On the average diameter of directed loop networks

Transcription

1 34 2 Vol.34 No Journal on Communications February 203 doi:0.3969/j.issn x L- 4 a b p q 2 L- TP393.0 B X(203) On the average diameter of directed loop networks CHEN Ye-bin, LI Ying 2, ZHENG Xiao, CHEN Tao (. School of Computer Science, Anhui University of Technology, Ma anshan , China; 2. Department of Physics and Engineering, Ma anshan Teacher s College, Ma anshan 24304, China) Abstract: The formula of average diameter about directed double-loop networks was presented according to the relationship between average diameter and four geometric parameters (a, b, p and q) of its minimum distance diagram (L-shaped tile) and a rapid algorithm to compute the average diameter was presented. novel A method to construct the minimum distance diagram (equivalent tree) of triple loop networks was presented, and the relationships between the layer of the equivalent tree and the shortest path of any two points in triple loop networks was studied, and a formula and a algorithm to compute its average diameter was provided. The experimental results show that average diameter approaches to a half of the diameter at the same network, in one infinite family, the diameter must obtain minimum value when average diameter obtains minimum value, but the inverse is not. Studies show that average diameter is the better parameter to measure transmission efficiency about loop networks instead of diameter, so average diameter should become one of important bases to design the optimal networks. Key words: average diameter; diameter; double-loop networks; triple-loop networks; L-shaped tile; shortest path; minimum distance diagram SDH synchronous digital hierarchy KJ202A262, KJ203A058 20DFB6 530 Foundation Items: The National Science Fund of Distinguished Young Scholars(60033) The Main Pro ject of the Education Department of Anhui Province (KJ202A262, KJ203A058); The International Science and Technology Cooperation Projects (20DFB6530)

2 2 39 DHARMASENA [0] [] 2009 avg d(n;r,s)= n d i /N i n- n- d N- N N 20 [3] ) 2) (N; s, s 2, s 3 ) 2 2 3) 2 (a) s r <s <N r, s [] L- Wong (b) L- a b p q 4 Coppersmith [] L a b p q a,b 2 0 p<a 0 Chen [2,3] q<b L(N; a,b,p,q) [] 3 L- N r s gcd(n;r,s)= [4~6] [2] G(N;r,s) G(N;r,s) N 4 r [7] 2007 N N GOMEZ [8] G(N;r,s) r 0 r < s < N s r N- CHEN [9] i =

3 40 34 G(2;2,5) L- 2 ) ab=n p=q=0 d (N;r,s) 2 0 N- d(n;r,s)=max{d u,v, (0 u? v N - )} d u,v 2 0 u v N- 3 ab d (N;r,s)= (a + b 2) N d (N;r,s)= N 2 di, j (0 N j = N = (a + b 2) i? j N- ) d i,j i j N 2 2) ab<n 0 a b 2009 [2] f 0 p q d L- O(N 2 ) d (N; r, s)= (j d ) L- N G(N; r, s) N r s ab pq L- 4 a b p q = (a + b 2) (p + q 2) pq(a +b p q) N ab pq = (a + b 2) (2a + 2b p q 2) N 2 2 L- 2.2 [8] d(n; r, s)=max{a+b- q- 2, a+b- p- 2} G(N;r,s) L a, b, p, d (N; r, s) L- q) 4 2 L- 4 L- 4 L (N; a, b, p, q) d (N;r,s) 2 N ) ab=n d (N;r,s)= (a + b 2) N 2 2) ab<n ab, )= ( 2) pq d (N;r s a +b ( 2a+ 2b p q 2) N 2 2 L- L- 2 ab=n L (N; a, b, p, q) ab < N L(N; a, b, p, q) L G(N;r,s) u v 0 v- u L(N; a,b,p,q) 0 G(N;r,s) r= 2 ) l - =N l 0 0 l 0 N rl 0 +s 0(mod N) l i q i l i - 2 =q i l i - +l i (0 l i l i -, i m+, l m+ =0) 2) U - =0 U 0 = U i+ =q i+ U i +U i- (0 i

4 2 4 m) l i i G(N; s,s 2,s3) N s s 2 s 3 l m+ =0 U i i U m+ =N N s? s 2? s 3 < N N l l l l s s 2 s 3 v 3 v 0= m + < m < L < 0 < 8 U m + U m U 0 U v s (mod N) v v s 2 (mod N) v v s 3 (mod N) [+s ] [+s 2 ] [+s 3 ] lu 3) u > v N s s 2 s 3 U u l U u u = L- l + U u + u [5] 4) L- 4 a=l u - vl u+ [6] b=u u +(v+)u u+ p=l u - (v+) l u+ q= U u - vu u+ 5) d (N;r,s) L- O( N ) 2 ) a= min{j ks=jr(mod N), j > k 0, j=2, N- } q={k ks=ar(mod N), k 0} j 2 N- k j- ks(mod N)=jr(mod N) a=j q= k 2) b= min{k ks=jr(mod N), k j 0, k=2, N- } p={j bs=jr(mod N), j 0} k 2 N- j k- ks(mod N)=jr(mod N) b=k p=j 3) d (N;r,s) O(2 N ) N N G(N;s,s 2,s 3 ) s s 2 0 s <s 2 <N s 3 s 2 N- s = s 3 =N- s 2 s s s 3 )=max{d u,v, (0 u? v N- )} d u,v [4] 0,,2,,N [8] 2 G(2;,3,4) 2 d(n; s, s 2,

5 42 34 u v 6 x y 0 y - x(x<y) N d (N;s,s 2,s 3 )= d i, j (0 i?j N- ) N j= 0 0 d i,j i j 5 6 d (N;s,s 2,s 3 )=max{d 0,v,(0<v N- )} d (N; N 2 2 s,s 2,s 3 )= d ( i N - ) N 0,i 2 i= T (N;s,s 2,s 3 ) T(N;s,s 2,s 3 ) ( 0) ) 0 p=m[+s ]+n[+s 2 ]+k[+s 3 ] (m 0 0, n 0, k 0 m n k 0) 2) [+ s ] [+s 2 ] [+s 3 ] v(v? 0) s s 2 s ) i i v [+s ] [+s 2 ] [+s 3 ] v+ s (mod ( 0) N) v+s 2 (mod N) v+s 3 (mod N) v(v? 0) v(v? i+ v =ms +ns 2 +ks 3 (mod N) (m 0, n 0, i+ k 0) v(v? 4) 3) p, p=m[+s ]+n [+s 2 ]+k [+s 3 ] (m 0, n 0, k 0 m n k 0) 7 2 T(N;s,s 2,s 3 ) G(2;,3,4) T(2;,3,4) p p=m[+s ]+n [+s 2 ]+k [+s 3 ] 3 v(v? 0) L p =m+n+k L p 2 G(N;s,s 2,s 3 ) 3 T(2;,3,4) 8 G(N;s,s 2,s 3 ) 0 v(v? 0) T(N;s,s 2,s 3 ) p=m[+s ]+n[+s 2 ]+k[+s 3 ] (m 0, n 0, k 0 m n k 0) p v p L p =m+n+k p ' p' v L p ' <L p v T(N;s,s 2,s 3 ) v 2 v 7 G(N;s,s 2,s 3 ) T(N;s,s 2,s 3 ) N s s 2 s 3 gcd(n, s, p v L p s 2, s 3 ) = N s s 2 s 3 4 v

6 T(N;s,s 2,s 3 ) v l L p 2 T(N;s,s 2,s 3 ) v L p v v=ms +ns 2 +ks 3 (mod layer= sum=3 2) l v v+s (mod N) v+s 2 (mod N) v+s 3 (mod N) 3 N) (m 0, n 0, k 0 m n k 0) l l 2 v L p =m+n+k 7 l l T(N;s,s 2,s 3 ) v=ms +ns 2 +ks 3 (mod N) 3) l 2 m+n+k v l 2 k layer ++ v sum=sum+layer*k l 2 2) l=m+n+k v l l 2 L p 3 T(N;s,s 2,s 3 ) G(N; s,s 2,s 3 ) 2 T(N;s,s 2,s 3 ) 4) 4 d (N;s,s 2,s 3 ) 4 v l L p 974 Wong C K Coppersmith D G(N;r,s) lb= 3N - 2 [], T(N;s,s 2,s 3 ) G(N;s,s 2,s 3 ) x x G(N;r,s) d(n;r,s)= 3N d (N;s,s 2,s 3 ) T(N;s,s 2,s 3 ) l G(N;r,s) L n l d (N;s,s 2,s 3 )= (ln l ) ( N l = l L) 3 T(N;s,s 2,s 3 ) 0 0 N- T(N;s,s 2,s 3 ) l=,2,3,, L L Java SQL Server 2005 n l 0 L (ln l ) d (N;s,s 2,s 3 )= l = ) N 4~ 000 L (ln l ) N l = 4~ 000 N ~ i+00 G(N;s,s 2,s 3 ) 2) ) 3 l l l 2 l 0 l 3 s r ~N/2 s r~n- s s 2 ~N- 2 s 3 s 2 s 3 l 2 N-

7 44 34 G(50;,s) 3) N lb= 3N - 2= = r s s s 2 s 3 d d G(50;,4) G(50;,5) G (50;,7) G G(50;,s),9) G(50;,22) G(50;,23) G(50;,s,49) 2 d 2 (50) d 2 (50) d 3 (50) d 3 (50) 5.9 G(50;,5) G(50;,9) d(50;,s) d (50;,s) d(50;,s,49) d (50;,s,49) G(50;,22) 4 5 G(50;,s) G(50;,s,49) d (50;,7)- s d 2 (5 0) d 2 (5 0) d 3 (50 ) d 3 (50 ) d (50;,5)= = G(N;s, s ,s 3 ) 4 N 86 d(n;s,s 2,s 3 ) N - N> G(50;,s,49) lb= 3 3N - =8 G(50;,8,49) G(50;, 0,49) G(50;,2,49) G(50;,4,49) G(50;,20,49) G(50;, 22,49) 8 d (50;,20,49)=4.4 4 G(50;3,s) 5 G(50;,s,49) 4 5 G(50;,s) s 2~49 G(50;,s,49) s 2~48 4 G(50;,s)

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10 34 2 Vol.34 No Journal on Communications February 203 doi:0.3969/j.issn x ,2,2,2, TP39 A X(203) Review on learning mobile user preferences for mobile network services MENG Xiang-wu,2, SHI Yan-cui,2, WANG Li-cai,2, ZHANG Yu-jie,2 (. Beijing Key Laboratory of Intelligent Telecommunica ions Software and Multimedia, Beijing University of Posts and Telecommunications, Bei ing 00876, China; 2. School of Computer Science, Beijing University of Posts and Telecommunications, Beijing 00876, China) Abstract: Recently, in order to alleviate the problem of mobile information overload, the learning of mobile user preferences has become a new hotspot of research in the personalized services. The research progress of learning mobile user preferences were summaried in recent years. The representation, acquisition method, adaptive learni evaluation about learning mobile user preferences were contrasted and analyzed. Finally, some future directions and development trends of learning mobile user preferences in the mobile network service were pointed out. Key words: mobile network services; mobile user preferences; preference acquisition; preference adaptation; personalized service [2] [3] [] ( ) Foundation Items: The National Natural Science Foundation of China ( ); The Program of the Co-construction with Beijing Municipal Commission of Education