The Complexity of the Evolution of Graph Labelings Geir Agnarsson Raymond Greenlaw Sanpawat Kantabutra.

The Complexity of the Evolution of Graph

LabelingsGeir Agnarsson

Raymond GreenlawSanpawat Kantabutra

The Complexity of the Evolution of Graph Labelings – Agnarsson, Greenlaw, and Kantabutra - 2

Outline

• Introduction• Preliminaries and Problem Definitions• Relating Vertex and Edge Relabeling• Tight Bounds for the Relabeling

Problem• Relabeling with Privileged Labels• Open Problems• References


Outline




Introduction

• Graph labeling is a well-studied subject in computer science and mathematics.

• Problem that has widespread applications, including in many other disciplines.

• Graph Relabeling Problem is a variant of graph labeling.

• See next slide for example.


Introduction

• Two configurations, a “start” and “goal” are given. Transform the “start” to the “goal” configuration with certain conditions on the movement of labels.


Introduction

• Presentation of new results and the extension of existing results; in particular,– NC1 reduce the Vertex Relabeling problem to

the Edge Relabeling Problem and vice versa.– Provide upper and lower bounds on the

complexity of the Vertex and Edge Relabeling Problems.

– Provide precise characterizations of when instances of relabeling with privileged labels are solvable.


Introduction

• A number of puzzles can be viewed as relabeled graphs.

• One of the most famous is the 15-Puzzle which consists of 15 tiles on a 4 x 4 board with one position empty.

• Graph labeling has been studied (and continues to have ongoing research) in the context of cartography, codings, colorings, and rankings.


Introduction

• Applications in bioinformatics, networks, and VLSI; also appear in unexpected places.

• Graph Relabeling Problem can model a wormhole routing in processor networks.

• Well-known Pancake Flipping Problem can be modeled as a special case of our problem.


Outline

• Introduction• Preliminaries and Problem

Definitions• Relating Vertex and Edge Relabeling• Tight Bounds for the Relabeling



Preliminaries and Problem Definitions

• Let SV, SE N = {1,2,…}.

• A labeling LV of V is a mapping LV: V SV.

• A labeling LE of E is a mapping LE: E SE.

• SV = {1,2,…,n} and SE = {1,2,…,m}.

• Graphs are associated with labelings using angle bracket notation.

• A mutation or flip function f maps triples <G, LV, LE> to triples <G, L′V, L′E>.



• A consecutive vertex mutation function is defined where f maps a pair <G, LV> to a pair <G, L′V> with the following conditions:– LV = L′V, except on two vertices u and w.

– {u, w} E.

– LV(u) = L′V(w) and LV(w) = L′V(u).

– SV = {1,2,…,n}.

– f is a bijection.



Vertex Relabeling Problem

• Instance: A graph G, labelings LV and L′V, and t N.

• Question: Can labeling LV evolve into L′V in t or fewer vertex mutations?



Edge Relabeling Problem

• Instance: A graph G, labelings LE and L′E, and t N.

• Question: Can labeling LE evolve into L′E in t or fewer edge mutations?


Outline

• Introduction• Preliminaries and Problem Definitions• Relating Vertex and Edge

Relabeling• Tight Bounds for the Relabeling



Relating Vertex and Edge Relabeling

Theorem (Vertex/Edge Relabeling Related)

• Vertex Relabeling Problem is NC1 many-one reducible to the Edge Relabeling Problem, and the Edge Relabeling Problem is NC1 many-one reducible to the Vertex Relabeling Problem.


Outline




Tight Bounds for the Relabeling Problem

Theorem (Vertex Relabeling Upper Bound)

• Let G = (V,E) be a graph, LV and L′V vertex labelings, and t = n(n — 1)/2, then the answer to the Vertex Relabeling Problem is YES. That is, any labeled graph can evolve into any other labeled graph in at most n(n — 1)/2 mutations.



Theorem (Vertex Relabeling Upper Bound)• Proof:

– Consider the number of mutations required to change an arbitrary labeling LV into an arbitrary labeling L′V.

– Construct a spanning tree T of G. Let p1 p2 … pn be vertex numbers (not labels) that denote the Prüfer code order when the leaves of T are deleted during the process of constructing a Prüfer code.

– Note, pj {vi | 1 ≤ i ≤ n} for 1 ≤ j ≤ n.



Theorem (Vertex Relabeling Upper Bound)• Proof (cont.):

– Not interested in the actual Prüfer code itself but rather just the leaf elimination order.

– Mutate labels from LV into their positions in L′V in the order specified by the pi’s and along the path in the spanning tree from their starting position in LV to their final position in L′V .



Theorem (Vertex Relabeling Upper Bound)• Proof (cont.):

– To move L′V(p1) from the initial labeling to its final position can take at most n — 1 mutations. Note, p1 is an initial leaf in T, and T contains at most n – 1 edges.

– To move L′ (p2) from the initial labeling to its final position, we need at most n — 2 mutations, since L′ (p1) is already in its rightful place.




• Proof (cont.): – After k iterations, where all of the labels L

′V(p1) through and including L′ (pk) are in their correct places, then, to move L′V(p(k+1)) to its correct place, we need at most n – k – 1 mutations, since the remaining spanning tree induced by the vertex set, V(T) — {pi | 1 ≤ i ≤ k} has at most n — k — 1 edges.




• Proof (cont.): – No mutations are performed in locations of

the tree that have already been completed.

– The total number of mutations is therefore at most n(n — 1)/2. □



Corollary (Edge Relabeling Upper Bound)

• Let G = (V,E) be a graph, LE and L′E edge labelings, and t = m(m — 1)/2, then the answer to the Edge Relabeling Problem is YES. That is, any labeled graph can evolve into any other labeled graph in at most m(m — 1)/2 mutations.



Theorem (Lower Bounds for Relabeling Graphs) [Muir 1882]

• There is a graph G = (V,E), labelings LV

and L′V, and t = (n(n — 1)/2) — 1 such that the Vertex Relabeling Problem has an answer of NO. That is, there exist two labelings that require n(n — 1)/2 mutations to evolve one into the other. There is a graph H = (V′,E′), labelings LE′

and L′E′, and t = (m(m — 1)/2) — 1 such that the Edge Relabeling Problem has an answer of NO.



Theorem (Lower Bounds for Relabeling Graphs)

• Proof: We provide two vertex labelings on a path that require n(n — 1)/2 mutations to be evolved from one to the other.

• For convenience, a labeling is represented by a permutation of {1,2,…,n}.

• There exists a permutation, namely, n(n — 1)…1, where at least n(n — 1)/2 mutations are needed to obtain it from the permutation 1 2 … n.




• Proof (cont.): If we use fewer than n(n — 1)/2 mutations on 1 2 … n, we can not end up with n(n — 1)…1.

• View the permutation a1a2…an as a string s.

• Continued…




• Attach a unique parameter p(s) to each string by letting p(s) be the number of occurrences ai and aj, where ai > aj among all i < j, so p(s) = |{{i,j} : 1 ≤ i < j ≤ n and ai > aj}|.

• p(s) is the number of inversions of the permutation.




• Proof (cont.): There are exactly n(n — 1)/2 pairs i < j if i and j are among 1,2,…,n, so for every such strings s, we clearly have 0 ≤ p(s) ≤ n(n — 1)/2.

• Each mutation reduces or increases the value of p(.) by exactly one. If s′ is the string obtained from s, then |p(s′)–p(s)| = 1, the absolute value of p(s′)–p(s) equals 1.




• Proof (cont.): Since p(1 2…n) = 0 and p(n(n — 1)…1) = n(n — 1)/2, we need at least n(n — 1)/2 mutations to obtain n(n — 1)…1 from 1 2…n. In fact, we can use exactly n(n — 1)/2 mutations to obtain n(n — 1)…1 from 1 2…n.

• This shows there are graphs and vertex labelings that require n(n — 1)/2 mutations.

□



Theorem (Tight Bound on Path Relabeling Complexity) [Muir 1882]

• Let Pn be the path on n vertices, LV and L′V vertex labelings, and t N. Then the answer to the Vertex Relabeling Problem is YES if and only if t ≥ p(LV, L′V).

• Proof: For the sake of simplicity, assume we are to evolve s into s′.



Theorem (Tight Bound on Path Relabeling Complexity)

• Proof (cont.): A mutate sequence is a sequence of strings with s0 = s, sm = s′, and where si+1 is obtained from si by a single mutation, 0 ≤ i ≤ m — 1.

• In this case, we have for an arbitrary labeling s = a1 a2 … an that the parameter p(.) satisfies

miis 0

1

0

1

1

0

10 )()()()(m

i

ii

m

i

iim mspspspspspspsp




• Proof (cont.): We now argue by using induction on n that p(s) mutations suffice to evolve s into s′. This is clearly true for n = 2.

• Assume this assertion is true for length (n — 1)-strings, and let s = a1a2…an be such that n = ai, for a fixed i, 1 ≤ i ≤ n. In this case, we have p(s) = n — i + p(ŝ), where ŝ = a1…ai-1ai+1…an.




• Proof (cont.): Clearly in s we can move n = ai to the rightmost position by precisely n — i mutations.

• By induction, we can obtain 1 2 … (n — 1) from ŝ by p(ŝ) mutations. Hence, we are able to evolve s into s′ using p(s) mutations.




• Proof (cont.): If we have two vertex labelings LV and L′V of the vertices of the path Pn, we can define the corresponding relative parameter p(LV, L′V), as p(s), where s is the unique permutation obtained from LV by renaming the labels in L′V from left-to-right as 1,2,…,n. We note that p(LV, L′V) = p(L′V, LV). □




• Let Pn be the path on n vertices, LV and L′V vertex labelings, and t N. Then we can evolve the labeling LV into L′V using t mutations if and only if t = p(LV, L′V) + 2k for some nonnegative integer k.




• Proof: By the previous theorem we can always evolve LV into L′V using the minimum of p(LV, L′V) mutations. Repeating the last mutation 2k times is not going to alter L′V, since repeating a fixed mutation an even number of times corresponds to the identity (or neutral) relabeling. For any nonnegative integer k one can always evolve LV into L′V using t = p(LV, L′V) + 2k mutations.




• Proof (cont.): We show that t — p(LV, L′V) must be even. Assume LV is given by the string s = a1a2…an and L′V by the string s′ = 1 2 … n.

• Now, let and be two mutation sequences with s0 = s′0 = s and sm = s′m′ = s′.

miis 0 '

0' m

iis




• Proof (cont.): Since p(s0) = p(s′0) = p(s) and p(sm) = p(s′m′) = 0, we have

and (cont. next slide)

m

iiim PPspspspspsp

010 ))()(()()()(

'

01'0 ''))'()'(()'()'()(

m

iiim PPspspspspsp




• Proof (cont.): where

• = |{i {1,…,m} : p(si) – p(si+1) = 1}|,

• = |{i {1,…,m} : p(si) – p(si+1) = -1}|,

• = |{i {1,…,m′ } : p(s′i) – p(s′i+1) = 1}|, and

• = |{i {1,…,m′ } : p(s′i) – p(s′i+1) = -1}|.

• We have .

• m = and m′ = and we obtain…

PPPP ''

P

P

'P

'P

PP '' PP




• Proof (cont.): • m′ — m =

, and thus m and m′ must

have the same parity.

• This shows that if LV is evolved into L′V in exactly t mutations, then t — p(s) must be even. □

• This theorem re-establishes a known result in the theory of permutations that each permutation has unique parity.

)'(2)'( PPPP

)'()()''( PPPPPP


Outline




Relabeling with Privileged Labels

Definition (Vertex Relabeling with Privileged Labels Problem)

• Instance: A graph G, labelings LV and L′V, a nonempty set S {1,2,…,n} of privileged labels, and t N.

• Question: Can labeling LV evolve into L′V in less than t restricted vertex mutations?



Definition (Edge Relabeling with Privileged Labels Problem)

• Instance: A graph G, labelings LE and L′E, a nonempty set S {1,2,…,n} of privileged labels, and t N.

• Question: Can labeling LE evolve into L′E in less than t restricted edge mutations?



Theorem (General Unsolvability, Privileged Labels)

• Among all connected vertex labeled graphs on n vertices with k privileged labels where k {0,1,…,n — 2}, the Vertex Relabeling with Privileged Labels Problem is, in general, unsolvable. Among all connected edge labeled graphs on m edges with k privileged labels where k {0,1,…,m — 2}, the Edge Relabeling with Privileged Labels Problem is, in general, unsolvable.



Definition ((n x n)-Puzzle Problem)• Instance: Two n x n board

configurations B1 and B2, and k N.

• Question: Is there a sequence of at most k moves that transforms B1 into B2?



Theorem (Intractability, Privileged Labels)• The Vertex Graph Relabeling with

Privileged Labels Problem is NP-complete.• Proof: Reduce the (n x n)-Puzzle Problem

to the Vertex Graph Relabeling with Privileged Labels Problem by taking G as an n x n mesh, LV corresponding to B1, L'V corresponding to B2, S = {n2} corresponding to the blank space, and t = k.



Theorem (Intractability, Privileged Labels)

• Proof (cont.): The instance of the (n x n)-Puzzle Problem is YES if and only if the answer to the constructed instance of the Vertex Graph Relabeling with Privileged Labels Problem is also YES.



Theorem (2-Connected Unsolvability, Privileged Labels)

• Among all 2-connected vertex labeled graphs on n vertices with k privileged labels where k {0,1,…,n — 3}, the Vertex Relabeling with Privileged Labels Problem is, in general, unsolvable. Among all 2-connected edge labeled graphs on m edges with k privileged labels where k {0,1,…, m — 3}, the Edge Relabeling with Privileged Labels Problem is, in general, unsolvable.



Theorem (2-Connected Unsolvability, Privileged Labels)

• Proof: Let n ≥ 3 and consider two vertex labelings LV and L′V of the cycle Cn, where we have precisely k privileged labels p1,…,pk, where k {0,1,…,n — 3}. For a fixed planar embedding of Cn, assume the labelings are given cyclically in clockwise order: LV : (p1,…,pk,1,2,3,…,n – k), and L′V : (p1,…,pk,2,1,3,…,n – k).



Claims• If a simple graph is neither a path nor a

cycle, then it has a spanning tree that is not a path (and hence contains a vertex of degree of at least three).

• Among vertex labeled trees, which are not paths, with exactly two non-privileged labels, any two labels can be swapped using restricted mutations.



Lemma• Among vertex labeled trees, which are

not paths, with exactly two non-privileged labels, the Vertex Relabeling with Privileged Labels Problem is solvable and in P.



Theorem (Vertex Solvability, Two Privileged Labels)

• Among all connected vertex labeled graphs G on n ≥ 4 vertices with all but two vertex labels privileged, the Vertex Relabeling with Privileged Labels Problem is solvable if and only if G is not a path.



Theorem (Edge Solvability, Two Privileged Labels)

• Among all connected edge labeled graphs G on n ≥ 4 edges with all but two edge labels privileged, the Edge Relabeling with Privileged Labels Problem is solvable if and only if G is not a path.


Outline




Open Problems

• Study other types of mutation functions where, for example, labels along an entire path are mutated, or where labels can be reused.

• In the parallel setting, compute the sequence of mutations required for the evolution of one labeling into another. The parallel time for computing the sequence could be much smaller than the sequential time to execute the mutation sequence.


Open Problems

• For various classes of graphs determine the probability of one labelings evolving naturally into another. Such an evolution of a labeling could be used to model mutation periods.

• Study the properties of the graphs of all labelings. In this graph all labelings of a given graph are vertices and two vertices are connected if they are one mutation apart. Other conditions for edge placement may also be worthwhile to examine.


Open Problems

• Determine if there is a version of the Edge Relabeling with Privileged Labels Problem that is NP-complete.

• Define the cost of a mutation sequence to be the sum of the weights on all edges that are mutated. Determine mutation sequences that minimize the cost of evolving one labeling into another. Explore other cost functions.


Outline




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The Complexity of the Evolution of Graph Labelings Geir Agnarsson Raymond Greenlaw Sanpawat Kantabutra.

Documents

v of v

introduction graph

problem definitions

problem definitions

problem instance

graph g

variant of graph

e of e