Tracking Mobile Users in Wiewless Communications Networks

1. Tracking Mobile Users in Wiewless Communications Networks Amotz Bar-Noy and Ilan Kessler IEEE Trans. On Info. Theo. VOL.39,NO.6,Nov 1993,p1877-1886 Speaker : Cheng-Chung,Li

2. Outline • Introduction • Problem Defined • Weighted Graph • Lines • Trees • Arbitrary Graphs(Approx.Algorithms) • Discussion

3. Introduction • A important issue in wireless networks is the design and analysis of tracking mobile users • The users are mobile and could be anywhere within network area • In this paper , the issue considered is the cost of utilizing the wireless links for the tracking mobile users in cellular networks • Update • Find

4. Two extreme strategies • Always-Update • Never-Update • But how to minimized the total (update+find) cost ? • Increasing one cost leads to a decrease in the other one

5. We will construct a model called Mobility Graph and defined the problem called reporting center problem , and try to solve above question

6. Problem Defined • The mobility graph G of the network is the graph in which each vertex corresponds to a different cell , and two vertices are connected by an edge if and only if the corresponding cells overlap • Each vertex i of the mobility graph has a weight wi > 0

7. 1 2 3 cell Mobility graph 6 5 4 9 7 8 2,4,6,8 are reporting centers

8. Let I be a set of vertices , referred to as centers . The vicinity of center v is the set of all vertices not in I that are reachable from v by a path containing no centers • By definition , the vicinity of center v includes v • The weight of I is w(I)=iIwi,and the size of the largest vicinity in the graph is denoted by z(I)

9. The Reporting Centers Problem-C(G,Z):Given a weighted graph G and integer Z , select a set of centers S such that z(S)Z and w(S)w(S’) for all S’ such that z(S’)Z • We are so greedy ! We want to find the min.(update+find) solution

10. An import special case of C(G,Z) is the case un which all the weighted are equal to one – But we just concert weighted graph in this report • C(G,Z) is an NP-Complete problem for any Z2

11. Weighted Graph-Lines • Given an integer 1<Z<n , the goal is to find a set of centers S such that the following hold • (a)The largest vicinity contains at most Z vertices • (b)w(S)=minI{1,-,n}{w(I)|(a)holds for I} • We denote this problem by C(n,Z) 1 2 n wn w1 w2

12. The modified problem is to find for a given integer 0kZ a set of centers Sk , such that the following hold • (a)the set Sk contains the vertex n-k and does not contain the vertices n-k+1,…,n • (b)the largest vicinity contains at most Z vertices • (c)w(Sk)= minI{1,-,n}{w(I)|(a)and(b)holds for I}. • We denote the modified problem by Ck(N,Z) 1 2 n-k n n-k+1

13. Clearly , at least one of the sets {Sk,k=0,1,…,Z-1} is a solution to C(n,Z) • Let k’ be an index for which w(Sk’)=min0k<Zw(Sk) . Then Sk’ is a solution to C(n,Z)

14. For every i=0,…,Z-1 and j=1,…,n-1 , letSi(j) to be a solution to Ci(j,Z) • For every 0k<Z , let rk be an index for which w(Srk(n-k-1))=min0r<Z-kw(Sr(n-k-1)) • Then Sk(n)=(n-k)Srk(n-k-1) is a solution to Ck(n,Z) 2 1 n-k n n-Z n-k-1 n-k+1

15. For all 0iZ-1 and Z-i <jn-iw(Si(i+j))=min{wj+w(SZ-i-1(j-1)),w(Si+1(j+i+1))} • The above algorithms can solve C(n,Z) in O(nZ) time Z values 1 2 j j+i j-Z+1 j-1 j+1

16. Weighted Graph-Trees • We describe the simple binary tree T first • For any set of vertices I in the tree T , consider the connected components that are obtained when all vertices of I are removed from T • We denote by a(I) the size of the connected component that contains the root of T , i.e. a(i)=0 if I contains the root

17. The modified problem is to find for given nonnegative integers k and l such that k+lZ-1 , a set of centers Slk such that the following hold • a.a(Slk)=k • b.the largest vicinity contains at most Z vertices • c.the largest vicinity that contains the root has at most Z-l vertices (l:external vertices) • D.w(Slk)=minIT{w(T)|a,b,c hold for I}

18. a.For every lZ-1 , let i’ and j’ be indexes for which w(S0i’(TL))+w(S0j’(TR))=mini+j+lZ-1 w(S0i(TL))+w(S0j(TR))Then Sl0(T)={r}S0i’(TL)S0j’(TR) is a solution to Cl0(T,Z) • b.For every k>0 and l 0 such that k+lZ-1 , let i’ and j’ be indexes for which w(S1+l+j’i’(TL))+w(S1+l+i’j’(TR))=mini+j+1=kw(S1+l+ji(TL))+wS1+l+ij(TR))Then Slk(T)=S1+l+j’i’(TL) S1+l+i’j’(TR) is a solution to Clk(T,Z)

19. Each min operation in the equation a. can be done by O(Z2) operations , and it computed for at most Z values of k • Each min operation in the equation b. can be done by O(Z) operations , and it computed for at most Z2 values of k and l • So the above solution for C(T,Z) is O(nZ3) • If T is an arbitrary tree , the solutions of T are computed after computing the solutions for all subtrees of T , time complexity is sill O(nZ3)

20. Weighted Graph-Arbitrary Graphs • Initially, all vertices are designated as centers • Then the centers are checked in an order of decreasing weights , and if making a center a noncenter vertex does not create a vicinity larger than Z , then this center is made a noncenter vertex

21. Approx.Algorithms • Let the vertices of the graph be denoted by 1,…,n and w.l.og. Assume that w1w2…wn • 1.S={1,…,n} • 2.x=1 • 3.If z(S-x)Z then S=S-{x} • 4.x=x+1 • 5.If xn the go to step 3 • 6.Return the set S

22. Two centers are said to be siblings if their vicinities are not disjoint • Let  is the maximum degree of the graph • Each center has at most Z sibilings

23. For any greedy center xS, there exists an opt. center yR such that wywx , and x is either in the vicinity of y or in the vicinity of a sibling of y (where the vicinities are with respect tp the opt. set R) • We can prove it by contradiction

24. xSwxZ2yRwy • Let x be a greedy center and let y(x) to be an opt. center such that wy(x)x and x is either in the vicinity of y(x) or x is in the vicinity of a sibling of y(x) • So xSwxxSwy(x) • But the largest vicinity has at most Z vertices , that each opt. center appears at most Z2 times • Therefore xSwxZ2yRwy

25. Discussion • As for weighted graphs , important special cases other than trees could be almost-trees and planar graphs • Also we believe there exists a better approx. algorithm for arbitrary graphs