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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)

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Tracking mobile users in wiewless communications networks

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
Outline

  • Introduction

  • Problem Defined

  • Weighted Graph

    • Lines

    • Trees

    • Arbitrary Graphs(Approx.Algorithms)

  • Discussion


Introduction
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



Problem defined
Problem Defined the problem called reporting center problem , and try to solve above question

  • 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 the problem called reporting center problem , and try to solve above question

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 the problem called reporting center problem , and try to solve above questionvicinity 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)


  • The the problem called reporting center problem , and try to solve above questionReporting 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



Weighted graph lines
Weighted Graph-Lines the weighted are equal to one

  • 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 0 the weighted are equal to one 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


  • Clearly , at least one of the sets {S the weighted are equal to one k,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)


  • For every i=0, the weighted are equal to one …,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


  • For all 0 the weighted are equal to one 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


Weighted graph trees
Weighted Graph-Trees the weighted are equal to one

  • 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+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}


  • a.For every l integers k and l such that k+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)


  • Each min operation in the equation a. can be done by O(Z integers k and l such that k+l2) 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
Weighted Graph-Arbitrary Graphs integers k and l such that k+l

  • 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
Approx.Algorithms integers k and l such that k+l

  • 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


  • Two centers are said to be integers k and l such that k+lsiblings 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 x integers k and l such that k+l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


  • integers k and l such that k+lx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


Discussion
Discussion integers k and l such that k+l

  • 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


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