Asymptotic

analysis

of

flooding

in

CSMA-based

large

scale

ad-hoc

wireless

networks

In

this

article,

we

study

the

asymptotic

behavior

of

flooding

in

large

scale

wireless

networks.

Specifically,

we

derive

an

upper

bound

on

the

coverage

of

flooding

when

the

number

of

nodes

n

in

the

network

goes

to

infinity.

We

consider

two

different

regimes

of

transmission

radii:

first,

the

case

of

constant

transmission

radius

r

where

the

percentage

of

covered

nodes

scales

as

O(nr

e

−K

S

nr

for

a

constant

K

S

In

this

case,

as

an

important

result,

we

observe

that

the

percentage

of

covered

nodes

is

upper

bounded

by

a

decreasing

function,

vanishing

as

the

network

size

grows.

Second,

the

case

of

vanishing

r

n

(i.e.,

r

decreases

as

n

increases)

is

considered

where

it

is

shown

in

the

literature

that

the

minimum

value

of

r

n

which

maintains

connectivity

is

log

n/π

n.

In

this

case,

a

coverage

percentage

of

at

most

O(n

−K

S

log

n)

is

expected

for

a

constant

value

of

K

S

leading

to

an

infinite

number

of

covered

nodes.

In

such

case,

the

rate

at

which

the

network

coverage

is

decreased

can

be

controlled

and

be

considerably

reduced

by

a

proper

choice

of

network

parameters

(K

S

Consequently,

this

result

shows

that

flooding

is

a

suitable

strategy

even

for

large

networks.

Flooding

is

the

simplest

and

most

widely

used

form

of

broadcasting

in

wireless

networks

where

each

node

simply

retransmits

received

data

once

to

its

neighboring

nodes

Data

dissemination

and

route

discovery

process

in

many

applications

are

based

on

flooding

Wide

adoption

of

flooding

for

broadcasting

traffic

is

mainly

due

to

its

simplicity.

In

addition,

it

does

not

require

any

centralized

information

about

the

network.

In

probabilistic

flooding-a

variant

of

flooding

introduced

in

-each

node

retransmits

received

data

with

a

pre-set

probability

called

forwarding

factor

in

order

to

reduce

unnecessary

retransmissions.

After

the

seminal

study

of

Gupta

and

Kumar

in

the

scaling

laws

and

fundamental

limits

of

large

wireless

networks

have

received

much

attention.

However,

to

the

best

of

authors'

knowledge,

no

study

has

been

conducted

in

the

literature

with

the

goal

of

studying

the

scaling

laws

of

flooding.

The

importance

of

such

Full

list

of

author

information

is

available

at

the

end

of

the

article

analysis

lies

on

the

aforementioned

popularity

of

flooding

for

data

dissemination

and

route

discovery

in

many

routing

protocols

Based

on

this

fact,

the

main

motivation

of

this

study

is

to

effectively

answer

the

question

that

whether

flooding

is

suitable

and

applicable

to

large

scale

networks

or

not..

Previous

related

studies

in

proposed

a

tight

upper

bound

on

the

coverage

of

flooding

in

a

multi-hop

network

using

carrier

sense

multiple

access

(CSMA)

as

the

MAC

layer

strategy

for

any

finite

network

size.

CSMA

is

a

well-known

multiple

access

algorithm

for

broadcasting

in

wireless

networks

It

should

be

noted

that

implementing

other

MAC

layer

algorithms

that

require

centralized

information

of

the

network

nodes

is

not

feasible

in

ad-hoc

wireless

networks.

In

addition,

utilizing

collision

avoidance

techniques

in

MAC

layer

for

broadcasting

traffic

is

not

practical,

since

it

generally

requires

a

large

amount

of

control

message

overhead

to

ensure

that

all

neighbors

successfully

receive

the

broadcasted

packet..

In

this

article,

we

take

a

step

further

by

investigating

the

scaling

law

and

asymptotic

behavior

of

flooding

in

ad-hoc

wireless

networks

based

on

the

framework

introduced

in

The

major

contribution

of

this

article

in

comparison

with

the

results

presented

in

is

providing

an

analytic

approach

to

asymptotic

behavior

of

flooding.

Although

provides

important

results

on

the

coverage

of

flooding,

its

results

are

only

obtained

numerically

through

a

recursive

scheme.

On

the

other

hand,

the

analytic

framework

proceeded

in

this

article

leads

to

closed

form

representations

of

coverage

upper

bound

in

large

networks.

Therefore,

the

derived

closed

form

results

clearly

demonstrate

the

behavior

of

coverage

and

provide

deeper

insight

on

the

performance

of

flooding

when

the

number

of

network

nodes

grows..

In

order

to

analyze

the

coverage

of

flooding,

first,

we

assume

that

only

one

of

the

nodes

intends

to

broadcast

its

packets.

Subsequently,

a

discussion

on

the

case

of

multiple

sources

is

presented.

Our

analysis

includes

two

different

cases:

the

case

where

the

transmission

radius

is

constant,

and

the

case

where

the

transmission

radius

r

approaches

zero

as

the

number

of

nodes

goes

to

infinity.

Although

flooding

is

in

general

known

to

be

a

simple

yet

inefficient

broadcasting

method

we

obtain

the

following

interesting

results

throughout

the

article:.

Under

the

assumption

of

constant

transmission

radius,

network

coverage

is

upper

bounded

by

a

decreasing

function

rapidly

vanishing

as

the

network

size

grows.

Therefore,

flooding

is

not

a

proper

broadcasting

scheme

in

such

conditions.

When

the

transmission

radius

approaches

zero

as

network

size

grows,

we

show

that

a

coverage

bound

(i.e.,

the

percentage

of

covered

nodes)

close

to

can

be

obtained

in

realistic

large

networks.

This

result,

along

with

the

tightness

of

the

bound,

demonstrates

that

flooding

is

a

suitable

and

applicable

broadcasting

scheme

in

terms

of

network

coverage

even

for

large

networks..

It

should

also

be

noted

that,

in

addition

to

the

aforementioned

interesting

results,

the

analytic

framework

presented

in

this

article

can

provide

proper

basis

for

asymptotic

analysis

of

other

broadcasting

schemes.

A

number

of

more

complex

broadcast

methods

are

already

proposed

in

the

literature,

such

as

counter-based,

distance-based,

location-based

broadcasting

and

multi

point

relaying

(MPR)

with

the

goal

of

improving

overall

network

efficiency.

Although

the

analysis

of

such

schemes

appears

to

be

more

complicated

than

simple

flooding,

we

believe

that

this

study,

as

the

first

research

in

this

field,

can

be

extended

to

analyze

them

as

well.

It

is

important

to

note

that

no

complete

and

comprehensive

analytic

model

for

performance

analysis

of

such

broadcasting

schemes

in

the

presence

of

a

practical

MAC

layer

protocol

currently

exists,

even

in

the

case

of

finite

networks..

The

rest

of

the

article

is

organized

as

follows:

in

Section

'Notations

and

system

model'

we

introduce

the

network

model,

MAC

layer

model,

and

an

exact

definition

of

flooding

and

network

coverage.

Then,

we

restate

briefly

the

results

obtained

in

in

Section

'Coverage

upper

bound'

The

asymptotic

behavior

of

the

upper

bound

on

the

coverage

is

derived

in

Section

'Asymptotic

analysis'

along

with

a

discussion

on

the

obtained

results.

Finally,

Section

'Conclusions'

concludes

the

article.

In

this

article,

the

scaling

laws

of

the

coverage

of

flooding

in

CSMA-based

large

wireless

networks

have

been

studied.

An

analysis

has

been

performed

for

both

cases

of

constant

transmission

radius

r

and

vanishing

radius

r

n

when

the

number

of

nodes

n

goes

to

infinity.

For

the

case

of

constant

r,

the

results

show

that

network

coverage

rapidly

approaches

zero

when

n

increases.

However,

the

rate

of

decrement

of

network

coverage

in

case

of

vanishing

r

n

is

much

smaller

compared

to

the

case

of

constant

radius.

In

addition,

the

rate

can

be

controlled

by

proper

choice

of

network

parameters.

Consequently,

it

makes

flooding

an

appropriate

and

practical

scheme

for

data

dissemination,

even

in

large

networks.

The

number

of

covered

nodes

also

approaches

zero

under

fixed

radius

assumption

while

an

infinite

number

of

nodes

can

be

covered

in

the

case

of

r

n

Endnotes

a

In

we

propose

two

upper

bounds

on

the

network

coverage.

Since

the

asymptotic

analysis

for

the

derivation

of

the

two

upper

bounds

is

similar,

we

focus

on

deriving

the

asymptotic

result

for

the

first

one

in

this

article.

b

The

Hölder

inequality

is

a

fundamental

inequality

between

integrals.

Let

a,

b

with

a

−1

b

−1

Consequently,

for

all

measurable

functions

f

and

g

on

a

measure

space,

||fg||

||f

a

||g||

b

In

we

have

assumed

that

a

and

consequently,

we

have

that

b

B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Word-Level Equivalence Checking in Bit-Level Accuracy by Synthesizing Designs onto Identical Datapath Equivalence checking is one of the most important issues in VLSI design to guarantee that bugs do not enter designs during optimization steps or synthesis steps. In this paper, we propose a new word-level equivalence checking method between two models before and after highlevel synthesis or behavioral optimization. Our method converts two given designs into RTL models which have same datapaths so that behaviors by identical control signals become the same in the two designs. Also, functional units become common to the two designs. Then word-level equivalence checking techniques can be applied in bit-level accuracy. In addition, we propose a rule-based equivalence checking method which can verify designs which have complicated control structures faster than existing symbolic simulation based methods. Experimental results with realistic examples show that our method can verify such designs in practical periods. System-level design methodology plays an important role to improve VLSI design productivity and shorten design periods. C or C++ based design languages such as SpecC [1] or SystemC [2] are uniformly used to describe both hardware and software parts.. A typical VLSI design flow starting from system-level is shown in Fig. 1 . First, the specification of a design is described without distinguishing the hardware and software parts. Then, they are transformed into a software program and an RTL design, respectively, through hardware/software partitioning and high-level synthesis processes. In this design flow, designs are gradually refined in step-by-step. Therefore, design bugs can be newly inserted during the refinements. It is important to detect such bugs in early steps in order to prevent design modifications in later steps, since they take much higher costs. Equivalence checking gives a solution for that goal. When two designs are proved to be equivalent and one of them has no bugs, the other is also guaranteed to have no bugs. We can apply equivalence checking to designs before and after each refinement step, which enables us to avoid bug insertions when the original design has been verified sufficiently. One of the simplest equivalence checking methods is proposed in [3] . It translates designs into Boolean formulas, and checks the equivalence of those formulas with BDD or SAT. However, large designs cannot be verified, since the complexity of such a bit-level analysis increases exponentially with the size of designs.. To avoid bit-level analysis as much as possible, wordlevel symbolic simulation [4] - [6] which treats each variable or operator as a symbol is applied. However, since the complexity of symbolic simulation is doubled for each conditional branch, it is still not applicable to entire designs. Also, loops are not acceptable, and they must be unrolled in advance.. Therefore, only textually different parts of two designs are compared in [5] . This method can handle large designs when compared designs are similar. Also in [6] , equivalences of paths between conditional branches are checked, and the results are gathered to prove the entire equivalence. To apply this divide-and-conquer approach, correspondences of intermediate variables or registers between two designs must be known or given by users (e.g. Names of variables or registers in two designs are same).. However, in practical refinement steps, it is usual that the entire structure of a design is changed or correspondences of intermediate variables or registers are unknown (e.g. between two designs before and after automated highlevel synthesis). Also, since bit-width or sign are not taken into account in symbolic simulation, bit-level accuracy is not considered in [4] - [6] .. Based on the arguments above, in this paper, we propose a new equivalence checking method between two models before and after a refinement step, such as high-level synthesis or behavioral optimization. In this method, we focus on a feature that designs after automated high-level synthesis are usually composed of controllers and datapaths. In such a design, computations of the design are executed at the datapath, and the controller determines computations executed at each clock cycle. In our method, two designs are converted into RTL models which have same datapaths. Then, we can get advantages of abstracting computations of the datapaths, and concentrate on the verification of the controllers. Concretely, since the datapaths are identical, the functional units in those RTL models become the same. Also, same control signals from the controllers represent same behaviors, and then the behaviors are equivalent in bit-level accuracy. Therefore, existing word-level methods, such as symbolic simulation, can be easily applied in bitlevel accuracy. In this paper, we proposed a word-level equivalence checking method in bit-level accuracy with synthesizing two designs with the same datapath. We also proposed a new wordlevel rule-based comparison method, and the experimental results show that our method is fast and it can verify some designs which cannot be verified by symbolic simulation. Since our method is a rule-based method, we can extend the range of verifiable designs by introducing additional rules. We are planning the following future works.. • Utilizing potentially equivalent states or sequences of state transitions In the current implementation, the explorations of the four rules are done exhaustively. If we can determine potentially equivalent states or sequences of state transitions by some sort of simulations, then the search domains can be reduced drastically.

O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Delay-Efficient Algorithm for Data Aggregation in Multihop Wireless Sensor Networks Abstract-Data aggregation is a key functionality in wireless sensor networks (WSNs). This paper focuses on data aggregation scheduling problem to minimize the delay (or latency). We propose an efficient distributed algorithm that produces a collision-free schedule for data aggregation in WSNs. We theoretically prove that the delay of the aggregation schedule generated by our algorithm is at most 16R þ Á À 14 time slots. Here, R is the network radius and Á is the maximum node degree in the communication graph of the original network. Our algorithm significantly improves the previously known best data aggregation algorithm with an upper bound of delay of 24D þ 6Á þ 16 time slots, where D is the network diameter (note that D can be as large as 2R). We conduct extensive simulations to study the practical performances of our proposed data aggregation algorithm. Our simulation results corroborate our theoretical results and show that our algorithms perform better in practice. We prove that the overall lower bound of delay for data aggregation under any interference model is maxflog n; Rg, where n is the network size. We provide an example to show that the lower bound is (approximately) tight under the protocol interference model when r I ¼ r, where r I is the interference range and r is the transmission range. We also derive the lower bound of delay under the protocol interference model when r < r I < 3r and r I ! 3r. W IRELESS sensor networks (WSNs) have drawn a considerable amount of research interest for their omnipresent applications such as environmental monitoring, spatial exploration, and battlefield surveillance. To design and deploy successful WSNs, many issues need to be resolved such as deployment strategies, energy conservation, routing in dynamic environments, localization, and so on. All the issues essentially correlate to collecting data from a set of targeted wireless sensors to some sink node(s) and then performing some further analysis at sink node(s), which can be termed as many-to-one communication. In-network data aggregation [16] is one of the most common many-to-one communication patterns used in these sensor networks; thus, it becomes a key field in WSNs and has been well studied in recent years.. We consider the problem of designing a schedule for data aggregation from within networks to sink node(s) with minimum time slot delay. Some of previous research works on in-network aggregation did not consider the collision problem and left it to the MAC layer. Resolving collisions in MAC layer could incur a large amount of energy consumption and a large delay during aggregation. Thus, in this paper, we mainly concentrate on the TDMA scheduling problem above the MAC layer. To define the problem formally, consider a WSN G formed by n wireless nodes V ¼ fv 1 ; . . . ; v n g deployed in a two-dimensional region. v s 2 V is the sink node that will collect the final aggregation result. Each node v i has a transmission range r and interference range r I ¼ ÂðrÞ. A node v i can send data correctly to another node v j if and only if 1) v j is within v i 's transmission range and (2) v j is not within interference range r I of any other transmitting node. Every node v i has an ability to monitor the environment, and collect some data (such as temperature), i.e., v i has a set of raw data A i . Let A ¼ [ analysis, we use the protocol interference model and assume that r I ¼ r. As an illustration, we first present an efficient centralized algorithm that will build a TDMA schedule of nodes based on the aggregation tree, which is build distributively. Our schedule uses a bottom-up approach: schedule nodes level by level starting from the lowest level. We theoretically prove that the delay of the aggregation schedule generated by our algorithm is at most 16R þ Á À 14 time slots. Note that for general r I , our algorithm will produce a collision-free schedule for aggregation whose delay is at most Âðð rI r Þ 2 R þ ÁÞ time slots. We then present an efficient distributed algorithm that builds an aggregation tree and gives a schedule for each node. For simplicity, our distributed method assumes that the clocks of all nodes are synchronized. Unlike our centralized algorithm, our distributed algorithm will not explicitly produce a schedule for nodes in the aggregation tree. The schedule for nodes is implicitly generated in the process of data aggregation. Our distributed scheduling algorithm thus works well in dynamic networks, as long as the constructed backbone of the network by our algorithm remains unchanged. Obviously, when r I ¼ r, for a network G with radius R and the maximum node degree Á, the delay by any data aggregation algorithm is at least R. This implies that our algorithm is within a small constant factor of the optimum. We then conduct extensive simulations to study the practical performance of our proposed data aggregation method. The simulation results corroborate our theoretical results and show that our method performs better in practice. We find that data aggregation by our distributed method has delay close to R. Besides, we prove that the overall lower bound of delay for data aggregation under any interference model is maxflog n; Rg. We provide an example to show that the lower bound is (approximately) tight under the protocol interference model when r I ¼ r. We also analyze the lower bound of delay under the protocol interference model when r < r I < 3r and r I ! 3r.. The rest of the paper is organized as follows: Section 2 formulates the problem. We present our centralized and distributed scheduling algorithms in Section 3 and analyze their performance and prove the overall lower bound in Section 4. Section 5 discusses the results in other interferences models. Section 6 presents the simulation results. Section 7 outlines the related work. Finally, Section 8 concludes the paper. Data aggregation is critical to the network performance in WSNs and aggregation scheduling is a feasible way of improving the quality. In this paper, we study the problem of distributed aggregation scheduling in WSNs and propose a distributed scheduling method with an upper bound on delay of 16R þ Á À 14 time slots. This is a nearly constant approximate algorithm, which significantly reduces the aggregation delay. The theoretical analysis and the simulation results show that our method outperforms the previous methods.. In addition, we provide the overall lower bound on delay for data aggregation under any interference model with formal proofs and give an example to show that the lower bound is (approximately) tight under the protocol interference model when r I ¼ r, where r is the transmission range and r I is the interference range. We also derive the lower bound on delay under the protocol interference model when r < r I < 3r and r I ! 3r.. Xiaohua Xu received the BS degree from ChuKochen Honors College at Zhejiang University, P.R. China, in 2007. He is currently working toward the PhD degree in computer science at Illinois Institute of Technology. His research interests include algorithm design and analysis, optimization in mesh network, and energy efficiency and security in wireless network. He is a student member of the IEEE. His research interests include wireless ad hoc and sensor networks, noncooperative computing, computational geometry, and algorithms. He was a guest editor of special issues for the ACM Mobile Networks and Applications, the IEEE Journal on Selected Areas in Communications, and editor of the Networks journal. He is a member of the ACM and a senior member of the IEEE. Data aggregation in sensor networks has been well studied recently [2] , [12] , [15] , [25] . In-network aggregation means computing and transmitting partially aggregated data rather than transmitting raw data in networks, thus, reducing the energy consumption [16] .. There are a lot of existing researches on in-network aggregation in the literature [6] , [17] . Suppression scheme and model-driven methods were proposed in [5] , [7] toward reducing communication cost. The trade-off between energy consumption and time delay was considered in [25] . A heuristic algorithm for both broadcast and data aggregation was designed in [1] . Another heuristic algorithm for data aggregation was proposed [18] , aiming at reducing time delay and energy consumption. Kesselman and Kowalski [13] proposed a randomized and distributed algorithm for aggregation in WSNs with an expected delay of Oðlog nÞ. Their method is based on two assumptions: One is that sensor nodes can adjust their transmission range without any limitation. The other is that each sensor node has the capability of detecting whether a collision occurs after transmitting data. Both assumptions pose some challenges for hardware design and are impractical when the network scales. A collision-free scheduling method for data collection is proposed in [14] , aiming at optimizing energy consumption and reliability. All these works did not discuss the minimal delay aggregation scheduling problem.. In addition, the minimum delay of data aggregation problem was proved NP-hard and a ðÁ À 1Þ-approximation algorithm was proposed in [4] , where Á is the maximum degree of the network graph. Another aggregation scheduling algorithm was proposed in [11] , which has a delay bound of 23R þ Á þ 18, where R is the network radius and Á is the maximum degree. Recently, Wan et al. [21] proposed three novel centralized data aggregation methods for networks when nodes have the same transmission radius and interference radius, which achieve schedules of latency 15R þ Á À 4, 2R þ Oðlog RÞ þ Á, and ð1 þ Oð log R ffiffiffi R 3 p ÞÞR þ Á, respectively. Recently, Xu et al. [24] studied aggregation with multiple queries in WSNs. All the algorithms mentioned above are centralized. In many cases, centralized algorithms are not practical, especially when the network topology changes often in a large sensor network.. The distributed algorithms for convergecast scheduling were proposed in [3] , [8] , [13] . [8] , [13] focused on the scheduling problem for data collection in sensor networks. In data collection, since data cannot be merged, the sink must receive N packets from all the nodes, where N is the number of sensor nodes in the network. Thus, the lower bound of delay is N. The upper bound of the time delay of this algorithm is maxð3n k À 1; NÞ, where n k is the number of nodes in the largest one-hop subtree. Bo Yu and Li [3] proposed a distributed scheduling algorithm generating collision-free schedules that has a delay bound of 24D þ 6Á þ 16, where D is the network diameter.

O O O O O B O O O B O O B O O O O O O B O O O O O O O B B O O O O B O O O O O O O O O O O O O O O B O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B B O O O O O O O O O O O O O O O O O B B O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B B B O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O


An Entropy-based Stability Algorithm for Regulating the Movement of MANET Nodes This paper proposes an algorithm that enables mobile nodes to implement self-regulated movements in mobile ad-hoc networks (MANETs). It is important for mobile nodes to maintain a certain level of network-based stability by harmonizing these nodes' movements autonomously due to their limited transmission range and dynamic topology. Entropy methods based on relative position are suggested, as a means for mobile nodes to regulate their local movements. Simulations show that an early warning mechanism is suitable to maintain movement-based stability. Isolation can be reduced by 99%, with an increased network cost of 12% higher power consumption, using the proposed algorithm. Ad hoc wireless communication has emerged as a new research topic due to the demand for mainly focuses on finding the route correctness to avoid incorrect routes that cause severe traffic drifting in the network and over consumption of valuable network resources. Optimizing MAC protocol parameters to meet the targeted overall network utilization [5] can be categorized as a logical-oriented approach, since the MAC protocol parameter gives the decision information to the network layer and the network layer has the main responsibility to control network topology.. In contrast, examples of the physical-oriented approach follow. A mobility-aware topology control strategy has been proposed in [6] , in which the system changes its behavior in response to, or by predicting, node movements. The key point of the physical-oriented approach is that nodes change their locations autonomously to optimize certain performance metrics. For example, a mobile sensor node can move about a sensor field to collect data from other sensor nodes, reducing energy consumption at those nodes. This strategy has been proposed as a means to recover a disconnected topology, increase sensor surveillance coverage, as well as reduce total energy consumption. Controlled mobility can reduce energy consumption up to 50 percent compared to the energy consumption of random mobility [6] . The failure of a critical node through intermediate relay nodes can result in a network partition, which can be termed as a network failure, since every node in an ad hoc network is responsible for forwarding data packets to other nodes [7] . In [6] , after receiving a data packet, an intermediate node calculates its new target location according to the current mobility strategy and starts moving toward that location to provide seamless data packet flows. A flow node periodically broadcasts hello messages containing the residual energy and location information regarding itself and its flow neighbors to achieve this. It is possible to complement node mobility strategies with approaches that account for energy management, as ad hoc nodes are used in disaster releif areas (search and rescue) or battlefields, as in a fleet of microrobots; some or all of the nodes in the networks are mobile. The particle swarm optimization based algorithm (PSOA) [8] is a physical-oriented approach that regards a node in the network as a swarm, and recognizes the nodes by the particle swarm optimization algorithm.. This paper proposes an entropy-based movement regulation algorithm as one of methods of the physical-oriented approach, whereby mobile nodes can easily implement a stable autonomous physical movement using MANETs. An entropy metric from information theory is introduced to define and measure the stability of node movements. The difference change of relative node positions will strongly affect the service quality of MANETs, where it is important for a node to be consistently located within the transmission range of another node to provide the best quality network service. The entropy metric is extended to regulate the relative positions of nodes that vary every second. This paper proposed an algorithm for network-based mobile nodes to implement a decentralized self-regulated stable movement system. In addition, the mathematical background to achieve node movement stability was presented.. An entropy metric is used to regulate the future movements of mobile nodes based on the current entropy. It has a decision metric to determine whether a node is stable based on past accumulated entropy. It was shown that the effect by the guidance controlled by the direction change of the entire network (global) does not strongly impact each node to revert to a stable status compared to the effect by the guidance controlled by the direction change of F n members (local). An early warning system is an effcient method to obtain stable movements. The proposed algorithm can reduce isolation by 99% with an increased network cost of 12% greater power consumption compared to that of the conventional BMA algorithm. Based on the guided movement regulations among nodes, as well as the construction method of stable node groups, the self-administrating, self-organizing and self-configuring characteristics of mobile nodes can be easily deployed in a network-based multi-node area. They can ultimately support cooperative tasks, such as rescue missions in an emergent disaster situation. In [7] , the authors introduced a movement control algorithm to realize fault-tolerant ad hoc sensor networks, in which every sensor node includes its location information (GPS coordinates or indoor relative location information) whenever this node broadcasts a link status update (LSU) to the rest of the network. When LSUs arrived at node R from other nodes, R extracts the location information from each LSU and calculates the geographic center C for the entire network. After calculating C, each node moves toward C. [9] expresses that the most commonly used mobility model in the ad hoc networking research is the random waypoint (RWP) model, in which each node chooses uniformly at random a destination point (waypoint) in a rectangular deployment region. A node moves to the latter destination with a velocity v chosen uniformly at random in the interval [v min , v max ].. After reaching the destination, the node pauses for a random amount of time, and the entire cycle is repeated by selecting a new destination. Fig. 1 shows the movement of ten nodes with respect to RWP in a 1 km 2 network. Marks, such as the triangle, circle and dot, illustrate the nodes' positions at time Δt, 2Δt, and xΔt, respectively.. The ability to analytically characterize the spatial topology distribution of nodes plays a key role in understanding fundamental network QoS measure, such as throughput, probability of successful transmission, and connectivity. Accordingly, Alparslan et al proposed a generalized random mobility model sufficiently flexible to capture different topology scenarios, and provide its long-run location and speed distibutions by a closed form expression for one-dimensional mobility terrains. In addition, they classified the existing mobility models for wireless ad hoc networks, briefly summarizing their assumptions [1] . Tang et al. studied the energy optimization problem that accounts for energy costs associated with both communication and physical node movement. They reviewed the theoretical foundations on how to reduce total communication energy consumption, as well as increase system lifetime, by combining node movement and transmission power [6] . The authors of [10] introduced a logical connectivity algorithm, in which a strong partitionable group with safe distance can guarantee the prevention of the occurrence of unannounced disconnection of ad hoc applications, to have a consistent application view among mobile ad hoc nodes.. MANETs are used as the method of the wireless communication network to control the movement of mobile robots in [7] . This indicates that traditionally researchers provided the use of server-client based network, in which all robot nodes communicate with a central controller (base station) over a wireless medium. However, in most application scenarios, it is difficult to guarantee the presence of a base station that coordinates the flow of information between any two nodes. Moreover, node movement can be severely restricted to stay in communication range of the base station, which ultimately causes the failure of wireless communication. They implemented the block movement algorithm (BMA) using the MANET characteristic of self-forming, self-organizing multi-hop communication, where the nodes' minimal possible movement to new locations yields the desired topography.. In PSOA, all nodes have fitness values, evaluated by the fitness function to be optimized. It is initialized with a group of random solutions and then searches for optima by updating generations. In every iteration, each node is updated by following the two best factors. The first one is the best fitness it has achieved so far; thus, it is stored in memory. Another best value obtained so far by any particle in the network is a global best. The best fitness and the global best are used as the parameters of the fitness function. Therefore, the current best position can be determined by recalling the previous best position [8] . However, when a leader, followed by hundreds or thousands of nodes in a group, loses a destination, all the following nodes may lose the same destination.. The literature review reveals that it is important for mobile nodes to obtain and keep a certain level of movement-oriented stability, when nodes are moving and relaying their packets due to limited transmission range and dynamic topology, since the multi-hop wireless connection can be easily broken in the middle of control and data packet transmission. Therefore, network-based communication might not perform well in a self-organized network. Therefore, this paper proposes an entropy-based stability algorithm to enable mobile nodes to auto-regulate their own movements, and eventually to maintain a stable wireless connection that results in high network performance, even when there is an external harmful environment, such as wireless interference. The position difference (P(n, m, t)) at t seconds of nodes n and m can be expressed as follows.

O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A HYBRID GENETIC ALGORITHM FOR PARTITIONING OF DATA MODEL IN DISTRIBUTION MANAGEMENT SYSTEMS Abstract. In this paper, we propose a Hybrid Genetic Algorithm for data model partitioning of power distribution network. Analytical functions are the core of Distribution Management Systems (DMSs). Efficient calculation of the functions is of the utmost importance for the DMS users; the necessary preconditions for the efficient calculation are optimal load balancing of processors and data model partitioning among processors. The proposed algorithm is applied to different real models of power distribution systems. It obtains better results than classical evolutionary algorithms (Genetic Algorithm and Particle Swarm Optimization). The Hybrid Genetic Algorithm also achieves better results than multilevel algorithm (METIS) in cases of small graphs. In order to effectively manage modern adaptive applications within the multiprocessor system, large amounts of data need to be optimally processed in parallel. For optimal calculations to be carried out on certain groups of data, the following conditions must be satisfied: the data are necessarily partitioned across the processors and load balancing of the processors must be achieved. It is necessary to make an optimal distribution of data across the processors in accordance with pre-defined criteria that determine the calculation functions so that the calculations and the overall system functioning is efficient.. The Integrated Smart Grid power system largely increases the amount of data in each of its subsystems. It integrates systems such as [1] : Geographic Information System, Energy Management System, Distribution Management System (DMS), Outage Management System, etc. We treat parallel calculations in the DMS as one subsystem of Smart Grid system. DMSs have the supervision, management, planning and visualization of the distribution network in two modes: on-line (real-time) and simulation. Analytical DMS functions are the central component of the DMS. They allow monitoring and manage the power distribution network.. The most important DMS functons are: Load Flow (LF), State Estimation (SE), Fault Calculation, Performance Indices, Volt/Var Control, etc. This paper describes the optimal partitioning of power distribution network data model by using the most common DMS functions -Load Flow and State Estimation. Also, the real power distribution networks and the corresponding functions are analyzed. The LF [2] and SE function algorithms are based on the weakly meshed radial data network. SE requires multiple LF execution and will thus not be discussed further. The network architecture makes the data models suitable for partitioning and parallel calculations [3, 4] .. Depending on an application context of the DMS, dynamic data can be monitored either on-line by the SCADA system (real-time context), or off-line -defined and analyzed by the user (within the planning, simulation or testing contexts). Calculation time of the functions above becomes critical with increasing amounts of data involved in calculations (the present network contains tens of millions of data). For DMS functions optimization, only real-time (RT) context is critical for time calculation.. The initial partitioning of the DMS data model is studied. The partitioning is based on the initial values of objects' attributes and the relationships between the data. As a result, groups of data necessary for calculation are made. Dynamic changes of certain attributes (usually the relations among the data objects) can affect the data; and consequently, calculations. Thus we introduce the optimization criterion by which a potential need of using the data stored in other partitions is lowered. The structure of the power distribution network enables the transformation of data into a graph that can be successfully partitioned. In other words, the problem of DMS data model partitioning can be reduced to the problem of graph partitioning.. http://dx.doi.org/10.5755/j01.itc.40. 4.981 This paper reports the results of experiments which show that hybrid Genetic Algorithm (HGA) can be successfully applied for initial partitioning and that it always provides better results than simple Genetic Algorithm (GA) [5] and Particle Swarm Optimization (PSO) [6] algorithm.. The paper is organized in the following way: next section describes related work on algorithms for partitioning graphs. Section 3 describes the problem and defines terms used in the paper. It also includes the details of data model and definitions of the optimization problem. The HGA for initial partitioning is presented in Section 4. Section 5 describes experimental setup, presents and discusses the results. Section 6 is a conclusion. Graph partitioning is classified as an NP hard optimization problem [7] . Many suboptimal graph partitioning solutions have been proposed. Two commonly used classes of algorithms are multilevel and evolutionary algorithms.. Multilevel algorithms [8, 9] consist of three phases: coarsening -the matching of vertices per levels, partitioning -the partition graph consisting of matched vertices at a certain level and refinement -improvement of the partitioning at various levels. Most of these algorithms use KL [10] or its variant FM [11] algorithms, which are specialized for local improvement of solutions by exchanging vertices between the individual partitions. In addition, the FM algorithm moves a vertex from one partition to another in each iteration, and the KL algorithm exchanges vertices between partitions. Most commonly used multilevel algorithm is METIS algorithm [9] .. In our studies [5, 6, 12] , evolutionary algorithms are used for the initial partitioning. This allows the use of somewhat slower algorithms since it is done prior to startup. However, before the algorithms are applied, the coarsening of data is required in order to reduce the problem dimensionality and make a smaller number of data groups. The aim of the study reported in this paper is to improve previously obtained results and determine the applicability of evolutionary algorithms to large data model partitioning in power distribution systems.. Evolutionary algorithms have been successfully used for graph partitioning. The most commonly used algorithms are PSO [6] and GA [5, [13] [14] [15] .. GA is used for graph partitioning: various size graphs or unweighted ones [13] . Furthermore, parallel GA [14, 15] was developed to speed up the algorithm. If the accuracy of a simple GA is not satisfied, then variants of hybrid GA are recommended [16] [17] [18] [19] . GA finds a solution close to the optimum and some other algorithm (often KL/FM) is used for local refinement [13] . This paper compares hybrid GA with previously used GA [5, 12] and PSO [6, 12] algorithms for initial graph partitioning. The DMS data model partitioning is described by analyzing the target DMS functions. It is assumed that only one function [ is running in the system. This is done in order to optimally partition datasets and finish data processing as quickly as possible. DMS data model (Q) contains all elements Į i that are included in a calculation. Conducting equipment and transformers are elements usually used for the DMS functions.

O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Simulated

unbound

structures

for

benchmarking

of

protein

docking

in

the

Dockground

resource

Background:

Proteins

play

an

important

role

in

biological

processes

in

living

organisms.

Many

protein

functions

are

based

on

interaction

with

other

proteins.

The

structural

information

is

important

for

adequate

description

of

these

interactions.

Sets

of

protein

structures

determined

in

both

bound

and

unbound

states

are

essential

for

benchmarking

of

the

docking

procedures.

However,

the

number

of

such

proteins

in

PDB

is

relatively

small.

A

radical

expansion

of

such

sets

is

possible

if

the

unbound

structures

are

computationally

simulated.

Results:

The

DOCKGROUND

public

resource

provides

data

to

improve

our

understanding

of

protein-protein

interactions

and

to

assist

in

the

development

of

better

tools

for

structural

modeling

of

protein

complexes,

such

as

docking

algorithms

and

scoring

functions.

A

large

set

of

simulated

unbound

protein

structures

was

generated

from

the

bound

structures.

The

modeling

protocol

was

based

on

ns

Langevin

dynamics

simulation.

The

simulated

structures

were

validated

on

the

ensemble

of

experimentally

determined

unbound

and

bound

structures.

The

set

is

intended

for

large

scale

benchmarking

of

docking

algorithms

and

scoring

functions.

Conclusions:

A

radical

expansion

of

the

unbound

protein

docking

benchmark

set

was

achieved

by

simulating

the

unbound

structures.

The

simulated

unbound

structures

were

selected

according

to

criteria

from

systematic

comparison

of

experimentally

determined

bound

and

unbound

structures.

The

set

is

publicly

available

at

The

DOCKGROUND

public

resource

provides

data

to

improve

our

understanding

of

protein-protein

interactions

and

to

assist

in

development

of

docking

algorithms

and

scoring

functions.

Sets

of

protein

structures

determined

in

both

bound

and

unbound

states

are

essential

for

benchmarking

docking

procedures.

However,

the

number

of

such

proteins

in

PDB

is

relatively

small.

A

radical

expansion

of

such

sets

is

possible

if

the

unbound

structures

are

computationally

simulated.

Such

simulated

unbound

protein

set

was

generated

for

the

DOCK-GROUND

resource.

The

modeling

protocol

was

based

on

ns

Langevin

dynamics

simulation.

Simulated

unbound

structure

was

selected

according

to

criteria

from

systematic

comparison

of

experimentally

determined

bound

and

unbound

structures.

The

set

is

publicly

available

at

Proteins

play

an

important

role

in

biological

processes

in

living

organisms.

Many

protein

functions

are

based

on

interaction

with

other

proteins.

The

structural

information

is

essential

for

adequate

description

of

these

interactions.

Protein

interaction

is

characterized

by

structural

and

physicochemical

recognition

factors

and

conformational

changes

upon

binding

Computational

approaches

to

the

structural

modeling

of

protein

interactions

are

important,

given

the

limitations

of

experimental

techniques

A

significant

progress

in

the

computational

prediction

of

protein-protein

complexes

(protein-protein

docking)

has

been

reflected

in

the

community-wide

assessment

The

original

steric

complementaritybased

algorithms

paved

the

way

to

knowledge-based

approaches

including

those

based

on

similarity

to

existing

co-crystallized

complexes,

low-resolution

(coarse-grained)

techniques,

and

proteome-wide

applications

The

docking

algorithms

are

generally

based

on

the

concept

of

structure

complementarity,

observed

in

experimentally

determined

complexes.

Thus,

most

docking

procedures

perform

better

when

the

bound

(co-crystallized)

protein

structures

are

used,

assuring

the

perfect

match

between

the

structures.

Such

bound

docking

allows

one

to

neglect

the

internal

degrees

of

freedom

(structural

flexibility),

providing

for

an

effective

search

of

the

six-dimensional

rigid-body

space

of

the

external

coordinates.

However,

in

the

real-case

scenario,

the

bound

structures

of

the

participating

proteins

are

unknown,

and

one

has

to

rely

on

the

unbound

(e.g.

crystallized

separately)

proteins.

Because

of

the

huge

number

of

potentially

relevant

internal

degrees

of

freedom,

the

problem

of

unbound

docking

is

far

from

being

solved..

The

rigid-body

docking

of

unbound

proteins

results

in

structural

mismatches

at

the

putative

interfaces.

Thus,

one

approach

to

the

unbound

docking

lowers

the

resolution

of

the

structures,

alleviating

the

difference

between

unbound

and

bound

structures,

and

decreasing

the

structural

overlap

The

downside

of

such

an

approach

is

a

lesser

(low-resolution)

precision

of

the

predicted

structure

of

the

complex.

An

alternative

paradigm

is

to

use

sophisticated

scoring

schemes

to

evaluate

a

large

number

(e.g.

hundreds

of

thousands)

of

high-resolution

rigid-body

predictions,

in

anticipation

that

it

would

capture

the

native

interface

containing

structural

mismatches

(thus

having

high

energy)

Docking

approaches

that

explicitly

search

the

internal

coordinates

are

being

developed

However,

their

success

in

the

unbound

docking

is

still

limited

Template-based

docking

approaches

(structure

or

sequence-based)

generally

are

based

on

the

backbone

alignment

(followed

by

the

repacking

of

the

side

chains

for

the

final

prediction).

Thus,

in

principle,

they

should

not

depend

on

the

bound/unbound

difference

in

the

side

chains

conformations.

However,

the

difference

in

the

backbone

may

affect

the

performance

of

the

procedure..

For

the

development

of

docking

techniques

applicable

to

the

unbound

proteins,

it

is

essential

to

learn

the

experimentally

determined

difference

between

bound

and

unbound

states.

A

number

of

proteins

have

structures

experimentally

determined

in

both

unbound

and

bound

states

In

most

proteins

of

complexes)

conformational

change

upon

binding

is

Å

all

atoms

RMSD

A

significant

number

of

complexes

with

larger

RMSD

have

a

domain

shift,

where

conformational

changes

in

the

domains

themselves

are

small.

Still,

the

other

cases

of

large

RMSD

involve

interface

loops,

which

change

conformation

significantly

upon

binding.

Thus,

our

ability

to

adequately

address

conformational

changes

in

docking

is

important..

The

utility

of

the

unbound

docking

approaches

is

tested

in

the

CAPRI

blind

experiment

To

provide

consistent

sets

for

validation

of

docking

and

scoring

procedures,

benchmark

sets

of

protein-protein

complexes

were

compiled

However,

the

number

of

known

representative

protein

pairs

with

experimentally

determined

structures

in

both

bound

and

unbound

states

is

relatively

small

(e.g.,

in

the

Weng's

Benchmark

At

the

same

time,

the

number

of

co-crystallized

complexes

is

much

larger.

A

key

feature

in

the

DOCKGROUND

resource

is

flexibility,

which

allows

users

to

build

the

datasets

according

to

their

own

requirements.

Such

datasets

can

involve

thousands

of

complexes

and

thus

can

be

used

for

truly

large-scale

benchmarking

of

docking

methodologies.

Simulating

the

unbound

structure

from

the

bound

one

provides

such

an

opportunity.

Our

earlier

set

of

simulated

unbound

structures

based

on

an

older

version

of

DOCKGROUND,

was

generated

by

changing

the

side

chain

conformations

according

to

the

rotamer

library

In

the

current

paper

we

describe

a

much

larger

set

obtained

by

Langevin

dynamics

simulation

and

based

on

a

systematic

analysis

of

the

experimentally

determined

bound/unbound

structural

differences.

The

set

is

a

valuable

resource

for

benchmarking

docking

procedures

and

development

of

docking

methodologies.

Protein

complexes

were

selected

from

the

Bound

part

of

DOCKGROUND

with

the

following

criteria:

mean

area

buried

Å

include

alternative

binding

modes,

homo/hetero

n-mers,

and

oligomers,

and

the

redundancy

cutoff

The

resulting

set

contained

proteinprotein

complexes.

Program

Profix

from

the

Jackal

package

php/Software)

was

used

to

build

the

disordered

residues

and

missing

atoms..

It

was

expected

that

dynamic

simulation

of

separate

bound

protein

structures,

without

the

interacting

partner,

would

relax

interface

side-chain

conformations

constrained

by

the

interacting

partner

and

thus

approximate

the

unbound

form

of

the

protein.

To

speed

up

the

calculations,

we

chose

the

Langevin

Dynamics

(LD)

simulation

in

CHARMM

force

field),

with

electrostatics

by

Generalized

Born

approximation,

performed

on

each

bound

structure

without

its

interacting

partner.

Prior

to

LD

simulation,

the

initial

structures

from

PDB

files

were

minimized

(by

steps

of

steepest

descent

minimization

followed

by

steps

of

adopted

basis

Newton-Raphson

minimization)..

In

the

simulation,

the

backbone

atoms

of

helices

and

strands

were

constrained

with

a

force

constant

of

kcal/

mol,

the

temperature

was

set

to

K,

the

bond

lengths

were

fixed

using

shake

with

tolerance

the

friction

force

fbeta

was

and

the

time

of

simulation

was

ns,

with

snapshots

saved..

Protein

structures

that

crashed

during

the

simulation

were

removed.

The

simulation

yielded

protein

structures.

The

number

of

resulting

complexes

with

both

proteins

simulated

was

In

complexes

only

one

partner

was

simulated.

The

structure

with

the

largest

all

atoms

RMSD

from

the

bound

structure

was

designated

as

the

simulated

unbound

structure.

Among

the

simulated

proteins

there

were

non-obligate

and

obligate

complexes,

according

to

NOXClass

procedure

O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Bayesian object matching Abstract Matching of object refers to the problem of inferring unknown co-occurrence or alignment between observations or samples in two data sets. Given two sets of equally many samples, the task is to find for each sample a representative sample in the other set, without prior knowledge on a distance measure between the sets. Given a distance measure, the problem would correspond to a linear assignment problem, the problem of finding a permutation that re-orders samples in one set to minimize the total distance. When no such measure is available, we need to consider more complex solutions. Typical approaches maximize statistical dependency between the two sets, whereas in this work we present a Bayesian solution that builds a joint model for the two sources. We learn a Bayesian canonical correlation analysis model that includes a permutation parameter for re-ordering the samples in one of the sets. We provide both variational and sampling-based inference for approximative Bayesian analysis, and demonstrate on three data sets that the resulting methods outperform the earlier solutions. We introduced a variational Bayesian solution for the object matching problem introduced by Jebara (2004) and popularized by Haghighi et al. (2008) , Quadrianto et al. (2010) , Tripathi et al. (2011) and Yamada and Sugiyama (2011) for solving alignment tasks for example in natural language processing and computational biology. By learning together a Bayesian canonical correlation analysis model (Klami et al. 2013 ) and a permutation matrix re-ordering the samples in one of the sets, we obtained matching accuracies better than those of any earlier solution.. We presented two alternative inference strategies, one based on approximative Gibbs sampling and the other on variational approximation, and derived the computational details necessary for approximating the posterior over the permutations for both. The resulting algorithms were applied on three benchmark data sets and further illustrated on artificially generated data, to confirm that the proposed algorithms produce accurate matches. In particular, we outperformed all the earlier variants by a comfortable margin. For image matching we improved from 64 % to 85 %, for metabolite alignment we improved from 35 % to 39 %, and in two document alignment tasks we improved from 29-37 % to 100 %. These improvements correspond to real practical gains, and in particular the last one represents a qualitative change where the new method is able to perfectly solve a problem for which the earlier solutions were not satisfactory.. The Gibbs sampler was found to be the better of the two inference solutions, since it can more effectively explore the posterior space. We additionally showed that for sufficiently low-dimensional data the true posterior is so wide that it is actually better to concentrate on some local region of the posterior space. For such setups the Gibbs sampler reduces to almost random guessing, and the variational inference is the best matching solution. Most earlier solutions to the matching problem maximize statistical dependency between the two sets. The idea is that statistical dependency should not arise by coincidence, but instead a high degree of dependency should be indicative of having found the correct match. A random permutation will make any two sets independent, and hence maximizing the dependency will at least allow escaping that extreme. The practical methods can be divided into two categories based on the dependency measure. The first category optimizes canonical correlation between the sets, whereas the other category maximizes a kernel-based dependency called Hilbert-Schmidt Independence Criterion (HSIC; Smola et al. 2007) or some other kernel-based measure. The former require access to real-valued feature vectors for the two sets, whereas for the latter it is sufficient to provide kernels representing pairwise-distances within each set.. The matching canonical correlation analysis (MCCA) method was introduced by Haghighi et al. (2008) for constructing bilingual dictionaries from monolingual corpora, by matching the individual words in two languages. The basic idea of the algorithm is that it finds a linear subspace that maximizes the correlation between the sets (CCA; canonical correlation analysis). Explicit representation for the subspace allows computing distances between the samples in the two sets, and hence a LAP can be used for finding the match. Since the subspace itself depends on the match, the algorithm alternates between these two steps until convergence. The original algorithm is defined for semi-supervised matching that requires an initial seeding with some known pairs, but Tripathi et al. (2009) presented independently almost the same algorithm for fully unsupervised matching of probes of gene expression platforms, and Tripathi et al. (2010) extended it to use kernel CCA while also presenting the semi-supervised matching problem where some initial seed pairs are given. Later, Tripathi et al. (2011) extended the CCA-based formulation to setups where the task is to find a consensus of multiple matching problem solutions, and Sysi-Aho et al. (2011) applied it to finding correspondence between metabolic profiles of different species. The basic idea is to merge several matching problem solutions by learning one more LAP to find a permutation that best agrees with the initial solutions.. The other category of dependency-maximizing matching solutions builds on the kernelized sorting (KS) idea initially presented by Jebara (2004) . Instead of finding an explicit representation that allows computing distances (and hence solving the matching as a LAP), the idea in kernelized sorting is to directly optimize a dependency measure that only depends on kernels computed for the two sets. Quadrianto et al. (2010) introduced the standard KS algorithm that maximizes the Hilbert-Schmidt Independence Criterion. The HSIC is defined as the trace-norm of KL, where K and L are properly centered kernel matrices for the two sets, and KS solves the matching problem by introducing a permutation matrix in that cost. The resulting cost corresponds to a quadratic assignment problem (QAP), which is NP complete (Burkard 1984) . Quadrianto et al. (2010) solve the QAP by iteratively applying a LAP solver. Later Jagarlamudi et al. (2010) improved the algorithm by proposing an improved initialization scheme and various other tricks to improve the robustness of KS solutions. They also consider application-specific details for an important domain of matching problems, natural language processing with specific tasks such as document alignment. Yamada and Sugiyama (2011) introduced another variant that replaces the HSIC measure with alternative kernel-based dependency measures, normalized cross-covariance operator and least-squares mutual information, using the same iterative learning algorithm as Quadrianto et al. (2010) .. Recently, Djuric et al. (2012) provided an alternative optimization algorithm for kernelized sorting. Instead of directly optimizing the QAP, they relax the optimization problem by replacing the optimization space of permutation matrices with that of the doubly-stochastic matrices; positive matrices with unit row and column sums. The resulting cost is convex and hence they can find the global optimum of the relaxed cost. However, the solution does not in general correspond with the solution of the original cost function, and even though the algorithm does not need to solve assignment problems it is still computationally very demanding as it performs constrained gradient-based optimization over a N 2 -dimensional parameter space. Also, while the model produces soft assignments between the samples, it does not correspond to a proper distribution over the permutations; it is merely the optimal solution over doubly-stochastic matrices. In the empirical sections, we will compare the proposed models primarily with CKS, since Djuric et al. (2012) showed that it typically outperforms other KS algorithms.. As mentioned already in the Introduction, the MCCA method by Haghighi et al. (2008) can also be interpreted as maximizing the joint likelihood of the two data sets, based on the probabilistic interpretation of CCA (Bach and Jordan 2005) . In fact, the MCCA model is identical to our formulation (1) except that it does not have priors for any of the model parameters and it uses maximum likelihood estimation. The model by Tripathi et al. (2011) can also be interpreted in similar fashion, since the maximum likelihood solution for probabilistic CCA is equivalent to the classical CCA solution. These two methods are hence the closest alternatives to ours, and in the empirical comparisons we will use them to demonstrate that the improved accuracy of the Bayesian solutions is because of the posterior inference. For running these comparisons we use the optimization algorithm as implemented by Tripathi et al. (2011) since it does not require initial seed pairing, and use the abbreviation CCA-ML to remind that the method corresponds to maximum likelihood estimation of a CCA-based matching method.. Another example of a method maximizing the join likelihood is the multilingual topic model by Boyd-Graber and Blei (2009). They learn a topic model for two languages by matching their vocabularies based on a maximum a posteriori estimation of a permutation matrix. While their eventual task is in learning the topic model itself, the matching solution is an integral part of the model.. Besides the above two criteria (maximal dependency and optimal joint model) for defining the right match, one can solve the matching problem also by explicitly constructing a distance between the two sets. This results in a non-iterative algorithm that merely needs to compute the distances once, since given the distances a single LAP solver will find the match. For example, Tripathi et al. (2011) used the manifold alignment method by Wang and Mahadevan (2009) to compute the distances by aligning local neighborhoods for the two sets in order to solve the matching problem. The application of Wang and Mahadevan (2009) also constitutes a good example of a potential application domain; they seek to match protein structures.. Some related work has also been done on posterior inference over permutations. While these works have not considered the matching problem as such, they are relevant background information. Kondor et al. (2007) considered exact variational inference over permutations by using Fourier transformations, and Plis et al. (2011) mapped the permutations to a highdimensional hypersphere to do the same. These approaches are, however, only applicable to small N , at most tens, and hence would not be sufficient for our scenarios where N is in the order of hundreds. Leskovec et al. (2010) , in turn, proposed a Metropolis sampler for permutations based on swaps of pairs. Their sampler is effectively equivalent to the one Gibbs-subset uses for sampling the posteriors, assuming we use J = 2; for larger J we can consider more complex operations than mere swaps of two pairs. They also provide illustrative characterizations of the properties of the permutation space, showing how only a tiny fraction of the permutations have non-negligible probability; this matches exactly our findings. We start the experiment with an artificial data experiment, used to demonstrate the characteristics of the solution. In particular, we will show how the two alternative inference strategies have very different strength and weaknesses. We also demonstrate empirically the quality of the approximation for the Gibbs-hard sampler.. After the demonstration, we compare the proposed methods with earlier matching problem solutions, using data collections analyzed by the earlier authors. We perform three different comparisons. The first is an image matching task from Quadrianto et al. (2010) , the second a metabolite matching task from Tripathi et al. (2011) , and the last a cross-lingual document alignment task from Djuric et al. (2012) . In all cases we compare the proposed methods with the leading kernelized sorting variant CKS by Djuric et al. (2012) and the CCA-ML method by Tripathi et al. (2011) which correspond to finding the maximum likelihood solution of our model. The purpose of these comparisons is to show that the proposed solution is more accurate than the earlier solutions, while also demonstrating that the improvement comes from the Bayesian treatment of the model.

O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


MULTI-CHANNEL SAMPLING ON SHIFT-INVARIANT SPACES WITH FRAME GENERATORS Abstract. Let ϕ be a continuous function in L 2 (R) such that the sequence {ϕ(t − n)} n∈Z is a frame sequence in L 2 (R) and assume that the shift-invariant space V (ϕ) generated by ϕ has a multi-banded spectrum σ(V ). The main aim in this paper is to derive a multi-channel sampling theory for the shift-invariant space V (ϕ). By using a type of Fourier duality between the spaces V (ϕ) and L 2 [0, 2π] we find necessary and sufficient conditions allowing us to obtain stable multi-channel sampling expansions in V (ϕ). As a natural extension of the classical Shannon sampling theorem, Papoulis introduced in [17] generalized sampling for arbitrary multi-channel sampling in Paley-Wiener spaces P W πσ of band-limited signals: In many common situations the available data are samples of some filtered versions of the signal itself. Following [17] , there have been many generalizations and applications of the multi-channel sampling. See, for example, [6, 7, 16, 19, 20] and references therein.. Although Shannon's sampling theory has had an enormous impact, it has a number of problems, as pointed out by Unser in [18] : It relies on the use of ideal filters; the bandlimited hypothesis is in contradiction with the idea of a finite duration signal; the bandlimiting operation generates Gibbs oscillations; and finally, the sinc function has a very slow decay, which makes computation in the signal domain very inefficient.. Moreover, many applied problems impose different a priori constraints on the type of functions. For this reason, sampling and reconstruction problems have been investigated in spline spaces, wavelet spaces, and general shift-invariant spaces. Indeed, in many practical applications, signals are assumed to belong to some shift-invariant space of the form: V (ϕ) := span L 2 (R) {ϕ(t−n) : n ∈ Z} where the function ϕ in L 2 (R) is called the generator of V (ϕ). In most of cases in the mathematical literature, it is supposed that the sequence {ϕ(t − n)} n∈Z forms a Riesz basis for V (ϕ). See, for instance, [1, 2, 3, 4, 12, 15, 18, 21, 22] and the references therein. Throughout this paper we assume that the sequence {ϕ(t − n)} n∈Z is a frame for V (ϕ) and that the spectrum of V (ϕ) is multi-banded in [0, 2π] (see Section 3 infra).. On the other hand, suppose that N linear time-invariant systems (filters) L j , j = 1, 2, . . . , N , are defined on the shift-invariant subspace V (ϕ) of L 2 (R). In mathematical terms we are dealing with continuous operators which commute with shifts. The recovery of any function f ∈ V (ϕ) from samples of the functions L j f , j = 1, 2, . . . , N , leads to a generalized sampling in V (ϕ).. Our challenge problem is the following: Given r, N positive integers and N real numbers 0 ≤ a j < r for 1 ≤ j ≤ N , find multi-channel sampling expansions like. (L j f )(a j + rn)S j,n (t) , t ∈ R , valid for any f ∈ V (ϕ), where the sequence of sampling functions S j,n (t) : 1 ≤ j ≤ N, n ∈ Z forms a frame or a Riesz basis for V (ϕ). Recently, García et al. ([8, 9, 10] ) introduced a novel idea for developing a sampling theory on a shift-invariant space V (ϕ) by using an analogous of the Fourier duality between the spaces V (ϕ) and L 2 [0, 2π]. In particular, García and Pérez-Villalón [9] (see also [14] ) developed a multi-channel sampling procedure on a shift-invariant space V (ϕ), where ϕ is a continuous Riesz generator. Unlike the author's claim (see section 4.1 in [9] ), the arguments used in [9] for the case of Riesz generator cannot be directly extended to the case of a frame generator.. In the present paper, by assuming that the sequence {ϕ(t − n)} n∈Z is a frame for V (ϕ) and that the spectrum of V (ϕ) is multi-banded in [0, 2π], and allowing more general filters than those used in [9] , we obtain necessary and sufficient conditions under which there exists a stable multi-channel sampling expansion on V (ϕ) like that in (1.1). We also provide some illustrating examples. All these tasks will be carried out throughout the remaining sections.

B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O