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  • What is wrong with my logic for the divide and conquer algorithm for Closest pair problem?

    - by Programming Noob
    I have been following Coursera's course on Algorithms and came up with a thought about the divide/conquer algorithm for the closest pair problem, that I want clarified. As per Prof Roughgarden's algorithm (which you can see here if you're interested): For a given set of points P, of which we have two copies - sorted in X and Y direction - Px and Py, the algorithm can be given as closestPair(Px,Py): Divide points into left half - Q, and right half - R, and form sorted copies of both halves along x and y directions - Qx,Qy,Rx,Ry Let closestPair(Qx,Qy) be points p1 and q1 Let closestPair(Rx,Ry) be p2,q2 Let delta be minimum of dist(p1,q1) and dist(p2,q2) This is the unfortunate case, let p3,q3 be the closestSplitPair(Px,Py,delta) Return the best result Now, the clarification that I want is related to step 5. I should say this beforehand, that what I'm suggesting, is barely any improvement at all, but if you're still interested, read ahead. Prof R says that since the points are already sorted in X and Y directions, to find the best pair in step 5, we need to iterate over points in the strip of width 2*delta, starting from bottom to up, and in the inner loop we need only 7 comparisions. Can this be bettered to just one? How I think is possible seemed a little difficult to explain in plain text, so I drew a diagram and wrote it on paper and uploaded it here: Since no one else came up with is, I'm pretty sure there's some error in my line of thought. But I have literally been thinking about this for HOURS now, and I just HAD to post this. It's all that is in my head. Can someone point out where I'm going wrong?

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  • Beginners guide to developing optimization software

    - by Florenc
    I am novice in "serious" programming i.e. applications that deal with real-life applications and software projects that go beyond school assignments. My interests include optimization, operations research, algorithms and lately i discovered how much I do like software design/development/engineering. I have already developed some simple desktop applications for some "famous" problems like TSP using heuristc approaches, a VRP solver (in progress) and so on. While developing this kind of software I actually used basic concepts taught at school such as object-orientation analysis and design. But, I found these courses rather elementary and quite boring (for my expectations). So I decided to go a little further and start developing "real" software (and this is where I realized how important and interesting software engineering/design is.) Now, here's my issue: I can not find a "study guide" for developing software of this kind. Currently, there are numerous resources out there (books, websites, tutorials) in designing and developing complex IS, web applications, smartphone apps but I can't find a book for example entitled "optimization software development". Definetly, someone could claim that "design patterns apply to software in general" but that's not my point. My point is that I could simply use my imagination for "simple" implementations, but what happens, when my imagination can not go further? In other words I'm looking for a guide/path to bridge the gap between: Mathematics-Algorithm Design-Software Engineering-Optimization-Software development

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  • Trying to sort the coefficients of the polynomial (z-a)(z-b)(z-c)...(z-n) into a vector

    - by pajamas
    So I have a factored polynomial of the form (z-a)(z-b)(z-c)...(z-n) for n an even positive integer. Thus the coefficient of z^k for 0 <= k < n will be the sum of all distinct n-k element products taken from the set {a,b,...,n} multiplied by (-1)^k, I hope that makes sense, please ask if you need more clarification. I'm trying to put these coefficients into a row vector with the first column containing the constant coefficient (which would be abc...n) and the last column containing the coefficient for z^n (which would be 1). I imagine there is a way to brute force this with a ton of nested loops, but I'm hoping there is a more efficient way. This is being done in Matlab (which I'm not that familiar with) and I know Matlab has a ton of algorithms and functions, so maybe its got something I can use. Can anyone think of a way to do this? Example: (z-1)(z-2)(z-3) = z^3 - (1 + 2 + 3)z^2 + (1*2 + 1*3 + 2*3)z - 1*2*3 = z^3 - 6z^2 + 11z - 6. Note that this example is n=3 odd, but n=4 would have taken too long to do by hand. Edit: Let me know if you think this would be better posted at TCS or Math Stack Exchange.

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  • Count unique visitors by group of visited places

    - by Mathieu
    I'm facing the problem of counting the unique visitors of groups of places. Here is the situation: I have visitors that can visit places. For example, that can be internet users visiting web pages, or customers going to restaurants. A visitor can visit as much places as he wishes, and a place can be visited by several visitors. A visitor can come to the same place several times. The places belong to groups. A group can obviously contain several places, and places can belong to several groups. Given that, for each visitor, we can have a list of visited places, how can I have the number of unique visitors per group of places? Example: I have visitors A, B, C and D; and I have places x, y and z. I have these visiting lists: [ A -> [x,x,y,x], B -> [], C -> [z,z], D -> [y,x,x,z] ] Having these number of unique visitors per place is quite easy: [ x -> 2, // A and D visited x y -> 2, // A and D visited y z -> 2 // C and D visited z ] But if I have these groups: [ G1 -> [x,y,z], G2 -> [x,z], G3 -> [x,y] ] How can I have this information? [ G1 -> 3, // A, C and D visited x or y or z G2 -> 3, // A, C and D visited x or z G3 -> 2 // A and D visited x or y ] Additional notes : There are so many places that it is not possible to store information about every possible group; It's not a problem if approximation are made. I don't need 100% precision. Having a fast algorithm that tells me that there were 12345 visits in a group instead of 12543 is better than a slow algorithm telling the exact number. Let's say there can be ~5% deviation. Is there an algorithm or class of algorithms that addresses this type of problem?

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  • Applying the Knuth-Plass algorithm (or something better?) to read two books with different length and amount of chapters in parallel

    - by user147133
    I have a Bible reading plan that covers the whole Bible in 180 days. For the most of the time, I read 5 chapters in the Old Testament and 1 or 2 (1.5) chapters in the New Testament each day. The problem is that some chapters are longer than others (for example Psalm 119 which is 7 times longer than a average chapter in the Bible), and the plan I'm following doesn't take that in count. I end up with some days having a lot more to read than others. I thought I could use programming to make myself a better plan. I have a datastructure with a list of all chapters in the bible and their length in number of lines. (I found that the number of lines is the best criteria, but it could have been number of verses or number of words as well) I then started to think about this problem as a line wrap problem. Think of a chapter like a word, a day like a line and the whole plan as a paragraph. The "length" of a word (a chapter) is the number of lines in that chapter. I could then generate the best possible reading plan by applying a simplified Knuth-Plass algorithm to find the best breakpoints. This works well if I want to read the Bible from beginning to end. But I want to read a little from the new testament each day in parallel with the old testament. Of course I can run the Knuth-Plass algorithm on the Old Testament first, then on the New Testament and get two separate plans. But those plans merged is not a optimal plan. Worst-case days (days with extra much reading) in the New Testament plan will randomly occur on the same days as the worst-case days in the Old Testament. Since the New Testament have about 180*1.5 chapters, the plan is generally to read one chapter the first day, two the second, one the third etc... And I would like the plan for the Old Testament to compensate for this alternating length. So I will need a new and better algorithm, or I will have to use the Knuth-Plass algorithm in a way that I've not figured out. I think this could be a interesting and challenging nut for people interested in algorithms, so therefore I wanted to see if any of you have a good solution in mind.

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  • Generic Adjacency List Graph implementation

    - by DmainEvent
    I am trying to come up with a decent Adjacency List graph implementation so I can start tooling around with all kinds of graph problems and algorithms like traveling salesman and other problems... But I can't seem to come up with a decent implementation. This is probably because I am trying to dust the cobwebs off my data structures class. But what I have so far... and this is implemented in Java... is basically an edgeNode class that has a generic type and a weight-in the event the graph is indeed weighted. public class edgeNode<E> { private E y; private int weight; //... getters and setters as well as constructors... } I have a graph class that has a list of edges a value for the number of Vertices and and an int value for edges as well as a boolean value for whether or not it is directed. The brings up my first question, if the graph is indeed directed, shouldn't I have a value in my edgeNode class? Or would I just need to add another vertices to my LinkedList? That would imply that a directed graph is 2X as big as an undirected graph wouldn't it? public class graph { private List<edgeNode<?>> edges; private int nVertices; private int nEdges; private boolean directed; //... getters and setters as well as constructors... } Finally does anybody have a standard way of initializing there graph? I was thinking of reading in a pipe-delimited file but that is so 1997. public graph GenereateGraph(boolean directed, String file){ List<edgeNode<?>> edges; graph g; try{ int count = 0; String line; FileReader input = new FileReader("C:\\Users\\derekww\\Documents\\JavaEE Projects\\graphFile"); BufferedReader bufRead = new BufferedReader(input); line = bufRead.readLine(); count++; edges = new ArrayList<edgeNode<?>>(); while(line != null){ line = bufRead.readLine(); Object edgeInfo = line.split("|")[0]; int weight = Integer.parseInt(line.split("|")[1]); edgeNode<String> e = new edgeNode<String>((String) edges.add(e); } return g; } catch(Exception e){ return null; } } I guess when I am adding edges if boolean is true I would be adding a second edge. So far, this all depends on the file I write. So if I wrote a file with the following Vertices and weights... Buffalo | 18 br Pittsburgh | 20 br New York | 15 br D.C | 45 br I would obviously load them into my list of edges, but how can I represent one vertices connected to the other... so on... I would need the opposite vertices? Say I was representing Highways connected to each city weighted and un-directed (each edge is bi-directional with weights in some fictional distance unit)... Would my implementation be the best way to do that? I found this tutorial online Graph Tutorial that has a connector object. This appears to me be a collection of vertices pointing to each other. So you would have A and B each with there weights and so on, and you would add this to a list and this list of connectors to your graph... That strikes me as somewhat cumbersome and a little dismissive of the adjacency list concept? Am I wrong and that is a novel solution? This is all inspired by steve skiena's Algorithm Design Manual. Which I have to say is pretty good so far. Thanks for any help you can provide.

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  • Multidimensional multiple-choice knapsack problem: find a feasible solution

    - by Onheiron
    My assignment is to use local search heuristics to solve the Multidimensional multiple-choice knapsack problem, but to do so I first need to find a feasible solution to start with. Here is an example problem with what I tried so far. Problem R1 R2 R3 RESOUCES : 8 8 8 GROUPS: G1: 11.0 3 2 2 12.0 1 1 3 G2: 20.0 1 1 3 5.0 2 3 2 G3: 10.0 2 2 3 30.0 1 1 3 Sorting strategies To find a starting feasible solution for my local search I decided to ignore maximization of gains and just try to fit the resources requirements. I decided to sort the choices (strategies) in each group by comparing their "distance" from the multidimensional space origin, thus calculating SQRT(R1^2 + R2^2 + ... + RN^2). I felt like this was a keen solution as it somehow privileged those choices with resouce usages closer to each other (e.g. R1:2 R2:2 R3:2 < R1:1 R2:2 R3:3) even if the total sum is the same. Doing so and selecting the best choice from each group proved sufficent to find a feasible solution for many[30] different benchmark problems, but of course I knew it was just luck. So I came up with the problem presented above which sorts like this: R1 R2 R3 RESOUCES : 8 8 8 GROUPS: G1: 12.0 1 1 3 < select this 11.0 3 2 2 G2: 20.0 1 1 3 < select this 5.0 2 3 2 G3: 30.0 1 1 3 < select this 10.0 2 2 3 And it is not feasible because the resources consmption is R1:3, R2:3, R3:9. The easy solution is to pick one of the second best choices in group 1 or 2, so I'll need some kind of iteration (local search[?]) to find the starting feasible solution for my local search solution. Here are the options I came up with Option 1: iterate choices I tried to find a way to iterate all the choices with a specific order, something like G1 G2 G3 1 1 1 2 1 1 1 2 1 1 1 2 2 2 1 ... believeng that feasible solutions won't be that far away from the unfeasible one I start with and thus the number of iterations will keep quite low. Does this make any sense? If yes, how can I iterate the choices (grouped combinations) of each group keeping "as near as possibile" to the previous iteration? Option 2: Change the comparation term I tried to think how to find a better variable to sort the choices on. I thought at a measure of how "precious" a resource is based on supply and demand, so that an higer demand of a more precious resource will push you down the list, but this didn't help at all. Also I thought there probably isn't gonna be such a comparsion variable which assures me a feasible solution at first strike. I there such a variable? If not, is there a better sorting criteria anyways? Option 3: implement any known sub-optimal fast solving algorithm Unfortunately I could not find any of such algorithms online. Any suggestion?

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  • Parallelism in .NET – Part 12, More on Task Decomposition

    - by Reed
    Many tasks can be decomposed using a Data Decomposition approach, but often, this is not appropriate.  Frequently, decomposing the problem into distinctive tasks that must be performed is a more natural abstraction. However, as I mentioned in Part 1, Task Decomposition tends to be a bit more difficult than data decomposition, and can require a bit more effort.  Before we being parallelizing our algorithm based on the tasks being performed, we need to decompose our problem, and take special care of certain considerations such as ordering and grouping of tasks. Up to this point in this series, I’ve focused on parallelization techniques which are most appropriate when a problem space can be decomposed by data.  Using PLINQ and the Parallel class, I’ve shown how problem spaces where there is a collection of data, and each element needs to be processed, can potentially be parallelized. However, there are many other routines where this is not appropriate.  Often, instead of working on a collection of data, there is a single piece of data which must be processed using an algorithm or series of algorithms.  Here, there is no collection of data, but there may still be opportunities for parallelism. As I mentioned before, in cases like this, the approach is to look at your overall routine, and decompose your problem space based on tasks.  The idea here is to look for discrete “tasks,” individual pieces of work which can be conceptually thought of as a single operation. Let’s revisit the example I used in Part 1, an application startup path.  Say we want our program, at startup, to do a bunch of individual actions, or “tasks”.  The following is our list of duties we must perform right at startup: Display a splash screen Request a license from our license manager Check for an update to the software from our web server If an update is available, download it Setup our menu structure based on our current license Open and display our main, welcome Window Hide the splash screen The first step in Task Decomposition is breaking up the problem space into discrete tasks. This, naturally, can be abstracted as seven discrete tasks.  In the serial version of our program, if we were to diagram this, the general process would appear as: These tasks, obviously, provide some opportunities for parallelism.  Before we can parallelize this routine, we need to analyze these tasks, and find any dependencies between tasks.  In this case, our dependencies include: The splash screen must be displayed first, and as quickly as possible. We can’t download an update before we see whether one exists. Our menu structure depends on our license, so we must check for the license before setting up the menus. Since our welcome screen will notify the user of an update, we can’t show it until we’ve downloaded the update. Since our welcome screen includes menus that are customized based off the licensing, we can’t display it until we’ve received a license. We can’t hide the splash until our welcome screen is displayed. By listing our dependencies, we start to see the natural ordering that must occur for the tasks to be processed correctly. The second step in Task Decomposition is determining the dependencies between tasks, and ordering tasks based on their dependencies. Looking at these tasks, and looking at all the dependencies, we quickly see that even a simple decomposition such as this one can get quite complicated.  In order to simplify the problem of defining the dependencies, it’s often a useful practice to group our tasks into larger, discrete tasks.  The goal when grouping tasks is that you want to make each task “group” have as few dependencies as possible to other tasks or groups, and then work out the dependencies within that group.  Typically, this works best when any external dependency is based on the “last” task within the group when it’s ordered, although that is not a firm requirement.  This process is often called Grouping Tasks.  In our case, we can easily group together tasks, effectively turning this into four discrete task groups: 1. Show our splash screen – This needs to be left as its own task.  First, multiple things depend on this task, mainly because we want this to start before any other action, and start as quickly as possible. 2. Check for Update and Download the Update if it Exists - These two tasks logically group together.  We know we only download an update if the update exists, so that naturally follows.  This task has one dependency as an input, and other tasks only rely on the final task within this group. 3. Request a License, and then Setup the Menus – Here, we can group these two tasks together.  Although we mentioned that our welcome screen depends on the license returned, it also depends on setting up the menu, which is the final task here.  Setting up our menus cannot happen until after our license is requested.  By grouping these together, we further reduce our problem space. 4. Display welcome and hide splash - Finally, we can display our welcome window and hide our splash screen.  This task group depends on all three previous task groups – it cannot happen until all three of the previous groups have completed. By grouping the tasks together, we reduce our problem space, and can naturally see a pattern for how this process can be parallelized.  The diagram below shows one approach: The orange boxes show each task group, with each task represented within.  We can, now, effectively take these tasks, and run a large portion of this process in parallel, including the portions which may be the most time consuming.  We’ve now created two parallel paths which our process execution can follow, hopefully speeding up the application startup time dramatically. The main point to remember here is that, when decomposing your problem space by tasks, you need to: Define each discrete action as an individual Task Discover dependencies between your tasks Group tasks based on their dependencies Order the tasks and groups of tasks

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  • Map Generation Algorithms for Minecraft Clone

    - by Danjen
    I'm making a Minecraft clone for the sake of it (with some inspriation from Dwarf Fortress) and had a few questions about the way the world generation is handled. Things I want it to cover: Biomes such as hills, mountains, forests, etc. Caves/caverns/tunnels Procedural (so it stretches to infinity... is wrap-around a possibility?) Breaking the map into smaller chunks Moddable (ie, new terrain types) Multiplayer compatible In particular, I've seen things such as Perlin Noise, Heightmaps, and Marching Cubes thrown around. These are like different tools to use, but I don't know when or why I would use them. Are there any other techniques that are useful for map generation? I realize this is borderline subjective and open-ended, but I am looking for some more insight into the processes involved.

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  • Recommened design pattern to handle multiple compression algorithms for a class hierarchy

    - by sgorozco
    For all you OOD experts. What would be the recommended way to model the following scenario? I have a certain class hierarchy similar to the following one: class Base { ... } class Derived1 : Base { ... } class Derived2 : Base { ... } ... Next, I would like to implement different compression/decompression engines for this hierarchy. (I already have code for several strategies that best handle different cases, like file compression, network stream compression, legacy system compression, etc.) I would like the compression strategy to be pluggable and chosen at runtime, however I'm not sure how to handle the class hierarchy. Currently I have a tighly-coupled design that looks like this: interface ICompressor { byte[] Compress(Base instance); } class Strategy1Compressor : ICompressor { byte[] Compress(Base instance) { // Common compression guts for Base class ... // if( instance is Derived1 ) { // Compression guts for Derived1 class } if( instance is Derived2 ) { // Compression guts for Derived2 class } // Additional compression logic to handle other class derivations ... } } As it is, whenever I add a new derived class inheriting from Base, I would have to modify all compression strategies to take into account this new class. Is there a design pattern that allows me to decouple this, and allow me to easily introduce more classes to the Base hierarchy and/or additional compression strategies?

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  • most efficient AABB vs Ray collision algorithms

    - by Asher Einhorn
    Is there a known 'most efficient' algorithm for AABB vs Ray collision detection? I recently stumbled accross Arvo's AABB vs Sphere collision algorithm, and I am wondering if there is a similarly noteworthy algorithm for this. One must have condition for this algorithm is that I need to have the option of querying the result for the distance from the ray's origin to the point of collision. having said this, if there is another, faster algorithm which does not return distance, then in addition to posting one that does, also posting that algorithm would be very helpful indeed. Please also state what the function's return argument is, and how you use it to return distance or a 'no-collision' case. For example, does it have an out parameter for the distance as well as a bool return value? or does it simply return a float with the distance, vs a value of -1 for no collision? (For those that don't know: AABB = Axis Aligned Bounding Box)

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  • Why Are the Search Algorithms Still So Stupid?

    Anyway, it was back in '96 that I first tried Google (except I think they called it something else back then). It was as if the clouds broke open and the sun shone through. The results were stunningly similar to the object of my query. Prior to Google, you had these directories, the biggest among them Yahoo!

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  • Algorithms or patterns for a linked question and answer cost calculator

    - by kmc
    I've been asked to build an online calculator in PHP (and the Laravel framework). It will take the answers to a series of questions to estimate the cost of a home extension. For example, a couple of questions may be: What is the lie of your property? Flat, slightly inclined, heavily inclined. (these suggestive values could have specific values in the underlying calculator like, 0 degrees, 5 degrees, 10 degrees. What is your current flooring system? Wooden, or concrete? These would then impact the results of other questions. Once the size of the extension has been input, the lie of the land will affect how much site works will cost, and how much rubbish collection will cost. The second question will impact the cost of the extensions flooring, as stumping and laying floorboards is a different cost to laying foundations and a concrete slab. It will also influence what heating and cooling systems are available in the calculator. So it's VERY interlinked. The answer to any question can influence the options of other questions, and the end result. I'm having trouble figuring out an approach to this that will allow new options and questions to be plugged in at a later stage without having things too coupled. The Observer pattern, or Laravel's events may be handy, but currently the sheer breadth of the calculator has me struggling to gather my thoughts and see a sensible implementation. Are there any patterns or OO approaches that may help? Thanks!

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  • Algorithms for positioning rectangles evenly spaced with unknown connecting lines

    - by MacGyver
    I'm new to game development, but I'm trying to figure out a good algorithm for positioning rectangles (of any width and height) in a given surface area, and connecting them with any variation of lines. Two rectangles will never have more than one line connecting them. Where would I begin working on a problem like this? This is only a 2 dimensional surface. I read about graph theory, and it seems like this is a close representation of that. The rectangles would be considered a node, and the lines connecting them would be considered an edge in graph theory.

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  • .NET Developer Basics – Recursive Algorithms

    Recursion can be a powerful programming technique when used wisely. Some data structures such as tree structures lend themselves far more easily to manipulation by recursive techniques. As it is also a classic Computer Science problem, it is often used in technical interviews to probe a candidate's grounding in basic programming techniques. Whatever the reason, it is well worth brushing up one's understanding with Damon's introduction to Recursion.

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  • Good book for THINKING in terms of algorithms?

    - by chrisgoyal
    Before you mark this is a duplicate, let me explain why this is different. Most of the books on algorithms are more of a reference. You basically have a list of algorithms at your disposal. But what happens when you need to create a new algorithm for something? These books don't teach how to think in terms of algorithms. So I'm looking for books that will teach me the thinking-process of creating algorithms. Any good suggestions?

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  • Beginner's resources/introductions to classification algorithms.

    - by Dirk
    Hi, everybody. I am entirely new to the topic of classification algorithms, and need a few good pointers about where to start some "serious reading". I am right now in the process of finding out, whether machine learning and automated classification algorithms could be a worthwhile thing to add to some application of mine. I already scanned through "How to Solve It: Modern heuristics" by Z. Michalewicz and D. Fogel (in particular, the chapters about linear classifiers using neuronal networks), and on the practical side, I am currently looking through the WEKA toolkit source code. My next (planned) step would be to dive into the realm of Bayesian classification algorithms. Unfortunately, I am lacking a serious theoretical foundation in this area (let alone, having used it in any way as of yet), so any hints at where to look next would be appreciated; in particular, a good introduction of available classification algorithms would be helpful. Being more a craftsman and less a theoretician, the more practical, the better... Hints, anyone?

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  • please help me to choose good boks on algorithms

    - by davit-datuashvili
    i want to help me to choose good books on algorithms many people from this site say me that show me your code and now i ask u to help me to choose good books on algorithms please i have not books on algorithms and in case i decide to buy it of course must buy book which has high quality yes? so please any ideas ?links everything

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  • How to know whether to create a general system or to hack a solution

    - by Andy K
    I'm new to coding , learning it since last year actually. One of my worst habits is the following: Often I'm trying to create a solution that is too big , too complex and doesn't achieve what needs to be achieved, when a hacky kludge can make the fit. One last example was the following (see paste bin link below) http://pastebin.com/WzR3zsLn After explaining my issue, one nice person at stackoverflow came with this solution instead http://stackoverflow.com/questions/25304170/update-a-field-by-removing-quarter-or-removing-month When should I keep my code simple and when should I create a 'big', general solution? I feel stupid sometimes for building something so big, so awkward, just to solve a simple problem. It did not occur to me that there would be an easier solution. Any tips are welcomed. Best

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  • understanding evaluation function

    - by mish
    I am developing a chess program. And have made use of an alpha beta algorithm and a static evaluation function. I have successfully implemented both but I want to improve the evaluation function by automatically tuning the weights assigned to its features. At this point am totally confused about the policy suitable for updating the weights of the function. One policy I have in mind is to check whether a move is good or bad before updating weights but I really know how to implement it. Thus I need ideas and pseudo code please.

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