Constant Time Generation of Trees Constant Time Generation of Trees with Specified Diameterwith Specified Diameter

Shin-ichi NakanoShin-ichi Nakano 　　　　　　　　 Takeaki Uno Takeaki Uno 　 Gunma University, 　 　　　　 National Institute of

　 　 　 Japan Informatics, Japan

　　　　　　　　21/Jun/2004 WG2004

BackgroundBackground

Recently, Enumeration algorithms are interestingRecently, Enumeration algorithms are interesting

・・ There are still many unsolved nice problems

(unlike to ordinal discrete algorithms)

・・ Recent increase of computer power makes

many enumeration problems practically solvable.

⇒⇒ many applications have been appearing,

such as, genome, data mining, clustering, so on.

Problem & ResultProblem & Result

Problem:Problem: enumerate all (un-rooted) trees of nn vertices and diameter kk without duplications (no two isomorphic trees are allowed)

Result:Result: an algorithm with

time complexitytime complexity: : constant constant for each (delaydelay:: constantconstant)

space complexityspace complexity: : O(n)O(n)

(delay:delay: max. time between two consecutive outputs)

Existing StudiesExisting StudiesR.A.Wright,B.Richmond,A.Odlyzko and B.D.McKay 1986

time complexitytime complexity: : constant for each

space complexityspace complexity: : O(n)

delaydelay:: polynomialpolynomial

Our advantages:Our advantages:

・・ constant delay

・・ simple algorithm

・・ specified diameter

・・ incremental enumeration is possible

(enumerate trees at most nn vertices, s.t. a tree of mm vertices is outputted after outputting its subtree of m-1m-1 vertices )

← motivated by data mining

Recent Related ResearchesRecent Related Researches

Nakano '99

Nakano, Uno '03Arimura, Asai, '01, etc.

Arimura, Asai, Nakano, Uno '03, etc

Kawano, Nakano '04

Nakano, Uno '04

rooted ordered trees

Rooted ordered treeswith labels rooted (un-ordered) trees

Rooted (un-ordered) treeswith labels

trees with labels

(un-rooted, un-(un-rooted, un-ordered)ordered) trees oftrees of specified diameter specified diameter

terminal fixed two-connected series-parallel graphs

this work

DifficultyDifficulty

・・ Enumeration problems are often solved by

backtracking algorithmbacktracking algorithm

Ex. Ex.

- - start from a singlesingle vertex,

- - add a vertex to be a leaf one-by-one,

until the tree becomes n vertices and diameter k,

- - backtrack, and add a leaf at another position.

⇒⇒ Naturally, there occur

so many duplications

Enumerate Rooted Ordered TreesEnumerate Rooted Ordered Trees

・・ Based on backtracking algorithms

・・ Add a vertex to be the rightmost leaf

((Add a vertex at the right side of the rightmost path))

⇒ ⇒ enumerated in constant time for each

⇒ ⇒ no duplication occurs

Enumerate Rooted (Enumerate Rooted (un-orderedun-ordered) Trees) Trees

・・ Many duplications occur, in the same way as ordered trees

　←　← because, many non-isomorphic rooted ordered trees

correspond to the same rooted (un-ordered) tree.

・・ Define the canonical form among these isomorphic ordered trees

・・ Enumerate only rooted ordered trees of canonical forms

Depth SequenceDepth Sequence

Depth sequence Depth sequence (of ordered tree) :

pre-order of the left-first depth-first search

(visiting order of depth-first search:

starting from the root, visiting children from left to right)

・ Two rooted ordered trees are isomorphic

⇔ having the same depth sequence

0,1,2,3,3,2,2,1,2,3 0,1,2,2,3,3,2,1,2,3 0,1,2,3,1,2,3,3,2,2

Left Heavy EmbeddingLeft Heavy Embedding

・ ・ We obtain many ordered trees from a rooted (un-ordered) tree,

by giving orders of children to each vertex

Left heavy embeddingLeft heavy embedding: lexicographically maximizer of depth sequence among them

　 (obtained by sorting children by lexicographic order of depth sequences of their subtrees)

・・ Two rooted (un-ordered) trees are isomorphic

　⇔⇔ left heavy embeddings are isomorphic, in terms of ordered trees

0,1,2,3,3,2,2,1,2,3 0,1,2,2,3,3,2,1,2,3 0,1,2,3,1,2,3,3,2,2

Parent of Left Heavy Embedding Parent of Left Heavy Embedding

Parent of left-heavy embedding of m m vertices :

removal of the rightmost leaf

　⇒⇒ parent is also a left-heavy embedding of m-1m-1 vertices

0,1,2,3,3,2,1,2,3,2,11 0,1,2,3,3,2,1,2,3,22 0,1,2,3,3,2,1,2,33

T parent of T

parent ofparent

Family Tree of Rooted Family Tree of Rooted (un-ordered) Trees Trees

Parent-child relation of left-heavy embeddings

Child of Left Heavy Embedding Child of Left Heavy Embedding

・・　 Any child of left-heavy embedding is obtained by adding a vertex to be the rightmost leaf, at the level no lower than copy copy depthdepth

Copy depth Copy depth : deepest level s.t. left-heavy condition is not violated

by adding the rightmost leaf on the level.

・ ・ Copy depth can be updated in constant time

　⇒⇒ Enumeration can be done in constant time for eachT parent

of Tparent ofparent

Enumerate Un-rooted TreesEnumerate Un-rooted Trees

・・ An un-rooted tree has no root, so depth sequence is not defined

　　⇒ 　　⇒ define the root by its centercenter(s)

CenterCenter: vertex minimizing the distance to the furthest leaf

・・ The diameter is even ⇒⇒ the center is unique

・・ We consider the case of even diameter

Parent of Left Heavy EmbeddingParent of Left Heavy EmbeddingParent of a left-heavy embedding of m m vertices and diameter kk ： removal of the rightmost leaf,

removal of the second rightmost leaf if its diameter changes

⇒⇒ Parent is a left-heavy embedding of mm-1-1 vertices and diameter kk

・ ・ Diameter changes ⇒⇒ rightmost leaf ∈∈ leftmost spinespine (longest path)

・ ・ Leftmost spine does not change by removing the leaf

⇒⇒ parent and child shares leftmost spine

⇒⇒ constant time for each tree of at most nn vertices diameter kk

T parent parent ofparent

Family Tree of Family Tree of (Un-rooted) TreesTrees

diameter 4

Generating Only Trees of Generating Only Trees of nn Vertices Vertices

Active leaf: rightmost leaf s.t. not a child of root and

whose removal does not change the diameter

Parent of a left-heavy embedding of nn vertices and diameter k k :

remove active leaf and add a leaf to the root

⇒⇒ also a left-heavy embedding of nn vertices and diameter kk

T parent parent ofparent

Family Tree of Trees of Family Tree of Trees of nn Vertices Vertices

7 verticesdiameter 4

Constant DelayConstant Delay

・ Output before recursive calls in odd levels

Output after recursive calls in even levels

At least one iteration in consecutive three iterations

Recursion structure

Similar algorithm by considering two centers as the single root vertex(the order of children of the root has to be an exception)

Odd DiameterOdd Diameter

・ ・ There are always two centers

ConclusionConclusion

・ Proposed an algorithm for enumerating un-rooted trees of n vertices and diameter k

・ Time complexity is constant for each, and delay is also constant

Future workFuture work ：：・ Enumeration of plane trees, plane graphs, and other classes of graphs with simple ideas