Lines Tangent to Four Unit Balls problem 2

The Golden Partition Conjecture is motivated by conjectures whose root is a famous sorting problem. Imagine that a finite set S with cardinality n has a hidden total order ≺ that we would like to uncover by making comparisons between pairs of elements from S. Comparing a pair u, v ∈ S means uncovering either u ≺ v or v ≺ u. How many comparisons are needed in the worst case? The classical answer is Θ(n log2(n)) comparisons are needed and ‘merge sort’, for example, gives an algorithm to achieve this bound. How many comparisons are needed if some partial information about ≺ is already known? Partial information about ≺ can be summarized by a poset P (≤, S). Let C(P ) denote the number of comparisons required to find the hidden linear extension ≺, in the worst case, starting from partial information P? Clearly C(P ) ≥ log2(e(P )), since each comparison can reduce the number of linear extensions by a factor of at most 2. The Golden Partition Conjecture gets its name because Peczarski [Pec06] has shown that it implies C(P ) ≤ logφ(e(P )), where φ = 1+ √ 5 2 ≈ 1.618033988 is the golden ratio. As Peczarski states, “informally this [bound] means that during the sorting process the number of linear extensions can be decreased in every comparison on average by at least the golden ratio φ.” Linial [Lin84] has constructed a sequence of posets that show that, if true, this bound would be tight. The survey article by Brightwell [Bri99] is highly recommended. Notes: