rounded upwards, such that on average, the quantization error is close to zero. The diffusion coefficients have the property that if the original pixel values are exactly halfway in between the nearest available colors, the dithered result is a checkerboard pattern. For example, 50% grey data could be dithered as a black-and-white checkerboard pattern. For optimal dithering, the counting of quantization errors should be in sufficient accuracy to prevent rounding errors from affecting the result. For correct results, all values should be linearized first, rather than operating directly on sRGB values as is common for images stored on computers. In some implementations, the horizontal direction of scan alternates between lines; this is called "serpentine scanning" or boustrophedon transform dithering. The algorithm described above is in the following pseudocode. This works for any approximately linear encoding of pixel values, such as 8-bit integers, 16-bit integers or real numbers in the range [0, 1]. for each y from top to bottom do for each x from left to right do oldpixel := pixels[x][y] newpixel := find_closest_palette_color(oldpixel) pixels[x][y] := newpixel quant_error := oldpixel - newpixel pixels[x + 1][y ] := pixels[x + 1][y ] + quant_error × 7 / 16 pixels[x - 1][y + 1] := pixels[x - 1][y + 1] + quant_error × 3 / 16 pixels[x ][y + 1] := pixels[x ][y + 1] + quant_error × 5 / 16 pixels[x + 1][y + 1] := pixels[x + 1][y + 1] + quant_error × 1 / 16 When converting grayscale pixel values from a high to a low bit depth (e.g. 8-bit grayscale to 1-bit black-and-white), find_closest_palette_color() may perform just a simple rounding, for example: find_closest_palette_color(oldpixel) = round(oldpixel / 255) The pseudocode can result in pixel values exceeding the valid values (such as greater than 255 in 8-bit grayscale images). Such values should ideally be handled by the find_closest_palette_color() function, rather than clipping the intermediate values, since a subsequent error may bring the value back into range. However, if fixed-width integers are used, wrapping of intermediate values would cause inversion of black and white, and so should be avoided. The find_closest_palette_color() implementation is nontrivial for a palette that is not evenly distributed, however small inaccuracies in selecting the correct palette color have minimal visual impact due to error being propagated to future pixels. A nearest neighbor search in 3D is frequently used.

Teaching dimension

In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).

Confirmatory blockmodeling

Confirmatory blockmodeling is a deductive approach in blockmodeling, where a blockmodel (or part of it) is prespecify before the analysis, and then the analysis is fit to this model. When only a part of analysis is prespecify (like individual cluster(s) or location of the block types), it is called partially confirmatory blockmodeling. This is so-called indirect approach, where the blockmodeling is done on the blockmodel fitting (e.g., a priori hypothesized blockmodel). Opposite approach to the confirmatory blockmodeling is an inductive exploratory blockmodeling.

Nearest neighbor search

Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M {\displaystyle q\in M} , find the closest point in S to q. Donald Knuth in volume 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. == Applications == The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Vector databases, where nearest-neighbor lookup over embeddings is used to retrieve semantically similar records Retrieval-augmented generation systems, where nearest-neighbor retrieval over embeddings is used to fetch candidate passages or documents before generation Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning == Methods == Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. === Exact methods === ==== Linear search ==== The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. ==== Space partitioning ==== Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. === Approximation methods === An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most c {\displaystyle c} times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. ==== Greedy search in proximity neighborhood graphs ==== Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs G ( V , E ) {\displaystyle G(V,E)} in which every point x i ∈ S {\displaystyle x_{i}\in S} is uniquely associated with vertex v i ∈ V {\displaystyle v_{i}\in V} . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph G ( V , E ) {\displaystyle G(V,E)} . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex v i ∈ V {\displaystyle v_{i}\in V} by computing the distances from the query q to each vertex of its neighborhood { v j : ( v i , v j ) ∈ E } {\displaystyle \{v_{j}:(v_{i},v_{j})\in E\}} , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet syst

E-gree (app)

E-gree is a legal app that became well known in 2020. It was the first app of its kind to protect users against a number of dating-related issues, including revenge porn. == Background == The app was co-founded by Araz Mamet, Keith Fraser and Ilya Flaks. The app focuses on privacy, with users being able to set up various contracts to protect themselves following a breakup, or while dating. This notably included signing an NDA when sexting. The app received investment from a number of notable people and companies, including Natalia Vodianova.

Correlation clustering

Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the