# Search Results

• ### Algorithm to reduce a bitmap mask to a list of rectangles?

##### - by mos
Before I go spend an afternoon writing this myself, I thought I'd ask if there was an implementation already available --even just as a reference. The first image is an example of a bitmap mask that I would like to turn into a list of rectangles. A bad algorithm would return every set pixel as a 1x1 rectangle. A good algorithm would look like the second image, where it returns the coordinates of the orange and red rectangles. The fact that the rectangles overlap don't matter, just that there are only two returned. To summarize, the ideal result would be these two rectangles (x, y, w, h): [ { 3, 1, 2, 6 }, { 1, 3, 6, 2 } ]

• ### Why hill climbing is called anytime algorithm?

##### - by crucified soul
From wikipedia, Anytime algorithm In computer science an anytime algorithm is an algorithm that can return a valid solution to a problem even if it's interrupted at any time before it ends. The algorithm is expected to find better and better solutions the more time it keeps running. Hill climbing Hill climbing can often produce a better result than other algorithms when the amount of time available to perform a search is limited, such as with real-time systems. It is an anytime algorithm: it can return a valid solution even if it's interrupted at any time before it ends. Hill climbing algorithm can stuck into local optima or ridge, after that even if it runs infinite time, the result won't be any better. Then, why hill climbing is called anytime algorithm?

• ### Dijkstra’s algorithm and functions

##### - by baris_a
Hi guys, the question is: suppose I have an input function like sin(2-cos(3*A/B)^2.5)+0.756*(C*D+3-B) specified with a BNF, I will parse input using recursive descent algorithm, and then how can I use or change Dijkstra’s algorithm to handle this given function? After parsing this input function, I need to execute it with variable inputs, where Dijkstra’s algorithm should do the work. Thanks in advance. EDIT: May be I should ask also: What is the best practice or data structure to represent given function?

• ### Efficient algorithm for finding largest eigenpair of small general complex matrix

##### - by mklassen
I am looking for an efficient algorithm to find the largest eigenpair of a small, general (non-square, non-sparse, non-symmetric), complex matrix, A, of size m x n. By small I mean m and n is typically between 4 and 64 and usually around 16, but with m not equal to n. This problem is straight forward to solve with the general LAPACK SVD algorithms, i.e. gesvd or gesdd. However, as I am solving millions of these problems and only require the largest eigenpair, I am looking for a more efficient algorithm. Additionally, in my application the eigenvectors will generally be similar for all cases. This lead me to investigate Arnoldi iteration based methods, but I have neither found a good library nor algorithm that applies to my small general complex matrix. Is there an appropriate algorithm and/or library?

• ### Algorithm for Text Wrapping Within a Shape

##### - by devongovett
I am looking for an algorithm to wrap text within a non-rectangular shape, preferably based on the Knuth and Plass algorithm. The hardest part of this is that the lines may have different heights due to differing font sizes in the text. The image below is an example of what the algorithm should be able to generate. Thanks for any help!

• ### Clustering [assessment] algorithm with distance matrix as an input

##### - by Max
Can anyone suggest some clustering algorithm which can work with distance matrix as an input? Or the algorithm which can assess the "goodness" of the clustering also based on the distance matrix? At this moment I'm using a modification of Kruskal's algorithm (http://en.wikipedia.org/wiki/Kruskal%27s_algorithm) to split data into two clusters. It has a problem though. When the data has no distinct clusters the algorithm will still create two clusters with one cluster containing one element and the other containing all the rest. In this case I would rather have one cluster containing all the elements and another one which is empty. Are there any algorithms which are capable of doing this type of clustering? Are there any algorithms which can estimate how well the clustering was done or even better how many clusters are there in the data? The algorithms should work only with distance(similarity) matrices as an input.

• ### Dijkstra's algorithm: why does it work? (not how)

##### - by BeeBand
I understand what Dijkstra's algorithm is but I don't understand why it works. When selecting the next vertice to examine, why does Dijkstra's algorithm select the one with the smallest weight? Why not just select a vertex arbitrarily, since the algorithm visits all vertices anyway?

• ### Algorithm for matching partially filled words

Hello All, I am writing a game which when given a partially filled word, searches a dictionary and returns all the matching words. To that effect, I am trying to find an algorithm that can be used for the said purpose. For example, given - - a -, the algorithm will search a dictionary for all the words which have length 4 and have 'a' as the third letter. Is there such an algorithm already? If not, can somebody given a rough idea of how to design such an algorithm? Thanks in Advance.

• ### isometric drawing order with larger than single tile images - drawing order algorithm?

##### - by Roger Smith
I have an isometric map over which I place various images. Most images will fit over a single tile, but some images are slightly larger. For example, I have a bed of size 2x3 tiles. This creates a problem when drawing my objects to the screen as I get some tiles erroneously overlapping other tiles. The two solutions that I know of are either splitting the image into 1x1 tile segments or implementing my own draw order algorithm, for example by assigning each image a number. The image with number 1 is drawn first, then 2, 3 etc. Does anyone have advice on what I should do? It seems to me like splitting an isometric image is very non obvious. How do you decide which parts of the image are 'in' a particular tile? I can't afford to split up all of my images manually either. The draw order algorithm seems like a nicer choice but I am not sure if it's going to be easy to implement. I can't solve, in my head, how to deal with situations whereby you change the index of one image, which causes a knock on effect to many other images. If anyone has an resources/tutorials on this I would be most grateful.

• ### What algorithm to use to fill a KenKen square board with cages?

##### - by JimmyBoh
I am working on recreating KenKen, a popular math puzzle involving a blank grid that is divided into "cages". Each cage is just a collection of adjacent squares and has a clue which is generally a number and an operand, shown below: What type of algorithm would be best to fill the square with cages? Assume the maximum number of cells per cage would be 3 and the board is 4x4 in size, like in the example above.

• ### Lawler's Algorithm Implementation Assistance

##### - by Richard Knop
Here is my implemenation of Lawler's algorithm in PHP (I know... but I'm used to it): <?php \$jobs = array(1, 2, 3, 4, 5, 6); \$jobsSubset = array(2, 5, 6); \$n = count(\$jobs); \$processingTimes = array(2, 3, 4, 3, 2, 1); \$dueDates = array(3, 15, 9, 7, 11, 20); \$optimalSchedule = array(); foreach (\$jobs as \$j) { \$optimalSchedule[] = 0; } \$dicreasedCardinality = array(); for (\$i = \$n; \$i >= 1; \$i--) { \$x = 0; \$max = 0; // loop through all jobs for (\$j = 0; \$j < \$i; \$j++) { // ignore if \$j already is in the \$dicreasedCardinality array if (false === in_array(\$j, \$dicreasedCardinality)) { // if the job has no succesor in \$jobsSubset if (false === isset(\$jobs[\$j+1]) || false === in_array(\$jobs[\$j+1], \$jobsSubset)) { // here I find an array index of a job with the maximum due date // amongst jobs with no sucessor in \$jobsSubset if (\$x < \$dueDates[\$j]) { \$x = \$dueDates[\$j]; \$max = \$j; } } } } // move the job at the end of \$optimalSchedule \$optimalSchedule[\$i-1] = \$jobs[\$max]; // decrease the cardinality of \$jobs \$dicreasedCardinality[] = \$max; } print_r(\$optimalSchedule); Now the above returns an optimal schedule like this: Array ( [0] => 1 [1] => 1 [2] => 1 [3] => 3 [4] => 2 [5] => 6 ) Which doesn't seem right to me. The problem might be with my implementation of the algorithm because I am not sure I understand it correctly. I used this source to implement it: http://www.google.com/books?id=aSiBs6PDm9AC&pg=PA166&dq=lawler%27s+algorithm+code&lr=&hl=sk&cd=4#v=onepage&q=&f=false The description there is a little confusing. For example, I didn't quite get how is the subset D defined (I guess it is arbitrary). Could anyone help me out with this? I have been trying to find some sources with simpler explanation of the algorithm but all sources I found were even more complicated (with math proofs and such) so I am stuck with the link above. Yes, this is a homework, if it wasn't obvious. I still have few weeks to crack this but I have spent few days already trying to get how exactly this algorithm works with no success so I don't think I will get any brighter during that time.

• ### Shuffling algorithm with no "self-mapping"?

##### - by OregonTrail
To randomly shuffle an array, with no bias towards any particular permutation, there is the Knuth Fischer-Yeats algorithm. In Python: #!/usr/bin/env python import sys from random import randrange def KFYShuffle(items): i = len(items) - 1 while i > 0: j = randrange(i+1) # 0 <= j <= i items[j], items[i] = items[i], items[j] i = i - 1 return items print KFYShuffle(range(int(sys.argv[1]))) There is also Sattolo's algorithm, which produces random cycles. In Python: #!/usr/bin/env python import sys from random import randrange def SattoloShuffle(items): i = len(items) while i > 1: i = i - 1 j = randrange(i) # 0 <= j <= i-1 items[j], items[i] = items[i], items[j] return items print SattoloShuffle(range(int(sys.argv[1]))) I'm currently writing a simulation with the following specifications for a shuffling algorithm: The algorithm is unbiased. If a true random number generator was used, no permutation would be more likely than any other. No number ends up at its original index. The input to the shuffle will always be A[i] = i for i from 0 to N-1 Permutations are produced that are not cycles, but still meet specification 2. The cycles produced by Sattolo's algorithm meet specification 2, but not specification 1 or 3. I've been working at creating an algorithm that meets these specifications, what I came up with was equivalent to Sattolo's algorithm. Does anyone have an algorithm for this problem?

• ### An implementation of Sharir's or Aurenhammer's deterministic algorithm for calculating the intersect

##### - by RGrey
The problem of finding the intersection/union of 'N' discs/circles on a flat plane was first proposed by M. I. Shamos in his 1978 thesis: Shamos, M. I. “Computational Geometry” Ph.D. thesis, Yale Univ., New Haven, CT 1978. Since then, in 1985, Micha Sharir presented an O(n log2n) time and O(n) space deterministic algorithm for the disc intersection/union problem (based on modified Voronoi diagrams): Sharir, M. Intersection and closest-pair problems for a set of planar discs. SIAM .J Comput. 14 (1985), pp. 448-468. In 1988, Franz Aurenhammer presented a more efficient O(n log n) time and O(n) space algorithm for circle intersection/union using power diagrams (generalizations of Voronoi diagrams): Aurenhammer, F. Improved algorithms for discs and balls using power diagrams. Journal of Algorithms 9 (1985), pp. 151-161. Earlier in 1983, Paul G. Spirakis also presented an O(n^2) time deterministic algorithm, and an O(n) probabilistic algorithm: Spirakis, P.G. Very Fast Algorithms for the Area of the Union of Many Circles. Rep. 98, Dept. Comput. Sci., Courant Institute, New York University, 1983. I've been searching for any implementations of the algorithms above, focusing on computational geometry packages, and I haven't found anything yet. As neither appear trivial to put into practice, it would be really neat if someone could point me in the right direction!

• ### average case running time of linear search algorithm

Hi all. I am trying to derive the average case running time for deterministic linear search algorithm. The algorithm searches an element x in an unsorted array A in the order A[1], A[2], A[3]...A[n]. It stops when it finds the element x or proceeds until it reaches the end of the array. I searched on wikipedia and the answer given was (n+1)/(k+1) where k is the number of times x is present in the array. I approached in another way and am getting a different answer. Can anyone please give me the correct proof and also let me know whats wrong with my method? E(T)= 1*P(1) + 2*P(2) + 3*P(3) ....+ n*P(n) where P(i) is the probability that the algorithm runs for 'i' time (i.e. compares 'i' elements). P(i)= (n-i)C(k-1) * (n-k)! / n! Here, (n-i)C(k-1) is (n-i) Choose (k-1). As the algorithm has reached the ith step, the rest of k-1 x's must be in the last n-i elements. Hence (n-i)C(k-i). (n-k)! is the total number of ways of arranging the rest non x numbers, and n! is the total number of ways of arranging the n elements in the array. I am not getting (n+1)/(k+1) on simplifying.

• ### Adaptive user interface/environment algorithm

##### - by WowtaH
Hi all, I'm working on an information system (in C#) that (while my users use it) gathers statistical data on what pieces of information (tables & records) each user is requesting the most, and what parts of the interface he/she uses most. I'm using this statistical data to make the application adaptive to the user's needs, both in the way the interface presents itself (eg: tab/pane-ordering) as in the way of using the frequently viewed information to (eg:) show higher in search results/suggestion-lists. What i'm looking for is an algorithm/formula to determine the current 'hotness'/relevance of these objects for a specific user. A simple 'hitcounter' for each object won't be sufficient because the user might view some information quite frequently for a period of time, and then moving on to the next, making the old information less relevant. So i think my algorithm also needs some sort of sliding/historical principle to account for the changing popularity of the objects in the application over time. So, the question is: Does anybody have some sort of algorithm that accounts for that 'popularity over time' ? Preferably with some explanation on the parameters :) Thanks! PS I've looked at other posts like http://stackoverflow.com/questions/32397/popularity-algorithm but i could't quite port it to my specific case. Any help is appreciated.

• ### Algorithm for creating a school timetable.

##### - by cand
Hello all. I've been wondering if there are known solutions for algorithm of creating a school timetable. Basically, it's about optimizing "hour-dispersion" (both in teachers and classes case) for given class-subject-teacher associations. We can assume that we have sets of classes, lesson subjects and teachers associated with each other at the input and that timetable should fit between 8AM and 4PM. I guess that there is probably no accurate algorithm for that, but maybe someone knows a good approximation or hints for developing it. P.S. I know, there was http://stackoverflow.com/questions/1259686/school-timetable-generation-algorithm-closed , but unlike in that case, I'm not looking for actual implementation, rather for thoughts on how to do it or why it's impossible.

• ### Algorithm to generate all possible letter combinations of given string down to 2 letters

##### - by Alan
Algorithm to generate all possible letter combinations of given string down to 2 letters Trying to create an Anagram solver in AS3, such as this one found here: http://homepage.ntlworld.com/adam.bozon/anagramsolver.htm I'm having a problem wrapping my brain around generating all possible letter combinations for the various lengths of strings. If I was only generating permutations for a fixed length, it wouldn't be such a problem for me... but I'm looking to reduce the length of the string and obtain all the possible permutations from the original set of letters for a string with a max length smaller than the original string. For example, say I want a string length of 2, yet I have a 3 letter string of “abc”, the output would be: ab ac ba bc ca cb. Ideally the algorithm would produce a complete list of possible combinations starting with the original string length, down to the smallest string length of 2. I have a feeling there is probably a small recursive algorithm to do this, but can't wrap my brain around it. I'm working in AS3. Thanks!

• ### Algorithm for digit summing?

##### - by Joe
I'm searching for an algorithm for Digit summing. Let me outline the basic principle: Say you have a number: 18268. 1 + 8 + 2 + 6 + 8 = 25 2 + 5 = 7 And 7 is our final number. It's basically adding each number of the whole number until we get down to a single (also known as a 'core') digit. It's often used by numerologists. I'm searching for an algorithm (doesn't have to be language in-specific) for this. I have searched Google for the last hour with terms such as digit sum algorithm and whatnot but got no suitable results. Any help would be great, thanks.

• ### Stable random color algorithm

##### - by Olmo
Here we have an interesting real-world algorithm requirement involving colors. 1) Nice random colors: In ordeeing to draw a beautifull chart (i.e: pie chart) we need to pick a random set of Colors that: a) are different enought b) Play nicely Doesnt Look hard. For example u fix bright and saturation and divide hue in steps of 360/Num_sectors 2) Stable: given Pie1 with sectors with labes ('A','B','C') and Pie2 with sector with labels ('B','C','D'), will be nice if color('B',pie1)= color('B',pie2) and the same for 'C' and so on, so people don't get crazy when seeing similar updated charts, even if some sectors appear some dissapeared or the number of sectors changed. The label is the only stable thing. 3) hard-coded colors: the algorithm allows hardcoded label-color relationships as an input but stills doing a good work (1 & 2) for the rest of free labels. I think this algorithm, even if it looks quite ad-hoc, will be usefull in more then one situation. Any ideas?

• ### Efficient algorithm to generate all solutions of a linear diophantine equation with ai=1

##### - by Ben
I am trying to generate all the solutions for the following equations for a given H. With H=4 : 1) ALL solutions for x_1 + x_2 + x_3 + x_4 =4 2) ALL solutions for x_1 + x_2 + x_3 = 4 3) ALL solutions for x_1 + x_2 = 4 4) ALL solutions for x_1 =4 For my problem, there are always 4 equations to solve (independently from the others). There are a total of 2^(H-1) solutions. For the previous one, here are the solutions : 1) 1 1 1 1 2) 1 1 2 and 1 2 1 and 2 1 1 3) 1 3 and 3 1 and 2 2 4) 4 Here is an R algorithm which solve the problem. library(gtools) H<-4 solutions<-NULL for(i in seq(H)) { res<-permutations(H-i+1,i,repeats.allowed=T) resum<-apply(res,1,sum) id<-which(resum==H) print(paste("solutions with ",i," variables",sep="")) print(res[id,]) } However, this algorithm makes more calculations than needed. I am sure it is possible to go faster. By that, I mean not generating the permutations for which the sums is H Any idea of a better algorithm for a given H ?

• ### Using Dijkstra's algorithm with negative edges?

##### - by Riddler
Most books explain the reason the algorithm doesn't work with negative edges as nodes are deleted from the priority queue after the node is arrived at since the algorithm assumes the shortest distance has been found. However since negative edges can reduce the distance, a future shorter distance might be found; but since the node is deleted it cannot be updated. Wouldn't an obvious solution to this be to not delete the node? Why not keep the node in the queue, so if a future shorter distance is found, it can be updated? If I am misunderstanding the problem, what is preventing the algorithm from being used with negative edges?

• ### What algorithm can I use to detect simple shapes in a 4x4 matrix?

##### - by ion
I'm working on a simple multiplayer game that receives a random 4x4 matrix from a server and extracts a shape from it. For example: XXOO OXOO XXOX XXOO XOOX and XOOO XXXX OXXX So in the first matrix the shape I want to parse is: oo o oo and the 2nd: oo oo ooo I know there must be an algorithm for this because I saw this kind of behavior on some puzzle games but I have no idea how to go about to detecting them or even where to start. So my question is: How do I detect what shape is in the matrix and how do I differentiate between multiple colors? (it doesn't come only in X and O, it comes in a maximum of 4). Additionally, the shape must be a minimum of 4 blocks.

• ### Extreme Optimization – Numerical Algorithm Support

##### - by JoshReuben
Function Delegates Many calculations involve the repeated evaluation of one or more user-supplied functions eg Numerical integration. The EO MathLib provides delegate types for common function signatures and the FunctionFactory class can generate new delegates from existing ones. RealFunction delegate - takes one Double parameter – can encapsulate most of the static methods of the System.Math class, as well as the classes in the Extreme.Mathematics.SpecialFunctions namespace: var sin = new RealFunction(Math.Sin); var result = sin(1); BivariateRealFunction delegate - takes two Double parameters: var atan2 = new BivariateRealFunction (Math.Atan2); var result = atan2(1, 2); TrivariateRealFunction delegate – represents a function takes three Double arguments ParameterizedRealFunction delegate - represents a function taking one Integer and one Double argument that returns a real number. The Pow method implements such a function, but the arguments need order re-arrangement: static double Power(int exponent, double x) { return ElementaryFunctions.Pow(x, exponent); } ... var power = new ParameterizedRealFunction(Power); var result = power(6, 3.2); A ComplexFunction delegate - represents a function that takes an Extreme.Mathematics.DoubleComplex argument and also returns a complex number. MultivariateRealFunction delegate - represents a function that takes an Extreme.Mathematics.LinearAlgebra.Vector argument and returns a real number. MultivariateVectorFunction delegate - represents a function that takes a Vector argument and returns a Vector. FastMultivariateVectorFunction delegate - represents a function that takes an input Vector argument and an output Matrix argument – avoiding object construction  The FunctionFactory class RealFromBivariateRealFunction and RealFromParameterizedRealFunction helper methods - transform BivariateRealFunction or a ParameterizedRealFunction into a RealFunction delegate by fixing one of the arguments, and treating this as a new function of a single argument. var tenthPower = FunctionFactory.RealFromParameterizedRealFunction(power, 10); var result = tenthPower(x); Note: There is no direct way to do this programmatically in C# - in F# you have partial value functions where you supply a subset of the arguments (as a travelling closure) that the function expects. When you omit arguments, F# generates a new function that holds onto/remembers the arguments you passed in and "waits" for the other parameters to be supplied. let sumVals x y = x + y     let sumX = sumVals 10     // Note: no 2nd param supplied.     // sumX is a new function generated from partially applied sumVals.     // ie "sumX is a partial application of sumVals." let sum = sumX 20     // Invokes sumX, passing in expected int (parameter y from original)  val sumVals : int -> int -> int val sumX : (int -> int) val sum : int = 30 RealFunctionsToVectorFunction and RealFunctionsToFastVectorFunction helper methods - combines an array of delegates returning a real number or a vector into vector or matrix functions. The resulting vector function returns a vector whose components are the function values of the delegates in the array. var funcVector = FunctionFactory.RealFunctionsToVectorFunction(     new MultivariateRealFunction(myFunc1),     new MultivariateRealFunction(myFunc2));  The IterativeAlgorithm<T> abstract base class Iterative algorithms are common in numerical computing - a method is executed repeatedly until a certain condition is reached, approximating the result of a calculation with increasing accuracy until a certain threshold is reached. If the desired accuracy is achieved, the algorithm is said to converge. This base class is derived by many classes in the Extreme.Mathematics.EquationSolvers and Extreme.Mathematics.Optimization namespaces, as well as the ManagedIterativeAlgorithm class which contains a driver method that manages the iteration process.  The ConvergenceTest abstract base class This class is used to specify algorithm Termination , convergence and results - calculates an estimate for the error, and signals termination of the algorithm when the error is below a specified tolerance. Termination Criteria - specify the success condition as the difference between some quantity and its actual value is within a certain tolerance – 2 ways: absolute error - difference between the result and the actual value. relative error is the difference between the result and the actual value relative to the size of the result. Tolerance property - specify trade-off between accuracy and execution time. The lower the tolerance, the longer it will take for the algorithm to obtain a result within that tolerance. Most algorithms in the EO NumLib have a default value of MachineConstants.SqrtEpsilon - gives slightly less than 8 digits of accuracy. ConvergenceCriterion property - specify under what condition the algorithm is assumed to converge. Using the ConvergenceCriterion enum: WithinAbsoluteTolerance / WithinRelativeTolerance / WithinAnyTolerance / NumberOfIterations Active property - selectively ignore certain convergence tests Error property - returns the estimated error after a run MaxIterations / MaxEvaluations properties - Other Termination Criteria - If the algorithm cannot achieve the desired accuracy, the algorithm still has to end – according to an absolute boundary. Status property - indicates how the algorithm terminated - the AlgorithmStatus enum values:NoResult / Busy / Converged (ended normally - The desired accuracy has been achieved) / IterationLimitExceeded / EvaluationLimitExceeded / RoundOffError / BadFunction / Divergent / ConvergedToFalseSolution. After the iteration terminates, the Status should be inspected to verify that the algorithm terminated normally. Alternatively, you can set the ThrowExceptionOnFailure to true. Result property - returns the result of the algorithm. This property contains the best available estimate, even if the desired accuracy was not obtained. IterationsNeeded / EvaluationsNeeded properties - returns the number of iterations required to obtain the result, number of function evaluations.  Concrete Types of Convergence Test classes SimpleConvergenceTest class - test if a value is close to zero or very small compared to another value. VectorConvergenceTest class - test convergence of vectors. This class has two additional properties. The Norm property specifies which norm is to be used when calculating the size of the vector - the VectorConvergenceNorm enum values: EuclidianNorm / Maximum / SumOfAbsoluteValues. The ErrorMeasure property specifies how the error is to be measured – VectorConvergenceErrorMeasure enum values: Norm / Componentwise ConvergenceTestCollection class - represent a combination of tests. The Quantifier property is a ConvergenceTestQuantifier enum that specifies how the tests in the collection are to be combined: Any / All  The AlgorithmHelper Class inherits from IterativeAlgorithm<T> and exposes two methods for convergence testing. IsValueWithinTolerance<T> method - determines whether a value is close to another value to within an algorithm's requested tolerance. IsIntervalWithinTolerance<T> method - determines whether an interval is within an algorithm's requested tolerance.

• ### Algorithm for autocomplete?

##### - by StackUnderflow
I am referring to the algorithm that is used to give query suggestions when a user type a search term in google. I am mainly interested in how google algorithm is able to show: 1. Most important results (most likely queries rather than anything that matches) 2. Match substrings 3. Fuzzy matches I know you could use Trie or generalized trie to find matches but it wouldn't meet the above requirements... Similar questions asked earlier here Thanks