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This paper explores the design and analysis of tracking mobile users in wireless networks. The cost of utilizing wireless links for tracking mobile users in cellular networks is considered, and strategies to minimize the total cost are discussed. The paper introduces the Mobility Graph model and the reporting center problem for solving the cost optimization.
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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
Outline • Introduction • Problem Defined • Weighted Graph • Lines • Trees • Arbitrary Graphs(Approx.Algorithms) • Discussion
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
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
We will construct a model called Mobility Graph and defined the problem called reporting center problem , and try to solve above question
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
1 2 3 cell Mobility graph 6 5 4 9 7 8 2,4,6,8 are reporting centers
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)=iIwi,and the size of the largest vicinity in the graph is denoted by z(I)
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
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 Z2
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
The modified problem is to find for a given integer 0kZ 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
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’)=min0k<Zw(Sk) . Then Sk’ is a solution to C(n,Z)
For every i=0,…,Z-1 and j=1,…,n-1 , letSi(j) to be a solution to Ci(j,Z) • For every 0k<Z , let rk be an index for which w(Srk(n-k-1))=min0r<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
For all 0iZ-1 and Z-i <jn-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
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
The modified problem is to find for given nonnegative integers k and l such that k+lZ-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)=minIT{w(T)|a,b,c hold for I}
a.For every lZ-1 , let i’ and j’ be indexes for which w(S0i’(TL))+w(S0j’(TR))=mini+j+lZ-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+lZ-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)
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)
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
Approx.Algorithms • Let the vertices of the graph be denoted by 1,…,n and w.l.og. Assume that w1w2…wn • 1.S={1,…,n} • 2.x=1 • 3.If z(S-x)Z then S=S-{x} • 4.x=x+1 • 5.If xn the go to step 3 • 6.Return the set S
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
For any greedy center xS, there exists an opt. center yR such that wywx , 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
xSwxZ2yRwy • 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 xSwxxSwy(x) • But the largest vicinity has at most Z vertices , that each opt. center appears at most Z2 times • Therefore xSwxZ2yRwy
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