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  • How to recalculate all-pairs shorthest paths on-line if nodes are getting removed?

    - by Pavel Shved
    Latest news about underground bombing made me curious about the following problem. Assume we have a weighted undirected graph, nodes of which are sometimes removed. The problem is to re-calculate shortest paths between all pairs of nodes fast after such removals. With a simple modification of Floyd-Warshall algorithm we can calculate shortest paths between all pairs. These paths may be stored in a table, where shortest[i][j] contains the index of the next node on the shortest path between i and j (or NULL value if there's no path). The algorithm requires O(n³) time to build the table, and eacho query shortest(i,j) takes O(1). Unfortunately, we should re-run this algorithm after each removal. The other alternative is to run graph search on each query. This way each removal takes zero time to update an auxiliary structure (because there's none), but each query takes O(E) time. What algorithm can be used to "balance" the query and update time for all-pairs shortest-paths problem when nodes of the graph are being removed?

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  • How to perform spatial partitioning in n-dimensions?

    - by kevin42
    I'm trying to design an implementation of Vector Quantization as a c++ template class that can handle different types and dimensions of vectors (e.g. 16 dimension vectors of bytes, or 4d vectors of doubles, etc). I've been reading up on the algorithms, and I understand most of it: here and here I want to implement the Linde-Buzo-Gray (LBG) Algorithm, but I'm having difficulty figuring out the general algorithm for partitioning the clusters. I think I need to define a plane (hyperplane?) that splits the vectors in a cluster so there is an equal number on each side of the plane. [edit to add more info] This is an iterative process, but I think I start by finding the centroid of all the vectors, then use that centroid to define the splitting plane, get the centroid of each of the sides of the plane, continuing until I have the number of clusters needed for the VQ algorithm (iterating to optimize for less distortion along the way). The animation in the first link above shows it nicely. My questions are: What is an algorithm to find the plane once I have the centroid? How can I test a vector to see if it is on either side of that plane?

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  • Suggestions of the easiest algorithms for some Graph operations

    - by Nazgulled
    Hi, The deadline for this project is closing in very quickly and I don't have much time to deal with what it's left. So, instead of looking for the best (and probably more complicated/time consuming) algorithms, I'm looking for the easiest algorithms to implement a few operations on a Graph structure. The operations I'll need to do is as follows: List all users in the graph network given a distance X List all users in the graph network given a distance X and the type of relation Calculate the shortest path between 2 users on the graph network given a type of relation Calculate the maximum distance between 2 users on the graph network Calculate the most distant connected users on the graph network A few notes about my Graph implementation: The edge node has 2 properties, one is of type char and another int. They represent the type of relation and weight, respectively. The Graph is implemented with linked lists, for both the vertices and edges. I mean, each vertex points to the next one and each vertex also points to the head of a different linked list, the edges for that specific vertex. What I know about what I need to do: I don't know if this is the easiest as I said above, but for the shortest path between 2 users, I believe the Dijkstra algorithm is what people seem to recommend pretty often so I think I'm going with that. I've been searching and searching and I'm finding it hard to implement this algorithm, does anyone know of any tutorial or something easy to understand so I can implement this algorithm myself? If possible, with C source code examples, it would help a lot. I see many examples with math notations but that just confuses me even more. Do you think it would help if I "converted" the graph to an adjacency matrix to represent the links weight and relation type? Would it be easier to perform the algorithm on that instead of the linked lists? I could easily implement a function to do that conversion when needed. I'm saying this because I got the feeling it would be easier after reading a couple of pages about the subject, but I could be wrong. I don't have any ideas about the other 4 operations, suggestions?

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  • Permuting a binary tree without the use of lists

    - by Banang
    I need to find an algorithm for generating every possible permutation of a binary tree, and need to do so without using lists (this is because the tree itself carries semantics and restraints that cannot be translated into lists). I've found an algorithm that works for trees with the height of three or less, but whenever I get to greater hights, I loose one set of possible permutations per height added. Each node carries information about its original state, so that one node can determine if all possible permutations have been tried for that node. Also, the node carries information on weather or not it's been 'swapped', i.e. if it has seen all possible permutations of it's subtree. The tree is left-centered, meaning that the right node should always (except in some cases that I don't need to cover for this algorithm) be a leaf node, while the left node is always either a leaf or a branch. The algorithm I'm using at the moment can be described sort of like this: if the left child node has been swapped swap my right node with the left child nodes right node set the left child node as 'unswapped' if the current node is back to its original state swap my right node with the lowest left nodes' right node swap the lowest left nodes two childnodes set my left node as 'unswapped' set my left chilnode to use this as it's original state set this node as swapped return null return this; else if the left child has not been swapped if the result of trying to permute left child is null return the permutation of this node else return the permutation of the left child node if this node has a left node and a right node that are both leaves swap them set this node to be 'swapped' The desired behaviour of the algoritm would be something like this: branch / | branch 3 / | branch 2 / | 0 1 branch / | branch 3 / | branch 2 / | 1 0 <-- first swap branch / | branch 3 / | branch 1 <-- second swap / | 2 0 branch / | branch 3 / | branch 1 / | 0 2 <-- third swap branch / | branch 3 / | branch 0 <-- fourth swap / | 1 2 and so on... Sorry for the ridiculisly long and waddly explanation, would really, really apreciate any sort of help you guys could offer me. Thanks a bunch!

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  • Splitting a set of object into several subsets of 'similar' objects

    - by doublep
    Suppose I have a set of objects, S. There is an algorithm f that, given a set S builds certain data structure D on it: f(S) = D. If S is large and/or contains vastly different objects, D becomes large, to the point of being unusable (i.e. not fitting in allotted memory). To overcome this, I split S into several non-intersecting subsets: S = S1 + S2 + ... + Sn and build Di for each subset. Using n structures is less efficient than using one, but at least this way I can fit into memory constraints. Since size of f(S) grows faster than S itself, combined size of Di is much less than size of D. However, it is still desirable to reduce n, i.e. the number of subsets; or reduce the combined size of Di. For this, I need to split S in such a way that each Si contains "similar" objects, because then f will produce a smaller output structure if input objects are "similar enough" to each other. The problems is that while "similarity" of objects in S and size of f(S) do correlate, there is no way to compute the latter other than just evaluating f(S), and f is not quite fast. Algorithm I have currently is to iteratively add each next object from S into one of Si, so that this results in the least possible (at this stage) increase in combined Di size: for x in S: i = such i that size(f(Si + {x})) - size(f(Si)) is min Si = Si + {x} This gives practically useful results, but certainly pretty far from optimum (i.e. the minimal possible combined size). Also, this is slow. To speed up somewhat, I compute size(f(Si + {x})) - size(f(Si)) only for those i where x is "similar enough" to objects already in Si. Is there any standard approach to such kinds of problems? I know of branch and bounds algorithm family, but it cannot be applied here because it would be prohibitively slow. My guess is that it is simply not possible to compute optimal distribution of S into Si in reasonable time. But is there some common iteratively improving algorithm?

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  • How to check if a number is a power of 2

    - by configurator
    Today I needed a simple algorithm for checking if a number is a power of 2. The algorithm needs to be: Simple Correct for any ulong value. I came up with this simple algorithm: private bool IsPowerOfTwo(ulong number) { if (number == 0) return false; for (ulong power = 1; power > 0; power = power << 1) { // this for loop used shifting for powers of 2, meaning // that the value will become 0 after the last shift // (from binary 1000...0000 to 0000...0000) then, the for // loop will break out if (power == number) return true; if (power > number) return false; } return false; } But then I thought, how about checking if log2x is an exactly round number? But when I checked for 2^63+1, Math.Log returned exactly 63 because of rounding. So I checked if 2 to the power 63 is equal to the original number - and it is, because the calculation is done in doubles and not in exact numbers: private bool IsPowerOfTwo_2(ulong number) { double log = Math.Log(number, 2); double pow = Math.Pow(2, Math.Round(log)); return pow == number; } This returned true for the given wrong value: 9223372036854775809. Does anyone have any suggestion for a better algorithm?

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  • How to find whole graph coverage path in dynamic state-flow diagram?

    - by joseph
    Hello, As I've been researching algorithms for path finding in graph, I found interesting problem. Definition of situation: 1)State diagram can have p states, and s Boolean Fields, and z Int Fields 2)Every state can have q ingoing and r outgoing transitions, and h Int fields (h belongs to z - see above) 3)Every transition can have only 1 event, and only 1 action 4)every action can change n Boolean Fields, and x Int Fields 5)every event can have one trigger from combination of any count of Boolean Fields in diagram 6)Transition can be in OPEN/CLOSED form. If the transition is open/closed depends on trigger2 compounded from 0..c Boolean fields. 7) I KNOW algorithm for finding shortest paths from state A to state B. 8) I KNOW algorithm for finding path that covers all states and transitions of whole state diagram, if all transitions are OPEN. Now, what is the goal: I need to find shortest path that covers all states and transitions in dynamically changing state diagram described above. When an action changes some int field, the algorithm should go through all states that have changed int field. The algorithm should also be able to open and close transition (by going through transitions that open and close another transitions by action) in the way that the founded path will be shortest and covers all transitions and states. Any idea how to solve it? I will be really pleased for ANY idea. Thanks for answers.

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  • Point covering problem

    - by Sean
    I recently had this problem on a test: given a set of points m (all on the x-axis) and a set n of lines with endpoints [l, r] (again on the x-axis), find the minimum subset of n such that all points are covered by a line. Prove that your solution always finds the minimum subset. The algorithm I wrote for it was something to the effect of: (say lines are stored as arrays with the left endpoint in position 0 and the right in position 1) algorithm coverPoints(set[] m, set[][] n): chosenLines = [] while m is not empty: minX = min(m) bestLine = n[0] for i=1 to length of n: if n[i][0] <= m and n[i][1] > bestLine[1] then bestLine = n[i] add bestLine to chosenLines for i=0 to length of m: if m <= bestLine[1] then delete m[i] from m return chosenLines I'm just not sure if this always finds the minimum solution. It's a simple greedy algorithm so my gut tells me it won't, but one of my friends who is much better than me at this says that for this problem a greedy algorithm like this always finds the minimal solution. For proving mine always finds the minimal solution I did a very hand wavy proof by contradiction where I made an assumption that probably isn't true at all. I forget exactly what I did. If this isn't a minimal solution, is there a way to do it in less than something like O(n!) time? Thanks

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  • Whats the best way to design this database scenario?

    - by ankimal
    I want to setup 2 MySQL databases which differ in schema in that, one is normalized and the other is flat for quicker reads and writes. The information being stored in both DBs is the same, but the representation is obviously different owing to the different design approaches. I need to find a robust solution to sync information in real time from my normalized version to my flatter version.

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  • What method do you use to identify the Aggregate Roots in Domain Drive Design?

    - by Robert
    When applying Domain Driven Design to a project, how do you identify the Aggregate Roots? For example, in a standard E-Commerce website, you might say that the Order is one, and the User is the other. But what if your Users belong to a Company? Does that make your Company the aggregate root? I'm interested in hearing people's approaches to working out the Aggregate roots, and how to identify poorly chosen aggregate roots.

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