On generalized Frame-Stewart numbers

For the multi-peg Tower of Hanoi problem with $k \geqslant 4$ pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: $\mathrm{S}\_k(n)=\min\_{1 \leqslant t \leqslant n} \left\{2 \cdot \mathrm{S}\_k(n-t) + \mathrm{S}\_{k-1}(t)\right\}$, $\mathrm{S}\_3(n) = 2^n -- 1$. In this paper, we generalize this recurrence relation to $\mathrm{G}\_k(n) = \min\_{1\leqslant t\leqslant n}\left\{ p\_k\cdot \mathrm{G}\_k(n-t) + q\_k\cdot \mathrm{G}\_{k-1}(t) \right\}$, $\mathrm{G}\_3(n) = p\_3\cdot \mathrm{G}\_3(n-1) + q\_3$, for two sequences of arbitrary positive integers $\left(p\_i\right)\_{i \geqslant 3}$ and $\left(q\_i\right)\_{i \geqslant 3}$ and we show that the sequence of differences $\left(\mathrm{G}\_k(n)- \mathrm{G}\_k(n-1)\right)\_{n \geqslant 1}$ consists of numbers of the form $\left(\prod\_{i=3}^{k}q\_i\right) \cdot \left(\prod\_{i=3}^{k}{p\_i}^{\alpha\_i}\right)$, with $\alpha\_i\geqslant 0$ for all $i$, arranged in nondecreasing order. We also apply this result to analyze recurrence relations for the Tower of Hanoi problems on several graphs.Comment: 13 pages ; 3 figure

Loopless Gray Code Enumeration and the Tower of Bucharest

We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer

On the restricted Hanoi Graphs

Consider the restricted Hanoi graphs which correspond to the variants of the famous Tower of Hanoi problem with multiple pegs where moves of the discs are restricted throughout the arcs of a movement digraph whose vertices represent the pegs of the puzzle and an arc from vertex $p$ to vertex $q$ exists if and only if moves from peg $p$ to peg $q$ are allowed. In this paper, we gave some notes on how to construct the restricted Hanoi graphs as well as some combinatorial results on the number of arcs in these graphs.Comment: 8 pages, 2 figure

Distance-uniform graphs with large diameter

An ϵ-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ϵ-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the follow- ing conjecture of independent interest: that every ϵ-distance-uniform graph (and, in fact, a broader class of ϵ-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal prob- lem. Speci-cally, for 1/n ≤ ϵ ≤ 1 /log n , we construct ϵ-distance-uniform graphs with critical distance 2ω(log n/log ϵ-1). We also prove an upper bound on the critical distance of the form 2O(log n/log ϵ-1) for all ϵ and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest.Peer ReviewedPostprint (author's final draft

On the Treewidth of Hanoi Graphs

The objective of the well-known Towers of Hanoi puzzle is to move a set of disks one at a time from one of a set of pegs to another, while keeping the disks sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs