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• Correct formulation of the A* algorithm

  - by Eli Bendersky

  Hello, I'm looking at definitions of the A* path-finding algorithm, and it seems to be defined somewhat differently in different places. The difference is in the action performed when going through the successors of a node, and finding that a successor is on the closed list. One approach (suggested by Wikipedia, and this article) says: if the successor is on the closed list, just ignore it Another approach (suggested here and here, for example) says: if the successor is on the closed list, examine its cost. If it's higher than the currently computed score, remove the item from the closed list for future examination. I'm confused - which method is correct ? Intuitively, the first makes more sense to me, but I wonder about the difference in definition. Is one of the definitions wrong, or are they somehow isomorphic ?

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• Algorithm for condensing/consolidating number combinations

  - by user1404383

  Using a horse race betting scenario, say I have a number of separate bets for predicting the first 4 finishers of the race (superfecta). The bets are as follows... 1/2/3/4 1/2/3/5 1/2/4/3 1/2/4/5 1/2/5/3 1/2/5/4 What I want to do is combine or condense these separate combinations as much as possible. For the bets above, they can be all condensed into 1 line... 1/2/3,4,5/3,4,5 What would algorithm look like for this?

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• Neural Network Basics

  - by Stat Onetwothree

  I'm a computer science student and for this years project, I need to create and apply a Genetic Algorithm to something. I think Neural Networks would be a good thing to apply it to, but I'm having trouble understanding them. I fully understand the concepts but none of the websites out there really explain the following which is blocking my understanding: How the decision is made for how many nodes there are. What the nodes actually represent and do. What part the weights and bias actually play in classification. Could someone please shed some light on this for me? Also, I'd really appreciate it if you have any similar ideas for what I could apply a GA to. Thanks very much! :)

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• Explain this O(n log n) algorithm for the Cat/Egg Throwing Problem

  - by ripper234

  This problem (How many cats you need to throw out of a building in order to determine the maximal floor where such a cat will survive. Quite cruel, actually), has an accepted answer with O(n^3) complexity. The problem is equivalent to this Google Code Jam, which should be solvable for N=2000000000. It seems that the O(n^3) solution is not good enough to solve it. From looking in the solutions page, jdmetz's solution (#2) seems to be O(n log n). I don't quite understand the algorithm. Can someone explain it? Edit

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• Parallelizing a serial algorithm

  - by user643813

  Hej folks, I am working on porting a Text mining/Natural language application from single-core to a Map-Reduce style system. One of the steps involves a while loop similar to this: Queue<Element>; while (!queue.empty()) { Element e = queue.next(); Set<Element> result = calculateResultSet(e); if (!result.empty()) { queue.addAll(result); } } Each iteration depends on the result of the one before (kind of). There is no way of determining the number of iterations this loop will have to perform. Is there a way of parallelizing a serial algorithm such as this one? I am trying to think of a feedback mechanism, that is able to provide its own input, but how would one go about parallelizing it? Thanks for any help/remarks

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• elastic / snaking line algorithm

  - by vhdirk

  Hi everyone I am making a graphics application in which I can edit a polyline by dragging the control point of it. However, I'd like to make it a bit easier to use by making it elastic; When dragging a control point, instead of moving a single point, I'd like the points within a certain distance of that point to be moved as well, depending on how hard the control point is 'pulled'. Does anyone know a simple algorithm for this? It may be quite rudimentary, as the primary requirement is speed. Actually, knowing how to call such behaviour would also be nice, so I can look it up on google. I tried 'snaking' line, but that seems to refer to active contours, which isn't what I'm looking for. Thanks

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• algorithm to combine data for linear fit?

  - by BoldlyBold

  I'm not sure if this is the best place to ask this, but you guys have been helpful with plenty of my CS homework in the past so I figure I'll give it a shot. I'm looking for an algorithm to blindly combine several dependent variables into an index that produces the best linear fit with an external variable. Basically, it would combine the dependent variables using different mathematical operators, include or not include each one, etc. until an index is developed that best correlates with my external variable. Has anyone seen/heard of something like this before? Even if you could point me in the right direction or to the right place to ask, I would appreciate it. Thanks.

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• Algorithm of JavaScript "sort()" Function

  - by Knowledge Craving

  Recently when I was working with JavaScript "sort()" function, I found in one of the tutorials that this function does not sort the numbers properly. Instead to sort numbers, a function must be added that compares numbers, like the following code:- <script type="text/javascript"> function sortNumber(a,b) { return a - b; } var n = ["10", "5", "40", "25", "100", "1"]; document.write(n.sort(sortNumber)); </script> The output then comes as:- 1,5,10,25,40,100 Now what I didn't understand is that why is this occurring & can anybody please tell in details as to what type of algorithm is being used in this "sort()" function? This is because for any other language, I didn't find this problem where the function didn't sort the numbers correctly. Any help is greatly appreciated.

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• Render rivers in a grid.

  - by Gabriel A. Zorrilla

  I have created a random height map and now i want to create rivers. I've made an algorithm based on a* to make rivers flow from peaks to sea and now i'm in the quest of figuring out an elegant algorithm to render them. It's a 2D, square, mapgrid. The cells which the river pases has a simple integer value with this form :rivernumber && pointOrder. Ie: 10, 11, 12, 13, 14, 15, 16...1+N for the first river, 20,21,22,23...2+N for the second, etc. This is created in the map grid generation time and it's executed just once, when the world is generated. I wanted to treat each river as a vector, but there is a problem, if the same river has branches (because i put some noise to generate branches), i can not just connect the points in order. The second alternative is to generate a complex algorithm where analizes each point, checks if the next is not a branch, if so trigger another algorithm that take care of the branch then returns to the main river, etc. Very complex and inelegant. Perhaps there is a solution in the world generation algorithm or in the river rendering algorithm that is commonly used in these cases and i'm not aware of. Any tips? Thanks!!

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• Resultant Vector Algorithm for 2D Collisions

  - by John

  I am making a Pong based game where a puck hits a paddle and bounces off. Both the puck and the paddles are Circles. I came up with an algorithm to calculate the resultant vector of the puck once it meets a paddle. The game seems to function correctly but I'm not entirely sure my algorithm is correct. Here are my variables for the algorithm: Given: velocity = the magnitude of the initial velocity of the puck before the collision x = the x coordinate of the puck y = the y coordinate of the puck moveX = the horizontal speed of the puck moveY = the vertical speed of the puck otherX = the x coordinate of the paddle otherY = the y coordinate of the paddle piece.horizontalMomentum = the horizontal speed of the paddle before it hits the puck piece.verticalMomentum = the vertical speed of the paddle before it hits the puck slope = the direction, in radians, of the puck's velocity distX = the horizontal distance between the center of the puck and the center of the paddle distY = the vertical distance between the center of the puck and the center of the paddle Algorithm solves for: impactAngle = the angle, in radians, of the angle of impact. newSpeedX = the speed of the resultant vector in the X direction newSpeedY = the speed of the resultant vector in the Y direction Here is the code for my algorithm: int otherX = piece.x; int otherY = piece.y; double velocity = Math.sqrt((moveX * moveX) + (moveY * moveY)); double slope = Math.atan(moveX / moveY); int distX = x - otherX; int distY = y - otherY; double impactAngle = Math.atan(distX / distY); double newAngle = impactAngle + slope; int newSpeedX = (int)(velocity * Math.sin(newAngle)) + piece.horizontalMomentum; int newSpeedY = (int)(velocity * Math.cos(newAngle)) + piece.verticalMomentum; for those who are not program savvy here is it simplified: velocity = v(moveX² + moveY²) slope = arctan(moveX / moveY) distX = x - otherX distY = y - otherY impactAngle = arctan(distX / distY) newAngle = impactAngle + slope newSpeedX = velocity * sin(newAngle) + piece.horizontalMomentum newSpeedY = velocity * cos(newAngle) + piece.verticalMomentum My Question: Is this algorithm correct? Is there an easier/simpler way to do what I'm trying to do?

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• Is there a perfect algorithm for chess?

  - by Overflown

  Dear Stack Overflow community, I was recently in a discussion with a non-coder person on the possibilities of chess computers. I'm not well versed in theory, but think I know enough. I argued that there could not exist a deterministic Turing machine that always won or stalemated at chess. I think that, even if you search the entire space of all combinations of player1/2 moves, the single move that the computer decides upon at each step is based on a heuristic. Being based on a heuristic, it does not necessarily beat ALL of the moves that the opponent could do. My friend thought, to the contrary, that a computer would always win or tie if it never made a "mistake" move (however do you define that?). However, being a programmer who has taken CS, I know that even your good choices - given a wise opponent - can force you to make "mistake" moves in the end. Even if you know everything, your next move is greedy in matching a heuristic. Most chess computers try to match a possible end game to the game in progress, which is essentially a dynamic programming traceback. Again, the endgame in question is avoidable though. -- thanks, Allan Edit: Hmm... looks like I ruffled some feathers here. That's good. Thinking about it again, it seems like there is no theoretical problem with solving a finite game like chess. I would argue that chess is a bit more complicated than checkers in that a win is not necessarily by numerical exhaustion of pieces, but by a mate. My original assertion is probably wrong, but then again I think I've pointed out something that is not yet satisfactorily proven (formally). I guess my thought experiment was that whenever a branch in the tree is taken, then the algorithm (or memorized paths) must find a path to a mate (without getting mated) for any possible branch on the opponent moves. After the discussion, I will buy that given more memory than we can possibly dream of, all these paths could be found.

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• Accurate least-squares fit algorithm needed

  - by ggkmath

  I've experimented with the two ways of implementing a least-squares fit (LSF) algorithm shown here. The first code is simply the textbook approach, as described by Wolfram's page on LSF. The second code re-arranges the equation to minimize machine errors. Both codes produce similar results for my data. I compared these results with Matlab's p=polyfit(x,y,1) function, using correlation coefficients to measure the "goodness" of fit and compare each of the 3 routines. I observed that while all 3 methods produced good results, at least for my data, Matlab's routine had the best fit (the other 2 routines had similar results to each other). Matlab's p=polyfit(x,y,1) function uses a Vandermonde matrix, V (n x 2 matrix) and QR factorization to solve the least-squares problem. In Matlab code, it looks like: V = [x1,1; x2,1; x3,1; ... xn,1] % this line is pseudo-code [Q,R] = qr(V,0); p = R\(Q'*y); % performs same as p = V\y I'm not a mathematician, so I don't understand why it would be more accurate. Although the difference is slight, in my case I need to obtain the slope from the LSF and multiply it by a large number, so any improvement in accuracy shows up in my results. For reasons I can't get into, I cannot use Matlab's routine in my work. So, I'm wondering if anyone has a more accurate equation-based approach recommendation I could use that is an improvement over the above two approaches, in terms of rounding errors/machine accuracy/etc. Any comments appreciated! thanks in advance.

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• Algorithm to get through a maze

  - by Sam

  Hello, We are currently programming a game (its a pretty unknown language: modula 2), And the problem we encountered is the following: we have a maze (not a perfect maze) in a 17 x 12 grid. The computer has to generate a way from the starting point (9, 12) to the end point (9, 1). I found some algorithms but they dont work when the robot has to go back: xxxxx x => x x xxx or: xxxxx x xxxxxx x x x x x xxxxxx x => x xxxxxxxxx I found a solution for the first example type but then the second type couldn't be solved and the solution I made up for the second type would cause the robot to get stuck in the first type of situation. It's a lot of code so i'll give the idea: WHILE (end destination not reached) DO { try to go right, if nothing blocks you: go right if you encounter a block, try up until you can go right, if you cant go up anymore try going down until you can go right, (starting from the place you first were blocked), if you cant go down anymore, try one step left and fill the spaces you tested with blocks. } This works for the first type of problem ... not for the second one. Now it could be that i started wrong so i am open for better algorithms or solutions specificaly to how i could improve my algorithm. Thanks a lot!!

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• Unexpected result in C algebra for search algorithm.

  - by Rhys

  Hi, I've implemented this search algorithm for an ordered array of integers. It works fine for the first data set I feed it (500 integers), but fails on longer searches. However, all of the sets work perfectly with the other four search algorithms I've implemented for the assignment. This is the function that returns a seg fault on line 178 (due to an unexpected negative m value). Any help would be greatly appreciated. CODE: 155 /* perform Algortihm 'InterPolationSearch' on the set 156 * and if 'key' is found in the set return it's index 157 * otherwise return -1 */ 158 int 159 interpolation_search(int *set, int len, int key) 160 { 161 int l = 0; 162 int r = len - 1; 163 int m; 164 165 while (set[l] < key && set[r] >= key) 166 { 167 168 printf ("m = l + ((key - set[l]) * (r - l)) / (set[r] - set[l])\n"); 169 170 printf ("m = %d + ((%d - %d) * (%d - %d)) / (%d - %d);\n", l, key, set[l], r, l, set[r], set[l]); 171 m = l + ((key - set[l]) * (r - l)) / (set[r] - set[l]); 172 printf ("m = %d\n", m); 173 174 #ifdef COUNT_COMPARES 175 g_compares++; 176 #endif 177 178 if (set[m] < key) 179 l = m + 1; 180 else if (set[m] > key) 181 r = m - 1; 182 else 183 return m; 184 } 185 186 if (set[l] == key) 187 return l; 188 else 189 return -1; 190 } OUTPUT: m = l + ((key - set[l]) * (r - l)) / (set[r] - set[l]) m = 0 + ((68816 - 0) * (100000 - 0)) / (114836 - 0); m = -14876 Thankyou! Rhys

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• Using Traveling Salesman Solver to Decide Hamiltonian Path

  - by Firas Assaad

  This is for a project where I'm asked to implement a heuristic for the traveling salesman optimization problem and also the Hamiltonian path or cycle decision problem. I don't need help with the implementation itself, but have a question on the direction I'm going in. I already have a TSP heuristic based on a genetic algorithm: it assumes a complete graph, starts with a set of random solutions as a population, and works to improve the population for a number of generations. Can I also use it to solve the Hamiltonian path or cycle problems? Instead of optimizing to get the shortest path, I just want to check if there is a path. Now any complete graph will have a Hamiltonian path in it, so the TSP heuristic would have to be extended to any graph. This could be done by setting the edges to some infinity value if there is no path between two cities, and returning the first path that is a valid Hamiltonian path. Is that the right way to approach it? Or should I use a different heuristic for Hamiltonian path? My main concern is whether it's a viable approach since I can be somewhat sure that TSP optimization works (because you start with solutions and improve them) but not if a Hamiltonian path decider would find any path in a fixed number of generations. I assume the best approach would be to test it myself, but I'm constrained by time and thought I'd ask before going down this route... (I could find a different heuristic for Hamiltonian path instead)

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• Resizing image algorithm in python

  - by hippocampus

  So, I'm learning my self python by this tutorial and I'm stuck with exercise number 13 which says: Write a function to uniformly shrink or enlarge an image. Your function should take an image along with a scaling factor. To shrink the image the scale factor should be between 0 and 1 to enlarge the image the scaling factor should be greater than 1. This is not meant as a question about PIL, but to ask which algorithm to use so I can code it myself. I've found some similar questions like this, but I dunno how to translate this into python. Any help would be appreciated. I've come to this: import image win = image.ImageWin() img = image.Image("cy.png") factor = 2 W = img.getWidth() H = img.getHeight() newW = int(W*factor) newH = int(H*factor) newImage = image.EmptyImage(newW, newH) for col in range(newW): for row in range(newH): p = img.getPixel(col,row) newImage.setPixel(col*factor,row*factor,p) newImage.draw(win) win.exitonclick() I should do this in a function, but this doesn't matter right now. Arguments for function would be (image, factor). You can try it on OP tutorial in ActiveCode. It makes a stretched image with empty columns :.

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• Video-codec rater by image comparison algorithm?

  - by Andreas Hornig

  Hi, perhaps anyone knows if this is possible. comparing image quality is almost imposible to describe without subjective influences. When someone rates an image quality as good there is at least one person, that doesn't think so. human preferences are always different. So, I would like to know if there is away to "rate" the image quality by an algorithm that compares the original image to the produced one in following issues colour change(difference pixel by pixel blur rate artifacts and macroblocking the first one would be the easiest one because you could check just the diffeence in colours and can give 3 values in +- of each hex-value both last once I don't know if this is possible, but the blocking could be detected by edge-finding. and the king's quest would be to do that for more then just one image, because video is done with several frames. perhaps you expert programmers could tell me, if such an automated algo can be done to bring some objective measurement divice into rating image quality. this could perhaps calm down some h.264 is better than x264 and better than vp8 and blaaah people :) Andreas 1st posted here http://www.hdtvtotal.com/index.php?name=PNphpBB2&file=viewtopic&p=9705

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• Bitap algorithm in Java [closed]

  - by davit-datuashvili

  The following is the bitap algorithm according to Wikipedia. Can someone translate this to Java? #include <string.h> #include <limits.h> const char *bitap_bitwise_search(const char *text, const char *pattern) { int m = strlen(pattern); unsigned long R; unsigned long pattern_mask[CHAR_MAX+1]; int i; if (pattern[0] == '\0') return text; if (m > 31) return "The pattern is too long!"; /* Initialize the bit array R */ R = ~1; /* Initialize the pattern bitmasks */ for (i=0; i <= CHAR_MAX; ++i) pattern_mask[i] = ~0; for (i=0; i < m; ++i) pattern_mask[pattern[i]] &= ~(1UL << i); for (i=0; text[i] != '\0'; ++i) { /* Update the bit array */ R |= pattern_mask[text[i]]; R <<= 1; if (0 == (R & (1UL << m))) return (text + i - m) + 1; } return NULL; }

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• Average performance of binary search algorithm?

  - by Passionate Learner

  http://en.wikipedia.org/wiki/Binary_search_algorithm#Average_performance BinarySearch(int A[], int value, int low, int high) { int mid; if (high < low) return -1; mid = (low + high) / 2; if (A[mid] > value) return BinarySearch(A, value, low, mid-1); else if (A[mid] < value) return BinarySearch(A, value, mid+1, high); else return mid; } If the integer I'm trying to find is always in the array, can anyone help me write a program that can calculate the average performance of binary search algorithm? I know I can do this by actually running the program and counting the number of calls, but what I'm trying to do here is to do it without calling the function. I'm not asking for a time complexity, I'm trying to calculate the average number of calls. For example, the average number of calls to find a integer in A[2], it would be 1.67 (5/3).

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• Open space sitting optimization algorithm

  - by Georgy Bolyuba

  As a result of changes in the company, we have to rearrange our sitting plan: one room with 10 desks in it. Some desks are more popular than others for number of reasons. One solution would be to draw a desk number from a hat. We think there is a better way to do it. We have 10 desks and 10 people. Lets give every person in this contest 50 hypothetical tokens to bid on the desks. There is no limit of how much you bid on one desk, you can put all 50, which would be saying "I want to sit only here, period". You can also say "I do not care" by giving every desk 5 tokens. Important note: nobody knows what other people are doing. Everyone has to decide based only on his/her best interest (sounds familiar?) Now lets say we obtained these hypothetical results: # | Desk# >| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 1 | Alise | 30 | 2 | 2 | 1 | 0 | 0 | 0 | 15 | 0 | 0 | = 50 2 | Bob | 20 | 15 | 0 | 10 | 1 | 1 | 1 | 1 | 1 | 0 | = 50 ... 10 | Zed | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | = 50 Now, what we need to find is that one (or more) configuration(s) that gives us maximum satisfaction (i.e. people get desks they wanted taking into account all the bids and maximizing on the total of the group. Naturally the assumption is the more one bade on the desk the more he/she wants it). Since there are only 10 people, I think we can brute force it looking into all possible configurations, but I was wondering it there is a better algorithm for solving this kind of problems?

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• priority queue with limited space: looking for a good algorithm

  - by SigTerm

  This is not a homework. I'm using a small "priority queue" (implemented as array at the moment) for storing last N items with smallest value. This is a bit slow - O(N) item insertion time. Current implementation keeps track of largest item in array and discards any items that wouldn't fit into array, but I still would like to reduce number of operations further. looking for a priority queue algorithm that matches following requirements: queue can be implemented as array, which has fixed size and _cannot_ grow. Dynamic memory allocation during any queue operation is strictly forbidden. Anything that doesn't fit into array is discarded, but queue keeps all smallest elements ever encountered. O(log(N)) insertion time (i.e. adding element into queue should take up to O(log(N))). (optional) O(1) access for *largest* item in queue (queue stores *smallest* items, so the largest item will be discarded first and I'll need them to reduce number of operations) Easy to implement/understand. Ideally - something similar to binary search - once you understand it, you remember it forever. Elements need not to be sorted in any way. I just need to keep N smallest value ever encountered. When I'll need them, I'll access all of them at once. So technically it doesn't have to be a queue, I just need N last smallest values to be stored. I initially thought about using binary heaps (they can be easily implemented via arrays), but apparently they don't behave well when array can't grow anymore. Linked lists and arrays will require extra time for moving things around. stl priority queue grows and uses dynamic allocation (I may be wrong about it, though). So, any other ideas?

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• Algorithm to determine which points should be visible on a map based on zoom

  - by lgratian

  Hi! I'm making a Google Maps-like application for a course at my Uni (not something complex, it should load the map of a city for example, not the whole world). The map can have many layers, including markers (restaurants, hospitals, etc.) The problem is that when you have many points and you zoom out the map it doesn't look right. At this zoom level only some points need to be visible (and at the maximum map size, all points). The question is: how can you determine which points should be visible for a specified zoom level? Because I have implemented a PR Quadtree to speed up rendering I thought that I could define some "high-priority" markers (that are always visible, defined in the map editor) and put them in a queue. At each step a marker is removed from the queue and all it's neighbors that are at least D units away (D depends on the zoom levels) are chosen and inserted in the queue, and so on. Is there any better way than the algorithm I thought of? Thanks in advance!

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• Number Algorithm

  - by James

  I've been struggling to wrap my head around this for some reason. I have 15 bits that represent a number. The bits must match a pattern. The pattern is defined in the way the bits start out: they are in the most flush-right representation of that pattern. So say the pattern is 1 4 1. The bits will be: 000000010111101 So the general rule is, take each number in the pattern, create that many bits (1, 4 or 1 in this case) and then have at least one space separating them. So if it's 1 2 6 1 (it will be random): 001011011111101 Starting with the flush-right version, I want to generate every single possible number that meets that pattern. The # of bits will be stored in a variable. So for a simple case, assume it's 5 bits and the initial bit pattern is: 00101. I want to generate: 00101 01001 01010 10001 10010 10100 I'm trying to do this in Objective-C, but anything resembling C would be fine. I just can't seem to come up with a good recursive algorithm for this. It makes sense in the above example, but when I start getting into 12431 and having to keep track of everything it breaks down.

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• Algorithm to see if keywords exist inside a string

  - by rksprst

  Let's say I have a set of keywords in an array {"olympics", "sports tennis best", "tennis", "tennis rules"} I then have a large list (up to 50 at a time) of strings (or actually tweets), so they are a max of 140 characters. I want to look at each string and see what keywords are present there. In the case where a keyword is composed of multiple words like "sports tennis best", the words don't have to be together in the string, but all of them have to show up. I've having trouble figuring out an algorithm that does this efficiently. Do you guys have suggestions on a way to do this? Thanks! Edit: To explain a bit better each keyword has an id associated with it, so {1:"olympics", 2:"sports tennis best", 3:"tennis", 4:"tennis rules"} I want to go through the list of strings/tweets and see which group of keywords match. The output should be, this tweet belongs with keyword #4. (multiple matches may be made, so anything that matches keyword 2, would also match 3 -since they both contain tennis). When there are multiple words in the keyword, e.g. "sports tennis best" they don't have to appear together but have to all appear. e.g. this will correctly match: "i just played tennis, i love sports, its the best"... since this string contains "sports tennis best" it will match and be associated with the keywordID (which is 2 for this example).

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• Algorithm for finding the best routes for food distribution in game

  - by Tautrimas

  Hello, I'm designing a city building game and got into a problem. Imagine Sierra's Caesar III game mechanics: you have many city districts with one market each. There are several granaries over the distance connected with a directed weighted graph. The difference: people (here cars) are units that form traffic jams (here goes the graph weights). Note: in Ceasar game series, people harvested food and stockpiled it in several big granaries, whereas many markets (small shops) took food from the granaries and delivered it to the citizens. The task: tell each district where they should be getting their food from while taking least time and minimizing congestions on the city's roads. Map example Sample diagram Suppose that yellow districts need 7, 7 and 4 apples accordingly. Bluish granaries have 7 and 11 apples accordingly. Suppose edges weights to be proportional to their length. Then, the solution should be something like the gray numbers indicated on the edges. Eg, first district gets 4 apples from the 1st and 3 apples from the 2nd granary, while the last district gets 4 apples from only the 2nd granary. Here, vertical roads are first occupied to the max, and then the remaining workers are sent to the diagonal paths. Question What practical and very fast algorithm should I use? I was looking at some papers (Congestion Games: Optimization in Competition etc.) describing congestion games, but could not get the big picture. Any help is very appreciated! P. S. I can post very little links and no images because of new user restriction.

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