Necklace with Colored Beads Cutting Problem

Necklace with Colored Beads Cutting Problem

Victor Kostyuk

Advisor: Michael Capalbo

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.

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?

Notes and Prospects

• Goldberg and West (1985) proved that such a partition is always possible with k+1 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.

Any questions?