A Linearithmic Time Algorithm for a Shortest Vector Problem in Compute-and-Forward Design

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1 A Linearithmic Time Algorithm for a Shortest Vector Problem in Compute-and-Forward Design Jing Wen and Xiao-Wen Chang ENS de Lyon, LIP, CNRS, ENS de Lyon, Inria, UCBL), Lyon 69007, France, jwen@math.mcll.ca School of Computer Science, McGill University, Montreal, H3A A7, Canada, chang@cs.mcll.ca Abstract We modify the algorithm proposed by Sahraei et al. in 05, resulting an algorithm with expected complexity of On log n) arithmetic operations to solve a special shortest vector problem arising in computer-and-forward design, n is the dimension of the channel vector. This algorithm is more efficient than the best known algorithms with proved complexity. Index Terms Shortest vector problem, Compute-and-forward, linearithmic time algorithm. I. INTRODUCTION In this paper, we consider solving the following quadratic integer programg problem arising in compute-and-forward CF) design: P a Z n \0} at Ga, G = I + P h hht, ) P, a constant, is the transmission power, h R n is a random channel vector following the normal distribution N 0, I) and h = h T h) /. In relay networks, CF is a promising relaying strategy that can offer higher rates than traditional ones such as amplifyand-forward and decode-and-forward, especially in the moderate SNR reme. To find the optimal coefficient vector that maximizes the computation rate at a relay in CF scheme, we need to solve ). Is easy to verify that the matrix G in ) is symmetric positive definite, so it has the Cholesky factorization G = R T R, R is an upper triangular matrix. Then we can rewrite ) as a shortest vector problem SVP): a Z n \0} Ra. ) The general SVP arises in many applications, including cryptography and communications, and there are different algorithms to solve it see, e.g., [], []). Although it has not been proved that the general SVP is NP-hard, it was shown in [3] that the SVP is NP-hard for randomized reductions. However, since the SVP ) is special due to the structure of G, efficient algorithms can be designed to solve it. Various methods have been proposed for solving ), including the branch-and-bound algorithm [4] which did not This work was supported by NSERC of Canada grant 79-, Programme Avenir Lyon Saint-Etienne de l Université de Lyon in the framework of the programme Inverstissements d Avenir ANR--IDEX-0007) and by ANR through the HPAC project under Grant ANR BS0 03. use the properties of G), the algorithm proposed in [5] and its improvement [6], which has the best known proved expected complexity of On.5 log n) In this paper, the complexity is measured by the number of arithmetic operations.), the sphere decoding based algorithm ven in [7], whose expected complexity is approximately On.5 ). There are also some suboptimal algorithms, see [8], [9] and [0]. In this paper, we will modify the algorithm proposed in [6] for solving ) to reduce the expected complexity to On log n). The rest of the paper is organized as follows. In Section II, we review the algorithms proposed in [6] and [7] for solving ). Then, in Section III, we propose a new algorithm. To compare these three algorithms computationally, we ve numerical results in Section IV. Finally, conclusions are ven in Section V. Notation. We use R n and Z n to denote the spaces of the n dimensional column real vectors and integer vectors, respectively, and R m n to denote the spaces of the m n real matrices. Boldface lowercase letters denote column vectors and boldface uppercase letters denote matrices. For a vector t, t denotes its l -norm. We use e k to denote the k-th column of an n n identity matrix I, e to denote the n dimensional vector with all of its elements being one and 0 denote the n dimensional zero vector. For x R, we use x and x respectively denote the smallesnteger thas not smaller than x and the largesnteger thas not larger than x. For t R n, we use t to denote its nearesnteger vector, i.e., each entry of s rounded to its nearesnteger. For a set Φ, we use Φ to denote its cardinality. II. EXISTING ALGORITHMS In this section, we review the algorithms proposed in [6] and [7] for solving ). A. The Algorithm of Sahraei and Gastpar In this subsection, we review the algorithm of Sahraei and Gastpar proposed in [6], which has the complexity Onψ lognψ)), ψ = + P h. 3) We will show that the expected complexity is On.5 log n) when h, as is assumed in computer-and-forward design,

2 follows the normal distribution N 0, I). For the sake of convenience, we will refer to the algorithm as Algorithm SG. Given h, we assume is a nonzero vector. Otherwise the problem ) becomes trivial. We first simplify notation as [6] does. Let v = P h h, α = h + P h. 4) Then v = and we can rewrite ) as a Z n \0} at Ga, G = I αvv T. 5) The algorithm in [6] is based on the following two theorems which were ven in [] and [5], respectively. Theorem : The solution a to the SVP problem 5) satisfies a ψ, ψ is defined in 3). Theorem : The solution a to 5) is a standard unit vector, up to a sign, i.e., a = ±e i for some i,..., n}, or satisfies for some x R, leading to a e < vx < a + e 6) a = vx. 7) Define ax) = vx, then for any v i 0, a i x) = v i x is a piecewise constant function of x and so is the objective function fax)) = ax) T Gax). Thus, fax)) can be represented as p k, if ξ k < x < ξ k+, k =,,... fax)) =, 8) q k, if x = ξ k, k =,,... ξ k are sorted real numbers denoting the discontinuity points of fax)). By 8), fax)) is a constant for x ξ k, ξ k+ ). Thus, by 6), fa ) = f k=,,... a ξk + ξ ) ) k+. 9) To reduce the computational cost, [6] looks at only part of the discontinuity points. Is easy to see that the discontinuity points of a i x) are x = c v i v i 0) for any c Z, which are also the discontinuity points of the objective function fax)). Notice thaf a is a solution, then a is also a solution. This fact was used in [] to reduce the search cost.) Using this fact, [6] just considers only positive discontinuity points, i.e., only positive candidates for c are considered. Furthermore, from Theorem, v i x ψ. Thus one needs only to consider those c satisfying 0 < c ψ + / this bound was ven in [6] and is easy to see that can be replaced by a slightly tighter bound ψ + /). Therefore, if 6) holds, from 9), we have fa ) = f ξ k,ξ k+ Ψ a ξk + ξ ) ) k+, 0) Ψ = n Ψ i, ) vi = 0 Ψ i = c 0 < c ψ +, c Z} v i 0. ) v i The algorithm proposed in [6] for solving 5) calculates the set Ψ and sorts its elements in increasing order and then computes the right hand side of 0), and then compares it with i f±e i ), which is equal to i αvi ), to get the solution. By 5), for a Z n, fa) = a T Ga = a i α a i v i ). According to [6], since the discontinuity points of f are sorted and at each step only one of the a i s change, n a i and α n a iv i ) can be updated in constant time. Therefore, fa) can also be calculated in constant time. Here we make a remark. Actually different Ψ i may have the same elements. But they can be regarded as different quantities when fa) is updated. In order to remember which a i needs to be updated at each step, a label is assigned to every element of Ψ to indicate which Ψ j it orinally belonged to. In the new algorithm proposed in Section III, we will ve more details about this idea. Now we describe the complexity analysis ven in [6] for the algorithm. By ), the number of elements of Ψ i is upper bounded by ψ +, so the number of elements of Ψ is upper bounded by n ψ + ) note thaf v i 0 for i n, this is the exact number of elements). From the above analysis, the complexity of the algorithm is detered by the sorting step, which has the complexity of Onψ lognψ)), ψ is defined in 3). In the following, we derive the expected complexity of Algorithm SG when h N 0, I) by following [7]. Since h N 0, I), h χ n). Therefore, E[ h ] = n. Since + P x is a concave function of x, by Jensen s Inequality, [ ] E [ψ] = E + P h + np. 3) Thus, the expected complexity of Algorithm SG is On.5 log n). B. The Algorithm of Wen, Zhou, Mow and Chang In this subsection, we review the algorithm of Wen et al ven in [7], an improvement of the earlier version ven in []. Its complexity is approximated by Onlog n+ψ)) based on the Gaussian heuristic, ψ is defined in 3). By 3), the expected complexity is approximately On.5 ) when h N 0, I). For the sake of convenience, we will refer to the algorithm as Algorithm WZMC. Again we want to simplify the matrix G in ). Define P t = + P h h = αv. 4)

3 Then, ) can be rewritten as a Z n \0} at Ga, G = I tt T. 5) Since h 0, t 0. Obviously, if a is a solution to 5), then so is a. Thus, for simplicity, only the solution a such that t T a 0 was considered in [] and [7]. We also use this restriction throughout the rest of this paper. In [7], 5) is first transformed to the standard form of the SVP ) by finding the Cholesky factor R of G i.e., G = R T R) based on the following theorem. Theorem 3: The Cholesky factor R of G in 5) is ven by r ij = g i, j = i n t j g i g i, i < j n, g 0 = and for i n, g i = i k= t k. Instead of forg the whole R explicitly, only the diagonal entries of R were calculated, so is easy to check that the complexity of this step is only On). It was showed in [4] thaf t t... t n 0, 6) then there exists a solution a to 5) satisfying a a... a n 0. 7) Given t, if < 0 for some i, we can change it to without channg anything else. To have the order 6), we can permute the entries of t. This step costs On log n). To decrease the computational cost, the following n + ) dimensional vector p was introduced in [7]: Define p n+ = 0, p i = p i+ + a i, i = n, n,...,. d n = 0, d i = r ii j=i+ r ij a j, i = n,...,. Then, by Theorem 3, d i = t i t j a j = p i+, i = n,...,. Thus, Ra = j=i+ n r iia i d i ) = r ii a i p i+ /g i ). The Schnorr-Euchner search algorithm [3] was modified to search the optimal solution satisfying 7) within the ellipsoid: riia i p i+ / ) < fe ) = t. If no nonzero integer poins found, e is the solution. The cost of the search process was approximately Onψ) based on the Gaussian heuristic. Thus, by the above analysis and 3), the expected complexity of Algorithm WZMC is approximated by On.5 ) when h N 0, I). III. A MODIFIED ALGORITHM In this section, we modify the SG algorithm, resulting an algorithm with complexity of O n + n, φ}φ) logn + n, φ}φ) ), φ = + P h max i n h i ). 8) By 3) and 3), the expected complexity is On log n) when h N 0, I). Recall that Algorithm SG checks some discontinuity points of the objective function to find the optimal solution. The main idea of our modified algorithm is to reduce the number of discontinuity points to be checked. In the following, we introduce a theorem, which can significantly reduce the number of discontinuity points to be checked in finding the optimal one. Theorem 4: Suppose that t satisfies t =... = t p > t p+... t q > t q+ =... = t n = 0, 9) integers p and q satisfying p, q n we define t n+ = 0 when p = n or q = n). Then the solution a to 5) satisfies either or for some x R satisfying a = ±e k, k p, a = ± tx t x µ, 0) t µ = i q i t Proof. Note that for k,..., p}, ) +. ) f±e i) = i n i n ) = t k = f±e k ). Is possible that ±e k for k =,..., p are solutions to 5). In the following proof we assume they are not. By 4) and Theorem, there exists x R such that the solution can be written as a = tx. Note thaf tx is a solution, so is tx. Thus we can just assume that x here is positive. Then by 9) we have a =... = a p a p+... a q a q+ =... = a n = 0. ) We must have a, otherwise a = e, contradicting with our assumption. Thus, t x a. Therefore, the firsnequality in 0) holds. In the following, we show that the second inequality in 0) holds. Since e is not an optimal solution, fa ) < fe ), i.e., a t T a ) < t.

4 Therefore, by the Cauchy-Schwarz inequality, By ), for i =,..., q, a < t t. 3) a ia i ). 4) Then, using the fact that a = tx and 3) and 4), we have x a i + t i t +. Since the aforementioned equality holds for all i =,..., q, the second inequality in 0) holds, completing the proof. Like Algorithm WZMC, our new algorithm first performs a transformation on n 5) such that 9) holds, costing On log n). We define ax) = tx cf. Section II-A). Then for any i =,..., q, q is defined in 9)), a i x) = x is a piecewise constant function of x and its discontinuity points are x = c c Z. To find the optimal discrete points, we need consider only a finite subset of those discrete points. In fact, by Theorem 4, we need to consider only those x = c/, c satisfies Thus, we define Φ i = t c µ, Φ = c Z q Φ i, 5) c, i ) t c µ, c Z }. 6) Then the optimal discontinuity point and its position in the vector ax) must be in Φ. Like in [6], we sort the first elements of the members of Φ in increasing order, then by 6), only one entry of a increase for each elemenn Φ note thaf some of the entries of t are the same, then the corresponding Φ i have the same x, but we can regard them as different quantities to update a and the corresponding fax))). By following [6], we can compute fax)) for each x by constant time. Specifically, denote T = n a i and T = n a i, then fax)) = T T. We start from T = T = 0 and a = 0, and for each x, i) Φ, we update a by setting a i = a i +, then we update T by setting T = T +a i, update T by setting T = T +, and update f by setting f = T T. During the enumeration process, we only keep the a which imizes f and the corresponding f. If f < t, then the final a is a, otherwise, a = e. By the above analysis, the algorithm can be summarized in Algorithm. Before analyzing the complexity of Algorithm, we look into the number of discontinuous points needed to be checked Algorithm Modified Algorithm Input: Channel vector h and transmission power P Output: a Initialization: : calculate t by 4) : perform a signed permutation on t such that the new t satisfies 9) 3: calculate µ by ) 4: let Φ =, f = t, a = e Phase : 5: for all i,..., q} do 6: for all c / Z such that /t ) c µ do 7: calculate x = c/ 8: Φ = Φ x, i) 9: end for 0: end for Phase : : sort Φ by the first element of the members in an increasing order : set T = 0, T = 0 and a = 0. 3: for every x, i) Φ do 4: a i = a i + 5: T = T + a i 6: T = T + 7: f = T T 8: if f < f then 9: set a = a 0: set f = f : end if : end for 3: return sign permuted a by Algorithm and Algorithm SG. By ), 4) and 8), for i,..., q}, µ t i t i = + P h max h j ) = φ, i j the max is involved because t is the largest among all t i after the permutation of t see 6)). Thus, by 6), Φ i φ/ i +. 7) By ), for i,..., n}, Ψ i = ψ +, 8) ψ is defined in 3). Thus, from 7), 8), 3) and 8), it follows that Φ i φ/ i + Ψ i <, i =,..., q. ψ + Note that φ can be arbitrarily smaller than ψ see 3) and 8)). Also when i is big enough, φ/ i = 0. Thus the modified

5 algorithm can significantly reduce the number of discontinuity points to be checked. Now we study the complexity of Algorithm. Clearly, ψ/ i = 0 when i > φ. Then, with k = q, φ } and by 7), we have Φ = q Φ i φ k i + q φ k = kφ + q n, φ}φ + n. i i x dx + q Thus the complexity of line of Algorithm is On + n, φ}φ) logn + n, φ}φ). Is easy to see it is actually the complexity of the whole algorithm. Then it follows from 3), 8) and 3) that the expected complexity of Algorithm is On log n) when h N 0, I). IV. NUMERICAL SIMULATIONS In this section, we present numerical results to compare the efficiency of our modified method, i.e., Algorithm denoted by Modified ) with those in [6] denoted by SG ) and [7] denoted by WZMC ). We do not compare Algorithm with the branch-and-bound algorithm in [4] since numerical tests in [7] show that the algorithm in [5] is faster, while the algorithm in [6] is an improved version of than [5]. All the numerical tests were done by MATLAB 00a on a laptop with IntelR) CoreTM) i5-5300u CPU@.30GHz. We set the dimension n of h being 0, 0,..., 80 and h N 0, I). For each ven n and P, we randomly generate 000 realizations of h and apply the three algorithms to solve ). Figure shows the total CPU time for P =. CPU Time SG Modified WZMC Dimension n Fig.. Total CPU time versus n for P = From Figure, we can see that the total CPU time for the modified method and WZMC are very close, So for comparisons we also ve Table I to show the total CPU time for P = 50. From Figure and Table I, we can see that our proposed algorithm is much faster than SG, and is also faster than WZMC if both n and P are not very large which means the new algorithm and WZMC have advantages in different n TABLE I TOTAL CPU TIME IN SECONDS VERSUS n FOR P = SG Modified WZMC settings, so both of them are useful. Algorithm has another advantage, i.e., its complexity can be rigorously analyzed, while the complexity for WZMC was based on Gaussian heuristic. V. CONCLUSIONS In this paper, we proposed an algorithm with the expected complexity of On log n) for a shortest vector problem arising in compute-and-forward network coding design, by modifying the algorithm in [6]. The complexity is lower than On.5 log n), the expected complexity of the latest algorithm in [6], and On.5 ), the approximated expected complexity of the algorithm proposed in [7]. Simulation results showed that the new algorithm is much faster than that ven in [6]. REFERENCES [] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, Closest point search in lattices, IEEE Transactions on Information Theory, vol. 48, no. 8, pp. 0 4, 00. [] G. Hanrot, X. Xavier Pujol, and D. Stehlé, Algorithms for the shortest and closest lattice vector problems, in Proceedings of the third international conference on coding and cryptology IWCC ). Springer- Verlag Berlin, Heidelberg, 0, pp [3] M. Ajtai, The shortest vector problem in l is np-hard for randomized reductions, in Proceedings of the thirtieth annual ACM symposium on Theory of computing. ACM, 998, pp [4] J. Richter, C. Scheunert, and E. Jorswieck, An efficient branch-andbound algorithm for compute-and-forward, in IEEE 3rd International Symposium on Personal Indoor and Mobile Radio Communications PIMRC ), Sept 0, pp [5] S. Sahraei and M. Gastpar, Compute-and-forward: Finding the best equation, in 04 5nd Annual Allerton Conference on Communication, Control, and Computing Allerton). IEEE, 04, pp [6], Polynomially solvable instances of the shortest and closest vector problems with applications to compute-and-forward, arxiv preprint arxiv: , 05. [7] J. Wen, B. Zhou, W. Mow, and X.-W. Chang, An efficient algorithm for optimally solving a shortest vector problem in compute-and-forward design, submitted to IEEE Transactions on Wireless Communications. [8] A. Sakzad, E. Viterbo, Y. Hong, and J. Boutros, On the ergodic rate for compute-and-forward, in 0 International Symposium on Network Coding NetCod), 0, pp [9] B. Zhou and W. Mow, A quadratic programg relaxation approach to compute-and-forward network coding design, in The 04 IEEE International Symposium on Information Theory ISIT 04), 04, pp [0] B. Zhou, J. Wen, and W. Mow, A quadratic programg relaxation approach to compute-and-forward network coding design, submitted to IEEE Transactions on Communications. [] B. Nazer and M. Gastpar, Compute-and-forward: Harnessing interference through structured codes, IEEE Transactions on Information Theory, vol. 57, no. 0, pp , 0. [] J. Wen, B. Zhou, W. Mow, and X.-W. Chang, Compute-and-Forward design based on improved sphere decoding, in 05 IEEE International Conference on Communications, ICC 05, London, United Kingdom, June 8-, 05, 05, pp [3] C. Schnorr and M. Euchner, Lattice basis reduction: improved practical algorithms and solving subset sum problems, Mathematical Programg, vol. 66, pp. 8 9, 994.