Paging Multiple Users in Cellular Network: Yellow ...

Paging Multiple Users in Cellular Network: Yellow Page and Conference Call Problems Amotz Bar-Noy Department of Computer Science City University of New York Joint Work with Panagiotis Cheilaris and Yi Feng

May 22, 2010

Bar-Noy (CUNY)

Paging Multiple Users

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Introduction

Hide-and-Seek

Hide-and-Seek

3 tokens are hidden in 5 boxes. Find one or all of them in at most 3 rounds. Open as few boxes as possible until desired token(s) are found.

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Introduction

Hide-and-Seek

Hide-and-Seek

3 tokens are hidden in 5 boxes. Find one or all of them in at most 3 rounds. Open as few boxes as possible until desired token(s) are found.

Bar-Noy (CUNY)


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Introduction

Hide-and-Seek

Hide-and-Seek

3 tokens are hidden in 5 boxes. Find one or all of them in at most 3 rounds. Open as few boxes as possible until desired token(s) are found. First round, no token is found, 1 box opened.

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Introduction

Hide-and-Seek

Hide-and-Seek

3 tokens are hidden in 5 boxes. Find one or all of them in at most 3 rounds. Open as few boxes as possible until desired token(s) are found. First round, no token is found, 1 box opened. Second round, one token is found, 3 boxes opened. Bar-Noy (CUNY)


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Introduction

Hide-and-Seek

Hide-and-Seek

3 tokens are hidden in 5 boxes. Find one or all of them in at most 3 rounds. Open as few boxes as possible until desired token(s) are found. First round, no token is found, 1 box opened. Second round, one token is found, 3 boxes opened. Third round, all tokens are found, 5 boxes opened. Bar-Noy (CUNY)


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Introduction

Problem Formulation

Problem Setting

M tokens are hidden in N boxes. The probability of hiding token m in box n is pm,n , all pm,n s are independent and known. The goal is to find one or all of the tokens by opening boxes in rounds. In each round, any subset of the boxes can be opened. The desired token(s) must be found within D rounds (1 ≤ D ≤ N). The searching process stops once the desired tokens (one or all) are found.

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Introduction

Problem Formulation

Optimization Objective

A searching strategy is an ordered D-partition of the N boxes, such that in the dth round boxes in part d are opened. The cost of a searching strategy is the expected number of opened boxes. Find a searching strategy with the minimum cost.

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Introduction

Problem Formulation

Two Algorithmic Challenges

Given a searching strategy, compute its cost (expected number of opened boxes). Given N and D, construct a searching strategy that minimizes the expected number of opened boxes for any set of M probability vectors.

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Cost Computation

Exhaustive Computation

Cost: Exhaustive Computation

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart.

Pm,n .5

.3

.2

.1

.4

.5

Yellow Page:

Conference Call:

Round 1

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Round 2


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Cost Computation




Pm,n .5

.3

.2

.1

.4

.5

Yellow Page: (.5 × .1) × 1

Conference Call: (.5 × .1) × 1

Round 1

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Round 2


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Cost Computation




Pm,n .5

.3

.2

.1

.4

.5

Round 1

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Round 2

Yellow Page: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 1

Conference Call: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 3


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Cost Computation




Pm,n .5

.3

.2

.1

.4

.5

Round 1

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Round 2

Yellow Page: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 1 +((.3 + .2) × .1) × 1 Conference Call: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 3 +(×(.3 + .2) × .1) × 3


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Cost Computation




Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

Round 2

Yellow Page: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 1 +((.3 + .2) × .1) × 1 +((.3 + .2) × (.4 + .5)) × 3 = 1.9 Conference Call: (.5 × .1) × 1 +(.5 × (.4 + .5)) × 3 +(×(.3 + .2) × .1) × 3 +((.3 + .2) × (.4 + .5)) × 3 = 2.9


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Cost Computation


Cost: Exhaustive Computation hA1 , . . . , AD i: the partition by searching strategy A. d = (d1 , . . . , dM ) ∈ {1, . . . , D}M : a vector encoding in which part each token is hidden (i.e., token m is hidden in part Adm ). Pm,dm : the probability of token m being hidden in part Adm . Sd : the number of boxes in the first d parts.

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Cost Computation


Cost: Exhaustive Computation hA1 , . . . , AD i: the partition by searching strategy A. d = (d1 , . . . , dM ) ∈ {1, . . . , D}M : a vector encoding in which part each token is hidden (i.e., token m is hidden in part Adm ). Pm,dm : the probability of token m being hidden in part Adm . Sd : the number of boxes in the first d parts. YP(A) =

CC(A) =

X

Smin{d1 ,...,dM } ·

d∈{1,...,D}M

m=1

X

M Y

Smax{d1 ,...,dM } ·

d∈{1,...,D}M

Bar-Noy (CUNY)

M Y


! Pm,dm

(1) !

Pm,dm

(2)

m=1

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Cost Computation


Cost: Exhaustive Computation hA1 , . . . , AD i: the partition by searching strategy A. d = (d1 , . . . , dM ) ∈ {1, . . . , D}M : a vector encoding in which part each token is hidden (i.e., token m is hidden in part Adm ). Pm,dm : the probability of token m being hidden in part Adm . Sd : the number of boxes in the first d parts. YP(A) =

CC(A) =

X

Smin{d1 ,...,dM } ·

M Y

d∈{1,...,D}M

m=1

X

M Y

Smax{d1 ,...,dM } ·

d∈{1,...,D}M

! Pm,dm

(1) !

Pm,dm

(2)

m=1

Time complexity: Θ(MN + (M + D)D M ). Bar-Noy (CUNY)


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Cost Computation

Exclusive Computation

Yellow Page Cost: Exclusive Computation

Pm,n .5

.3

.2

.1

.4

.5

Round 1

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Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. At least one token is found in round 1: (1 − ((1 − .5) × (1 − .1))) × 1

Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. At least one token is found in round 1: (1 − ((1 − .5) × (1 − .1))) × 1 No token is found in round 1: ((1 − .5) × (1 − .1)) × 3

Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

Round 2

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. At least one token is found in round 1: (1 − ((1 − .5) × (1 − .1))) × 1 No token is found in round 1: ((1 − .5) × (1 − .1)) × 3 Total cost: 1.9


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Cost Computation


Conference Call Cost: Exclusive Computation

Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)


Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. Both tokens are found in round 1: (.5 × .1) × 1

Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. Both tokens are found in round 1: (.5 × .1) × 1 At most one token is found in round 1: (1 − (.5 × .1)) × 3

Round 2


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Cost Computation



Pm,n .5

.3

.2

.1

.4

.5

Round 1

Bar-Noy (CUNY)

Round 2

2 tokens, 3 boxes, 2-round searching strategy {1|23}, pm,n in chart. Both tokens are found in round 1: (.5 × .1) × 1 At most one token is found in round 1: (1 − (.5 × .1)) × 3 Total cost: 2.9


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Cost Computation


Cost: Exclusive Computation hA1 , . . . , AD i: the partition by searching strategy A. Pm,d : the probability of token m being hidden in part Ad . P Rm,d = D i=d+1 Pm,i : the suffix probability. Pd Qm,d = i=1 Pm,i : the prefix probability. Sd : the number of boxes in the first d parts.

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Cost Computation


Cost: Exclusive Computation hA1 , . . . , AD i: the partition by searching strategy A. Pm,d : the probability of token m being hidden in part Ad . P Rm,d = D i=d+1 Pm,i : the suffix probability. Pd Qm,d = i=1 Pm,i : the prefix probability. Sd : the number of boxes in the first d parts. YP(A) =

CC(A) =

D X d=1

m=1

D X

M Y

d=1

Bar-Noy (CUNY)

Sd ·

M Y

Sd ·

Rm,d−1 −

M Y

!! Rm,d

(3)

m=1

Qm,d −

m=1


M Y

!! Qm,d−1

(4)

m=1

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Cost Computation


Cost: Exclusive Computation hA1 , . . . , AD i: the partition by searching strategy A. Pm,d : the probability of token m being hidden in part Ad . P Rm,d = D i=d+1 Pm,i : the suffix probability. Pd Qm,d = i=1 Pm,i : the prefix probability. Sd : the number of boxes in the first d parts. YP(A) =

CC(A) =

D X

Sd ·

M Y

d=1

m=1

D X

M Y

Sd ·

Rm,d−1 −

!! Rm,d

(3)

m=1

Qm,d −

m=1

d=1

M Y

M Y

!! Qm,d−1

(4)

m=1

Time complexity: Θ(MN + MD). Bar-Noy (CUNY)


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Cost Computation

Recursive Computation

Yellow Page Cost: Recursive Computation

YP(hAd , . . . , AD i): the expected cost of paging parts hAd , . . . , AD i given that no token is found in parts A1 , . . . , Ad−1 .

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Cost Computation



YP(hAd , . . . , AD i): the expected cost of paging parts hAd , . . . , AD i given that no token is found in parts A1 , . . . , Ad−1 . YP(hAD i) = |AD |

QM

Rm,d YP(hAd , . . . , AD i) = |Ad | + QMm=1 YP(hAd+1 , . . . , AD i) , m=1 Rm,d−1

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Cost Computation



YP(hAd , . . . , AD i): the expected cost of paging parts hAd , . . . , AD i given that no token is found in parts A1 , . . . , Ad−1 . YP(hAD i) = |AD |

QM

Rm,d YP(hAd , . . . , AD i) = |Ad | + QMm=1 YP(hAd+1 , . . . , AD i) , m=1 Rm,d−1

(5)

Time complexity: Θ(MN + MD)

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Cost Computation


Conference Call Cost: Recursive Computation CC(hA1 , . . . , Ad i): the conference call cost of finding all tokens in parts A1 , . . . , Ad .

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Cost Computation



CC(hA1 i) =

M Y

Qm,1 |A1 |

m=1

CC(hA1 , . . . , Ad i) = CC(hA1 , . . . , Ad−1 i)+

M Y m=1

Qm,d −

M Y

! Qm,d−1

|Ad |

m=1

(6)

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Cost Computation



CC(hA1 i) =

M Y

Qm,1 |A1 |

m=1

CC(hA1 , . . . , Ad i) = CC(hA1 , . . . , Ad−1 i)+

M Y m=1

Qm,d −

M Y

! Qm,d−1

|Ad |

m=1

(6) Time complexity: Θ(MN + MD) Bar-Noy (CUNY)


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Related Work

Motivation

Original Motivation: Cellular Networks

M mobile users roaming in N cells in a Cellular System. Only statistical knowledge about users’ locations. Goal: find one, some, or all of the users by paging cells in at most D paging rounds. Bandwidth is saved by minimizing the total number of paged cells until the desired users are found. Yellow Page: find one expert from a group of M experts, stop after finding the first expert. Conference Call: establish a conference call among M users, stop only when all participants are connected.

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Related Work

Motivation

Another Application: Sensor Networks

Sensors jointly collect data items. Users pull data items. Sensors do not push data items to save energy (sensors batteries). Goal: find one, some, or all of the data items by probing sensors in at most D probing rounds. Yellow Page: find a particular data item, stop after finding it. Conference Call: find a collection of M data items, stop after collecting all the data items

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Related Work

Motivation

Related Work I

Single Token [Goodman et. al, Mobile Networks 1996][Rose and Yates, Wireless Networks 1995][Krishnamachari et. al., Wireless Networks 2004][Bar-Noy et. al., INFOCOM 2007]: Dynamic Programming: Θ(DN 2 ) → Θ(DN). Some papers with heuristic solutions.

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Related Work

Motivation

Related Work II

Conference Call Problem [Bar-Noy and Malewicz, PODC 2002]: NP-Hardness, approximation.

e e−1

[Epstein and Levin, WAOA 2004]: PTAS for constant D. [Bar-Noy and Noar, Wireless Networks 2006][Gau and Hass, ToN 2004][Epstein and Levin, WAOA 2004]: other paging and cost models.

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Related Work

Motivation

Related Work III

Yellow Page Problem No existing work on this model to the best of our knowledge. [Weitzman, Econometrica 1979] Pandora Box Problem in continuous domain (in contrast to discrete domain). [Fiat et. al, SODA 2003]: A similar problem with different parameterizations on number of users. [Kaplan et. el, STOC 2005]: A harder problem is NP hard, 4-approximation for D = N rounds.

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Related Work

Motivation

Our Contribution Three ways to compute the searching cost: leading to more efficient algorithms. helping in proofs. verifying programming correctness.

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Related Work

Motivation


For a given order of opening the boxes, both problems can be solved via an efficient Dynamic Programming.

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Related Work

Motivation


For a given order of opening the boxes, both problems can be solved via an efficient Dynamic Programming. There exists a “duality” between the Conference Call Problem and the Yellow Page Problems.

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Related Work

Motivation


For a given order of opening the boxes, both problems can be solved via an efficient Dynamic Programming. There exists a “duality” between the Conference Call Problem and the Yellow Page Problems. Reduced (exponential) complexity for finding optimal searching strategies for some special cases: for sequential search (D = N). for disjoint tokens.

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Related Work

Motivation


For a given order of opening the boxes, both problems can be solved via an efficient Dynamic Programming. There exists a “duality” between the Conference Call Problem and the Yellow Page Problems. Reduced (exponential) complexity for finding optimal searching strategies for some special cases: for sequential search (D = N). for disjoint tokens.

“simple” and “fast” heuristics that provide good performance on synthetic and “real” data. Bar-Noy (CUNY)


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Types of Tokens

Types of Tokens

Tokens are identical, if p1,n = · · · = pM,n for any box. Tokens are uniform, if for any token, pm,n is either 0 or 1/k , where k is the number of non-zero entries in {pm,1 , . . . , pm,N }. Tokens are similar, if for all tokens m = 2, . . . , M, {pm,1 , . . . , pm,N } is some permutation of token 1’s probabilities {p1,1 , . . . p1,N }. Tokens are disjoint, if for each box there is exactly one non-zero entry pm,n .

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Types of Tokens

Types of Tokens Identical-Uniform 1/k ... 1/k 1/k 1/k ... 1/k 0

1/k 1/k

a a a

Identical b . .1/k . b . .1/k . b . .1/k .

Similar-Uniform-Disjoint ... 1/k 0 ... ... 0 1/k ...

0 1/k

Similar-Uniform Similar-Disjoint Uniform-Disjoint c ... 1/k 0 a b ... 01/k . . . . . . 0 1/k 0 c 0 ... 1/k 0 ... 0 a 0 b ...... 0 1/l c ... 1/k 0

a b 0

Similar b c a 0 a b

0 1/k c 1/h c

a c e

Uniform ... 1/k ... 1/h

b 0 f

Arbitrary 0 0 d 0 g 0

a 0 0 1/h 0

b 0 0

Disjoint 0 0 e 0 0 f

... ...

0 1/l

0 0 g

0 c a

Figure: Hierarchy among different token types; a, b, c, d, e, g, h ≤ 1 are positive real numbers, h, k , l ≤ N are positive integers. Bar-Noy (CUNY)


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Theoretical Results

Duality

Duality for Sequential Search

Let A be a searching strategy that looks for M tokens in N boxes in D = N rounds. Let an instance for the Yellow Page Problem be the probabilities pm,n . Let another instance for the Conference Call Problem be the probabilities qm,n = pm,N+1−n . Lemma YP(A, pm,n ) + CC(A, qm,n ) = N + 1.

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Theoretical Results

Optimal Solution w.r.t. Particular Order

Optimal Solution for a Given Order Lemma Given an order of the boxes, the optimal searching cost and the corresponding searching strategy that “respects” this order can be computed via an efficient Dynamic Programming. optimal for d rounds .....

... optimal for (d-1) rounds

Proof. Implied by the recursive computation of the searching cost. The Dynamic Programming is an adaptation of the known solution for the one token case.

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Theoretical Results

Optimal Sequential Search

An Optimal Strategy for Sequential Search

When D = N, a searching strategy is a permutation of the boxes. A naive optimal algorithm takes Ω(N!) time by examining all possible permutations. We reduce the complexity to O(N · 2N ). The more efficient algorithm requires exponential space.

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Theoretical Results

Optimal Sequential Search

An Optimal Strategy for Sequential Search Algorithm 1 D = N, arbitrary users: compute the optimal cost and strategy for yellow page using dynamic programming; opt = YPDP(A) for ∀A, that |A| = 1 do Best[A]cost ← 1 Best[A]strategy ← hAi end for for ∀A that |A| = 2 . . . N do QM

m=1

P

C ∈A\C

pm,i

i · Best[A \ Ci ]cost } Best[A]cost ← minCi ∈A {1 + QM Pn p m,i m=1 C ∈A n

Best[A]strategy ← arg min{Ci |Best[A \ Ci ]cost }, Best[A \ Ci ]strategy end for return {Best[A]| that |A| = N}

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Theoretical Results

Heuristics

Greedy Heuristics Order the boxes according to one of the following criteria and then find the optimal searching strategy for this order. Xn : The product of the M probabilities in box n. Yn : The probability of finding at least one token in box n. Zn : The maximum among the M probabilities in box n. Sn : The sum of the M probabilities in box n.

For each order consider the following three options: Order the boxes following the original probabilities. Best First (BF): first select the best box, continue recursively with normalized probabilities. Worst Last (WL): first select the worst box, continue recursively with normalized probabilities.

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Experimental Study

Setup

Experimental Study

Evaluate the performance of all the 12 heuristics. Compare their performance with the optimal solution when it can be computed. Evaluate the heuristics on synthetic data and “real” data. Evaluate the heuristics for different types of tokens.

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Experimental Study

Data

Data Regular

Disjoint

Small

Large

Exhaustive

Zipf

Random

D=2

D=N

Figure: Experiment data: every path in the graph

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Experimental Study

Results

Some Conclusions

General Comments Over 300 large tables were generated. Results are consistent across data types. The ratios to optimal of all the heuristics are reasonably good: < 2. The ratio to optimal of the best heuristic is very good < 1.06.

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Experimental Study

Results

Some Conclusions

General Comments Over 300 large tables were generated. Results are consistent across data types. The ratios to optimal of all the heuristics are reasonably good: < 2. The ratio to optimal of the best heuristic is very good < 1.06. Results Yellow Page: Y ≥ S > Z ≥ X , BFY > Y ≥ WLY Conference Call: Y ≥ S > Z ≥ X , Y > BFY ∼ WLY

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Experimental Study

Results

Running Time for the Yellow Page Problem Running time for sequential search Running time (in seconds)

10 1

N! naive approach: N = 11 instance solved in 36 minutes

G AG OPT

0.1 0.01 0.001 0.0001 1e-05 1e-06 4

6

8

10

12 N

14

16

18

20

Figure: Running time: greedy (Y ) vs. adaptive greedy (BFY ) vs optimal

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2N fast approach: N = 29 instance solved in 17 minutes (reached physical memory limit)

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Open Problems

Open Problems and Research Directions Prove NP-Hardness for the Yellow Page problem.

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Open Problems

Open Problems and Research Directions Prove NP-Hardness for the Yellow Page problem. Design an approximation-guaranteed algorithms for the Yellow Page Problem.

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Open Problems

Open Problems and Research Directions Prove NP-Hardness for the Yellow Page problem. Design an approximation-guaranteed algorithms for the Yellow Page Problem. Address the generalized problem of finding k out of the M users: k = 1 is the Yellow Page Problem k = M is the Conference Call Problem

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Open Problems


Assume different searching costs for opening different boxes: Congestions in Cellular Networks. Life time of batteries in Sensor Networks.

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Open Problems



Analyze adaptive strategies for the conference call problem. Select the next set of boxes to open depending on the tokens found in earlier rounds.

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Open Problems



Analyze adaptive strategies for the conference call problem. Select the next set of boxes to open depending on the tokens found in earlier rounds.

More experimental study with “real” data and for the different token types. Bar-Noy (CUNY)


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