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Minimum Combinations problem, n detectives with unique clues. Min. no. of phonecalls?

  1. Dec 27, 2012 #1
    1. The problem statement, all variables and given/known data
    We have n detectives. At the start each has a single unique clue.
    The aim is for all the n detectives to obtain all the n clues.
    If detective A knows clue 1 and 33; and B knows, 1, 2 and 4. If A phone calls B they share their clues. i.e. A will know 1, 2, 4 and 33 and B the same.

    Q: Whats the minimum number of phone calls for all detectives to know all clues?

    2. Relevant equations
    P(n)= minimum number of phone calls for n detectives.

    3. The attempt at a solution
    I represent this with a polyhedra with n vertices, and with links representing calls.
    A simple strategy is for A to call (n-1) people, then go in reverse order and call (n-2) people.
    P(n)≤ (n-1)+(n-2) =2n-3

    For a value under I use a bit of a guess of something that grows fast. Like this. So if we have 8 we get to 2 knowing all with 1+2+4 calls. 1+2+4=8-1 Therefore P(n)>n-1. (n≠2)

    How to solve:
    Find an algorithm for round numbers (2^m or even 2m). Introduce corrections for non round numbers.
    Prove the algorithm is the best with symmetry arguments mainly.
    This works I think but it looks a bit extensive.

    One algorithm I use is basically:
    Divide the people into n/2 pairs, with the left L and the right R person of pair p.
    Link L of pair p with R of pair p+1.
    Link L of p with R of p+2.
    Repeat: Link L of p with R of p+k. Repeat until all know all.

    For odd numbers an easy fix that's not necessarily the optimal is this: Take A away to make it an even group. Link A with a random element. Run even algorithm. Link A again with a random element.

    A list of values P(n) for n, some may be wrong:
    n____2_3_4_5_6_8__10
    P(n)._1_3_4_5_9_12_16
     
    Last edited: Dec 27, 2012
  2. jcsd
  3. Dec 27, 2012 #2

    haruspex

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    Re: Minimum Combinations problem, n detectives with unique clues. Min. no. of phoneca

    I couldn't figure out that list until I 'quoted' it. That revealed gaps at 7 and 9.
    The suggestion to look at powers of 2 is a good one. That reveals an alogorithm that takes n log2(n) / 2 if n is a power of 2, though I haven't proved it's optimal. Fits your P(8) = 12.
     
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