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k paths

6.5 any algorithm for a path ≥ 3/2. k paths. k=12 x=6. k-1 spaces:. x between same color. k-1-x between different colors. So far: any algorithm uses 2k ADMs. now – a short path at each gap of diffferent colors ( k-1-x such gaps). k=12, x=6, 12-1-6=5.

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k paths

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  1. 6.5 any algorithm for a path ≥ 3/2 k paths k=12 x=6 k-1 spaces: x between same color k-1-x between different colors

  2. So far: any algorithm uses 2k ADMs now – a short path at each gap of diffferent colors (k-1-x such gaps) k=12, x=6, 12-1-6=5 Any algorithm uses at least one more ADM for each (ALG uses exactly one) So: any algorithm ≥2k + (k-1-x) ADMs

  3. So far: use ≥2k + (k-1-x) ADMs now – two long paths at each of the k gap of same color Any algorithm must use 2 ADMs for each So: any algorithm ≥2k + (k-1-x) + 4x = =3k+3x-1 ADMs

  4. We showed: any algorithm uses ≥ 3k+3x-1 ADMs OPT: the short paths ≤ 2k ADMs for the long paths 2x ADMs OPT 2k+ 2x any algorithm/OPT  3/2 – 1/(2k)

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