Theorem 2 shows that the amortized running time of the Move-to-Front (MF) heuristic on linear lists is within a constant (i.e., 2) multiplicative factor of the best off-line strategy. Does the same result hold (possibly with a somewhat larger multiplicative constant factor) for the Move-Half-Way-to-Front (MHWF) strategy? Prove your claim. (The latter strategy moves the accessed/inserted item half-way towards the front. That is, if the item was at position i, then it is moved to position ⎡i/2⎤, without affecting the relative order of the other elements.)
[Hint: First compare MHWF with MF, then apply transitivity.]