Necklace with Colored Beads Cutting Problem

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Necklace with Colored Beads Cutting Problem

• Victor Kostyuk
• Advisor Michael Capalbo

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Problem Setup

• Consider a necklace (cycle) with 2n beads of k
  colors. There are 2ai beads of color i, and the
  beads are arranged on the necklace arbitrarily.

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Goal - Efficient Cutting Alg.

• Is there an O(nc) algorithm for making the
  least number of cuts between beads such that the
  resulting bead strings can be partitioned into
  two groups, with ai beads of color i per group?

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Notes and Prospects

• Goldberg and West (1985) proved that such a
  partition is always possible with k1 cuts.
• There is an O(nk-2) algorithm for finding least
  number of cuts, but no O(nc) algorithm is known
  where c is independent of k.
• While such O(nc) algorithm might not exist, an
  improvement on O(nk-2) is a possibility.
• Perhaps O(nc) algorithm exits for O(k) cuts.

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Any questions?