On the Parallel Tower of Hanoi Puzzle: Acyclicity and a Conditional Triangle Inequality

1 Introduction

1.1 Problem Description

The Tower of Hanoi Puzzle consists of $n (n \in N_{0})$ annular disks (no two disks of equal radius) and $p$ posts $(p \geq 2)$ attached to a fixed base. The puzzle begins with all of the disks stacked on a single post with no larger disk being stacked atop a smaller disk. The goal of the puzzle is to transfer the initial “tower” of disks to another post while adhering to the following rules:

one disk is transferred from the top of one stack to the top of another (possibly empty) stack at each stage of the puzzle, and
a larger disk cannot be transferred to a post occupied by a smaller disk.

See [2] for a complete summary and history of the Puzzle. One prevailing question that has been studied over the past decades is summarized as follows: assuming feasibility, how does one perform the task in the fewest number of disk transfers?

This article considers the same question in a generalization of the classic Puzzle where there are one or more “towers” of disks. We will introduce $t$ towers $(t \in N)$ on $p$ posts where $p \geq t + 2$ . Each tower of disks is assigned its own color, and each tower has $n$ disks of sizes $1, \dots, n$ where $i < j$ implies that disk $i$ is larger than disk $j$ . Similar to above, we require that no disk (of any color) is atop any smaller or equal-sized disk (of any color). We will define herein the rules of the parallel puzzle in an analogous manner:

one or more one disks (of any color) are transferred—each from the top its own stack—to the top of one or more (possibly empty) stacks at each stage of the puzzle, where no two transfers share the same destination stack, and
a larger disk cannot be transferred to a post occupied by a smaller or equal-sized disk.

For other parallel variants of the Tower of Hanoi Puzzle (on a single tower of disks, but where more than one disk can be transferred at each stage) see [8] and [4]

. We examine the question of finding walks of minimal length within our parallel context, and we establish two results—with constructive proofs—that prescribe necessary properties that minimal-length walks connecting configurations must possess. These results can be iteratively applied to a given a walk in the state space connecting arbitrary, valid configurations (e.g., a learning episode of a reinforcement learning procedure within the context of the Tower of Hanoi Puzzle; e.g, see

[3], [7], [1]); these results may 1.) potentially reduce the size of the search space for finding walks of minimal length connecting the configurations, and 2.) decide how this reduced search space should be traversed by a walk of minimal length.

1.2 Article Summary

The aim of this article is as follows:

It formally encapsulates the Parallel Tower of Hanoi Puzzle in a finite metric space on graphs, words, and sets of words (clusters).
Two theorems—a theorem on acyclicity, and a conditional triangle inequality—are presented. These results are generalizations of the results from [5]; and the prescribe necessary properties of minimal-length walks within this mathematical framework.

1.3 Notation

The domain of all variables is the set of non-negative integers. Throughout, we use $n$ , $p$ , and $t$ to denote the number of disks, posts, and towers, respectively.

For any fixed positive integer $m$ , we let $[m]$ denote the set of the first $m$ positive integers. We write $[m]_{0}$ to denote the set $[m] \cup {0} .$ Furthermore, we’ll write $[m)$ and $[m)_{0}$ to denote the sets $[m - 1]$ and $[m - 1]_{0}$ , respectively. We will define the set of $p$ posts to be $P$ .

Assume $(X_{i}, \dots, X_{i + k})$ is a subsequence of $k + 1$ contiguous elements within the sequence $(X_{1}, \dots, X_{m})$ (e.g., a subsequence of contiguous symbols in a string, a subsequence of adjacent disk configurations). Should the case arise where $k < 0$ , we will assume that the subsequence $(X_{i}, \dots, X_{i + k})$ is the empty sequence (e.g, the empty string $ϵ$ , the empty subsequence).

For a fixed $t$ , we will write $1^{t} = (1, \dots, 1) \in N^{t}$ , and for $j \in N_{0}$ , we will write $j \cdot 1^{t} = (j, \dots, j) \in N_{0}^{t}$ . Context permitting, we will omit the superscript.

Assume that $i, j \in N_{0}^{t}$ where $i = {(i_{u})}_{u \in [t]}$ and $j = {(j_{u})}_{u \in [t]}$ . We will say that $i ⪯ j$ (equivalently, $j ⪰ i$ ) if and only if $i_{u} \leq j_{u}$ for each $u \in [t]$ . We will also write $i ≺ j$ (equivalently, $j ≻ i$ ) if the inequality $i_{u} < j_{u}$ (equivalently, $i_{u} > j_{u}$ ) holds for at least one index $u$ .

We assume that each of the $t$ towers is assigned its own color. We will write $D^{u, j}$ to denote the $j$ -th largest disk of color $u$ ( $j \in [n], u \in [t]$ ). We will represent a configuration of $n t$ disks as $t \times n$ matrix over the set of posts $P$ : the matrix

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} a_{1, 1} & \dots & a_{1, n} ⋮ & ⋱ & ⋮ a_{t, 1} & \dots & a_{t, n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

encodes a configuration of $t$ towers, each with $n$ disks, on $p$ posts where the disk $D^{u, j}$ occupies the post $a_{u, j}$ ; the largest disks of each color occupy the posts in the in the first (left-most) column, and the post configurations of the smaller disks are listed by column in decreasing size from left to right.

2 The Parallel Tower of Hanoi Problem

In this section, we will establish the mathematical machinery to analyze the state space of disk configurations in a hierarchical manner: we will partition the state space of disk configurations into sets (called clusters) that are defined by the arrangement of a prescribed subset of the largest disks. Afterwards, as was originally done in [6], we will represent the parallel state space of configurations in the context of graphs. We will define the formalism to analyze walks within these graphs, as well as cluster walks: for a pair $(α, β)$ of arbitrary configurations within the state space, this machinery will allow us to decide in which subsets of configurations (i.e., the corresponding subsets of vertices in the state graph) that minimal walks connecting $α$ and $β$ must be contained.

Definition 2.1 (Parallel Clusters/Cluster Sets).

Assume that $g = (g_{1}, \dots, g_{t}) \in [n]_{0}^{t}$ .

A cluster (of grading $g$ ) is a set of disk configurations in which, for each $u \in [t]$ , the $g_{u}$ disks $D^{u, 1}, \dots, D^{u, g_{u}}$ are in a fixed configuration.

We denote the set of all clusters of g on $p$ posts as $S_{p}^{t, g}$ . We also define the cluster of grading $0 \cdot 1$ to be the set of all disk configurations, and we denote this set as $S_{p}^{t, n}$ .
Assume the cluster $A \in S_{p}^{t, g}$ . For all $h ⪰ g$ , we define the set of clusters

$S_{p}^{t, h} (A) = {B ∣ B \in S_{p}^{t, h}, B \subseteq A} .$

Context permitting, we will omit the superscript $t$ and subscript $p$ whenever possible.

For $j \in [n]$ , there are $p$ choices to place $D^{1, j}$ , $p - 1$ choices to place $D^{2, j}, \dots, p - t + 1$ choices to place $D^{t, j}$ . Thus, there are $p^{t -}$ choices to place the $t$ disks of index $j$ . Consequently, induction yields ${(p^{t -})}^{j}$ clusters of grading $j \cdot 1$ .

2.1 EREW-Adjacency (Exclusive Read, Exclusive Write Adjacency)

In the classical puzzle, two configurations were adjacent if and only if they differed by the valid transfer of a single disk. We now extend this definition into the parallel context as follows.

Definition 2.2 (Exclusive Read/Write Adjacency).

Let the configuration $α = (a_{u, y}) \in S_{p}^{t, n}$ .

A configuration $β = (a_{u, y}) \in S_{p}^{t, n}$ is EREW-adjacent to $α$ if and only if

the following conditions are satisfied:

Non-triviality: there exists $u \in [t]$ and $j_{u} \in [n]$ where $a_{u, j_{u}} \neq b_{u, j_{u}}$ (at least one disk has transferred), and
Stack Read/Write: if $a_{u, j_{u}} \neq b_{u, j_{u}}$ , then, for $v \in [t]$ and $y > j_{u}$ , the inequalities $a_{v, y} \neq a_{u, j_{u}}$ and $b_{v, y} \neq b_{u, j_{u}}$ hold.

One consequence of this definition is the exclusive read/write property for the parallel Puzzle: if $a_{u, j_{u}} \neq b_{u, j_{u}}$ and $a_{v, j_{v}} \neq b_{v, j_{v}}$ where $u \neq v$ , then $a_{u, j_{u}} \neq a_{v, j_{v}}$ and $b_{u, j_{u}} \neq b_{v, j_{v}}$ .

Equipped with this definition of adjacency, we will define the state graph of the parallel Puzzle as was done in [6] for the single-tower case (see Figure 1).

Definition 2.3.

We define the state graph of the parallel Puzzle to be the pair $(S_{p}^{t, n}, E)$ where

E={{α,β}∣α\textmdandβ\textmdare\slEREW−adjacentconfigurations},

and we will denote this graph as $H_{p}^{t, n}$ .

Figure 1: Illustrated Examples of the graphs

H_{4}^{1, 2}

and

H_{4}^{2, 1}

; only the edges on the hull of

H_{4}^{2, 1} \equiv K_{12}

are presented.

We now define a cluster walk, and we will equip walks in the parallel state space with a measure in the parallel state space that naturally aligns with the counting measure of the classic Puzzle. To this end, we will introduce an auxiliary graph object for the parallel Puzzle.

Definition 2.4.

Let configurations $α$ and $β$ be EREW-adjacent. We define the transition graph of the adjacent pair to be the directed, edge-labelled, graph $(P, j)$ on $P$ with edge-labels over $[t] \times [n]$ : the edge $(q, r)$ is in the edge set of the transition graph with label $(u, j_{u})$ if and only if the disk $D^{u, j_{u}}$ transfers from post $q$ to post $r$ .

The edge set $j$ is defined to be the

transfer vector

of the EREW pair

(α, β)

. We will say that the configurations are

j

-adjacent, and express this relation with the notation

{(α, β)}^{j}

We denote the number of edges in the transition graph of the pair ${(α, β)}^{j}$ to be $η (α, β)$ . If $α = β$ , then we define $η (α, β) = 0$ .

As per the rules of the parallel Puzzle, the in-degree and out-degree of any vertex in the transition graph is at most one.

We will now define adjaceny for clusters as follows.

Definition 2.5 (Parallel Cluster Pair).

Let clusters $A, B \in S_{p}^{t, g}$ for some fixed grading $g \in [n]^{t}$ where $A \neq B$ . The clusters $A$ and $B$ are EREW-adjacent if and only if there are configurations $α \in S^{n} (A)$ and $β \in S^{n} (B)$ where ${(α, β)}^{k}$ for some transfer vector $k$ .

Let $j$ denote the subset¹¹1As $A \neq B$ , we are assured that $j \neq \emptyset$ . of $k$ where

j={(q,r)∣\textmdtheedgelabelof(q,r)\textmdis(u,ju)\textmdwhereju≤g}.

We will say that $A$ and $B$ form a $j$ -adjacent cluster pair (with transfer vector $j$ ), denoted by ${(A, B)}^{j}$ .

Definition 2.6 (Parallel Cluster Walk).

Let $A, B \in S_{p}^{t, g}$ for some fixed grading $g \in [n]^{t}$ . A (cluster) walk (of grading $g$ ) connecting $A$ and $B$ , denoted as

π (A, B) = (A_{1}, \dots, A_{w}),

is a sequence of clusters of grading $g$ (a $g$ -walk) that satisfies the following properties: $A = A_{1}$ , $B = A_{w}$ , and, for $u \in [w)$ , either $A_{u} = A_{u + 1}$ or ${(A_{u}, A_{u + 1})}_{u}^{j}$ for some transfer vector $j$ .

We will also define a valid sequence of configurations $(α_{1}, \dots, α_{w})$ analogously (we allow repeated configurations in a valid sequence of configurations).

We will define both the sequence length and transfer length of configuration sequences. If $π = (α_{1}, \dots, α_{w})$ is a configuration sequence, then its sequence length is defined to be $| π | = w$ (the number of configurations in $π$ ), and its transfer length is defined to be $L (π) = \sum_{1 \leq v < w} η (α_{v}, α_{v + 1})$ (the sum of the number of edges in all transition graphs of $π$ ).

In order to establish results on the transfer length of configuration sequences, we will formally prescribe disk transfers as mappings on such sequences.

Definition 2.7 (Translative and Reflective Mappings).

Let the disk configuration $α = (a_{u, y}) \in S_{p}^{t, n}$ , and let $g \in [n]$ .

Let $g = g \cdot 1 \in [n]^{t}$ . Let the cluster $B \in S_{p}^{t, g}$ where the disk $D^{u, y}$ occupies the post $b_{u, y}$ for $u \in [t]$ and $y \in [g]$ . The translative map $T_{B} : S_{p}^{t, n} \to S_{p}^{t, n} (B)$ maps $α$ to the configuration $T_{B} (α)$ , where

$T_{B} (α) = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} b_{1, 1} & \dots & b_{1, g} & a_{1, g + 1} & \dots & a_{1, n} ⋮ & ⋱ & ⋮ & ⋮ & ⋱ & ⋮ b_{t, 1} & \dots & b_{t, g} & a_{t, g + 1} & \dots & a_{t, n} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ .$
Let $q, r \in P$ . The reflective map $R_{q ∣ r}^{g} : S_{p}^{t, n} \to S_{p}^{t, n}$ maps $α$ to the configuration $R_{q ∣ r}^{g} (α) = (b_{v, y})$ , where, for $v \in [t]$ and $y \geq g$ ,
1. if $a_{v, y} = q$ , then $b_{v, y} = r$ ;
2. if $a_{v, y} = r$ , then $b_{v, y} = q$ ;
3. if $a_{v, y} \notin {r, q}$ , then $b_{v, y} = a_{v, y}$ .

We will now identify a class of mappings that for which the images of valid configuration sequences are themselves valid.

Lemma 2.8 (Mappings on Configuration Sequences).

Let $π = (α_{1}, \dots, α_{w})$ be a valid sequence of configurations. Also, let $g \in [n]$ , and let $g = g \cdot 1 \in [n]^{t}$ .

For any cluster $B \in S_{p}^{t, g}$ , the sequence $T_{B} (π) = {(T_{B} (α_{v}))}_{v \in [w]}$ is a valid sequence of configurations contained in $B$ .
If there exists a cluster $B \in S_{p}^{t, g - 1}$ that contains $π$ , then, for $q, r \in P$ , the sequence $R_{π ∣ q}^{g} (r) = {(R_{α_{v} ∣ q}^{g} (r))}_{v \in [w]}$ is a valid sequence of configurations contained in $B$ .

Proof.

Assume the hypotheses and notation in the statement of the lemma. For each $v \in [w]$ , let $α_{v} = (a_{u, y}^{(v)})$ .

Assume that the cluster $B$ is such that the disk $D^{u, y}$ occupies the post $b_{u, y}$ for $u \in [t]$ and $y \in [i]$ .

If $α_{v} = α_{v + 1}$ , then we have the equality $T_{B} (α_{v}) = T_{B} (α_{v + 1})$ .

If ${(α_{v}, α_{v + 1})}_{v}^{j}$ for some transfer vector $j$ , then write $T_{B} (α_{v}) = (b_{u, y}^{(v)})$ where

$b_{u, y}^{(v)} = {\begin{matrix} a_{u, y}^{(v)} & y > g b_{u, y} & y \leq g . \end{matrix}$

The claim is that either $T_{B} (α_{v}) = T_{B} (α_{v + 1})$ or ${(T_{B} (α_{v}), T_{B} (α_{v + 1}))}_{B}^{i}$ for some transfer vector $i$ where $i \subseteq j$ .

Let i denote the subset of $j$ where

$i={(q,r)∣\textmdtheedgelabelof(q,r)\textmdis(u,ju)\textmdwhereju>g}.$

If $i$ is empty, then we have the equality $T_{B} (α_{v}) = T_{B} (α_{v + 1})$ . Otherwise, if $a_{u, j_{u}}^{(v)} \neq a_{u, j_{u}}^{(v + 1)}$ where $j_{u} > g$ , then $a_{w, y}^{(v)} \neq a_{w, j_{u}}^{(v)}$ and $a_{w, y}^{(v + 1)} \neq a_{u, j_{u}}^{(v + 1)}$ for $w \in [t]$ and $y > j_{u}$ (by the definition of the j-adacency of $α_{v}$ and $α_{v + 1}$ ). As $j_{u} > g$ , we have that $b_{u, j_{u}}^{(v)} = a_{u, j_{u}}^{(v)}$ and $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)}$ ; furthermore, as the equalities $b_{w, y}^{(v)} = a_{w, y}^{(v)}$ and $b_{w, y}^{(v + 1)} = a_{w, y}^{(v + 1)}$ hold for $w > j_{u}$ , it follows that $b_{w, y}^{(v)} \neq b_{w, j_{u}}^{(v)}$ and $b_{w, y}^{(v + 1)} \neq b_{u, j_{u}}^{(v + 1)}$ . Thus, the configurations $T_{B} (α_{v})$ and $T_{B} (α_{v + 1})$ form an i-adjacent pair ${(T_{B} (α_{v}), T_{B} (α_{v + 1}))}_{B}^{i}$ .
If $α_{v} = α_{v + 1}$ , then we have the equality $R_{α_{v} ∣ q}^{g} (r) = R_{α_{v + 1} ∣ q}^{i} (r)$ .

If $q = r$ , then $R_{q ∣ r}^{g} (π) = π$ . Thus, we will assume that $q \neq r$ .
If ${(α_{v}, α_{v + 1})}_{v}^{j}$ for some transfer vector j, then, under the assumption that $[π]^{g - 1} = (B)$ , it must be the case that each edge in $j$ has an edge label $(u, j_{u})$ where $j_{u} \geq g$ . Let $R_{q ∣ r}^{g} (α_{v}) = (b_{u, y}^{(v)})$ , where, for $y \geq g$ , we define

$b_{u, y}^{(v)} = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} r & a_{u, y}^{(v)} = q, q & a_{u, y}^{(v)} = r, a_{u, y}^{(v)} & a_{u, y}^{(v)} \notin {q, r} . \end{matrix}$

We will show that forms a cluster pair.
1. If $a_{u, j_{u}}^{(v)} \neq a_{u, j_{u}}^{(v + 1)}$ , then:
  1. if $a_{u, j_{u}}^{(v)} = q$ and $a_{u, j_{u}}^{(v + 1)} = r$ , then $b_{u, j_{u}}^{(v)} = r$ , and $b_{u, j_{u}}^{(v + 1)} = q$ ; an analogous argument holds for when $a_{u, j_{u}}^{(v)} = r$ and $a_{u, j_{u}}^{(v + 1)} = q$ ;
  2. if $a_{u, j_{u}}^{(v)} = q$ and $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , then $b_{u, j_{u}}^{(v)} = r$ , and $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ ; analogous arguments hold for when $a_{u, j_{u}}^{(v)} = r$ and $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , and when $a_{u, j_{u}}^{(v)} \notin {q, r}$ and $a_{u, j_{u}}^{(v + 1)} \in {q, r}$ ;
  3. if $a_{u, j_{u}}^{(v)} \notin {q, r}$ and $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , then $b_{u, j_{u}}^{(v)} = a_{u, j_{u}}^{(v)}$ , and $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)}$ .
  In all cases, the inequality $b_{u, j_{u}}^{(v)} \neq b_{u, j_{u}}^{(v + 1)}$ holds, and the non-triviality condition 2.2.1 is met.
2. If $a_{u, j_{u}}^{(v)} \neq a_{u, j_{u}}^{(v + 1)}$ , then $a_{s, y}^{(v)} \neq a_{u, j_{u}}^{(v)}$ and $a_{s, y}^{(v + 1)} \neq a_{u, j_{u}}^{(v + 1)}$ for $s \in [t]$ and $y > j_{u}$ , as per condition 2.2.2.
  1. if the post $a_{u, j_{u}}^{(v)} = q$ and the post $a_{u, j_{u}}^{(v + 1)} = r$ , then the post $b_{u, j_{u}}^{(v)} = r$ , and the post $b_{u, j_{u}}^{(v + 1)} = q$ ; furthermore, for $y > j_{u}$ , the post $b_{s, y}^{(v)} \neq r$ (otherwise, the post $a_{s, y}^{(v)} = q$ ), and the post $b_{s, y}^{(v + 1)} \neq q$ (otherwise, the post $a_{s, y}^{(v + 1)} = r$ ); an analogous argument holds for when $a_{u, j_{u}}^{(v)} = r$ and $a_{u, j_{u}}^{(v + 1)} = q$ ;
  2. if the post $a_{u, j_{u}}^{(v)} = q$ and the post $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , then the post $b_{u, j_{u}}^{(v)} = r$ and the post $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ . Furthermore, the post $b_{s, y}^{(v)} \neq r$ (otherwise, the post $a_{s, y}^{(v)} = q$ ), and the post
    
    $b_{s, y}^{(v + 1)} = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} q & a_{s, y}^{(v + 1)} = r r & a_{s, y}^{(v + 1)} = q a_{s, y}^{(v + 1)} & a_{s, y}^{(v + 1)} \notin {q, r}; \end{matrix}$
    
    in each case, as the post $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , we have the inequality $b_{s, y}^{(v + 1)} \neq b_{u, j_{u}}^{(v + 1)}$ . Analogous arguments hold for when the post $a_{u, j_{u}}^{(v)} = r$ and $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , and when the post $a_{u, j_{u}}^{(v)} \notin {q, r}$ and the post $a_{u, j_{u}}^{(v + 1)} \in {q, r}$ ;
  3. if $a_{u, j_{u}}^{(v)} \notin {q, r}$ and $a_{u, j_{u}}^{(v + 1)} \notin {q, r}$ , then $b_{u, j_{u}}^{(v)} = a_{u, j_{u}}^{(v)}$ , and $b_{u, j_{u}}^{(v + 1)} = a_{u, j_{u}}^{(v + 1)}$ ; as before, whether $a_{s, y}^{(v)} \in {q, r}$ or not (and whether $a_{s, y}^{(v + 1)} \in {q, r}$ or not), we have the inequalities $b_{s, y}^{(v)} \neq b_{u, j_{u}}^{(v)}$ and $b_{s, y}^{(v + 1)} \neq b_{u, j_{u}}^{(v + 1)}$ .
In all cases, the stack read/write condition 2.2.2 is met.

∎

3 Results

An elementary result of graph theory states that any minimal path in a graph connecting a pair of vertices must be acyclic. The first theorem extends the notion of acyclicity of minimal walks into the context of parallel cluster walks.

Theorem 3.1.

Let $g \in [n]$ , let $g = g \cdot 1 \in [n]^{t}$ , and let $A \in S_{p}^{t, g}$ . If the configurations $α, β \in S_{p}^{t, n} (A)$ , then every minimal configuration sequence .

Proof.

Assume the notation within the hypotheses of the theorem statement. We will show that for any walk $ν (α, β)$ where $ν ⊈ S^{n} (A)$ , there exists a walk $μ (α, β)$ with the property that $μ \subseteq S_{p}^{t, n} (A)$ and $L (ν) > L (μ)$ .

As $ν (α, β) ⊈ S^{n} (A)$ , there exists a unique sequence of clusters $(B_{1}, \dots, B_{m})$ of grading g where $B_{1} = B_{m} = A$ , and ${(B_{w}, B_{w + 1})}_{w}^{i_{w}}$ for some transfer vector $i_{w}$ for each $w \in [m)$ .

Thus, we write $ν$ as

ν = \sum B_{w} ν_{w},

where $ν_{w} \subseteq S_{p}^{t, n} (B_{w})$ , and

ν_{w} = (α_{w, 1}, \dots, α_{w, l (w)}) .

For $w \in [m)$ , the $j_{w}$ -adjacent configurations ${(α_{w, l (w)}, α_{w + 1, 1})}_{w, l (w)}^{j_{w}}$ satisfy the properties that $α_{w, l (w)} \in S_{p}^{t, n} (B_{w})$ and $α_{w + 1, 1} \in S_{p}^{t, n} (B_{w + 1})$ where ${(B_{w}, B_{w + 1})}_{w}^{i_{w}}$ .

By Definition 2.5, the transfer vector

iw={(q,r)∣\textmdtheedgelabelof(q,r)\textmdis(u,ju)\textmdwhereju≤g}≠∅;

thus, as per the proof in 2.8.1, we have the following inequality on the edge counts of the translation graphs:

η (T_{A} (α_{w, l (w)}), T_{A} (α_{w + 1, 1})) < η (α_{w, l (w)}, α_{w + 1, 1})

(as the transfer of any disk of index less than or equal to $g$ is removed from the set $i_{w}$ ).

Thus, the walk $μ = T_{A} (ν)$ is contained in $S_{p}^{t, n} (A)$ as per Lemma 2.8, and $L (μ) < L (ν)$ .

∎

The second theorem yields a conditional triangle inequality within the context of parallel clusters.

Theorem 3.2.

Fix $g \in [n]$ , and let $g = g \cdot 1 \in [n]^{t}$ . Fix $A_{0} \in S_{p}^{t, g - 1}$ where the disk $D^{u, y}$ occupies the post $a_{u, y}$ for $u \in [t]$ and $y \in [g)$ .

Let $a, b, c \in P$ be pairwise unequal post values. Fix $v \in [t]$ , and let $g_{v} = (g - 1 + δ_{v, 1}, \dots, g - 1 + δ_{v, t})$ where $δ_{v, u} = [u = v]$ for each $v \in [t]$ . Let the clusters $A, B$ and $C$ be elements of $S_{p}^{t, g_{v}} (A_{0})$ where the disk $D^{v, g}$ occupies the post $a, b$ and $c$ , respectively.

Let the configuration $α \in S_{p}^{t, n} (A)$ be such that the identity

R_{α ∣ b}^{g} (c) = α

holds.

Then, for any configuration $β \in S_{p}^{t, n} (B)$ , and for any configuration sequence $ν (α, β)$ contained in , there exists a configuration sequence $μ (α, β)$ contained in that satisfies the inequality

L (μ) < L (ν) .

Proof.

Assume the notation and hypotheses within the theorem statement.

By applying Theorem 3.1 to walks connecting configurations in $S_{p}^{t, n} (A_{0})$ , we can assume that the configuration sequence $ν$ is contained in the cluster $A_{0}$ . Furthermore, by applying Theorem 3.1 to walks connecting configurations within the clusters $A, B$ and $C$ , we assume that we can uniquely express the walk $ν$ as

ν = ν_{A} + ν_{C} + ν_{B},

where $ν_{A}, ν_{B}$ , and $ν_{C}$ are contained in $A, B$ , and $C$ , respectively.

Assume that the subwalk

ν_{A} = (α_{1}, \dots, α_{x}),

the subwalk

ν_{C} = (γ_{1}, \dots, γ_{z}),

and the subwalk

ν_{B} = (β_{1}, \dots, β_{y}) .

The following conditions hold within the images of $ν_{A}$ and $ν_{C}$ under the reflective map $R_{b ∣ c}^{g}$ :

The cluster $R_{α_{1} ∣ b}^{g} (c) = α_{1}$ by assumption.
The image $R_{ν_{C} ∣ b}^{g} (c)$ is contained in $S_{p}^{t, n} (B)$ .
The configurations $γ_{z}$ and $β_{1}$ are j-adjacent for some transfer vector j, and the pair $(c, b) \in j$ with the edge label $(v, g)$ .

We will now construct a transfer vector $i$ where $i ⊊ j$ and ${(R_{γ_{z} ∣ b}^{g} (c), β_{1})}_{1}^{i}$ .

Firstly, we express the configuration $γ_{z} = (c_{u, y})$ , the configuration $β_{1} = (b_{u, y})$ , and the configuration $R_{γ_{z} ∣ b}^{g} (c) = (c_{u, y}^{'})$ for $u \in [t]$ and $y \in [n]$ . We have that the configurations $γ_{z}, R_{γ_{z} ∣ b}^{g} (c)$ and $β_{1}$ are elements of $S_{p}^{t, n} (A_{0})$ ; thus, we have the equalities

c_{u, y} = b_{u, y} = c_{u, y}^{'}

for $y < g$ .

We now address the case of static disks: if the post $c_{u, y} = b_{u, y} = q$ for $y \geq g$ , then it must be the case that $q \notin {b, c}$ (otherwise, the transfer of disk $D^{v, g}$ is prohibited). Thus, under the assumption that the disk $D^{u, y}$ statically occupies post $q$ in ${(γ_{z}, β_{1})}_{z}^{j}$ , we have that $c_{u, y}^{'} = b_{u, y} = q$ .

We now address the case of active disks in ${(γ_{z}, β_{1})}_{z}^{j}$ : as the post $c_{v, g} = c$ , and the post $b_{v, g} = b$ , we have that the post $c_{v, g}^{'} = b_{v, g} = b$ , and the edge $(c, b) \notin i$ .

Let $(q, r) \in j$ with the edge label $(w, j_{w})$ where $(q, r) \neq (c, b)$ . As per Definition 2.2.2, it must be the case that $q \neq c$ and $r \neq b$ . Moreover, as $(c, b) \in j$ with edge label $(v, g)$ , it follows that $c_{u, y} \neq c$ and $b_{u, y} \neq b$ for $u \in [t]$ and $y > g$ .

Under these conditions, we condition by cases as follows. Assume that $h > j_{w}$ and $u \in [t]$ :

$q \neq b, r \neq c$ : The post $c_{w, j_{w}}^{'} = c_{w, j_{w}} = q$ . The post $c_{y, h} \neq q$ (as per Definition 2.2.2), and the post $c_{u, h}^{'} \neq q$ (whether or not $c_{y, h} = b$ ). The post $b_{u, h} \neq r$ ; thus, the edge $(q, r) \in i$ with with the edge label $(w, j_{w})$ .
$q = b, r \neq c$ : The post $c_{w, j_{w}}^{'} = c$ , and as the post $c_{u, z} \neq b$ , the post $c_{u, z}^{'} \neq c$ . The post $b_{u, h} \neq r$ ; thus, the edge $(c, r) \in i$ with with the edge label $(w, j_{w})$ .
$q \neq b, r = c$ : We have that $c_{w, j_{w}}^{'} = c_{w, j_{w}} = q$ , and $c_{u, z} \neq q$ ; as $q \notin {b, c}$ , it follows that $c_{u, z}^{'} \neq q$ (whether or not $c_{u, z} = b$ ). The post $b_{u}$

On the Parallel Tower of Hanoi Puzzle: Acyclicity and a Conditional Triangle Inequality

Authors

A New Inequality Related to Proofs of Strong Converse Theorems in Multiterminal Information Theory

A New Inequality Related to Proofs of Strong Converse Theorems for Source or Channel Networks

Combinatorial proofs of two theorems of Lutz and Stull

KPynq: A Work-Efficient Triangle-Inequality based K-means on FPGA

Combinatorics of nondeterministic walks of the Dyck and Motzkin type

State-Space Constraints Improve the Generalization of the Differentiable Neural Computer in some Algorithmic Tasks

Metrics and Ambits and Sprawls, Oh My

1 Introduction

1.1 Problem Description

1.2 Article Summary

1.3 Notation

2 The Parallel Tower of Hanoi Problem

Definition 2.1 (Parallel Clusters/Cluster Sets).

2.1 EREW-Adjacency (Exclusive Read, Exclusive Write Adjacency)

Definition 2.2 (Exclusive Read/Write Adjacency).

Definition 2.3.

Definition 2.4.

Definition 2.5 (Parallel Cluster Pair).

Definition 2.6 (Parallel Cluster Walk).

Definition 2.7 (Translative and Reflective Mappings).

Lemma 2.8 (Mappings on Configuration Sequences).

Proof.

3 Results

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

On the Parallel Tower of Hanoi Puzzle: Acyclicity and a Conditional Triangle Inequality

Authors

Related Research

A New Inequality Related to Proofs of Strong Converse Theorems in Multiterminal Information Theory

A New Inequality Related to Proofs of Strong Converse Theorems for Source or Channel Networks

Combinatorial proofs of two theorems of Lutz and Stull

KPynq: A Work-Efficient Triangle-Inequality based K-means on FPGA

Combinatorics of nondeterministic walks of the Dyck and Motzkin type

State-Space Constraints Improve the Generalization of the Differentiable Neural Computer in some Algorithmic Tasks

Metrics and Ambits and Sprawls, Oh My

This week in AI

1 Introduction

1.1 Problem Description

1.2 Article Summary

1.3 Notation

2 The Parallel Tower of Hanoi Problem

Definition 2.1 (Parallel Clusters/Cluster Sets).

2.1 EREW-Adjacency (Exclusive Read, Exclusive Write Adjacency)

Definition 2.2 (Exclusive Read/Write Adjacency).

Definition 2.3.

Definition 2.4.

Definition 2.5 (Parallel Cluster Pair).

Definition 2.6 (Parallel Cluster Walk).

Definition 2.7 (Translative and Reflective Mappings).

Lemma 2.8 (Mappings on Configuration Sequences).

Proof.

3 Results

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.