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Math Help - Set notation for a computer network

  1. #1
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    Set notation for a computer network

    Hi Everyone

    I want to write a set notation for the following problem.

    Suppose we have a computer network like the one in the Figure(attached here). Suppose we have 3 paths(indicated by Red,Green and Blue Lines) between the
    source node B and the destination node C. Through some method, we get the minimum value of links of all the 3 paths, which in this case are:
    1. 5 for the Green Path (5 is a minimum between 5 & 15)
    2. 5 for the Blue Path (5 is a minimum between 5 & 10)
    3. 25 for the Red Path (25 is the only number here)
    SO in the end we have a set which contains 3 values {5,5,25}


    Now i want to write a notation/function in Mathematical form that describes how the above is achieved.I was thinking of something like this:

    Y(P) = {Y|Y ϵ Z+ and Y contains minimum values of all the I number of paths}
    What i mean is that P is a path between source and destination nodes(B & C in this example), I is the number of paths that exist between
    two nodes(I=3 in the above case), Y contains set of minimum link values of all the I number of paths({5,5,25}) in this case, which belong to set of
    positive integers Z+.



    If anybody can suggest any improvement/alternate or identify/correct mistakes, in this regard, it is highly appreciated.

    Thanking You

    Best Regards
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  2. #2
    MHF Contributor
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    SO in the end we have a set which contains 3 values {5,5,25}
    In a set, the number of times an element occurs does not matter. If you want to keep track of the number of occurrences, you need a multiset.

    Y(P) = {Y|Y ϵ Z+ and Y contains minimum values of all the I number of paths}
    What i mean is that P is a path between source and destination nodes(B & C in this example), I is the number of paths that exist between
    two nodes(I=3 in the above case), Y contains set of minimum link values of all the I number of paths({5,5,25}) in this case, which belong to set of
    positive integers Z+.
    * It's not good to use the same variable name for two entities. In Y(P), Y is a function that takes a path, but in the right-hand side, Y is a number.

    * The left-hand side Y(P) takes a particular path P as an argument, so the right-hand side must be some characteristic of this path. However, the right-hand side is a property of all paths.

    * "Y ϵ Z+ and Y contains minimum values": If Y is a number, it cannot contain anything. Only sets can.

    * The right-hand side has a variable I that does not occur in the left-hand side. It is not clear what its value should be.

    I would first come up with a definition of a path (e.g., a sequence of edges with some properties). Then I would define a function MinEdge(P) for a path P. Then, given a network N and two nodes B and C, one can define

    Y(N,B,C) = {MinEdge(P) | P is a path in N between B and C}

    Alternatively,

    Y(N,B,C) = {n in Z^+ | n = MinEdge(P) for some path P in N between B and C}
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  3. #3
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    Dear "emakarov"

    Thank you very much for your time and help.

    Best Regards
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