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On Two Measures of Distance for

Fully-Labelled Trees

Giulia Bernardini1, Paola Bonizzoni1, Paweł

Gawrychowski2

1University of Milano - Bicocca, Italy

2University of Wrocław, Poland

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Why?

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Why?

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Ingredients

A ﬁnite set of n labels L = {♣ }

Two rooted trees fully labeled by L

Two operations: link&cut and permutation

T S

♣ ♣

♣♠♠

♠♥ ♥

♥ ♦ ♦

♦ ♥ ♦

♣

♠

[CPM’19] A rearrangement distance for fully-labelled trees

♣♠♠♥

♥ ♦ ♦

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Link&cut

♥| → = cut the edge ( , ♥ )

d

d

T S

♣ ♣♣ ♣

♣♠♠

♠♠ ♥ ♥

♥ ♦ ♦♦ ♥ ♦

♣

♠

♣ ♠♣

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Link&cut

d

d

T S

♣ ♣♣ ♣

♣♠♠

♠♠ ♥ ♥

♦ ♦ ♥♦ ♥ ♦

♣

♠

♥| → = cut the edge ( , ♥ ) and link ♥ to

♣ ♠♣♠

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Permutation

( ♦ )

d

d

T S

♣ ♣

♣♠♠

♠♠ ♥ ♥

♥ ♦ ♦♦ ♥ ♦

♣

♠

♠♠♣

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Permutation

d

d

T S

♣ ♣

♣♠♠

♠♠ ♥ ♥

♥ ♦ ♦♦ ♥ ♦

♣

♠

( ♦ )

♠♠♣

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Permutation

d

d

T S

♣ ♣

♣♠♠

♠♠ ♥ ♥

♥ ♦ ♦♦ ♥ ♦

♣

♠

( ♦ )

♠♠♣

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Permutation

d

d

T S

♣ ♣

♠♠

♠ a

♠ ♥ ♥

♥ ♦♦ ♥ ♦

♦♣♣

♠

( ♦ )

♠♠♣

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Operational distances

Permutation distance between two isomorphic trees S , T:

the smallest size of a permutation that transforms T into S

Rearrangement distance between any two trees S , T with

identical roots: the smallest size of any sequence of

link&cut and permutation operations that transforms T into

S without permuting the root

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Contributions

Permutation

distance

Rearrangement

distance

CPM 2019 O(n³) time algorithm

NP-hard:

constant-factor

approximation

algorithm for

binary trees

This work

Equivalent to

Bipartite Maximum

Matching.

Õ(n4/3+o(1)) time

algorithm

Constant-factor

approximation

algorithm for any

two trees

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Contributions

Permutation

distance

Rearrangement

distance

CPM 2019 O(n³) time algorithm

NP-hard:

constant-factor

approximation

algorithm for

binary trees

This work

Equivalent to

Bipartite Maximum

Matching.

Õ(n4/3+o(1)) time

algorithm

Constant-factor

approximation

algorithm for any

two trees

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Contributions

Permutation

distance

Rearrangement

distance

CPM 2019 O(n³) time algorithm

NP-hard:

constant-factor

approximation

algorithm for

binary trees

This work

Equivalent to

Bipartite Maximum

Matching.

Õ(n4/3+o(1)) time

algorithm

Constant-factor

approximation

algorithm for any

two trees

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Operational distances

Rearrangement distance revised: the smallest size of any

sequence of cut and permutation operations that

transforms T into a forest T’~S without permuting the root

T’ S

♣♠♠

♠♥ ♥

♥ ♦ ♦♦ ♥ ♦

♣

♠

♣ ♣

~

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Rearrangement distance: what is difﬁcult?

F1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Rearrangement distance: what is difﬁcult?

F1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Rearrangement distance: what is difﬁcult?

F1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Rearrangement distance: what is difﬁcult?

F1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Four steps: step i transforms F1

i-1 into F1

i with ALG( i )

operations: ALG( i ) =O( d(F1

i-1,F2) )

Constant-factor approximation algorithm

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make the nodes with diﬀerent children in F1 and F2

roots

Step 1: cut the grandparents

F1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make the nodes with diﬀerent children in F1 and F2

roots

ALG(1) ≤ 4d(F1 , F2)

Step 1: cut the grandparents

F1

1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two children of a node in F1

2 have

diﬀerent parents in F2

For each node of F1

1, each child vote for its representative

(its parent in F2). The majority wins, the rest is cut.

Step 2: let the children vote!

F1

1

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two children of a node in F1

2 have

diﬀerent parents in F2

ALG(2) ≤ 2d(F1

1 , F2)

Step 2: let the children vote!

F1

2

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two nodes in F1

2 have children with

the same representative

Among the parents of the nodes in F1

2 that have the same

representative, the one with more children wins.

Step 3: make the parents ﬁght!

F1

2

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two nodes in F1

2 have children with

the same representative

ALG(3) ≤ 2d(F1

2 , F2)

Step 3: let the parents ﬁght!

F1

3

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two nodes in F1

2 have the same

representative

Permute each node of F1

3 with the representative

of its children.

Step 4: permute the rest

F1

3

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Goal: make sure that no two nodes in F1

2 have the same

representative

ALG(4) ≤ 4d(F1

3 , F2)

Step 4: permute the rest

F1

4

F2

Giulia Bernardini On Two Measures of Distance for Fully-Labelled Trees CPM 2020

Future work

●Lower the constant factor of the approximation

algorithm

●Is there any approximation scheme for the

rearrangement distance?

Thank you for your attention