Degree-Constrained

Network

Spanners

With

Non-constant

Delay

Arthur

L.

Liestman

Thomas

C.

Shermer

CMPT

TR

School

of

Computing

Science

Simon

Fraser

University

Burnaby

B.C.

Canada

July

Abstract

A

spanning

subgraph

S

-LRB-

V

-RRB-

of

a

connected

simple

graph

G

-LRB-

V

E

-RRB-

is

a

f

-LRB-

x

-RRB-

spanner

if

for

any

pair

of

nodes

u

and

v

dS

-LRB-

u

v

-RRB-

f

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

where

dG

and

dS

are

the

usual

distance

functions

in

graphs

G

and

S

respectively

The

delay

of

the

f

-LRB-

x

-RRB-

spanner

is

f

-LRB-

x

-RRB-

x

We

find

-LRB-

x

-RRB-

spanners

for

two-dimensional

grids

with

maximum

degree

We

prove

that

the

delay

of

these

spanners

is

within

a

constant

factor

of

optimal

We

describe

a

-LRB-

x

k

x

-RRB-

spanner

of

the

X-tree

with

maximum

degree

We

prove

that

the

delay

of

this

spanner

is

within

a

constant

factor

of

optimal

We

describe

a

-LRB-

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

and

a

-LRB-

x

k

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

We

prove

that

the

delay

of

the

latter

spanner

is

within

a

constant

factor

of

optimal

AMS

-LRB-

MOS

-RRB-

Subject

Classifications

This

research

was

supported

by

the

Natural

Sciences

and

Engineering

Research

Council

of

Canada

under

Grants

No

and

Introduction

and

Definitions

There

are

several

popular

topologies

used

for

constructing

parallel

computers

Our

goal

is

to

determine

substructures

of

such

topologies

with

smaller

maximum

degree

We

require

also

that

these

substructures

called

spanners

have

the

property

that

the

distance

between

two

vertices

in

the

substructure

is

not

significantly

larger

than

the

corresponding

distance

in

the

original

structure

In

this

paper

we

consider

spanners

of

the

two-dimensional

grid

that

have

maximum

degree

spanners

of

the

X-tree

with

maximum

degree

and

spanners

of

the

pyramid

with

maximum

degrees

amd

Spanners

were

introduced

by

Peleg

and

Ullman

-LSB-

-RSB-

who

used

these

structures

in

order

to

efficiently

simulate

synchronous

networks

on

asynchronous

networks

A

second

motivation

for

studying

spanners

is

their

possible

use

as

network

topologies

A

spanner

can

be

used

as

a

substitute

for

a

desired

network

often

giving

a

much

sparser

network

with

a

similar

structure

and

only

slightly

larger

communication

costs

A

third

motivation

is

to

provide

a

relatively

inexpensive

way

to

improve

the

quality

of

a

network

For

example

if

one

wished

to

construct

a

network

using

specialized

links

-LRB-

perhaps

links

with

higher

bandwidth

or

higher

reliability

-RRB-

an

alternative

would

be

to

construct

a

spanner

with

these

links

and

to

use

normal

connections

for

the

remaining

links

in

the

network

A

network

is

represented

by

a

connected

simple

graph

G

-LRB-

V

E

-RRB-

We

use

the

notation

dG

-LRB-

u

v

-RRB-

to

denote

the

distance

between

vertices

u

and

v

in

graph

G

The

maximum

degree

of

G

is

denoted

G

and

the

average

degree

is

denoted

ffiG

The

subscript

G

may

be

omitted

if

it

is

clear

from

context

A

spanning

subgraph

S

of

a

connected

simple

graph

G

is

a

f

-LRB-

x

-RRB-

spanner

if

for

any

pair

of

nodes

u

and

v

dS

-LRB-

u

v

-RRB-

f

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

-LSB-

-RSB-

This

definition

has

natural

generalizations

for

edge-weighted

graphs

digraphs

and

hypergraphs

We

call

dS

-LRB-

u

v

-RRB-

dG

-LRB-

u

v

-RRB-

the

delay

between

vertices

u

and

v

in

S

denoted

-LRB-

u

v

-RRB-

For

an

f

-LRB-

x

-RRB-

spanner

S

we

let

f

-LRB-

x

-RRB-

f

-LRB-

x

-RRB-

x

and

refer

to

f

-LRB-

x

-RRB-

as

the

delay

of

the

spanner

Note

that

f

-LRB-

x

-RRB-

is

actually

an

upper

bound

on

the

maximum

delay

in

S

between

any

pair

of

vertices

at

distance

x

in

G

It

may

be

possible

to

express

the

delay

f

-LRB-

x

-RRB-

in

several

ways

In

particular

any

spanner

S

of

a

finite

graph

G

is

an

-LRB-

x

c

-RRB-

spanner

where

c

is

the

maximum

delay

between

any

pair

of

vertices

in

S

A

more

careful

analysis

of

S

may

reveal

a

closer

relationship

between

the

distance

in

G

and

the

delay

in

S.

For

example

the

-LRB-

x

c

-RRB-

spanner

mentioned

above

may

also

be

determined

to

be

a

In

general

we

prefer

to

express

f

-LRB-

x

-RRB-

in

a

manner

that

bounds

the

delay

as

clearly

as

possible

Peleg

and

Schaffer

-LSB-

-RSB-

investigated

the

existence

and

constructability

of

sparse

spanners

for

various

classes

of

graphs

They

showed

that

to

determine

for

a

given

graph

G

and

integer

m

whether

G

has

a

with

m

or

fewer

edges

is

NP-complete

Cai

-LSB-

-RSB-

showed

that

for

any

integer

t

to

determine

for

a

given

graph

G

and

integer

m

whether

G

has

a

tx-spanner

with

m

or

fewer

edges

is

NP-complete

Degree-constrained

spanners

were

studied

by

Richards

and

Liestman

-LSB-

-RSB-

-LRB-

for

pyramids

-RRB-

and

by

Liestman

and

Shermer

-LSB-

-RSB-

-LRB-

for

multidimensional

grids

-RRB-

In

-LSB-

-RSB-

we

provided

the

more

general

formulation

of

spanners

and

initiated

the

study

of

-LRB-

t

x

-RRB-

spanners

giving

constructions

for

low-degree

-LRB-

t

x

-RRB-

spanners

of

hypercubes

We

showed

-LSB-

-RSB-

that

for

any

integer

t

to

determine

for

a

given

graph

G

and

integer

m

whether

G

has

a

-LRB-

t

x

-RRB-

spanner

with

m

or

fewer

edges

is

NP-complete

In

the

same

paper

we

also

described

how

to

construct

low

degree

-LRB-

t

x

-RRB-

spanners

for

X-trees

pyramids

and

grids

-LRB-

both

finite

and

infinite

-RRB-

In

this

paper

we

return

to

the

study

of

degree-constrained

spanners

Section

details

our

results

on

spanners

of

two-dimensional

grids

We

begin

by

proving

a

lower

bound

on

the

delay

of

a

maximum

degree

spanner

of

the

grid

in

In

we

describe

the

construction

of

-LRB-

x

-RRB-

spanners

for

two-dimensional

grids

with

maximum

degree

The

delay

of

these

spanners

is

within

a

constant

factor

of

the

lower

bound

Section

contains

our

results

on

spanners

of

X-trees

and

pyramids

In

we

prove

a

lower

bound

on

the

delay

of

a

maximum

degree

spanner

of

the

X-tree

In

we

describe

the

construction

of

a

-LRB-

x

k

x

-RRB-

spanner

for

X-trees

with

maximum

degree

In

we

show

how

to

construct

of

a

-LRB-

x

-RRB-

spanner

for

pyramids

with

maximum

degree

In

we

prove

a

lower

bound

on

the

delay

of

a

maximum

degree

spanner

of

the

pyramid

In

we

present

a

construction

of

a

-LRB-

x

k

x

-RRB-

spanner

for

pyramids

with

maximum

degree

Spanners

of

Two-Dimensional

Grids

A

grid

where

has

the

vertex

set

V

f

-LRB-

-RRB-

Its

edges

are

between

pairs

of

vertices

whose

labels

differ

by

in

exactly

one

position

that

is

vertex

-LRB-

-RRB-

is

connected

to

vertex

-LRB-

-RRB-

for

all

and

and

vertex

-LRB-

-RRB-

is

connected

to

vertex

-LRB-

-RRB-

for

all

and

In

order

to

simplify

some

of

the

notation

and

terminology

we

assume

the

natural

embedding

of

these

grids

in

the

plane

Lower

Bound

on

Delay

In

this

section

we

show

that

any

finite

grid

spanner

with

maximum

degree

must

have

delay

at

least

Theorem

Any

spanner

S

with

maximum

degree

of

any

grid

with

for

any

even

k

has

a

pair

of

vertices

u

and

v

such

that

dG

-LRB-

u

v

-RRB-

and

-LRB-

u

v

-RRB-

qdG

-LRB-

u

v

-RRB-

PROOF

Let

S

be

any

spanner

of

with

maximum

degree

Consider

the

paths

in

G

and

in

S

between

pairs

of

vertices

-LRB-

ki

-RRB-

and

-LRB-

ki

-RRB-

where

i

In

G

the

shortest

path

between

such

a

pair

of

vertices

contains

exactly

vertical

edges

In

S

any

path

between

such

a

pair

of

vertices

u

and

v

must

contain

at

least

vertical

edges

and

possibly

some

horizontal

edges

Thus

the

delay

between

u

and

v

in

S

is

at

least

as

large

as

the

number

of

horizontal

edges

in

the

shortest

path

between

u

and

v

in

S

This

implies

that

the

shortest

path

between

any

such

pair

must

stay

in

the

subgrid

with

between

ki

and

ki

k

otherwise

the

number

of

horizontal

edges

-LRB-

and

thus

the

delay

between

these

vertices

-RRB-

is

at

least

k

and

we

are

done

Similarly

each

shortest

path

in

S

between

a

pair

of

vertices

-LRB-

kj

-RRB-

and

-LRB-

kj

-RRB-

where

j

must

have

delay

at

least

as

large

as

the

number

of

vertical

edges

and

thus

stay

in

the

subgrid

with

between

kj

and

kj

k.

Consider

the

subgrid

with

between

ki

and

ki

k

and

between

kj

and

kj

k

The

paths

between

-LRB-

ki

-RRB-

and

-LRB-

ki

-RRB-

and

between

-LRB-

kj

-RRB-

and

-LRB-

kj

-RRB-

must

intersect

in

this

subgrid

Since

these

paths

can

not

intersect

at

a

degree

vertex

at

least

one

of

the

paths

must

turn

upon

meeting

the

other

thereby

incurring

at

least

one

unit

of

delay

As

there

are

subgrids

of

this

type

the

paths

must

incur

a

total

of

at

least

units

of

delay

Thus

some

particular

path

P

must

incur

at

least

k

units

of

delay

proving

the

claim

Letting

u

and

v

be

the

endpoints

of

P

dG

-LRB-

u

v

-RRB-

and

-LRB-

u

v

-RRB-

k.

k

l

Figure

A

highway

spanner

Thus

a

spanner

S

of

a

sufficiently

large

grid

G

must

have

various

pairs

of

vertices

at

distances

x

in

G

that

have

delay

at

least

in

S.

Constructions

We

present

two

methods

to

construct

-LRB-

O

-LRB-

px

-RRB-

x

-RRB-

spanners

with

maximum

degree

for

finite

grids

In

-LSB-

-RSB-

we

constructed

highway

spanners

for

finite

grids

by

using

all

of

the

edges

of

every

lth

row

and

every

kth

column

-LRB-

referred

to

as

highways

-RRB-

to

subdivide

the

grid

into

k

l

rectangles

as

illustrated

in

Figure

To

complete

the

spanner

we

added

some

additional

edges

from

within

each

rectangle

These

edges

can

be

added

so

that

the

only

vertices

of

degree

are

the

intersections

of

the

highways

These

highway

spanners

can

be

modified

to

produce

spanners

of

maximum

degree

One

simple

modification

is

to

replace

each

highway

intersection

by

a

roundabout

as

shown

in

Figure

In

this

scheme

each

highway

is

now

contained

in

two

horizontal

or

two

vertical

lines

Let

u

and

v

be

two

vertices

at

distance

x

in

G

on

the

same

horizontal

-LRB-

or

vertical

-RRB-

highway

in

such

a

spanner

S

The

shortest

path

between

u

and

v

in

S

follows

that

highway

and

passes

through

approximately

xk

roundabouts

incurring

a

delay

of

at

least

unit

in

each

roundabout

Thus

the

delay

of

such

a

spanner

is

at

least

xk

or

O

-LRB-

x

-RRB-

We

note

that

this

delay

can

be

eliminated

in

one

of

the

dimensions

by

collapsing

the

roundabout

into

one

straight

path

and

one

path

with

a

one

unit

jog

as

shown

in

Figure

A

modified

highway

spanner

Figure

Collapsing

a

roundabout

Figure

The

resulting

spanner

illustrated

in

Figure

has

smaller

average

degree

and

a

constant

delay

for

pairs

of

vertices

that

are

separated

mainly

in

the

However

this

type

of

spanner

still

has

O

-LRB-

x

-RRB-

delay

for

pairs

of

vertices

that

are

separated

mainly

in

the

Roundabout

Spanners

A

smaller

maximum

delay

can

be

obtained

by

collapsing

some

of

the

roundabouts

horizontally

and

some

vertically

One

possible

method

of

doing

this

yields

the

roundabout

spanner

which

has

delay

within

a

constant

factor

of

optimal

To

construct

a

roundabout

spanner

for

the

square

grid

first

include

the

edges

along

the

border

of

this

grid

forming

two

horizontal

and

two

vertical

highways

These

highways

are

said

to

have

index

k.

Next

construct

a

roundabout

on

the

four

center

vertices

of

the

grid

and

extend

highways

in

all

four

directions

to

the

borders

as

shown

in

Figure

-LRB-

a

-RRB-

The

resulting

structure

consists

of

a

roundabout

with

four

highway

sections

radiating

from

it

These

sections

form

Figure

A

further

modified

highway

spanner

one

horizontal

and

one

vertical

highway

both

of

which

have

index

k

In

order

to

include

the

remaining

vertices

on

these

lines

in

the

spanner

we

add

some

additional

edges

as

shown

in

Figure

-LRB-

b

-RRB-

When

we

include

these

edges

in

our

discussion

we

refer

to

highways

as

augmented

highways

and

highway

sections

as

augmented

highway

sections

We

recursively

apply

this

construction

to

each

of

the

four

quadrants

of

the

grid

placing

in

each

a

roundabout

and

its

four

radiating

augmented

highway

sections

-LRB-

see

Figure

-LRB-

c

-RRB-

-RRB-

Note

that

although

we

have

just

added

augmented

highway

sections

we

have

only

added

augmented

highways

each

with

index

k

A

further

level

of

subdivision

is

shown

in

Figure

-LRB-

d

-RRB-

This

recursion

continues

with

index

decreasing

by

one

at

each

further

level

of

subdivision

until

the

remaining

quadrants

are

grids

The

last

augmented

highways

added

have

index

At

this

point

the

structure

forms

a

connected

spanning

subgraph

of

The

roundabout

spanner

is

shown

in

Figure

with

the

indices

of

vertical

highways

indicated

Consider

a

vertex

u

of

If

u

is

on

the

border

of

then

it

has

degree

at

most

in

the

grid

and

therefore

degree

at

most

in

the

spanner

Otherwise

the

vertex

u

is

first

encountered

during

the

construction

of

when

a

roundabout

or

augmented

highway

section

is

added

Note

that

u

is

included

in

and

is

a

border

vertex

of

one

quadrant

in

which

the

recursive

construction

is

applied

Being

a

border

vertex

it

could

receive

at

most

one

additional

incident

edge

from

this

recursive

construction

Thus

if

u

has

degree

at

most

at

this

point

of

the

construction

it

will

have

degree

at

most

when

the

construction

is

completed

If

u

is

a

roundabout

vertex

then

it

has

degree

at

this

level

of

the

construction

and

is

not

connected

to

during

any

subsequent

level

If

abcd

Figure

Building

a

roundabout

spanner

Figure

A

roundabout

spanner

u

is

a

degree

vertex

which

is

not

on

the

roundabout

then

it

is

on

the

clockwise

border

of

its

quadrant

when

viewed

from

the

roundabout

and

is

at

odd

distance

from

the

roundabout

However

the

recursive

construction

ensures

that

only

the

vertices

at

even

distance

on

this

border

are

given

additional

incident

edges

Thus

in

all

cases

the

degree

of

u

is

at

most

We

use

the

term

frame

to

refer

to

the

maximum

subgraph

of

this

-LRB-

or

any

-RRB-

spanner

The

frame

of

is

shown

in

Figure

The

natural

embedding

of

the

frame

divides

the

plane

into

areas

which

we

referred

to

as

tiles

in

-LSB-

-RSB-

We

use

the

term

rectangular

frame

to

refer

to

any

frame

whose

tiles

are

rectangles

Note

that

aside

from

the

corners

of

such

frames

all

degree

vertices

are

incident

on

either

two

horizontal

or

two

vertical

edges

in

the

frame

Consider

any

two

vertices

u

and

v

of

a

grid

G.

All

shortest

paths

between

u

and

v

are

contained

in

the

rectangle

defined

by

u

and

v

and

either

increase

or

decrease

monotonically

in

both

the

and

directions

Due

to

the

shape

of

such

paths

we

call

them

staircase

paths

Lemma

From

any

vertex

u

on

a

rectangular

frame

F

there

is

a

staircase

path

in

F

from

u

to

each

of

the

four

corners

of

F

PROOF

Let

v

be

the

upper

right

corner

of

F

By

symmetry

we

need

only

establish

that

there

Figure

the

frame

of

is

a

staircase

path

from

u

to

v

We

observe

that

for

any

vertex

w

of

F

there

must

be

a

vertex

adjacent

to

w

which

is

on

F

and

which

lies

either

above

or

to

the

right

of

w

unless

w

v.

Thus

we

can

construct

a

staircase

path

from

u

to

v

by

including

an

edge

to

a

neighbor

above

or

to

the

right

at

each

step

Lemma

Let

u

and

v

be

two

vertices

in

a

rectangular

frame

F

There

exists

a

vertex

in

F

such

that

and

v

have

either

the

same

or

the

same

dG

-LRB-

v

-RRB-

dG

-LRB-

u

v

-RRB-

and

-LRB-

v

-RRB-

-LRB-

u

v

-RRB-

PROOF

If

u

and

v

already

share

an

or

let

u.

Otherwise

by

symmetry

assume

that

v

is

above

and

to

the

right

of

u.

Find

a

staircase

path

from

u

to

the

upper

right

corner

of

F

This

path

must

intersect

either

the

upper

or

right

side

of

the

rectangle

defined

by

u

and

v.

Let

the

first

such

intersection

point

be

as

shown

in

Figure

This

v

u

u

upper

rightcorner

of

grid

u

-LRB-

-RRB-

Figure

Finding

the

intersection

point

vertex

satisfies

properties

and

Noting

that

dG

-LRB-

u

-RRB-

dF

-LRB-

u

-RRB-

as

there

is

a

staircase

path

from

u

to

we

get

-LRB-

u

v

-RRB-

dF

-LRB-

u

v

-RRB-

dG

-LRB-

u

v

-RRB-

dF

-LRB-

u

v

-RRB-

-LRB-

dG

-LRB-

u

-RRB-

dG

-LRB-

v

-RRB-

-RRB-

-LRB-

dF

-LRB-

u

-RRB-

dF

-LRB-

v

-RRB-

-RRB-

dG

-LRB-

u

-RRB-

dG

-LRB-

v

-RRB-

dF

-LRB-

v

-RRB-

dG

-LRB-

v

-RRB-

-LRB-

v

-RRB-

satisfying

property

This

lemma

implies

that

we

need

only

consider

co-horizontal

and

co-vertical

pairs

of

vertices

to

determine

the

delay

function

f

F

-LRB-

x

-RRB-

for

a

rectangular

frame

F

Corollary

a

Let

F

be

a

rectangular

frame

with

non-decreasing

delay

function

f

F

-LRB-

x

-RRB-

If

f

F

-LRB-

x

-RRB-

f

F

-LRB-

x

-RRB-

then

there

exists

a

pair

of

co-horizontal

or

co-vertical

vertices

u

and

v

with

dG

-LRB-

u

v

-RRB-

x

and

-LRB-

u

v

-RRB-

f

F

-LRB-

x

-RRB-

PROOF

Assume

by

way

of

contradiction

that

f

F

-LRB-

x

-RRB-

f

F

-LRB-

x

-RRB-

but

that

no

such

u

and

v

exist

There

must

be

a

pair

of

vertices

a

and

b

which

are

not

colinear

but

with

dG

-LRB-

a

b

-RRB-

x

and

-LRB-

a

b

-RRB-

f

F

-LRB-

x

-RRB-

By

Lemma

we

can

find

an

such

that

and

b

are

colinear

dG

-LRB-

b

-RRB-

dG

-LRB-

a

b

-RRB-

x

and

-LRB-

b

-RRB-

-LRB-

a

b

-RRB-

f

F

-LRB-

x

-RRB-

Let

u

and

v

b

If

dG

-LRB-

u

v

-RRB-

x

then

-LRB-

u

v

-RRB-

f

F

-LRB-

x

-RRB-

contradicting

our

assumption

Otherwise

dG

-LRB-

u

v

-RRB-

x

In

this

case

-LRB-

u

v

-RRB-

f

F

-LRB-

x

-RRB-

contradicts

either

the

property

that

f

F

-LRB-

x

-RRB-

is

non-decreasing

or

the

hypothesis

that

f

F

-LRB-

x

-RRB-

f

F

-LRB-

x

-RRB-

Lemma

If

u

and

v

are

two

vertices

on

a

vertical

-LRB-

or

horizontal

-RRB-

highway

with

index

i

then

-LRB-

u

v

-RRB-

bdG

-LRB-

u

v

-RRB-

c

PROOF

Without

loss

of

generality

we

consider

a

pair

u

v

on

a

vertical

highway

The

proof

for

u

v

on

a

horizontal

highway

is

similar

A

vertical

highway

with

index

i

is

composed

of

sections

consisting

of

a

roundabout

and

its

associated

vertical

highway

sections

These

sections

are

separated

by

horizontal

highways

of

index

i

The

intersections

with

these

horizontal

highways

consist

of

one

horizontal

and

one

vertical

edge

The

order

of

these

edges

on

the

vertical

highway

is

unimportant

but

depends

on

the

index

of

the

horizontal

highway

and

the

position

of

the

vertical

highway

The

net

effect

of

the

roundabouts

and

intersections

is

that

a

vertical

highway

of

index

i

consists

of

segments

of

vertical

edges

alternating

with

turn

sections

composed

of

one

horizontal

and

one

vertical

edge

in

some

order

Consider

the

path

from

u

to

v

along

this

highway

The

vertical

distance

from

u

to

v

is

at

most

x

We

are

interested

in

the

number

of

turn

sections

encountered

in

the

path

from

u

to

v

Since

both

u

and

v

may

be

in

turn

sections

and

since

one

turn

section

is

included

in

every

vertical

interval

of

length

the

total

number

of

turn

sections

encountered

is

dx

e

bx

c

Since

no

vertical

backtracking

occurs

the

delay

on

this

path

is

no

more

than

the

number

of

horizontal

edges

which

is

no

more

than

the

number

of

turn

sections

encountered

Thus

-LRB-

u

v

-RRB-

bx

c

Although

this

lemma

gives

us

a

bound

on

the

delay

between

the

two

vertices

in

general

we

may

be

able

to

achieve

a

smaller

delay

by

following

paths

through

highways

of

larger

index

Theorem

is

a

-LRB-

x

-RRB-

spanner

of

with

maximum

degree

and

average

degree

-LRB-

-RRB-

PROOF

By

Corollary

a

we

need

only

consider

co-vertical

-LRB-

or

co-horizontal

-RRB-

pairs

of

vertices

We

first

consider

the

case

where

u

and

v

are

co-vertical

vertices

on

the

frame

but

not

on

the

border

of

the

grid

Note

that

for

any

i

i

k

the

number

of

columns

between

augmented

highways

of

index

i

is

Thus

u

and

v

are

either

on

an

augmented

highway

of

index

i

or

lie

in

a

column

which

is

distance

from

an

augmented

highway

of

index

i

and

distance

from

an

augmented

highway

of

index

i.

Let

and

be

vertices

at

the

same

horizontal

positions

as

u

and

v

respectively

on

the

vertical

highway

of

index

i

Then

dG

-LRB-

u

-RRB-

and

dG

-LRB-

v

-RRB-

Let

be

the

path

from

to

along

that

vertical

highway

of

index

i.

v

u

v

u

Pv

Pu

w

a

v

u

v

u

Pv

Pu

v

u

a

a

b

Figure

Paths

between

u

and

v

Without

loss

of

generality

we

assume

that

u

is

below

v

and

that

-LRB-

and

-RRB-

are

to

the

left

of

u

and

v.

Consider

the

staircase

path

Pu

from

u

to

the

upper

left

corner

and

the

staircase

path

Pv

from

v

to

the

lower

left

corner

as

guaranteed

by

Lemma

If

Pu

and

Pv

intersect

at

a

vertex

w

to

the

right

of

then

the

path

P

constructed

by

following

Pu

from

u

to

w

and

then

Pv

from

w

to

v

gives

a

path

from

u

to

v

with

delay

-LRB-

see

Figure

-LRB-

a

-RRB-

-RRB-

Otherwise

Pu

and

Pv

must

intersect

at

vertices

and

respectively

-LRB-

see

Figure

-LRB-

b

-RRB-

-RRB-

In

this

case

the

path

P

from

u

to

v

following

Pu

from

u

to

from

to

and

Pv

from

to

v

gives

a

path

from

u

to

v

with

delay

-LRB-

-RRB-

the

delay

from

to

along

This

latter

quantity

is

at

most

the

delay

from

to

along

which

is

at

most

bdG

-LRB-

-RRB-

c

by

Lemma

Thus

the

delay

along

P

is

at

most

-LRB-

-RRB-

bdG

-LRB-

u

v

-RRB-

c

Now

we

consider

the

path

Q

along

the

augmented

highway

of

index

i

at

distance

from

u

and

v

A

similar

argument

establishes

that

the

delay

on

such

a

path

is

at

most

-LRB-

-RRB-

bdG

-LRB-

u

v

-RRB-

c

-LRB-

The

denominator

is

due

to

the

fact

that

this

highway

has

index

i

-RRB-

Thus

the

delay

between

u

and

v

is

at

most

the

minimum

of

the

delay

along

P

and

the

delay

along

Q.

To

find

an

upper

bound

on

this

delay

we

consider

to

be

a

continuous

variable

and

maximize

min

-LRB-

-LRB-

-RRB-

bdG

-LRB-

u

v

-RRB-

c

-LRB-

-RRB-

bdG

-LRB-

u

v

-RRB-

c

-RRB-

This

minimum

is

maximized

when

both

delays

are

the

same

or

when

x

and

the

delay

is

The

above

bound

on

delay

holds

for

any

i

between

and

k

inclusive

Considered

together

these

i

imply

an

upper

bound

of

min

i

k

on

the

delay

between

u

and

v

This

is

in

turn

bounded

above

by

for

any

particular

choice

of

i.

Consider

i

-LRB-

-RRB-

c.

-LRB-

Note

that

i

-LRB-

-RRB-

minimizes

the

continuous

function

-RRB-

This

gives

an

upper

bound

on

the

delay

of

-LRB-

-RRB-

c

-LRB-

-RRB-

c

which

we

claim

is

at

most

Let

y

Substituting

y

and

subtracting

from

both

sides

the

claim

becomes

yc

yc

y

Let

OE

log

y

blog

yc

Thus

OE

Then

yc

OE

y

Let

As

OE

we

have

that

and

yc

y

The

claim

now

becomes

y

y

y

y

y

which

is

true

for

Recalling

that

the

claim

is

proved

This

gives

an

upper

bound

of

for

pairs

of

vertices

on

the

frame

that

are

separated

by

distance

x

in

the

grid

Consider

a

pair

of

vertices

u

and

v

that

are

not

necessarily

on

the

frame

As

any

non-frame

vertex

is

at

distance

from

a

frame

vertex

in

S

we

can

find

and

both

on

the

frame

such

that

dS

-LRB-

u

-RRB-

and

dS

-LRB-

v

-RRB-

This

implies

dG

-LRB-

-RRB-

dG

-LRB-

u

v

-RRB-

and

dS

-LRB-

-RRB-

dG

-LRB-

-RRB-

-LRB-

-RRB-

This

means

that

dS

-LRB-

-RRB-

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

or

dS

-LRB-

u

v

-RRB-

dS

-LRB-

-RRB-

-LRB-

dG

-LRB-

u

v

-RRB-

-LRB-

dG

-LRB-

u

v

-RRB-

-RRB-

-RRB-

Simplifying

gives

a

delay

of

at

most

-LRB-

u

v

-RRB-

between

u

and

v.

To

calculate

the

average

degree

we

begin

by

summing

the

number

of

edges

in

highways

of

each

index

The

borders

of

the

grid

constitute

the

highways

of

index

k

There

are

-LRB-

-RRB-

edges

in

these

highways

A

single

roundabout

of

index

i

with

its

four

associated

augmented

highway

sections

contributes

edges

Since

there

are

i

roundabouts

of

index

i

there

are

a

total

of

i

-LRB-

-RRB-

edges

on

augmented

highways

of

index

i.

Summing

over

all

indices

this

gives

-LRB-

-RRB-

k

X

i

i

-LRB-

-RRB-

k

X

i

i

k

X

i

i

k

X

i

i

k

X

i

i

k

X

i

k

X

i

-LRB-

-RRB-

-LRB-

-RRB-

edges

in

the

spanner

The

average

degree

is

-LRB-

-RRB-

-LRB-

-RRB-

To

construct

a

-LRB-

x

-RRB-

spanner

for

with

maximum

degree

first

construct

where

k

is

chosen

to

be

as

small

as

possible

such

that

and

Add

all

of

the

edges

of

column

and

all

of

the

edges

of

row

Discard

the

portion

of

the

resulting

structure

that

lies

outside

of

to

obtain

the

spanner

The

delay

between

any

pair

of

vertices

in

the

resulting

spanner

can

not

exceed

the

delay

between

the

corresponding

vertices

of

Note

that

in

the

roundabout

spanner

we

have

collapsed

about

of

the

roundabouts

that

occurred

in

the

simple

construction

If

we

collapse

all

of

the

roundabouts

we

obtain

a

different

spanner

with

smaller

average

degree

Jogging

Spanners

To

construct

the

jogging

spanner

for

the

square

grid

first

include

the

edges

along

the

border

of

this

grid

Add

the

edges

of

row

and

alternate

edges

of

row

as

shown

in

Figure

-LRB-

a

-RRB-

This

creates

three

augmented

horizontal

and

two

augmented

vertical

highways

of

index

k.

Next

we

construct

an

augmented

vertical

highway

of

index

k

on

columns

and

The

highway

consists

of

the

column

edges

from

row

to

row

where

it

jogs

to

column

and

continues

to

row

In

addition

to

these

edges

the

augmented

highway

includes

every

second

edge

in

the

unused

portion

of

these

columns

as

shown

in

Figure

-LRB-

b

-RRB-

We

now

insert

two

augmented

horizontal

highways

of

index

k

in

the

same

fashion

For

every

adjacent

pair

of

existing

augmented

horizontal

highways

insert

a

new

augmented

horizontal

highway

on

the

two

rows

midway

between

the

existing

highways

To

insert

a

highway

begin

on

the

even

numbered

row

at

column

insert

horizontal

edges

across

to

column

jogging

to

the

other

row

of

the

highway

whenever

encountering

the

solid

section

of

a

vertical

highway

Include

every

second

edge

in

the

unused

portion

of

these

rows

as

before

to

form

augmented

highways

The

result

is

shown

in

Figure

-LRB-

c

-RRB-

Repeat

this

step

in

the

vertical

direction

to

achieve

the

structure

shown

in

Figure

-LRB-

d

-RRB-

this

figure

also

indicates

the

indices

of

the

augmented

highways

created

so

far

The

construction

continues

by

repeating

this

step

alternating

between

the

vertical

and

horizontal

augmented

highway

insertions

Each

set

of

insertions

creates

augmented

highways

with

index

one

less

than

the

previous

set

of

insertions

in

that

direction

A

full

construction

for

a

grid

is

shown

in

Figure

-LRB-

d

-RRB-

Theorem

is

a

-LRB-

x

-RRB-

spanner

of

with

maximum

degree

and

average

degree

Figure

Building

a

jogging

spanner

abcd

Figure

Final

steps

in

construction

of

jogging

spanner

PROOF

Note

that

since

has

a

rectangular

frame

Lemma

Lemma

and

Corollary

a

apply

By

our

assignment

of

index

numbers

to

highways

Lemma

also

holds

To

prove

the

upper

bound

on

the

delay

of

we

proceed

as

in

the

proof

of

Theorem

noting

that

the

number

of

columns

between

augmented

vertical

highways

of

index

i

is

-LRB-

as

in

the

previous

proof

-RRB-

while

the

number

of

rows

between

augmented

horizontal

highways

of

index

i

is

Following

the

analysis

of

the

proof

of

Theorem

gives

delay

between

co-vertical

pairs

of

vertices

of

at

most

while

the

delay

between

co-horizontal

pairs

of

vertices

is

smaller

To

calculate

the

average

degree

we

begin

by

summing

the

number

of

edges

in

augmented

highways

of

each

index

The

borders

of

the

grid

contribute

-LRB-

-RRB-

edges

The

other

index

k

augmented

highway

is

the

first

horizontal

highway

added

which

contributes

-LRB-

-RRB-

-LRB-

-RRB-

edges

Generally

each

of

the

i

index

i

augmented

horizontal

highways

contributes

-LRB-

-RRB-

i

-LRB-

-RRB-

edges

We

construct

augmented

horizontal

highways

for

each

index

from

to

k.

Similarly

each

of

the

i

index

i

augmented

vertical

highways

contributes

-LRB-

-RRB-

k?i

-LRB-

-RRB-

edges

We

construct

augmented

vertical

highways

for

each

index

from

to

k

Thus

the

total

number

of

edges

added

during

the

construction

of

is

-LRB-

-RRB-

kX

i

i

-LSB-

-LRB-

-RRB-

i

-LRB-

-RRB-

-RSB-

k

X

i

i

-LSB-

-LRB-

-RRB-

i

-LRB-

-RRB-

-RSB-

-LRB-

-RRB-

k

X

i

-LSB-

i

i

-RSB-

k

X

i

-LSB-

i

i

-RSB-

-LRB-

-RRB-

k

X

i

i

k

X

i

k

X

i

k

X

i

i

-LRB-

-RRB-

k

X

i

i

kX

i

i

k

X

i

i

-LRB-

-RRB-

k

X

i

k

X

i

k

X

i

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

The

average

degree

is

-LRB-

-RRB-

As

with

the

roundabout

spanner

we

can

construct

a

-LRB-

x

-RRB-

spanner

of

this

type

for

First

construct

where

k

is

chosen

to

be

as

small

as

possible

such

that

and

Add

all

of

the

edges

of

column

and

all

of

the

edges

of

row

Discard

the

portion

of

the

resulting

structure

that

lies

outside

of

to

obtain

the

spanner

The

delay

between

any

pair

of

vertices

in

the

resulting

spanner

can

not

exceed

the

delay

between

the

corresponding

vertices

of

This

spanner

has

maximum

degree

and

average

degree

less

than

that

of

the

spanner

of

derived

from

the

roundabout

construction

Spanners

of

X-trees

and

Pyramids

X-Tree

Spanners

with

Maximum

Degree

We

begin

our

investigation

of

spanners

of

X-trees

by

reviewing

the

previous

results

in

this

area

Richards

and

Liestman

-LSB-

-RSB-

investigated

degree

constrained

-LRB-

tx

-RRB-

spanners

of

X-trees

In

particular

they

constructed

a

with

maximum

degree

and

a

with

maximum

degree

Liestman

and

Shermer

-LSB-

-RSB-

constructed

a

-LRB-

x

-RRB-

spanner

of

the

X-tree

with

maximum

degree

In

this

section

we

are

interested

in

constructing

spanners

with

maximum

degree

and

improved

f

-LRB-

x

-RRB-

An

X-tree

consists

of

a

full

balanced

binary

tree

with

all

the

leaves

on

the

same

level

plus

some

additional

edges

Edges

are

added

to

the

tree

such

that

the

vertices

on

each

level

are

connected

from

left

to

right

in

a

path

We

will

use

the

term

horizontal

edge

to

denote

an

edge

of

such

a

path

Horizontal

edges

are

of

two

types

The

term

sibling

edge

denotes

a

horizontal

edge

that

connects

two

vertices

with

the

same

parent

and

the

term

cousin

edge

denotes

any

of

the

remaining

horizontal

edges

The

term

vertical

edge

denotes

a

tree

edge

Vertical

edges

from

a

parent

to

a

left

or

right

child

will

be

referred

to

as

left

or

right

vertical

edges

respectively

We

will

imagine

that

the

tree

is

positioned

with

the

root

on

level

at

the

top

and

will

refer

to

movement

up

and

down

the

tree

and

use

the

terms

above

and

below

with

that

orientation

in

mind

Note

that

an

X-tree

has

maximum

degree

a

non-boundary

vertex

has

degree

and

the

average

degree

of

a

vertex

in

the

X-tree

is

approximately

Let

height

-LRB-

G

-RRB-

denote

the

number

of

levels

in

graph

G

and

let

levelG

-LRB-

v

-RRB-

denote

the

level

associated

with

vertex

v

in

graph

G

where

G

is

either

an

X-tree

or

a

pyramid

We

will

use

Tu

to

denote

the

subtree

of

the

X-tree

rooted

at

vertex

u

The

following

result

-LRB-

from

-LSB-

-RSB-

-RRB-

will

be

useful

in

our

proofs

Lemma

Given

any

two

vertices

a

and

b

in

an

X-tree

there

is

some

shortest

path

between

a

and

b

that

contains

at

most

one

sibling

edge

and

at

most

two

cousin

edges

with

these

horizontal

edges

at

the

top

of

the

path

We

note

that

Richards

and

Liestman

-LSB-

-RSB-

produced

results

under

the

original

definition

of

spanner

which

is

concerned

only

with

the

delay

between

vertices

adjacent

in

the

original

graph

G.

Among

these

results

they

showed

that

-LRB-

in

their

terminology

-RRB-

of

sufficiently

large

X-trees

can

not

be

constructed

with

maximum

degree

Although

corresponds

to

in

our

terminology

their

lower

bound

proof

only

shows

that

f

-LRB-

x

-RRB-

spanners

of

sufficiently

large

X

trees

can

not

be

constructed

with

maximum

degree

and

f

-LRB-

-RRB-

-LRB-

The

upper

bounds

translate

readily

between

the

terminologies

their

is

a

-RRB-

Consequently

we

are

still

left

with

the

problem

of

constructing

good

spanners

of

the

X-tree

with

maximum

degree

Lower

Bound

for

X-tree

Spanners

with

Maximum

Degree

In

this

section

we

show

that

any

X-tree

spanner

with

maximum

degree

must

have

delay

that

is

linear

in

x.

Theorem

There

is

no

f

-LRB-

x

-RRB-

such

that

f

-LRB-

x

-RRB-

o

-LRB-

x

-RRB-

and

there

exist

f

-LRB-

x

-RRB-

spanners

Sh

with

maximum

degree

for

X-trees

of

every

height

h

PROOF

If

we

can

show

that

such

a

spanner

S

must

contain

some

horizontal

edges

we

observe

that

as

we

are

limited

to

maximum

degree

any

endpoint

of

an

included

horizontal

edge

must

be

incident

on

a

vertical

edge

that

is

not

included

in

S

We

now

show

that

some

horizontal

edges

must

be

included

in

S

The

funnel

of

a

non-leaf

vertex

u

denoted

F

-LRB-

u

-RRB-

is

the

subgraph

induced

by

u

and

all

of

the

endpoints

of

the

horizontal

edges

directly

below

u

in

the

usual

embedding

of

the

X-tree

as

shown

for

the

example

in

Figure

u

Figure

the

funnel

of

u

We

claim

that

given

a

funnel

F

-LRB-

u

-RRB-

of

such

a

spanner

S

there

must

be

a

horizontal

edge

of

S

in

that

funnel

within

f

-LRB-

-RRB-

levels

of

u.

Consider

the

horizontal

edge

-LRB-

a

b

-RRB-

of

F

-LRB-

u

-RRB-

that

is

df

-LRB-

-RRB-

e

levels

below

u

If

this

edge

or

any

horizontal

edge

above

it

in

F

-LRB-

u

-RRB-

is

contained

in

S

then

the

claim

is

satisfied

Otherwise

any

path

from

a

to

b

in

S

either

ascends

at

least

as

far

as

u

or

crosses

the

funnel

below

-LRB-

a

b

-RRB-

In

the

former

case

the

path

must

go

up

at

least

df

-LRB-

-RRB-

e

levels

and

then

down

at

least

df

-LRB-

-RRB-

e

levels

and

thus

be

at

least

f

-LRB-

-RRB-

edges

long

Since

S

is

assumed

to

be

an

f

-LRB-

x

-RRB-

spanner

a

shortest

path

from

a

to

b

must

cross

the

funnel

on

a

horizontal

edge

below

-LRB-

a

b

-RRB-

If

this

horizontal

edge

is

more

than

f

-LRB-

-RRB-

levels

below

-LRB-

a

b

-RRB-

then

this

path

is

of

length

greater

than

-LRB-

f

-LRB-

-RRB-

-RRB-

f

-LRB-

-RRB-

a

contradiction

Thus

in

this

case

there

must

be

a

horizontal

edge

within

f

-LRB-

-RRB-

levels

below

-LRB-

a

b

-RRB-

This

implies

that

there

is

a

horizontal

edge

within

df

-LRB-

-RRB-

e

f

-LRB-

-RRB-

f

-LRB-

-RRB-

levels

of

u

Since

these

horizontal

edges

must

be

included

we

know

that

some

corresponding

vertical

edges

are

not

included

in

S.

Let

u

and

v

be

two

vertices

of

an

X-tree

We

call

v

an

interior

descendant

of

u

if

v

is

a

descendant

of

u

and

the

vertical

path

from

u

to

v

includes

both

left

and

right

vertical

edges

If

v

is

an

interior

descendant

of

u

then

Tv

is

said

to

be

properly

nested

in

Tu

that

is

no

node

of

Tv

is

adjacent

to

a

node

that

is

not

in

Tu

Consider

a

shortest

path

in

the

X-tree

between

the

root

r

and

some

node

w

at

level

h

In

the

X-tree

this

path

consists

of

h

vertical

edges

In

a

spanner

of

the

X-tree

such

a

path

will

include

at

least

h

vertical

edges

and

may

also

include

some

horizontal

edges

Thus

the

delay

in

the

spanner

xi

a

b

li

Figure

candidates

for

xi

is

at

least

as

large

as

the

number

of

horizontal

edges

Suppose

now

that

u

is

an

ancestor

of

w

such

that

the

edge

from

u

to

its

parent

in

the

X-tree

is

not

included

in

the

spanner

As

this

is

the

only

vertical

edge

out

of

the

subtree

Tu

rooted

at

u

then

any

path

from

w

to

r

in

the

spanner

must

include

a

horizontal

edge

out

of

Tu

We

can

extend

this

argument

Suppose

that

we

have

a

sequence

of

vertices

r

xl

w

such

that

w

is

in

Txl

each

xi

is

an

interior

descendant

of

xi

and

the

edge

from

each

xi

to

its

parent

is

not

included

in

the

spanner

In

this

situation

any

path

from

w

to

r

in

the

spanner

must

contain

at

least

l

horizontal

edges

and

hence

have

delay

at

least

l.

-LRB-

Due

to

the

definition

of

interior

descendant

any

horizontal

edge

with

an

endpoint

in

Txi

has

its

other

endpoint

in

Txi

-RRB-

We

now

show

how

to

find

a

sequence

of

vertices

r

xl

w

as

described

above

with

l

b

h

f

-LRB-

-RRB-

c

in

any

f

-LRB-

x

-RRB-

spanner

with

maximum

degree

of

an

X-tree

of

height

h

Start

with

r

We

obtain

xi

from

xi

as

follows

Let

li

be

the

left

interior

grandchild

of

xi

i.e.

the

vertex

that

is

the

right

child

of

the

left

child

of

xi

Note

that

li

and

all

of

its

descendants

are

interior

descendants

of

xi

Consider

the

funnel

F

-LRB-

li

-RRB-

There

is

a

horizontal

edge

-LRB-

a

b

-RRB-

in

this

funnel

that

is

included

in

the

spanner

and

is

at

most

f

-LRB-

-RRB-

levels

below

li

The

endpoint

a

of

this

edge

must

have

degree

no

more

than

Thus

either

the

edge

from

a

to

its

parent

is

not

in

the

spanner

or

the

edge

from

a

to

one

of

its

children

is

not

in

the

spanner

as

indicated

in

Figure

In

the

former

case

let

xi

a

and

in

the

latter

case

let

xi

Thus

we

can

find

xi

which

is

at

most

f

-LRB-

-RRB-

levels

below

xi

-LRB-

The

vertex

li

is

two

levels

below

xi

a

is

at

most

f

-LRB-

-RRB-

levels

below

li

and

is

one

level

below

a.

-RRB-

If

the

X-tree

has

h

levels

we

may

repeat

this

process

b

h

f

-LRB-

-RRB-

c

times

We

therefore

let

l

b

h

f

-LRB-

-RRB-

c

and

have

obtained

xl

Choosing

w

xl

we

have

found

the

desired

sequence

in

the

spanner

By

the

preceding

argument

this

implies

that

the

path

from

w

to

r

in

the

spanner

has

delay

at

least

l

b

h

f

-LRB-

-RRB-

c.

Furthermore

the

path

from

r

to

any

descendant

of

w

also

has

delay

at

least

l.

Assume

by

way

of

contradiction

that

we

can

construct

an

f

-LRB-

x

-RRB-

spanner

Sh

of

any

X-tree

of

height

h

such

that

f

-LRB-

x

-RRB-

o

-LRB-

x

-RRB-

This

implies

that

for

any

constant

c

there

exists

such

that

f

-LRB-

x

-RRB-

cx

for

all

x

We

choose

c

f

-LRB-

-RRB-

By

our

assumption

some

exists

satisfying

f

-LRB-

x

-RRB-

x

f

-LRB-

-RRB-

for

all

x

Consider

the

spanner

Sh

of

an

X-tree

of

height

h

where

h

is

the

smallest

multiple

of

f

-LRB-

-RRB-

that

is

greater

than

By

the

above

argument

we

can

find

a

vertex

w

at

the

bottom

of

the

X-tree

-LRB-

i.e.

at

distance

greater

than

from

r

-RRB-

such

that

the

path

from

the

root

r

to

w

has

delay

at

least

l

h

f

-LRB-

-RRB-

x

f

-LRB-

-RRB-

This

contradicts

f

-LRB-

x

-RRB-

x

f

-LRB-

-RRB-

and

therefore

our

assumption

that

f

-LRB-

x

-RRB-

o

-LRB-

x

-RRB-

Construction

of

X-tree

Spanners

with

Maximum

Degree

Before

describing

some

new

spanners

of

this

type

we

note

that

the

of

Richards

and

Liestman

-LSB-

-RSB-

which

has

maximum

degree

can

be

shown

to

be

a

-LRB-

-RRB-

spanner

of

the

X-tree

The

proof

is

simple

and

is

omitted

here

A

k-level

shrub

is

a

spanner

of

an

X-tree

of

height

k

for

k

To

construct

a

k-level

shrub

connect

the

root

r

to

its

right

child

and

continue

adding

right

vertical

edges

to

form

a

path

of

k

right

vertical

edges

from

the

root

through

vertices

rk

At

each

vertex

ri

along

this

path

add

the

sibling

edge

to

its

left

sibling

li

Include

the

edges

of

a

complete

binary

tree

of

height

k

i

rooted

at

li

i

k

Add

all

of

the

cousin

edges

at

the

bottom

of

each

of

these

binary

trees

-LRB-

that

is

k

levels

below

r

-RRB-

and

those

cousin

edges

on

the

bottom

level

that

connect

these

binary

trees

Add

the

left

vertical

edge

from

rk

to

lk

and

the

sibling

edge

from

lk

to

rk

The

resulting

structure

is

a

k-level

shrub

Figure

shows

k-level

shrubs

for

k

Note

that

the

root

has

degree

and

the

vertices

on

level

k

below

the

root

have

degree

at

most

To

construct

a

k-level

shrub

spanner

of

an

X-tree

of

height

h

begin

by

constructing

a

k-level

Figure

and

level

shrubs

Figure

shrub

spanner

shrub

rooted

at

the

root

of

the

X-tree

As

long

as

k

levels

remain

below

the

level

of

the

lowest

vertices

included

in

the

structure

add

a

k-level

shrub

rooted

at

each

vertex

on

the

lowest

level

of

the

existing

structure

Connect

these

newly

added

k-level

shrubs

by

adding

cousin

edges

between

adjacent

vertices

from

different

shrubs

on

the

bottom

level

of

these

shrubs

Note

that

adding

these

edges

does

not

increase

the

degree

of

the

vertices

beyond

When

i

k

levels

remain

add

the

top

i

levels

of

a

k-level

shrub

rooted

at

each

vertex

on

the

lowest

level

of

the

existing

structure

Add

all

of

the

missing

cousin

edges

of

the

bottom

level

of

the

X-tree

Figure

shows

a

shrub

spanner

for

an

X-tree

of

height

Lemma

The

largest

face

on

a

k-level

shrub

spanner

is

bounded

by

at

most

edges

PROOF

The

largest

face

within

a

k-level

shrub

for

k

is

the

face

below

the

sibling

edge

between

and

which

is

bounded

by

-LRB-

k

-RRB-

vertical

and

horizontal

edges

in

total

edges

The

shrub

has

no

faces

Other

faces

are

formed

by

edges

from

two

consecutive

layers

of

k-level

shrubs

The

largest

face

of

this

type

is

the

face

lying

below

a

sibling

edge

between

vertices

lk

and

rk

of

some

k-level

shrub

This

face

is

bounded

by

-LRB-

k

-RRB-

vertical

and

horizontal

edges

in

total

edges

Due

to

the

inclusion

of

cousin

edges

at

the

bottom

of

each

layer

no

faces

are

bounded

by

edges

from

more

than

two

consecutive

layers

Theorem

The

k-level

shrub

spanner

S

is

a

-LRB-

x

k

x

-RRB-

spanner

of

the

X-tree

with

maximum

degree

and

average

degree

PROOF

From

Lemma

we

know

that

the

shortest

path

P

between

any

pair

of

vertices

a

and

b

in

an

X-tree

can

be

divided

into

two

vertical

sections

and

one

horizontal

section

Let

and

denote

the

number

of

edges

in

the

two

vertical

sections

and

h

denote

the

number

of

edges

in

the

horizontal

section

Given

a

k-level

shrub

spanner

S

we

note

that

a

vertical

path

of

length

v

can

touch

upon

at

most

bv

c

d

shrubs

Since

each

shrub

can

add

at

most

one

horizontal

edge

-LRB-

and

no

vertical

edges

-RRB-

to

a

vertical

path

the

maximum

number

of

horizontal

edges

added

to

a

vertical

path

of

length

v

is

d.

Thus

the

vertical

sections

of

the

path

P

in

the

X-tree

correspond

to

path

sections

in

the

spanner

containing

no

more

than

k

c

k

c

k

c

units

of

delay

Let

dh

denote

the

delay

in

S

between

the

endpoints

of

the

horizontal

section

of

P

The

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

k

c

dh

bx?h

k

c

dh

If

the

horizontal

section

of

P

consists

of

a

single

sibling

edge

which

is

missing

from

the

k-level

shrub

spanner

it

can

be

replaced

by

the

two

edges

to

the

parent

giving

dh

In

this

case

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx

k

c

x

xk

k

If

the

horizontal

section

of

P

consists

of

a

single

cousin

edge

which

is

missing

from

the

k-level

shrub

spanner

its

endpoints

are

on

a

face

of

no

more

than

edges

so

it

can

be

replaced

by

a

path

of

length

at

most

k

giving

dh

k

k

In

this

case

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx

k

c

-LRB-

k

-RRB-

x

k

xk

k

k

If

the

horizontal

section

of

P

consists

of

one

sibling

and

one

cousin

edge

and

both

are

missing

from

S

then

the

edges

to

the

parent

-LRB-

x

-RRB-

of

the

sibling

vertices

are

in

the

spanner

This

vertex

and

the

remaining

endpoint

-LRB-

w

-RRB-

of

the

cousin

edge

are

on

the

same

face

of

S.

Thus

there

is

a

path

between

w

and

x

of

no

more

than

k

edges

giving

delay

of

dh

-LRB-

k

-RRB-

k

In

this

case

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx

k

c

-LRB-

k

-RRB-

x

k

xk

k

k

Note

that

if

the

horizontal

section

of

P

is

of

this

form

and

only

one

of

the

edges

is

missing

then

one

of

the

previous

cases

applies

If

the

horizontal

section

of

P

consists

of

one

sibling

edge

-LRB-

v

w

-RRB-

and

two

cousin

edges

-LRB-

u

v

-RRB-

and

-LRB-

w

x

-RRB-

both

of

which

are

missing

from

S

then

one

of

the

cousin

edges

-LRB-

say

-LRB-

w

x

-RRB-

-RRB-

must

be

immediately

below

a

sibling

edge

Let

y

be

the

parent

of

v

and

w

and

let

z

the

parent

of

x

-LRB-

and

sibling

to

y

-RRB-

By

the

structure

of

k-level

shrubs

we

know

that

there

is

a

path

in

S

from

x

to

z

of

at

most

edges

There

is

also

a

path

in

S

from

z

to

y

of

at

most

edges

consisting

of

either

the

sibling

edge

between

them

or

the

two

edges

to

their

parent

Finally

there

is

a

path

in

S

from

y

to

u

of

at

most

k

edges

since

y

and

u

are

on

the

same

face

of

S.

Thus

there

is

a

path

in

S

from

u

to

x

of

length

at

most

k

giving

delay

of

dh

-LRB-

k

-RRB-

k

In

this

case

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx

k

c

-LRB-

k

-RRB-

x

k

xk

k

k

If

the

horizontal

section

of

P

is

of

this

form

and

either

of

the

cousins

is

present

then

one

of

the

previous

cases

applies

Thus

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

xk

k

k

so

f

-LRB-

x

-RRB-

k

k

x

k

To

bound

the

average

degree

we

note

that

most

of

the

vertices

that

are

not

at

the

bottom

level

of

the

spanner

are

of

degree

Consider

a

vertex

not

at

the

bottom

level

of

the

spanner

If

this

vertex

is

the

second

vertex

from

the

right

on

the

bottom

level

of

its

shrub

then

it

has

degree

if

it

is

the

root

of

either

the

leftmost

or

rightmost

shrub

on

its

level

then

it

has

degree

if

it

is

the

root

of

the

X-tree

then

it

has

degree

Otherwise

it

has

degree

As

the

vertices

of

degree

are

relatively

sparse

we

can

obtain

a

good

upper

bound

on

the

average

degree

by

assuming

that

all

vertices

not

at

the

bottom

level

of

the

X-tree

have

degree

Consider

now

the

vertices

at

the

bottom

of

the

X-tree

By

our

construction

at

most

one

such

vertex

in

each

shrub

-LRB-

or

partial

shrub

-RRB-

is

of

degree

The

other

vertices

at

this

level

are

of

degree

at

most

There

are

at

most

shrubs

-LRB-

or

partial

shrubs

-RRB-

at

this

level

As

there

are

vertices

at

the

bottom

level

and

vertices

not

on

the

bottom

level

the

average

degree

is

at

most

-LRB-

-RRB-

Pyramid

Spanners

We

begin

our

investigation

of

spanners

of

pyramids

by

reviewing

the

previous

results

in

this

area

Richards

and

Liestman

-LSB-

-RSB-

investigated

degree

constrained

-LRB-

tx

-RRB-

spanners

of

pyramids

In

particular

they

constructed

a

with

maximum

degree

a

with

maximum

degree

and

a

with

maximum

degree

Liestman

and

Shermer

-LSB-

-RSB-

constructed

a

-LRB-

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

In

this

section

we

are

interested

in

constructing

spanners

with

lower

maximum

degree

and

improved

f

-LRB-

x

-RRB-

A

pyramid

consists

of

a

series

of

levels

with

vertices

on

level

i

starting

with

vertex

on

level

The

vertices

on

each

level

are

arranged

in

a

square

grid

graph

or

mesh

where

each

vertex

is

connected

to

its

four

orthogonal

neighbors

Further

each

vertex

is

connected

to

the

four

corresponding

vertices

on

the

level

below

it

A

pyramid

is

shown

in

Figure

As

with

X-trees

we

can

partition

the

edges

of

the

pyramid

into

horizontal

and

vertical

edges

further

classifying

the

horizontal

edges

as

sibling

or

cousin

edges

The

horizontal

edges

also

naturally

fall

into

two

classes

of

parallel

edges

corresponding

to

dimensions

that

we

call

and

We

will

use

the

terms

left

and

right

to

refer

to

the

relations

and

in

the

dimension

and

similarly

use

the

terms

front

and

back

to

refer

to

the

relations

and

in

the

dimension

Let

P

be

a

maximum

length

path

in

one

dimension

-LRB-

or

-RRB-

in

the

bottom

level

of

the

pyramid

The

set

AP

consisting

of

nodes

of

P

and

all

of

their

ancestors

induce

an

X-tree

XP

in

the

pyramid

If

P

is

in

dimension

xi

we

call

such

an

X-tree

an

induced

X-tree

in

dimension

xi

An

induced

X-tree

is

shown

in

Figure

Let

height

-LRB-

G

-RRB-

denote

the

number

of

levels

in

graph

G

and

let

levelG

-LRB-

v

-RRB-

denote

the

level

associated

with

vertex

v

in

graph

G

where

G

is

either

an

X-tree

or

a

pyramid

We

will

use

Tu

to

denote

the

subpyramid

of

the

pyramid

rooted

at

vertex

u

The

following

result

-LRB-

from

-LSB-

-RSB-

-RRB-

will

be

useful

in

our

proofs

Lemma

Given

any

two

vertices

a

and

b

in

a

pyramid

there

is

some

shortest

path

between

a

and

b

that

contains

at

most

one

sibling

edge

and

at

most

three

cousin

edges

with

these

horizontal

edges

at

the

top

of

the

path

level

level

level

level

Figure

a

pyramid

Figure

an

induced

X-tree

As

discussed

above

the

lower

bounds

given

by

Richards

and

Liestman

-LSB-

-RSB-

do

not

directly

apply

with

our

definition

of

f

-LRB-

x

-RRB-

spanner

Their

bounds

show

that

there

is

no

f

-LRB-

x

-RRB-

spanner

of

sufficiently

large

pyramids

with

maximum

degree

and

f

-LRB-

-RRB-

maximum

degree

and

f

-LRB-

-RRB-

maximum

degree

and

f

-LRB-

-RRB-

O

-LRB-

-RRB-

Below

we

give

improved

bounds

on

f

-LRB-

x

-RRB-

for

degree-constrained

f

-LRB-

x

-RRB-

spanners

of

pyramids

Construction

of

Pyramid

Spanners

with

Maximum

Degree

Although

the

focus

of

this

paper

is

on

spanners

with

non-constant

delay

we

include

the

following

result

for

the

sake

of

completeness

In

-LSB-

-RSB-

we

showed

how

to

construct

a

-LRB-

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

A

constant

delay

spanner

with

maximum

degree

can

also

be

constructed

Theorem

For

any

pyramid

P

there

exists

a

-LRB-

x

-RRB-

spanner

S

with

and

ffi

PROOF

To

construct

such

a

spanner

S

omit

all

of

the

sibling

edges

from

P

Omit

all

of

the

cousin

edges

in

one

dimension

on

even

levels

and

all

of

the

cousin

edges

in

the

other

dimension

on

odd

levels

of

the

pyramid

All

of

the

cousin

edges

on

the

lowest

level

of

the

pyramid

can

remain

The

resulting

graph

has

maximum

degree

and

average

degree

less

than

To

prove

that

S

is

a

-LRB-

x

-RRB-

spanner

is

relatively

straightforward

and

can

be

done

as

in

the

proofs

of

-LSB-

-RSB-

We

omit

the

details

here

To

calculate

the

average

degree

note

that

the

spanner

contains

Phi

vertices

Phi

vertical

edges

and

Ph

i

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

cousin

edges

where

h

is

the

number

of

levels

in

the

pyramid

P

The

total

number

of

edges

in

S

is

hX

i

h

X

i

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

hX

i

h

X

i

h

X

i

hX

i

h

X

i

h

X

i

hX

i

-LRB-

hX

i

-RRB-

hX

i

hX

i

hX

i

hX

i

-LRB-

-RRB-

The

average

degree

is

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

Lower

Bound

for

Pyramid

Spanners

with

Maximum

Degree

In

this

section

we

show

that

any

pyramid

spanner

with

maximum

degree

must

have

delay

that

is

linear

in

x.

Theorem

There

is

no

f

-LRB-

x

-RRB-

such

that

f

-LRB-

x

-RRB-

o

-LRB-

x

-RRB-

and

there

exist

f

-LRB-

x

-RRB-

spanners

Sh

with

maximum

degree

for

pyramids

of

every

height

h

PROOF

If

we

can

show

that

such

a

spanner

S

must

contain

some

horizontal

edges

we

observe

that

as

we

are

limited

to

maximum

degree

any

endpoint

of

an

included

horizontal

edge

must

be

incident

on

a

vertical

edge

that

is

not

included

in

S

We

now

show

that

some

horizontal

edges

must

be

included

in

S

The

funnel

of

a

non-leaf

vertex

u

of

the

pyramid

denoted

F

-LRB-

u

-RRB-

is

the

subgraph

induced

by

u

and

the

four

innermost

descendants

of

u

on

each

level

below

u

A

funnel

is

shown

in

Figure

We

claim

that

there

must

be

a

horizontal

edge

of

S

in

Tu

within

f

-LRB-

-RRB-

levels

of

u.

Consider

a

horizontal

edge

-LRB-

a

b

-RRB-

of

F

-LRB-

u

-RRB-

that

is

df

-LRB-

-RRB-

e

levels

below

u.

Any

path

of

length

f

-LRB-

-RRB-

between

a

and

b

is

contained

in

Tu

If

-LRB-

a

b

-RRB-

or

any

horizontal

edge

at

least

as

high

as

-LRB-

a

b

-RRB-

in

Tu

is

contained

in

S

then

the

claim

is

satisfied

Otherwise

a

shortest

path

from

a

to

b

in

S

either

ascends

at

least

as

far

as

u

or

includes

a

horizontal

edge

parallel

to

-LRB-

a

b

-RRB-

below

-LRB-

a

b

-RRB-

In

the

former

case

u

Figure

the

funnel

of

u

the

path

must

go

up

at

least

df

-LRB-

-RRB-

e

levels

and

then

down

at

least

df

-LRB-

-RRB-

e

levels

and

thus

be

at

least

f

-LRB-

-RRB-

edges

long

This

is

a

contradiction

so

a

shortest

path

from

a

to

b

must

include

a

horizontal

edge

below

-LRB-

a

b

-RRB-

If

this

horizontal

edge

is

more

than

f

-LRB-

-RRB-

levels

below

-LRB-

a

b

-RRB-

then

this

path

is

of

length

greater

than

-LRB-

f

-LRB-

-RRB-

-RRB-

f

-LRB-

-RRB-

a

contradiction

Thus

there

must

be

a

horizontal

edge

within

df

-LRB-

-RRB-

e

levels

below

-LRB-

a

b

-RRB-

This

implies

that

there

is

a

horizontal

edge

within

df

-LRB-

-RRB-

e

f

-LRB-

-RRB-

f

-LRB-

-RRB-

levels

of

u

Since

these

horizontal

edges

must

be

included

we

know

that

some

corresponding

vertical

edges

are

not

included

in

S.

Given

any

f

-LRB-

x

-RRB-

spanner

with

maximum

degree

of

a

pyramid

of

height

h

we

may

proceed

as

in

the

proof

of

Theorem

to

construct

a

sequence

of

vertices

r

xl

w

with

l

b

h

f

-LRB-

-RRB-

c

such

that

w

is

in

Txl

each

xi

is

an

interior

descendant

of

xi

and

the

edge

from

each

xi

to

its

parent

is

not

included

in

the

spanner

-LRB-

In

a

pyramid

vertex

v

is

considered

to

be

an

interior

descendant

of

vertex

u

if

v

is

an

interior

descendant

of

u

in

induced

X-trees

in

both

the

and

dimensions

-RRB-

As

before

this

implies

that

the

path

from

r

to

w

-LRB-

or

any

descendant

of

w

-RRB-

in

the

spanner

has

delay

at

least

l

b

h

f

-LRB-

-RRB-

c.

Again

as

in

the

proof

of

Theorem

by

assuming

that

we

can

construct

an

f

-LRB-

x

-RRB-

spanner

Sh

of

any

pyramid

of

height

h

such

that

f

-LRB-

x

-RRB-

o

-LRB-

x

-RRB-

we

arrive

at

a

contradiction

Construction

of

Pyramid

Spanners

with

Maximum

Degree

A

k-level

hedge

is

a

spanner

of

a

pyramid

of

height

k

for

k

To

construct

a

k-level

hedge

connect

the

root

r

to

its

two

right

children

and

continue

adding

all

right

vertical

edges

to

form

a

binary

tree

of

height

k

on

the

right

face

of

the

pyramid

From

each

vertex

in

this

tree

add

the

sibling

edge

to

its

left

sibling

Under

each

such

left

sibling

construct

a

tree

by

including

all

of

the

vertical

edges

down

to

the

bottom

level

of

the

pyramid

Add

all

of

the

cousin

edges

at

the

bottom

of

each

of

these

trees

-LRB-

that

is

k

levels

below

r

-RRB-

and

those

cousin

edges

on

the

bottom

level

that

connect

these

trees

Note

that

the

binary

tree

constructed

on

the

right

face

of

the

pyramid

has

its

leaves

one

level

above

the

bottom

of

the

pyramid

From

each

of

these

leaves

l

add

the

vertical

edges

to

connect

to

the

two

left

children

From

each

of

these

children

add

the

sibling

edge

to

the

right

sibling

From

all

four

children

of

l

add

cousin

edges

in

the

dimension

-LRB-

from

front

to

back

-RRB-

The

resulting

structure

is

a

k-level

hedge

Figure

shows

the

vertical

and

sibling

edges

of

a

hedge

where

the

solid

areas

indicate

complete

trees

of

vertical

edges

with

no

horizontal

edges

present

except

those

horizontal

edges

explicitly

included

on

the

bottom

level

Figure

shows

the

cousin

and

sibling

edges

on

the

bottom

level

of

a

hedge

Note

that

the

root

has

degree

and

the

vertices

on

lowest

level

have

degree

at

most

To

construct

a

k-level

hedge

spanner

of

a

pyramid

of

height

h

begin

by

constructing

a

k-level

hedge

rooted

at

the

root

of

the

pyramid

As

long

as

k

levels

remain

below

the

level

of

the

lowest

vertices

included

in

the

structure

add

a

k-level

hedge

rooted

at

each

vertex

on

the

lowest

level

of

the

existing

structure

Connect

these

newly

added

k-level

hedges

by

adding

cousin

edges

between

adjacent

vertices

from

different

hedges

on

the

bottom

level

of

these

hedges

Note

that

adding

these

edges

does

not

increase

the

degree

of

the

vertices

beyond

When

i

k

levels

remain

add

the

top

i

levels

of

a

k-level

hedge

rooted

at

each

vertex

on

the

lowest

level

of

the

existing

structure

Add

all

of

the

missing

cousin

edges

of

the

bottom

level

of

the

pyramid

Consider

a

k-level

hedge

spanner

S

of

the

pyramid

We

note

that

by

our

construction

if

P

is

a

path

in

dimension

then

AP

induces

a

k-level

shrub

spanner

of

XP

An

induced

shrub

spanner

is

shown

in

Figure

If

P

is

a

path

in

dimension

then

AP

induces

a

structure

YP

in

XP

-LRB-

and

in

the

pyramid

-RRB-

YP

can

be

constructed

as

follows

The

vertices

of

AP

are

on

h

different

levels

of

XP

-LRB-

and

of

the

pyramid

-RRB-

From

the

top

of

XP

group

these

levels

into

dh

k

e

groups

of

k

levels

each

-LRB-

except

the

bottom

group

that

may

have

k

levels

-RRB-

From

each

of

these

groups

choose

at

most

one

level

Include

all

vertical

edges

of

XP

except

those

going

down

from

the

chosen

levels

Beginning

at

the

root

include

all

cousin

edges

at

each

kth

level

Finally

include

all

horizontal

edges

at

the

bottom

level

An

example

of

YP

in

a

single

hedge

is

shown

in

Figure

Note

that

the

vertical

edges

missing

from

YP

are

the

endpoints

of

a

path

of

length

in

the

spanner

S

consisting

of

a

left

to

right

sibling

edge

and

a

vertical

edge

Theorem

The

k-level

hedge

spanner

S

is

a

-LRB-

x

k

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

and

average

degree

PROOF

Consider

a

k-level

hedge

spanner

S.

From

Lemma

we

know

that

the

shortest

path

P

between

any

pair

of

vertices

a

and

b

in

a

pyramid

can

be

divided

into

two

ascending

vertical

sections

and

one

horizontal

section

Let

and

denote

the

number

of

edges

in

the

vertical

sections

and

Figure

vertical

and

sibling

edges

of

a

hedge

Figure

cousin

and

sibling

edges

on

the

bottom

level

of

a

hedge

Figure

an

induced

shrub

spanner

Figure

an

induced

YP

in

a

single

hedge

c

x

s

y

c

wcu

v

s

x

c

wc

u

s

x

u

c

x

v

c

xu

s

cx

u

s

v

cx

u

c

v

u

cx

v

s

w

c

u

c

v

w

c

x

s

u

v

Figure

possible

layouts

of

the

horizontal

section

h

denote

the

number

of

edges

in

the

horizontal

section

Note

that

any

ascending

vertical

path

is

contained

in

some

induced

X-tree

in

dimension

Since

this

X-tree

induces

a

k-level

shrub

spanner

the

analysis

in

the

proof

of

Theorem

holds

Thus

the

vertical

sections

of

the

path

P

in

the

pyramid

correspond

to

path

sections

in

the

spanner

S

containing

no

more

than

k

c

units

of

delay

Let

dh

denote

the

delay

in

S

between

the

endpoints

of

the

horizontal

section

of

P

The

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx?h

k

c

dh

The

horizontal

section

of

P

consists

of

at

most

one

sibling

edge

and

at

most

three

cousin

edges

Using

c

to

denote

a

cousin

edge

and

s

to

denote

a

sibling

edge

the

horizontal

section

of

the

path

must

be

of

one

of

the

following

forms

ccsc

csc

ccs

cc

sc

s

or

c

The

possible

layouts

of

these

forms

are

shown

in

Figure

These

layouts

are

either

entirely

in

the

or

dimension

or

consist

of

a

section

in

the

dimension

and

a

section

in

the

dimension

In

any

case

if

we

consider

only

the

sections

in

one

dimension

the

only

forms

that

appear

are

the

straight

layouts

of

csc

sc

s

and

c.

Consider

the

dimension

Any

horizontal

path

section

in

this

dimension

is

contained

in

an

induced

X-tree

in

dimension

From

the

proof

of

Theorem

we

know

that

the

delay

between

the

endpoints

of

this

section

in

the

induced

k-level

shrub

spanner

is

at

most

k

k

and

k

for

sections

of

forms

s

c

sc

and

csc

respectively

Consider

the

dimension

Any

horizontal

path

section

in

this

dimension

is

contained

in

a

structure

YP

as

described

above

If

the

horizontal

path

section

in

dimension

is

a

single

sibling

edge

then

either

the

two

edges

to

the

parent

vertex

are

in

S

or

these

two

edges

are

missing

In

the

former

case

the

delay

is

and

in

the

latter

case

the

delay

is

-LRB-

using

the

two

paths

of

length

to

the

parent

-RRB-

If

the

horizontal

path

section

in

dimension

is

a

single

cousin

edge

-LRB-

u

x

-RRB-

then

consider

YP

augmented

with

the

paths

of

length

that

substitute

for

the

missing

vertical

edges

The

two

endpoints

of

the

cousin

edge

are

on

a

cycle

in

this

structure

-LRB-

the

augmented

YP

-RRB-

of

length

at

most

The

cycle

includes

paths

from

u

and

x

upwards

in

the

structure

to

either

a

common

ancestor

or

up

to

the

next

level

on

which

cousin

edges

are

included

The

cycle

also

includes

paths

from

u

and

x

down

to

the

next

level

on

which

cousin

edges

are

included

These

paths

are

chosen

so

that

the

bottom

vertices

are

connected

by

a

cousin

edge

which

is

included

in

the

cycle

This

cycle

is

concluded

by

joining

these

upward

paths

at

the

common

ancestor

of

u

and

x

or

by

adding

a

cousin

edge

at

the

top

level

The

upward

and

downward

paths

consist

entirely

of

vertical

edges

except

for

there

being

possibly

two

horizontal

edges

included

as

part

of

the

length

paths

replacing

missing

vertical

edges

Thus

there

is

a

path

from

u

to

x

of

length

at

most

k

so

the

delay

is

at

most

k

If

the

horizontal

section

consists

of

one

sibling

edge

-LRB-

u

v

-RRB-

and

one

cousin

edge

-LRB-

v

x

-RRB-

then

the

parent

of

u

and

v

lies

on

the

same

cycle

of

length

as

v

and

x

-LRB-

as

described

in

the

preceding

case

-RRB-

There

must

be

a

path

in

S

of

length

at

most

from

u

to

the

parent

of

u

and

v.

Thus

there

is

a

path

of

length

at

most

k

between

u

and

x

and

the

delay

is

at

most

k

If

the

horizontal

section

consists

of

one

sibling

edge

-LRB-

v

w

-RRB-

and

two

cousin

edges

-LRB-

u

v

-RRB-

and

-LRB-

w

x

-RRB-

then

one

of

the

cousin

edges

-LRB-

say

-LRB-

w

x

-RRB-

-RRB-

must

be

immediately

below

a

sibling

edge

Let

y

be

the

parent

of

v

and

w

and

let

z

be

the

parent

of

x

-LRB-

and

sibling

of

y

-RRB-

We

now

consider

two

cases

depending

on

whether

cousin

edges

are

present

at

the

level

of

z

If

there

are

cousin

edges

at

this

level

then

we

obtain

a

path

of

length

at

most

by

going

from

u

to

its

parent

across

the

cousin

edge

to

y

from

y

to

its

parent

down

to

z

and

then

down

to

x.

By

the

structure

of

the

augmented

YP

each

of

these

vertical

steps

can

be

accomplished

with

at

most

two

edges

If

there

are

not

cousin

edges

at

this

level

then

we

obtain

a

path

of

length

at

most

k

As

in

the

analysis

of

the

sc

case

there

is

a

path

from

u

to

y

of

length

at

most

k

To

this

path

we

append

a

path

from

y

to

its

parent

down

to

z

and

then

down

to

x

In

this

case

however

at

most

of

these

appended

vertical

edges

are

missing

from

YP

and

thus

the

appended

path

is

of

length

at

most

giving

a

path

of

length

at

most

k

Because

k

we

have

in

either

case

that

the

length

of

the

path

from

v

to

x

in

S

is

at

most

k

giving

a

delay

of

at

most

k

We

now

consider

the

layouts

that

contain

edges

in

both

dimension

If

the

horizontal

section

consists

of

one

sibling

edge

-LRB-

u

v

-RRB-

and

one

cousin

edge

-LRB-

v

x

-RRB-

where

-LRB-

u

v

-RRB-

and

-LRB-

v

x

-RRB-

are

in

different

dimensions

then

let

y

be

the

parent

of

u

and

v

If

-LRB-

u

v

-RRB-

is

in

the

dimension

and

-LRB-

v

x

-RRB-

is

in

the

dimension

then

there

is

a

path

of

length

at

most

from

u

to

y

and

as

noted

above

there

must

be

a

path

from

y

to

x

of

length

at

most

k

Thus

the

delay

is

at

most

k

If

-LRB-

u

v

-RRB-

is

in

the

dimension

and

-LRB-

v

x

-RRB-

is

in

the

dimension

then

either

-LRB-

u

v

-RRB-

is

in

S

or

-LRB-

u

y

-RRB-

is

in

S

As

in

the

case

of

the

straight

layouts

in

the

dimension

x

y

and

v

all

lie

on

a

cycle

of

length

at

most

in

S.

Thus

there

is

a

path

from

u

to

x

of

length

at

most

k

giving

delay

at

most

k

Consider

the

layout

of

cc

where

one

cousin

edge

-LRB-

u

v

-RRB-

is

in

the

dimension

and

the

other

-LRB-

v

x

-RRB-

is

in

the

dimension

In

the

pyramid

there

are

ascending

paths

from

u

and

x

to

either

a

common

ancestor

or

up

to

the

first

level

on

which

cousin

edges

are

included

In

the

pyramid

there

are

also

descending

paths

from

u

and

x

-LRB-

such

that

at

each

level

the

vertices

on

these

paths

are

also

separated

by

two

cousin

edges

-RRB-

that

continue

down

to

the

first

level

encountered

on

which

cousin

edges

are

included

We

follow

the

convention

that

if

u

v

and

x

are

on

a

level

with

cousin

edges

then

the

upward

paths

are

empty

and

the

downward

paths

are

of

length

k

We

use

the

term

vertical

path

on

u

to

denote

the

union

of

the

upward

and

downward

paths

from

u

described

above

In

S

at

most

one

edge

of

the

vertical

path

on

u

-LRB-

or

x

-RRB-

is

missing

Such

a

missing

edge

corresponds

to

a

path

of

length

in

S

and

thus

the

path

in

S

corresponding

to

the

vertical

path

on

u

is

of

length

at

most

k

-LRB-

and

similarly

for

x

-RRB-

The

two

cousin

edges

separating

the

bottom

vertices

of

these

vertical

paths

are

present

in

S

as

the

only

missing

cousin

edges

on

this

level

are

directly

below

sibling

edges

The

top

vertices

of

these

paths

are

either

the

same

-LRB-

a

common

ancestor

of

u

and

x

-RRB-

or

they

are

separated

by

one

or

two

cousin

edges

in

the

pyramid

The

cousin

edges

are

present

unless

-LRB-

at

most

-RRB-

one

of

them

is

directly

below

a

sibling

edge

This

would

occur

if

the

missing

cousin

edge

was

a

rightmost

cousin

inside

a

k-level

hedge

This

missing

edge

corresponds

to

a

length

path

in

S

-LRB-

two

vertical

edges

and

one

sibling

-RRB-

and

thus

the

top

vertices

of

the

vertical

paths

on

u

and

v

are

connected

by

a

path

of

length

at

most

in

S

The

vertical

paths

on

u

and

x

in

S

together

with

the

two

cousin

edges

at

the

bottom

and

the

connection

at

the

top

comprise

a

cycle

of

length

at

most

in

S.

Thus

there

is

a

path

from

u

to

x

of

length

at

most

k

giving

a

delay

of

at

most

k

If

the

horizontal

section

consists

of

two

cousin

edges

edges

-LRB-

u

v

-RRB-

and

-LRB-

v

w

-RRB-

followed

by

a

sibling

edge

-LRB-

w

x

-RRB-

then

let

y

denote

the

parent

of

w

and

x

As

in

the

previous

case

we

can

find

a

cycle

in

S

of

length

no

more

than

including

vertices

u

and

w

If

no

cousin

edges

are

included

on

the

level

of

u

v

w

and

x

then

y

is

also

on

this

cycle

and

thus

a

path

from

u

to

y

of

at

most

k

edges

There

is

also

a

path

in

S

from

y

to

x

of

at

most

two

edges

giving

a

path

from

u

to

x

of

at

most

k

edges

and

a

delay

from

u

to

x

of

at

most

k

If

however

the

level

of

u

v

w

and

x

does

include

some

cousin

edges

then

one

of

-LRB-

u

v

-RRB-

and

-LRB-

v

w

-RRB-

is

present

in

S

If

-LRB-

u

v

-RRB-

is

in

S

then

this

reduces

to

the

case

of

the

straight

layout

of

sc

-LRB-

from

v

to

x

-RRB-

If

-LRB-

v

w

-RRB-

is

in

S

then

let

be

the

vertex

other

than

v

that

is

a

cousin

of

both

u

and

w.

Due

to

our

construction

-LRB-

u

-RRB-

must

also

be

in

S

and

this

reduces

to

the

case

of

the

bent

layout

of

sc

-LRB-

from

to

x

-RRB-

If

the

horizontal

section

of

the

path

consists

of

the

bent

form

of

csc

from

u

to

x

then

there

is

also

a

path

with

the

horizontal

section

from

u

to

x

of

form

ccs

and

the

previous

analysis

applies

Finally

we

consider

the

form

ccsc

Let

-LRB-

u

v

-RRB-

-LRB-

v

w

-RRB-

and

-LRB-

y

x

-RRB-

be

cousin

edges

and

-LRB-

w

y

-RRB-

be

a

sibling

edge

We

can

choose

one

of

the

two

single

bend

shortest

paths

from

u

to

x

so

that

w

and

x

have

a

common

grandparent

z.

Consider

the

case

when

cousin

edges

are

not

included

in

the

level

immediately

above

u

and

v

As

before

we

can

find

a

cycle

in

S

of

length

no

more

than

that

includes

u

and

w

This

cycle

will

include

z

and

therefore

there

is

a

path

of

length

at

most

k

between

u

and

z.

Furthermore

at

most

one

of

the

vertical

edges

in

the

path

from

z

to

x

is

not

included

in

S

giving

a

path

of

length

at

most

from

z

to

x

The

resulting

path

from

u

to

x

is

of

length

at

most

k

giving

a

delay

of

at

most

k

Consider

the

case

when

cousin

edges

are

included

in

the

level

immediately

above

u

and

v.

Let

and

be

the

parents

of

u

and

v

respectively

Let

be

the

parent

of

w

and

y.

Note

that

-LRB-

-RRB-

must

be

a

cousin

edge

and

that

-LRB-

-RRB-

may

be

either

a

cousin

edge

or

a

sibling

edge

Also

note

that

if

either

of

these

edges

is

a

cousin

and

is

not

present

in

S

then

the

length

path

between

its

endpoints

through

their

parents

is

included

in

S.

Consider

the

case

that

-LRB-

-RRB-

is

a

cousin

edge

If

-LRB-

-RRB-

and

-LRB-

-RRB-

are

both

in

S

then

there

is

a

path

from

u

to

x

consisting

of

a

vertical

path

from

u

to

the

edges

-LRB-

-RRB-

and

-LRB-

-RRB-

a

vertical

path

from

to

z

and

a

-LRB-

two

level

-RRB-

vertical

path

from

z

to

x

As

each

level

of

a

vertical

path

may

take

at

most

two

edges

the

total

length

of

this

path

is

at

most

giving

a

delay

of

If

-LRB-

-RRB-

is

in

S

and

-LRB-

-RRB-

is

not

in

S

then

there

is

a

path

from

u

to

x

consisting

of

a

vertical

path

from

u

to

the

edge

-LRB-

-RRB-

the

edge

from

to

its

parent

the

sibling

edge

from

the

parent

of

to

z

and

a

-LRB-

two

level

-RRB-

vertical

path

from

z

to

x

This

path

consists

of

edges

plus

levels

of

vertical

paths

or

at

most

edges

giving

a

delay

of

If

-LRB-

-RRB-

is

not

in

S

and

-LRB-

-RRB-

is

in

S

then

let

be

the

vertex

other

than

that

is

a

common

cousin

of

and

The

edge

-LRB-

-RRB-

is

in

S

and

-LRB-

-RRB-

is

not

in

S

There

is

a

path

from

u

to

x

consisting

of

a

vertical

path

from

u

to

the

edge

-LRB-

-RRB-

the

edge

from

to

its

parent

the

sibling

edge

from

the

parent

of

to

z

and

a

-LRB-

two

level

-RRB-

vertical

path

from

z

to

x

This

path

consists

of

edges

plus

levels

of

vertical

paths

or

at

most

edges

giving

a

delay

of

Now

consider

the

case

that

-LRB-

-RRB-

is

a

sibling

edge

Let

be

the

sibling

of

that

is

also

a

cousin

of

If

-LRB-

-RRB-

is

in

S

then

there

is

a

path

from

u

to

x

consisting

of

a

vertical

path

from

u

to

the

edge

-LRB-

-RRB-

a

vertical

path

from

to

z

and

a

-LRB-

two

level

-RRB-

vertical

path

from

z

to

x

This

path

consists

of

edge

plus

levels

of

vertical

paths

or

at

most

edges

giving

a

delay

of

If

-LRB-

-RRB-

is

not

in

S

then

there

is

a

path

from

u

to

x

consisting

of

a

vertical

path

from

u

to

the

edge

from

to

its

parent

the

sibling

edge

from

the

parent

of

to

z

and

a

-LRB-

two

level

-RRB-

vertical

path

from

z

to

x

This

path

consists

of

edges

plus

levels

of

vertical

paths

or

at

most

edges

giving

a

delay

of

Thus

the

layout

ccsc

gives

a

delay

of

at

most

or

k

since

k

In

the

cases

the

horizontal

paths

of

length

and

gave

maximum

delays

of

at

most

k

k

k

and

k

respectively

As

the

total

delay

along

the

path

in

S

corresponding

to

P

is

at

most

bx?h

k

c

dh

these

give

bx

c

k

bx

c

k

bx

c

k

and

bx

c

k

respectively

Each

of

these

quantities

is

at

most

xk

k

k

so

f

-LRB-

x

-RRB-

k

k

x

k

To

bound

the

average

degree

we

note

that

most

of

the

vertices

that

are

not

at

the

bottom

level

of

the

spanner

are

of

degree

Similarly

to

our

analysis

of

the

X-tree

shrub

spanner

relatively

few

vertices

that

are

not

at

the

bottom

level

of

the

pyramid

spanner

have

degree

We

can

therefore

obtain

a

good

upper

bound

on

the

average

degree

by

assuming

that

all

vertices

not

at

the

bottom

level

of

the

pyramid

have

degree

Consider

now

the

vertices

at

the

bottom

of

the

pyramid

By

our

construction

at

most

one

vertex

in

each

line

in

the

dimension

in

each

hedge

-LRB-

or

partial

hedge

-RRB-

is

of

degree

The

other

vertices

at

this

level

are

of

degree

Each

line

in

the

dimension

at

this

level

contains

vertices

that

all

have

the

same

degree

-LRB-

except

for

its

endpoints

which

have

lower

degree

-RRB-

Thus

the

proportion

of

degree

vertices

to

degree

vertices

on

this

level

is

the

same

as

the

proportion

along

any

line

in

the

dimension

at

this

level

-LRB-

except

for

the

two

lines

along

the

boundary

of

the

pyramid

-RRB-

Each

of

these

lines

in

the

dimension

encounters

at

most

vertices

of

degree

Thus

the

proportion

of

degree

to

degree

vertices

is

at

least

As

there

are

vertices

at

the

bottom

level

and

vertices

not

on

the

bottom

level

the

average

degree

is

at

most

Conclusions

We

showed

how

to

construct

two

different

types

of

-LRB-

x

-RRB-

spanner

with

maximum

degree

for

two-dimensional

grids

The

constructions

yield

different

average

degrees

Further

we

established

a

lower

bound

that

shows

that

the

delay

of

these

spanners

is

within

a

constant

factor

of

optimal

We

showed

how

to

construct

a

-LRB-

x

k

x

-RRB-

spanner

of

the

X-tree

with

maximum

degree

and

we

established

a

lower

bound

that

shows

that

the

delay

of

this

spanners

is

within

a

constant

factor

of

optimal

We

showed

how

to

construct

a

-LRB-

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

and

a

-LRB-

x

k

x

-RRB-

spanner

of

the

pyramid

with

maximum

degree

We

also

established

a

lower

bound

that

shows

that

the

delay

of

the

latter

spanner

is

within

a

constant

factor

of

optimal

References

-LSB-

-RSB-

Cai

L.

Tree

Simon

Fraser

University

Tech

Report

No

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Grid

Spanners

Networks

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Additive

Spanners

for

Hypercubes

Par

Proc

Lett

-LRB-

-RRB-

-LSB-

-RSB-

Liestman

A.

L.

and

T.

C.

Shermer

Additive

Graph

Spanners

Networks

to

appear

-LSB-

-RSB-

Peleg

D.

and

A.

A.

Schaffer

Graph

Spanners

J.

Graph

Theory

-LRB-

-RRB-

-LSB-

-RSB-

Peleg

D.

and

J.

D.

Ullman

An

Optimal

Synchronizer

for

the

Hypercube

Proc

ACM

Symp

on

Princ

Distr

Comp

-LRB-

-RRB-

-LSB-

-RSB-

Richards

D.

and

A.

L.

Liestman

Degree-Constrained

Pyramid

Spanners

J.

Par

Distr

Comp.

to

appear