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Tricky question about Tree Construction

  1. Oct 24, 2009 #1
    1. The problem statement, all variables and given/known data

    There are N signals that must be all ANDed together using two-input AND gates, and produce a result as fast as possible.

    The N signals arrive at different times (T1 to TN)

    An AND gate first waits for both input signals to arrive, and then takes 1 second to output the resultant signal.

    Prove that, no matter whatever scheme is used, the shortest calculation time satisfies :

    Total time >= log2( sum( 2^Ti) )

    And that the optimal scheme guarantees:

    Total time <= log2( sum(2^Ti) ) + 1

    3. The attempt at a solution

    Following a derivation similar to the derivation for the Huffman code, I have proven that the optimal scheme is to always combine the two earliest signals, at any one point in time.

    But I can't prove that this scheme satisfies the specified bound.

    Thanks for your help
    -Patrick
     
  2. jcsd
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