Research Article A Novel Algorithm of Quantum Random …downloads.hindawi.com/journals/jam/2014/818479.pdf · Research Article A Novel Algorithm of Quantum Random Walk in Server Traffic

Research ArticleA Novel Algorithm of Quantum Random Walk inServer Traffic Control and Task Scheduling

Dong Yumin1 and Xiao Shufen2

1 Network Center Qingdao Technological University Qingdao 266033 China2 College of Automobile and Transportation Qingdao Technological University Qingdao 266033 China

Correspondence should be addressed to Dong Yumin dym1188163com

Received 18 January 2014 Accepted 19 February 2014 Published 7 April 2014

Academic Editor Feng Gao

Copyright copy 2014 D Yumin and X Shufen This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A quantum random walk optimization model and algorithm in network cluster server traffic control and task scheduling isproposed In order to solve the problem of server load balancing we research and discuss the distribution theory of energyfield in quantum mechanics and apply it to data clustering We introduce the method of random walk and illuminate what thequantum random walk is Here we mainly research the standard model of one-dimensional quantum random walk For thedata clustering problem of high dimensional space we can decompose one 119898-dimensional quantum random walk into 119898 one-dimensional quantum random walk In the end of the paper we compare the quantum random walk optimization method withGA (genetic algorithm) ACO (ant colony optimization) and SAA (simulated annealing algorithm) In the same time we prove itsvalidity and rationality by the experiment of analog and simulation

1 Introduction

The server cluster technology may be connecting multipleindependent servers and in the same time it must provideservices as a whole by a cluster In the server cluster how tosolve the problem of server traffic control and task schedulingis very important

In order to reduce the access time optimize the overallperformance and achieve parallel program in a high effi-ciency the task request must be allocated to each on theserver So load balancing mechanism is the core of clustertechnologies

In the literature [1] it expands the analogies employedon the development of quantum evolutionary algorithms byputting forward quantum-inspired Hadamard walks calledQHW In order to solve combinatorial optimization prob-lems a quantum evolutionary algorithm abbreviatedHQEAis proposed From the results of the experiments carriedout on the knapsack problem HQEA performs noticeablybetter than a conventional genetic algorithm in terms ofconvergence speed and accuracy The literature [2] exploreshow a spectral technique suggested through coined quantum

walks can be used to differentiate between graphs whichare cospectral as for standard matrix representations Thisalgorithm runs in polynomial time it can differentiate manygraphs for there is no subexponential time algorithm whichis proven to be able to differentiate between them

By the literature [3] they propose a quantum algorithmto evaluate formulas by an extended gate set including two-and three-bit binary gatesThis algorithm is more optimal onread once formulas for that each gatersquos inputs are balanced ina certain sense It describes a very compact triaxial instru-ment in the literature [4] The triaxial instrument is basedon a rhombic dodecahedral geometry that can accommodatethree nonplanar ring light paths with orthogonal sensingaxes Component count can be substantially reduced by adischarge of layout to use a single cathode and two anodesrunning all three axes in balanced plasma currents TwoMonte Carlo-based approaches to assess parameter uncer-tainty with complex hydrologic models are considered inthe literature [5] The importance sampling has been carriedout in the generalized likelihood uncertainty estimationframework by Beven and Binley The metropolis algorithmis different from importance sampling which uses a random

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 818479 8 pageshttpdxdoiorg1011552014818479

2 Journal of Applied Mathematics

walk that adapts to the real probability distribution describingparameter uncertainty

Because existing search protocols for unstructured peer-to-peer systems to create huge burden on communicationsor cause long response time and result in unreliable per-formance In the literature [6] in order to discover serviceproviders it reports that an important function of a peer-to-peer system is a distributed message relaying They presentan incentive mechanism which not only relieves the free-riding problem but also accomplishes good system efficiencyin message relaying for peer discovery The passed alongmessage propagation process is promised rewards in themechanism

In the literature [7] it analyzes the discrete-time quan-tum walk by separating the quantum evolution equationinto Markovian and its interference terms Because of thisseparation it is possible to show analytically which quadraticincrease in the variation of the position of quantum walkerwith time is a direct aftermath of the coherence of thequantum evolution As expected the variation is shown toincrease linearly with time if the evolution is decoherent asin the classical caseMoreover it shows that the system has anevolving operator analogous to which of a resonant quantumkicked rotor At the same time the rotator can be describedby evolution of the quantum walker

Quantum random walks on a graph which is analogousto classical stochastic walk form the basis for many ofthe recent quantum algorithms that promise to obviouslyoutperform existing classical random walk algorithms Anumber of studies have been done on the many applicationsof quantum random walk to some important computingproblems There are two kinds of quantum random walkalgorithms continuous-time and discrete-time It is reportedthat a quantum arithmetic is defined by a sequence ofthe operations that runs on an actual model of quantumcomputation in the literature [8] It proposes quantum circuitdesigns for both kinds of random walk algorithms whichoperate on various graphs It considers two important prob-lems to which random walk arithmetic are applicable thetriangle finding problem and binary tree problem Because ofit a few research works that are related to quantum randomwalk circuit design on graphs exist the circuit designs theypresent here are the first of their kind At the same time theyalso provide an estimate of the quantum cost of the circuits ofquantum systems And it is based on the number of executioncycles and quantum operations

In the literature [9] the natural random walk causingBrownianmotion occurs to be always biased in a very delicateway emphasizing some possibilities by only approximativemaximal uncertainty principle It introduces a new methodof stochastic model and they use the merely maximizingentropy choice of transition probabilities

Berry and Wang show numerically that a discrete timequantum random walk of two irrelevant particles is able todifferentiate some nonisomorphic powerfully regular graphsfrom the same household in the literature [10] They analyt-ically show how it is possible for the walks to differentiatesuch graphs while the continuous time quantumwalks of twoirrelevant particles cannot

It is reported that the quantum walks are quantummechanical theory analog of random walks in the literature[11] By traversing the edges of a graph a quantum ldquowalkerrdquoprogresses between initial and final states They present ahybrid model for general quantum computing in which aquantum walker gets discrete steps of continuous evolution

Effective server traffic control can extend the ldquocapacityrdquoof the server and the task scheduling can improve systemthroughput In early research methods of it such as Min-Min algorithmMax-Min algorithm genetic algorithm (GA)round robin (RR) simulated annealing algorithm (SAA)dynamic feedback algorithm (DFA) and ant colony opti-mization algorithm (ACO) These arithmetics have someimprovements in different degree on the task scheduling

But these algorithms have this or that problem such aslocal premature problem and divergence problem

In order to overcome the instability above the algorithmsthe quantum random walk algorithm is proposed and it isproved better than above GA ACO and SAA by simulationexperiments

2 Quantum Random Walk

21 Random Walk Random walk is a mathematical methodto study the formation of trajectory by a random sequenceof continuous it is not only a means to study mathematicsbut also a basic tool in the natural sciences Any stage of therandom walk behavior is not limited to previous history ofmigration the process is also calledMarkov process Randomwalk can be simply described as follows

Suppose in a straight line there is a moving particle itis at the origin to move left or right one unit of distancethe probability is 119901 and 119902 = 1 minus 119901 respectively each timethe particle in accordance with the probability to move aunit distance to the left or right Here we assume that theprobability of the particle is equal to the left or right that is119901 = 119902 = 12 random variable can be used to represent theprobability 120590

119894 its value is as follows

120590119894=

minus1 119901 =

1

2

1 119902 =

1

2

(1)

If the particles every moment in a straight line positionconstitute an independent identical distribution of randomvariables sequence denoted 120590

1 1205902 120590

119905 is a sequence

of independent and identically distributed variables on meetEX1

= 0 Var(1205901) = 1 119878

119905= sum119905

119894=1120590119894is its first 119899 terms and

119878 = (119878119905)119905isin119879

is called a random walk After the particles move119899 steps the probability of it being found in position 119898 is

119901119899

0119898= (

119899

119899 + 119898

2

)119902(119899+119898)2

119901(119899minus119898)2

(2)

Among them (119899 + 119898)2 take only integer (119899 + 119898)2 isin

0 1 119899 other cases were 0

Journal of Applied Mathematics 3

Although the classical random walk has a broad applica-tion but compared with the quantum random walk it feelsmuch ashamed of its inferiority

Quantum random walk is a quantum computing modelproposed in recent years scholars have also become increas-ingly interested in research

22 Quantum Random Walk For discrete quantum randomwalk the system added an extra degree of freedom someliterature defines it as chirality that can build an adaptation toglobal local unitary process This walk is also called quantumHadamard walk The only possible remained unchanged inthe global process of unitary transformation is the onlymobile operator between adjacent lattice points to the left orthe right

Significantly different between quantum Hadamard ran-dom walk and classical quantum is the interference thediffusion rate of quantum walk square magnitude fasterthan classical square Due to the existence of quantumsuperposition states in quantum random walk positionof the particle from the probability distribution may beseen particles may be in several locations simultaneouslywith different probability Quantum random walk process isaccomplished by a unitary matrix transformation [12]

221 One-Dimensional Quantum Random Walk The ran-dom walk model sets up corresponding quantum algorithmsand does quantum information processing People com-monly used coined quantum walks it corresponds to Hilbertspace which can be expressed as follows

119867 = 119867119862

otimes 119867119881 (3)

where 119867V = span|V119894⟩|119881|

119894=1is the random walk of grid space

it corresponds to a classic case of 119889-degree regular graphs119866(119881 119864)119867

119862= 119862119889 which is a coin flip operator space (coin

space) The total unitary evolution matrix 119880 is by the twoindependent parts namely flipping a coin and conditionalreplacement

119880 = 119878 ∙ (119862 otimes 119868) (4)

The first step of the quantumwalk is to perform a rotationoperation 119862 in coin space equivalent to the classical randomwalk in a coin toss through this operation to get a coinsuperposition state [13] Then the replacement operator 119878

makes the particles by a coin to decide an edge vertex adjacentto move to the next Starting from the initial state |120595(0)⟩repeat the walk after 119899 steps and obtain the probabilitydistribution of each vertex as follows

119875 (V 119899) = V⟨V| 119873119903119888[1003816100381610038161003816120595 (119899)⟩ ⟨120595 (119899)

1003816100381610038161003816] |V⟩ V (5)

Quantum randomwalks of a variety of ways common areone-dimensional linear walk ring walk hypercube walk andso forth For one-dimensional linear walk 119867V = span|119909⟩

119909 isin 119885119867119888= span|119877⟩ |119871⟩ replacement operator applied to

the base is expressed as

119878 |119877 119909⟩ = |119877 119909 + 1⟩ 119878 |119871 119909⟩ = |119871 119909 minus 1⟩ (6)

Starting from the initial state |Φin⟩ = | darr⟩otimes|0⟩ continuousaction119880 = 119878∙(119862otimes119868) after each step the distribution of everypoint is as follows

1003816100381610038161003816Φin⟩

119880

997888rarr

1

radic2

(|uarr⟩ otimes |1⟩ minus |darr⟩ otimes |minus1⟩ )

119880

997888rarr

1

2

[|uarr⟩ otimes |2⟩ minus (|uarr⟩ minus |darr⟩ ) otimes |darr⟩ otimes |minus2⟩ ]

119880

997888rarr

1

2radic2

[|uarr⟩ otimes |3⟩ + |darr⟩ otimes |1⟩ + (|uarr⟩ minus 2 |darr⟩ )

otimes |minus1⟩ minus |darr⟩ otimes |minus3⟩ ]

(7)

Not only is the distribution situation different with theclassical random walk but one-dimensional linear walk isalso higher and faster than the classical random walk in thediffusion velocity

222 M-Dimensional Hypercube Quantum Random WalkFor the 119872-dimensional hypercube quantum random walkit has 2

119898 vertices each vertex can be marked by an 119898 binarystring

119867V = span|⟩119909 isin [0 2119899] Each vertex is 119898 degrees

therefore coin space 119867119888

= 119862119888 using |119889⟩ to mark the coin

space basis vectors 119889 isin [0 119899] it indicates the direction of thenext step each |119889⟩ corresponds to a 119872-dimensional vector|e119889⟩

1003816100381610038161003816

119890119889⟩ =

1003816100381610038161003816100381610038161003816100381610038161003816

0 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119889minus1)

10 sdot sdot sdot 0⏟⏟⏟⏟⏟⏟⏟⏟⏟

(119899minus119889)

⟩ (8)

| 119890119889⟩ the number of 119889 bit is 1 and that of the other bits is 0The

replacement operator walk on the hypercube is applied to thebase vectors and can be expressed as

119878 |119889 ⟩ =1003816100381610038161003816119889 oplus 119890

119889⟩ (9)

It represents two points marked by119872 quantum bits onlywhen they have only one bit is not the same when theyare connected directly via a side (eg 001101 and 011101 arecommunicating) There are many ways to select hypercubewalking coin flip operators however the following form isusually taken to maintain certain symmetry

119862120572120573

= (

120572 120573 120573 120573 120573

120573 120572 120573 120573 120573

120573 120573 120573 120572 120573

120573 120573 120573 120573 120572

) (10)

The operator of such forms has the characteristics that inall directions are a permutation invariant it retains hypercubereplacement invariance Such a form is a commonly usedGrover diffusion operator selection

119866 = minus119868119899+ 2

1003816100381610038161003816119904119888⟩ ⟨1199041198881003816100381610038161003816 (11)

Among them |119904119888⟩ = 1radic119898sum119898

119889=1|119889⟩ are equally weighted

superposition states in all directions Grover operator is one

4 Journal of Applied Mathematics

of the permutation invariant operators which is the farthestwith unit transformation 119868 It will effectivelymix all of the anygiven initial state into the superposition of them The totalrandom walk evolution operator can be expressed as

119880 = 119878 sdot (119866 otimes 119868) (12)

Define the Hamming distance starting from any point toanother point of the minimum number of edges experience(ie the required number of steps) with 119889

ℎ= 1 Marking

each vertex string the number of ldquo1rdquo is called Hammingweights for example Hammingweight 010 to 1 Starting from00 sdot sdot sdot 0 any Hamming weighing the same point total canreach at the samenumber of stepsWhen the coins are formedto the symmetry type such as (10) it has the same probabilitystarting from this point to reaching the point with the sameweight This allows us to put all Hamming weight of thesame point ldquoaccumulationrdquo to a point thereby reducing thesymmetry of the random walk the walk on the hypercube isbecoming walk in a straight line It is noteworthy that thiswalk difference on the straight is not unbiased it differs fromthe previously discussed one-dimensional linear walk Thenumber of total vertices after walking on the variable linear is119898 + 1 119898 is the hypercube dimension

By putting a hypercube walk into walk on straight linemany problems can be resolved to simplify and get resultsMoor and Russell found that when 119879 = 120587 sdot1198984 randomwalkis a balanced distribution Kempe through the research onthe hitting time found hypercube quantum random walk toreach vertical angles of time relative to the classic case whichis an exponential acceleration this shows that the quantumrandom walk has the potential to make quantum algorithmacceleration

3 The Model of Server Traffic Control andTask Scheduling

In cluster services the task scheduling can be described asfollows119873 tasks need to be allocated to119898 nodes (these nodesare the servers) with different handling capacity the goalis finding an optimization schedule to minimize the totalcompletion time The system model is shown as follows

We suppose there are 119898 nodes (or servers) and 119899 tasksEvery task should be assigned to only one node We use119875 = 119901

1 1199012 119901

119898 denoting the nodes (or servers) in this

paper where 119901119894denotes one of the nodes (or servers) 119871 =

1198971 1198972 119897119898 expresses the current load where 119897

119894expresses

the current load of node119901119894 For instance 119897

119894= 0means that the

node (server) 119901119894has a current load of 0 in other words the

node is idleHere 119899 tasks are expressed by119883 = 1199091 1199092 119909

119899

where 119909119895is one of the tasks A 119898 times 119899 matrix is built between

servers and tasks 119882119898119899 where 119882

119894119895is one of the elements So

there are two states as follows

119882119894119895

=

1 Task 119909119894is assigned on node 119901

119894

0 Task 119909119894is not assigned on node 119901

119894

(13)

where 119894 isin 1 2 119898 119895 isin 1 2 119899

We use 119905119894119895to express the time of processing on one task

in other words the time of task 119909119895processed on node 119901

119894 The

processing time is denoted as follows

119879119894119895

=

119905119894119895

Task 119909119894is processed on node 119901

119894

0 Task 119909119894is not processed on node 119901

119894

(14)

where 119894 isin 1 2 119898 119895 isin 1 2 119899It is not difficult to see that 119879

119894119895is also an 119898 times 119899 matrix

Here we define the optimal state occurring with theseconditions (a) the total system has a relative short time ofprocessing (b) the throughput of system is relatively largerin unite time We can describe this state using the followingequations

119884max =

119898

sum

119894

120596 (119909119894 119897119894 119902119894)

120596 (119909119894 119897119894 119902119894) = 1198881((

119898

sum

119894=1

119899

sum

119895=1

119905119894119895+

119898

sum

119894=1

119902119894119905119894) minus 1198882

119898

sum

119894=1

119897119894)

2

(15)

where 119883 = 1199091 1199092 119909

119899 is the new task 119871 = 119897

1 1198972 119897119898

is the current total load at the node 119902119894is the length of ready

queue at node 119901119894 119905119894is the average processing time at node 119901

119894

1198881 1198882are constant 120596(119909

119894 119897119894 119902119894) is a function which can show

the ability of node processing The system is on the optimalrunning state when the capacity of processing tasks (or taskscheduling) reaches the maximal matching at one node

4 The Method of Task SchedulingBased on Quantum Random Walk (QRW)Clustering Algorithm

In the paper we mainly research the standard model of one-dimensional quantum random walk For the data clusteringproblem of high dimensional space we can decomposeone 119898-dimensional quantum random walk into 119898 one-dimensional quantum random walk

41 Clustering Algorithm Based on One-DimensionalQuantum RandomWalk Referred to as QuantumRandomWalk Clustering Algorithm (QRWA)

Step 1 Assume an unlabeled data set 119883 = 1198831

0 1198832

0 119883

119899

0

where each data point with m features

Step 2 Each data point in the data set can be consideredas a particle that transfers in the space according to theprobability

Step 3 Establish clustering algorithm based on the one-dimensional quantum random walk

The clustering algorithm uses a distributed control strat-egy that is each data point of the data set only affected byits neighbor within the neighborhood The neighbor of datapoints available 119896-nearest neighbor method or the method ofdefault scope of 119877 to determine and use Γ

119905(119894) indicates the set

of neighbors of a data point 119883119894119905in 119905 time

Journal of Applied Mathematics 5

Step 4 Calculate the probability for each data point transferto all neighbors in the neighborhood 119901

119905(119894 119895) 119895 isin Γ

119905(119894) the

formula is as follows

119901119905(119894 119895) =

119886119905(119894 119895)

sum119895isinΓ119905(119894)

119886119905(119894 119895)

if 119895 isin Γ119905 (

119894)

0 otherwise

119886119905(119894 119895)

= ((

Deg119905(119895)

sum119895isinΓ119905(119894)Deg119905(119895)

)

times(

Deg0(119895)

sum119895isinΓ119905(119894)Deg0(119895)

))

times ((119889 (119883119894

119905 119883119895

119905)) times (119889 (119883

119894

0 119883119895

0)))

minus1

(16)

Among them Deg119905(sdot) and Deg

0(sdot) respectively represent

the degree of current and initial data points similarly119889(119883119894

119905 119883119895

119905) and 119889(119883

119894

0 119883119895

0) respectively represent the current

and initial data point distances between 119883119894 and 119883

119895

Step 5 Find the maximum transition probability 119901119905(119894 ℎ) and

the neighbors of greatest probability of metastasis 119883ℎ

119905 ℎ isin

Γ119905(119894) ℎ = 119894

42 The Server Traffic Control Clustering Method of QuantumRandom Walk As previously mentioned 119898-dimensionalquantum random walk is decomposed into 119898 one-dimensional quantum random walk For each dimensiondata points 119883

119894

119905can only move a step left or right 119897

119871or 119897119877

Therefore the maximum transition probability 119901119905(119894 ℎ) is

mapped to the interval 120588 = 119891(119901119905(119894 ℎ)) isin [05 1] so the

probability of transfer in the opposite direction is 1minus120588 When120588 = 05 and probability of metastasis in both directions isequal available aforementioned Hadamard transforms 119867

as a coin matrix However in normal conditions 120588 = 1 minus 120588therefore the coin matrix 119862 is used in the algorithm is

119862 = (radic120588 radic1 minus 120588

radic1 minus 120588 minusradic120588

)

120588 = 119891 (119901119905 (

119894 ℎ)) = 119891(max119895isinΓ119905(119894)

(119901119905(119894 119895)))

(17)

It is easy to verify that the matrix119862 is unitary matrix meetingthe reversibility requirements of quantum mechanics

Since the quantum randomwalk clustering algorithmwilluse 119903 consecutive transformation119880 = 119878sdot(119862otimes119868) for initial statethus every time it changes left and right transfer step 119897

119871and

119897119877to the original 12 that is 119897

119871119903 and 119897

119877119903 Then conditional

operator 119878 will press type structure

119878 = |uarr⟩ ⟨uarr| otimes sum

119887

10038161003816100381610038161003816100381610038161003816

119887 + 119897119877

119903

⟩ ⟨119887| + |darr⟩ ⟨darr|

otimes sum

119887

10038161003816100381610038161003816100381610038161003816

119887 minus 119897119871

119903

⟩ ⟨119887|

(18)

As is known in quantum mechanics each one ofsuperposition states can be seen as a position of particleand indicates the probability of finding the particle at thislocation If repeatedly used119880 transforms for initial state thenthe resulting superposition state |120595⟩ will contain more itemsthis increases possible appearing position of the particleand this is not present in the classical random walk It isthese possible positions that increase the searching rangeof solution space and provide an opportunity for betterresults To calculate the probability of multiple locations ofparticles and their appearance in the corresponding locationsa unitary operation is sufficient because of the quantumparallelism but in the classical world you need multipleoperations to complete it also reflects an aspect of quantumcomputing to accelerate the classical computing

If the initial state of the particle is |1205950⟩ = | uarr⟩ otimes |0⟩ then

after applying 119903 = 2 times transform 119880 get superpositionstate |120595⟩ which is

10038161003816100381610038161205950⟩

119880

997888rarr radic120588 |uarr⟩ otimes

10038161003816100381610038161003816100381610038161003816

119897119877

119903

⟩ + radic1 minus 120588 |darr⟩ otimes

10038161003816100381610038161003816100381610038161003816

minus

119897119871

119903

⟩

119880

997888rarr 120588 |uarr⟩ otimes1003816100381610038161003816119897119877⟩ + radic120588 (1 minus 120588) |darr⟩ otimes

100381610038161003816100381610038161003816100381610038161003816

(119897119877minus 119897119871)

119903

⟩

+ (1 minus 120588) |uarr⟩ otimes

100381610038161003816100381610038161003816100381610038161003816

(119897119877minus 119897119871)

119903

⟩

minus radic120588 (1 minus 120588) |darr⟩ otimes1003816100381610038161003816minus119897119871⟩ =

1003816100381610038161003816120595⟩

(19)

From (19) particles can be found not only with probability1205882 and 120588(1 minus 120588) at the same time appear in the probability of

119897119877and 119897119871appearing in another new position (119897

119877minus 119897119871)119903 could

be (1 minus 120588) At this point if projection measurement of thesuperposition state |120595⟩ it will collapse to one of these threepositions according to the probability then the componentof particle in the 119895 dimension is updated with the followingformula

119883119894

119905+1(119895) =

119883119894

119905(119895) + 119897

119877

After the measurement if the positionof the particle at 119897

119877

119883119894

119905(119895) +

(119897119877minus 119897119871)

119903

After the measurement if the position

of the particle at(119897119877minus 119897119871)

119903

(119897119877minus 119897119871) minus 119897119871

After the measurement if the positionof the particle at minus 119897

119871

119868 119897119877

= 119897119871= (119883ℎ

119905(119895) minus 119883

119894

119905(119895)) 119895 isin 1 2 119898

119868119868

119897119877

= 120588 times (119883ℎ

119905(119895) minus 119883

119894

119905(119895)) 119895 isin 1 2 119898

119897119871= (1 minus 120588) times (119883

ℎ

119905(119895) minus 119883

119894

119905(119895)) 119895 isin 1 2 119898

(20)

6 Journal of Applied Mathematics

Linux

RG-S6506

CISCO6509

Unix Solaris

Linux

OSPF

VRRP

Figure 1 The topological structure of the network servers

As data points are random walk in the space its positionand its nearest neighbor are constantly changing with timeTherefore in the process of walking the distance of the datapoints and the degree of it need to be recalculated Repeatthe entire process above until the sum of moving length of allthe particles is less than some preset threshold 120576 At this timesome separating section of the natural emergence in the spacecan be observed each section corresponds a separate cluster

5 Analog and Simulation Experiments

51 The Experimental Environment In order to comparequantum random walk clustering algorithm (QRWA)genetic algorithm (GA) ant colony optimization (ACO) andsimulated annealing algorithm (SAA) we select six serversas nodes In the experiments we select the number of taskfrom 0 to 2500 (or 3000) We compare the results of theseschemes by Matlab The correlation parameters of selectedservers for experiments are in Table 1

In Table 1 OS represents operation system NA representsnetwork adapter MS represents memory size SM representsspecifications and models

The topological structure of the network servers is asshown in Figure 1

52 Results Figure 2 shows the system flow control rate ofQRWA is better than GA SAA and ACO And the more the

Table 1 Parameters of selected servers

SM Model CPU MS NA OSSUN SPARCEnterpriseT5120

UltraSPARC T212 G 320G 2 lowast 1000M Solaris

IBM Systemp5520Q

POWER5+266G 320G 4 lowast 1000M Linux

HPrx4640

intel Itanium 215 G 320G 2 lowast 1000M Unix

task quantity is the closer the flow control rate is The taskquantity is from 0 to 2500

Figure 3 shows the server traffic of GA SAA ACO andQRWAThat is to say theQRWA is bigger thanGA SAA andACO

The Figure 5 shows that the throughput of QRWA isbigger than ACO GA and SAA

The Figure 6 shows that the throughput of QRWA isbigger than ACO GA and SAA

Figures 4 5 and 6 show the performances of QRWA arebetter than GA SAA and ACO

From the results it is clear that quantum random walkalgorithm (QRWA) is better in server traffic control andtask scheduling than genetic algorithm (GA) simulated

Journal of Applied Mathematics 7

10 20 30 40 50 60 70 800

250

500

750

1000

1250

1500

1750

2000

2250

2500

Flow control rate ()

Task

qua

ntity

GASAA

ACOQRWA

Figure 2 The flow control rate of SUN SPARC Enterprise T5120 inGA SAA ACO and QRWA

2 4 6 8 10 12 14 160

200

400

600

800

1000

1200

1400

1600

1800

Server traffic (MB)

Tim

e (s)

GASAA

ACOQRWA

Figure 3The server traffic of SUN SPARC Enterprise T5120 in GASAA ACO and QRWA

annealing algorithm (SAA) and ant colony optimization(ACO) QRWA is more effective in task scheduling

6 Conclusions

The paper gives a quantum random walk model and algo-rithm on server traffic control and task scheduling Wemainly research the standard model of one-dimensionalquantum random walk For the data clustering problemof high dimensional space we can decompose one 119898-dimensional quantum random walk into119898 one-dimensionalquantum random walk

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 050010001500200025003000

0

200

400

600

800

1000

1200

1400

1600

1800

Task

qua

ntity

Flow control rate ()

Tim

e (s)

GASAA

ACOQRWA

Figure 4 In SUN SPARC Enterprise T5120 the relationship of theflow control rate task quantity and time And the QRWA is betterthan GA SAA and ACO

0

1900 2200 0100 0400 0700 1000 1300 1600

ACOGA

SAAQRWA

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

Bits

(sK

)

Figure 5 The throughput of HP rx4640 in ACO GA SAA andQRWA during one day

0

ACOGA

SAAQRWA

80000

72000

64000

56000

48000

40000

32000

24000

16000

8000

Bits

(sK

)

February 01 2014 April 01 2014 May 01 2014 June 01 2014 August 01 2014

Figure 6The throughput of IBMSystemp5520Q inACOGA SAAand QRWA during one week

8 Journal of Applied Mathematics

The simulation results demonstrate the effectiveness andsuperiority of QRWA

The model and algorithm increases the throughput andefficiency of the system and it had some merits than tradi-tional model and arithmetics

We will research the two directions in the future Thefirst one is the effects of noise on the scheme and modelthe second one is the method of how to apply in the field ofintelligence

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (61173056)

References

[1] S Koppaka and A R Hota ldquoSuperior exploration-exploitationbalance with quantum-inspired Hadamard walksrdquo in Proceed-ings of the 12th Annual Genetic and Evolutionary ComputationConference (GECCO rsquo10) pp 2093ndash2094 Companion Publica-tion July 2010

[2] D Emms S Severini R CWilson and E R Hancock ldquoCoinedquantumwalks lift the cospectrality of graphs and treesrdquoPatternRecognition vol 42 no 9 pp 1988ndash2002 2009

[3] B W Reichardt and R Spalek ldquoSpan-Program-based quantumalgorithm for evaluating formulasrdquo in Proceedings of the 40thAnnual ACM Symposium on Theory of Computing (STOC rsquo08)pp 103ndash112 May 2008

[4] G JMartin S C Gillespie andCHVolk ldquoLitton 11 cm triaxialzero-lock gyrordquo in Proceedings of the IEEE Position Location andNavigation Symposium (PLANS rsquo96) pp 45ndash55 April 1996

[5] G Kuczera and E Parent ldquoMonte Carlo assessment of parame-ter uncertainty in conceptual catchmentmodels theMetropolisalgorithmrdquo Journal of Hydrology vol 211 no 1-4 pp 69ndash851998

[6] C Li B Yu and K Sycara ldquoAn incentive mechanism for mes-sage relaying in unstructured peer-to-peer systemsrdquo ElectronicCommerce Research and Applications vol 8 no 6 pp 315ndash3262009

[7] A Romanelli A C S Schifino R Siri G Abal A Auyuanetand R Donangelo ldquoQuantum random walk on the line as aMarkovian processrdquo Physica A Statistical Mechanics and itsApplications vol 338 no 3-4 pp 395ndash405 2004

[8] A Chakrabarti C Lin and N K Jha ldquoDesign of quantumcircuits for randomwalk algorithmsrdquo in Proceedings of the IEEEComputer Society Annual Symposium on VLSI (ISVLSI rsquo12) pp135ndash140 2012

[9] J Duda ldquoFrom maximal entropy random walk to quantumthermodynamicsrdquo Journal of Physics Conference Series vol 361no 1 Article ID 012039 2012

[10] K Rudinger J K Gamble E Bach M Friesen R Joynt andS N Coppersmith ldquoComparing algorithms for graph isomor-phism using discrete-and continuous-time quantum randomwalksrdquo Journal of Computational and Theoretical Nanosciencevol 10 no 7 pp 1653ndash1661 2013

[11] M S Underwood and D L Feder ldquoUniversal quantum com-putation by discontinuous quantum walkrdquo Physical Review AAtomic Molecular and Optical Physics vol 82 no 4 Article ID042304 2010

[12] L Jun Investigations on Quantum Random Walk Search Algo-rithm 2006

[13] N Shenvi J Kempe andK BWhaley ldquoQuantum random-walksearch algorithmrdquo Physical Review A Atomic Molecular andOptical Physics vol 67 no 5 Article ID 052307 2003

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