Demystifying the Master Thesis and Research in General: The Story of Some Master Theses

I don’t think that there is a generically good way of doing a master thesis (nor of doing research in general). Research (like other creative activities) evolves in unpredictable ways, and each research project has its own story. I will tell a few such stories, and naturally I will rely on stories I know from the inside (or close to that). The only lesson that I can offer is maintaining an openness towards ideas that may emerge. (Notes for a talk to be given on January 20, 2008.) 1 My own thesis (1981) In one of my first meeting with my predetermined interim supervisor, Shimon Even (who has later become my Master and Doctorate thesis advisor) tossed to my direction a Rubic Cube and asked if I can arrange it. A few days later, when I described to him a highly wasteful algorithm, the question of the minimum move sequence arose naturally. Phrased in general terms, this yields a computational problem regarding permutation groups, to be described next. A permutation group over a set D is represented by a set of generators; that is, the group generated by a set S of permutation (over D) is defined as 〈S〉 def = {g1 ◦ g2 ◦ · · · ◦ gl : g1, g2, ..., gl ∈ S} where ◦ denotes the composition of permutations. For example, in the case of the Rubic Cube, the set of generators corresponds to the 6 × 2 rotations that can be applied to the cube (where each rotation is determined by a rotating side of the cube and a direction of rotation). A shortest move sequence between two permutations π1, π2 ∈ 〈S〉 is the shortest sequence (g1, g2, ..., gl) over S such that π2 = gl ◦ · · · ◦ g2 ◦ g1 ◦ π1. A natural computational problem is finding, given S and π1, π2 ∈ 〈S〉, a shortest move sequence from π1 to π2. A computationally equivalent problem refers to finding the shortest sequence of permutations that generates a given permutation. A corresponding decision problem is presented next. Definition 1 (short generating sequence): Given a set of generators S, a permutation π ∈ 〈S〉, and in integer l presented in unary, determine whether or not there exists l′ ≤ l and g1, g2, ..., gl′ ∈ S such that π = gl′ ◦ · · · ◦ g2 ◦ g1.