# A New Nature-inspired Algorithm for Load Balancing

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3 r i e = (t Because 1 e x is a monotone increasing function and between 0 and 1, we choose the function to standardize the execution times as r i E =1 exp[ e ] = 1 exp [ ]; (3 x where E j represents the radial distance from particle A j to the circumference in force-field The radius of the force-field will always be 1 and E j will be in [0, 1] We define the personal utility function for particle (processor A j {A 1,,A n } as r i u =1 E =exp[ ]; (4 x where u j represents the radial distance from the origin to particle A j in the force-field The longer the execution times of a processor, the smaller the processor s personal utility, and the corresponding particle will be closer to the origin in the force-field We define the potential energy function of the force-field as P (t =k 2 ln (2 exp [ E2 ]; (5 2 Theorem 1: In equation 5, if k is very small, the decrease of the potential energy P (t of the force-field will cause a decrease of the processors maximal execution times Proof Supposing that H(t =maxej 2 (t, wehave j [e H(t 2 ] 2 [ E j 2 (t e 2 ] 2 [ne H(t 2 ] 2 (6 Taking the logarithm for both sides of Eq 6 leads to H(t 2 E j 2 (t ln[ e 2 ] H(t+ 2 ln n Because n is constant and k is very small, we have H(t =maxej 2 (t 2 E j 2 (t ln[ e 2 ]=2P (t j The potential energy function P (t of the force-field at time t turns out to represent the maximal execution times among E, j =1,,n So, decreasing P (t amounts to decreasing the processors maximal execution times We define the dynamic equation of subtask r i in the PM model as is reached By the PM algorithm, can be computed and updated in parallel without any information exchange, which is the foundation of the algorithm s parallelism The most important related factor that influences the update and iteration of is the potential energy function P (t According to differential equation theory, a variable s increment to make it minimum is equal to the sum of negative items from related factors differentiating the variable Thus we define the first item of r i as (t +1= λ 1 (8 (t where 0 <λ 1 < 1 Theorem 2: The update and iteration of according to Eq 8 will always cause a decrease of the processors maximal execution times Proof Consider the effect of only the potential energy P (t on (t +1; namely, let (t +1 be λ 1 (t (Eq 8 We determine the increment of the potential energy P (t in the unit time period as follows: P (t = d = λ So, the update and iteration of r i according to Eq 8 will reduce the potential energy P (t, with the intension strength being λ 1 By Theorem 1, the conclusion thus is straightforward The other important related factor that influences the update and iteration of is the utility function u Because u needs to maximize, we use its opposite function E to define the second item of (t +1 as E (t +1= λ 2 (9 (t Theorem 3: The update and iteration of according to Eq 9 will always cause a decrease of the processors execution times Proof Consider the effect of only the execution times function E on (t +1; namely, let (t +1 be λ 1 r λ ij(t Ej(t (t (Eqs 8, 9 Then, the changing rate of the personal execution times of processor A j is equal to E = de j = E j d E j = E j E j [ λ 1 λ 2 ] = E j E j E j [ λ 1 λ 2 ] E j (t +1=r i+ r i (7 The dynamic equation is seen as the PM evolution, which manipulates the update and iteration of until an equilibrium where = [λ 1 E j + λ 2 ] E j 2 291

4 E 2 j exp( = E 2 j E j 0 exp( E2 j Thus E 0 So, the personal execution times of processor A j will decrease In the PM physical model, by the theorem, we can conclude that all particles will try to move as close as possible to the circumference along their own radial orbits Combining Eq 8 and Eq 9, we have (t +1= λ 1 (t λ E 2 (10 (t In order to satisfy the constraints of the LBP, will be dealt with using the following two steps in the parallel computing process Nonnegativity: If min < 0, then let = min Normalization: Let = rij, in order to map all tasks to processors; that is, =1,,,m We can obtain the radial velocity of particle A j along its radial orbit to the circumference of the force-field by the equation v j = de j = E j d = u j d u j (11 When all v j (j =1,,n are equal to 0, the PM model reaches a stable state, and the solution to LBP is obtained The parallel PM algorithm for LBP 0 Input: s i, 1 Initialization: t 0 t, λ 1,λ 2,r i 2 while (v 0 do t t +1 Compute E according to equation 3 Compute r i according to equation 10 r i (t 1 + r i Compute v according to equation 11 If min (t < 0, then (t r i min (t r i rij(t (t At the end, when an equilibrium is reached, = s i is the solution to LBP V CONVERGENCE AND PARAMETERS ANALYSIS A Convergence analysis In this section, we construct a Lyapunov function, which is an energy-related positive definite function We can judge the stability of the PM model by analyzing if the Lyapunov function monotonically decreases with the elapsing time Lyapunov second theorem on stability Consider a function L(X such that L(X > 0 (positive definite; dl(x(t/ < 0 (negative definite Then L(X(t is called a Lyapunov function candidate and X is asymptotically stable in the sense of Lyapunov It is easier to visualize this method of analysis by considering the energy of the PM model If the PM model loses energy over time, then eventually the model must grind to a halt and reach some final resting state This final state is called the stable equilibrium state Theorem 4: If the condition (Eq 12 remains valid, then the PM model will converge to a stable equilibrium state Proof For the physical PM model, we define a Lyapunov function L(r i as L(r i t u j (y r i+2λ 2 dy 0 Obviously, L(r i > 0 de j(t dr i = u j(t dp (t dr i = E de dr i = E j u j r i = λ 1 (t λ E 2 (t dl(t = u j(t = d(t = u j(t [λ E exp( E λ 2 u 2 [ λ E exp( E λ 2 ] = r i+2λ 2 u 2 + λ 2 ] Because uj(t > 0, if the following Eq 12 remains valid, then dl(t/ < 0 E exp( E 2 λ 2 <λ 2 1 (12 292

5 Based on the Lyapunov second theorem on stability, as long as we properly select the parameters λ 1,λ 2 according to Eq 12, the convergence and stability can be guaranteed That is, when t, then all r i (constants B Parameters analysis There are only three parameters in the PM model and algorithm, k in Eq 5 and λ 1,λ 2 in Eq 10 k represents the strength of the gravitational force in the force-field The larger k is, the faster the particles would move away from the origin along their radial orbits; hence, k influences the convergence speed of the PM algorithm As required in Theorem 1, k must be small Usually, 0 <k<1 0 < E j(t exp( E 2 2 exp( E n < 1 According to Eq 12, λ 2 should be much smaller than λ 1 that is, λ 2 < λ 1 n where n is the number of processors in the LBP VI SIMULATIONS Simulations of the LBP verify our PM algorithm s advantages in terms of parallelism and convergence speed to an optimal solution The simulation used these parameters: k =08,λ 1 =09,λ 2 = 09 n Because the parallel computation of r i is the foundation of the algorithm s parallelism, the PM algorithm is scalable When the number of processors (n in the LBP is very large (eg, n is more than 10000, the PM algorithm can still perform well Here we will give the experimental results for the LBP (n = 900 in Figure 2 As shown in Figure 2(a, most of particles are far from the circumference that is, most E j are large, which represent most execution times e j of processors are long In Figure 2(b and (c, by some iterations, particles move away from the origin along their radial orbits, which represents that the execution times of most of the processors are being shortened In Figure 2(d, at the end of the PM algorithm, particles move as close as possible to the circumference in the force-field, which represents that all execution times of processors are balanced and substantially shortened VII CONCLUSION In this paper, we propose the PM model and algorithm for the LBP The PM algorithm is inspired by the physical model of particle dynamics Our future research direction include solving the Load Rebalancing Problem (LRP, where progresses of the computation may dynamically alter the load distribution, and is therefore more difficult than the LBP We will conceive a physical Water Flow model to model the LRP, and its corresponding mathematical model and algorithm Acknowledgment This project is supported in part by the Hong Kong General Research Fund (HKU E REFERENCES [1] Kleinberg G and Tardos E Algorithm Design, Cornell University, Spring 2004 [2] P Bruckner, Scheduling Algorithms, third ed, Springer-Verlag, 2001 [3] M Pinedo, Scheduling: Theory, Algorithms, and Systems, Prentice Hall, 2002 [4] SM Ross, Probability Models for Computer Science, Academic Press, 2002 [5] V Ungureanu, B Melamed, M Katehakis, PG Bradford, Deferred assignment scheduling in cluster-based servers, Cluster Computing 9 (1 (2006 (a t =0 (b t = 300 (d t = 600 (e t = 851 Fig 2 The trajectories of particles using the PM algorithm for the LBP (n =

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