independent sets and graph coloring with applications to the frequency allocation problem in...

Independent Sets and Graph Coloring with Applications to the Frequency Allocation Problem in

Wireless Networks

Evi Papaioannou

PhD Thesis

Department of Computer Engineering and InformaticsUniversity of Patras

Subject

• Wireless networks

– Frequency allocation problem in cellular wireless netowrks

– Call control problem in wireless networks with cellular, planar, arbitrary topology

• Networks of autonomous transmitters

– Maximum independent set problem

– Minimum coloring problem

Methodology

• On-line problems– Users/transmitters appear gradually and the

sequence can stop arbitrarily

– On-line algorithms

– Algorithms cannot change their choices

• Performance evaluation – Competitive analysis

– Metric = value of competitive ratio

Cellular wireless networks

• The geographical area is divided in regions (cells)• Each cell is the calling area of a base station• Base stations are interconnected via a high speed

network

Communication

• Communication between user and base station is always required

• Frequency Division Multiplexing (FDM) Technology : many users within the same cell can simultaneously communicate with their base station using different frequencies [Hale 80]

Interference graph

• Irregular networks

Interference graph

• Cellular networks– Reuse distance (k): the min distance between two cells

where the same frequency can be used

Frequency allocation

Input• A cellular network and users that wish to communicate

with their base station

Output• Frequency allocation to all users, so that:

– Users in the same or adjacent cells are assigned distinct frequencies

– The number of frequencies used in minimized

Graph coloring

Imagine:• Frequencies colors• Users that wish to communicate with their base station

nodes of the interference graph of the wireless network

Then:• Frequency allocation problem problem of

multicoloring the nodes of the interference graph• The interference graph is constructed gradually

– Nodes are added gradually as calls appear

Call control

Input• A cellular network supporting w frequencies and users

that wish to communicate with their base station

Output• Frequency allocation to some of the users, so that:

– Users in the same or adjacent cells are assigned distinct frequencies

– At most w frequencies are used– The number of the users served is maximized

Independent sets

Imagine:• Frequencies colors• Users that wish to communicate with their base station

nodes of the interference graph of the wireless network

Then:• Call control problem Maximum independent set

problem in the interference graph• The interference graph is constructed gradually

– Nodes are added gradually as calls appear

Competitive analysis

Frequency allocationCost:• Number of frequencies used

Competitive ratio:

Call ControlBenefit:• Number of users served

Competitive ratio:

) σ (C) σ (C

maxρOPT

A

σ

) σ (C)] σ (E[C

maxρOPT

A

σ

) σ (B) σ (B

maxρA

OPT

σ

)] σ (E[B) σ (B

maxρA

OPT

σ

Frequency allocation

Previous results

• Off-line algorithms

– 4/3-προσέγγιση [NS97, MR97, JKNS98]

• Even if the sequence of calls is know a priori, the frequency allocation problem cannot be solved optimally in polynomial time [MR97]

– Simple 3/2- and 17/12-approximation algorithms [JKNS98]

• On-line algorithms

– Fixed Allocation algorithm: competitive ratio 3 [JKNS98]

– No deterministic algorithm can have a competitive ratio smaller than 2 [JKNS98]

The Greedy algorithm

• Frequencies: positive integers 1, 2, 3, ...• When a call appears, it is assigned the smaller available

frequency, so that– There is no interference between calls in the same or

adjacent cells (according to reuse distance of the network)

• The greedy algorithm is at most 2.5- and at least 2.429-competitive, against off-line adversaries

• New [ΝΤ04] lower bound = 2.5 Tight analysis

Proof – Upper bound


D


...α0

...α1

...α2

...α3...α

4

...α5

...α6

D


...α0

...α1

...α2

...α3...α

4

...α5

...α6

D

a0 2.5D

Proof – Lower bound

Call control

Previous results

• Greedy algorithm, networks of maximum degree Δ that support one frequency [PPS97]

Greedy algorithm

Benefit = 1

...

Optimal algorithm

......

Benefit = Δ

Previous results

• «Classify and Randomly Select» paradigm for networks with chromatic number χ that support one frequency [ΑΑFLR96, PPS97]

Previous results

• «Classify and Randomly Select» paradigm for networks with chromatic number χ that support one frequency [ΑΑFLR96, PPS97]

Chromatic number = 4

4 times worse

...

Networks of max degree ΔChromatic number Δ+1 Δ+1 times worse

Previous results

• Lower bounds for arbitrary networks [BFL96]• Simple way of transforming an algorithm designed for

networks that support one frequency to an algorithm for networks that support arbitrarily many frequencies [AAFLR01]

• Upper bounds for networks with planar and arbitrary interference graphs using the «Classify and Randomly Select» paradigm [PPS02]

The Greedy algorithm

• The greedy algorithm in networks that support one frequency achieves a competitive ratio equal to the size of the maximum independent set of every node of the interference graph

...

Deterministic algorithms

• The greedy algorithm in cellular networks that support one frequency

• Optimal in the class of deterministic on-line algorithms• Competitive ratio: 3

Benefit = 1 Benefit = 3

Randomized algorithms

• Based on the «Classify and Randomly Select» paradigm• Competitive ratio = number of colors used for the

coloring of the interference graph• Competitive ratio for cellular networks: 3

Idea

Accept the call with probability p

(1-p)t0: w.h.p. one of the calls is accepted

Marking TechniqueIdea


Marking TechniqueIdea


(1-p)t0: w.h.p. one of the calls is accepted

Algorithm p-Random

• Initially all cells are unmarked

• For each new call c in a cell v

– If v is marked, reject c

– If there is an accepted call in cell v or in its adjacent cells, reject c

– Otherwise:

• With probability p, accept c

• With probability 1-p, reject c and mark cell v

Upper bounds

• Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

Upper bounds

c

cXB )()(


Upper bounds

)(')()(

))'(

)'()(()()(

ccAcAc cd

cXccbB


Upper bounds

• Best upper bound: 2.651


Upper bounds

• Algorithm p-Random achieves better competitive ratio than all deterministic algorithms in all networks that support one frequency– 27/28 Δ– Extend the analysis for sparse networks of degree 3 or 4

• Disadvantages– No improvement fro networks that support arbitrarily many

frequencies– Uses randomness proportional to the size of sequence of

calls

CRS-based algorithms

Objective • Randomized algorithms

• Arbitrarily many frequencies• Whatever reuse distance • Few randomness

• Weak random sources• Constant number of random bits

Given• «Classify and Randomly Select» paradigm

• Simple• Use randomness only once at the beginning• Behaves «well» independently of the number of supported

frequencies

Algorithm CRS-A

• Color the interference graph with 4 colors 0,1,2,3• Select one of the colors, ignore calls in cells colored with the

selected color and execute the greedy algorithm for all other calls

0 1 0 1 0 1 0 1 0 1

1 0 1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1 0 1

1 0 1 0 1 0 1 0 1 0

2 3 2 3 2 3 2 3 2

3 2 3 2 3 2 3 2 3

2 3 2 3 2 3 2 3 2

Algorithm CRS-A

0 1 0 1 0 1 0 1 0 1

1 0 1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1 0 1

1 0 1 0 1 0 1 0 1 0

2 3 2 3 2 3 2 3 2

3 2 3 2 3 2 3 2 3

2 3 2 3 2 3 2 3 2

3 3 3 3

3 3 3 3 3

3 3 3 3

• Color the interference graph with 4 colors 0,1,2,3• Select one of the colors, ignore calls in cells colored with the

selected color and execute the greedy algorithm for all other calls

Algorithm CRS-A: analysis

• The greedy algorithm will accept at least half of the optimal calls• Work on average on the 3/4 of the total calls• Competitive ratio = 8/3

1 0 1 1 0 1

2 2 2 22

1 0 1 1 0 1 1 0

0 0 0 1

0 0

1 0 1 1 0 1

2 2 2 22

1 0 1 1 0 1 1 0

0 0 0 1

0 0

2 2 22

CRS-based algorithms

• Network that supports w frequencies• CRS-based algorithms:

– Color the interference graph– Define v color classes from the colors used– Select equiprobably one out of v color classes– Execute the greedy algorithm only for cells colored with

colors from the selected color class

• If: – Each color belongs to at least λ different color classes, and– Each connected component of the subgraph of G

containing nodes colored with colors of the same color class is a clique

then, the CRS-based algorithm is v/λ-competitive against oblivious adversaries

Algorithm CRS-B

• Use 5 colors 0,1,2,3,4 for coloring the interference graph and define 5 color classes {0,1}, {1,2}, {2,3}, {3,4}, {4,0}

• The coloring and the color classes meet the conditions of the previous Lemma for v=5 and λ=2

• Competitive ratio = 5/2

0 1 2 43 4 0 1 2 3

0 1 2 3 4 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4

3 4 0 10 1 2 3 4

3 4 0 10 1 2 3 4

3 4 0 10 1 2 3 4

Algorithm CRS-C

• Use 7 colors 0,1,2,3,4,5,6 and 7 for coloring the interference graph and define 7 color classes {0,1,3}, {1,2,4}, {2,3,5}, {3,4,6}, {4,5,0}, {5,6,1}, {6,0,2}

• The coloring and the color classes meet the conditions of the previous Lemma for v=7 and λ=3

• Competitive ratio = 7/3

0 1 2 13 4 5 26 0

5 6 0 61 2 3 04 5

3 4 5 46 0 1 52 3

1 2 3 24 5 6 30 1

6 0 13 4 5 2 3 4

2 3 46 0 1 5 6 0

4 5 61 2 3 0 1 2

Use of random bits

• Random source: small number of random bits (fair coins)• For all ε > 0, use t=O(log 1/ε) random bits

• For 2tmod7 out of 2t of their outcomes do nothing• For the rest of their outcomes execute algorithm CRS-C

• At most 7/3+ε: • Randomized on –line algorithms, • Cellular networks of reuse distance 2, • Arbitrarily many frequencies, • O(log 1/ε) random bits

Algorithm CRS-k

• Use λ=3k2-3k+1 colors 0,1,… 3k2-3k and 3k2-3k+1 for coloring the interference graph and define color classes appropriately so that each color belongs to– v=3k2/4 color classes, if k even– v=(3k2+1)/4 color classes, if k odd

if k even

if k odd

• Algorithms with slightly worse competitive ratios for arbitrarily many frequencies using O(log 1/ε+log k) random bits

23k13k

14CR

13k3k

14CR 2

12 16 17 1814 1513 0 1 2 3 4 5 7 8 9 106

16 17 1814 15130 1 2 3 4 5 7 8 9 10 11 126

16 17 1814 15135 7 8 9 10 11 126 0 1 2 34

16 17 18 14130 1 2 3 4 5 7 8 9 10 11 126

5 6 73 429 10 11 12 13 15 16 17 18 0 1148

1

13

6

18

11

4

16 17 1814 15132 3 4 5 7 8 9 10 11 126

14 1512 13110 1 2 3 5 6 7 8 9 104

7 16 1714 15138 9 10 11 12

5 14 1512 13116 7 8 9 10

1614 1512 13

1816 1714 15 0 1 2 3 4 5 6 7

17 18 0 1 2 3 4 5

1 2 3

5

9 10

16 17

12

18

6 7

16 17 18 0

8 9 10 11

0 1 2 3

0 1 2 3 4 5 7 8 16 17 189 10 11 126 14 1513

654321036353433323130292827

333231302928272625242322212019181716 34

23222120191817161514131211109876

7

3332 34 3635 10987654321 131211

22

12

21

2827

17

76

33 34 3635 10987654321 131211

232221201918171615141312111098 24

2322

12

141312111098765432

3332313029282726252423 34 3635 210

11 151413 2928272625242322212019181716

1514131211109876543 181716

3332313029 34 3635 87654321

33323130292827262524232221201918 34

1 1918171615

151413 28272625242322212019181716 29

33323130292827262524 34 3635 3210

0

0

0

Lower bounds

• Using Minimax Principle for randomized algorithms– k = 2, 3, 4: competitive ratio 1,857 planar network: competitive ratio 2,086– k 5: competitive ratio 25/12– k 12: competitive ratio 127/60

Networks of autonomous transmitters

Disk graphs

Disk graphs

Unit disk graphs

Disk graphs

σ-bounded disk graphsUnit disk graphs

Independent Sets

Representation Lower bound Upper bound

σ-bounded disk graphs

Yes Ω(min{n, logσ})O(min{n, logσ})

O(min{n, Πj=1log*σ log(j)σ})

No Ω(min{n, σ2}) O(min{n, σ2})

Unit disk graphs

Yes 2.5 4,41

No 3 5

R/4R/2

R

2R

4R

Algorithm Classify

R/4R/2

R

2R

4R

Algorithm Classify

R/4R/2

R

2R

4R

2

1

σ = 2

24

2

3

1 7

9

8

6

1011

12

13

14

155

R/4R/2

R

2R

4R

Algorithm Classify

R/4R/2

R

2R

4R

Independent Sets






Unit disk graphs

Yes 2.5 4,41

No 3 5

R/4R/2

R

2R

4R

Algorithm Guess

Tries to guess σ

Independent Sets






Unit disk graphs

Yes 2.5 4,41

No 3 5

• We can construct bad instances where no randomized algorithm can be better than Ω(min{n, logσ}) even if the disk representation is given as part of the input


Benefit = log σ


Benefit = log σ Ε[Benefit] = 2

Independent Sets






Unit disk graphs

Yes 2.5 4,41

No 3 5

Adversary

Ε1

Adversary

Ε1

Ε2

Adversary

Ε1

Ε2

Ε3

Adversary

Ε1

Ε2

Ε3

Εκ-1

Adversary

Ε1

Ε2

Ε3

Εκ-1

Εκ

OptimalAlgorithm

Benefit = κ+1

Adversary

Ε1

Ε2

Ε3

Εκ-1

Εκ

Optimal Algorithm

Benefit = κ+1

Adversary

Ε1

Ε2

Ε3

Εκ-1

Εκ

Randomized Algorithm

Ε[Benefit] 2

Disk graph for κ=Ω(min{n, σ2})

Independent Sets






Unit disk graphs

Yes 2.5 4,41

No 3 5

Algorithm Filter

• IDEA: – Use a 2-dimensional geometric construction, place it

uniformly at random on the plane and work only on disks from specific parts of it

(x,y)(x’,y’)

32

4

1 2 3 4 5 6 7 8 9 10 11 12

3

34

33

32

3-

3-2

3-3

3-4

1 2 3 4 5 6 7 8 9 10 11 12

3

34

33

32

3-

3-2

3-3

3-4

Ο

C

A

B

1 2 3 4 5 6 7 8 9 10 11 12

3

34

33

32

3-

3-2

3-3

3-4

1 2 3 4 5 6 7 8 9 10 11 12

3

34

33

32

3-

3-2

3-3

3-4

π 12

4x23

4.41

1

Independent Sets






Unit disk graphs

Yes 2.5 4,41

No 3 5

1

2

3

4

5

Minimum coloring


Yes Ω(min{logn, loglogσ}) O(min{logn, logσ})

No O(min{logn, logσ})

Unit disk graphs No 2 5

Algorithm Layered

At most logσ+1 layers

First Fit Layered

Combination of Layered and First Fit

First Fit Layered

D1


First Fit Layered

D1 d1


First Fit Layered

D1 d1

Didi-1


First Fit Layered

D1 d1

Didi-1

logn OPT logσ OPT


Open problems

• Close the gap between upper and lower bounds– Call control in cellular networks– Maximum independent set problem on unit disk graphs– Maximum independent set problem in σ-bounded disk

graphs when the disk representation os given as part of the input but σ is not given

– Minimum coloring of σ-bounded disk graphs

Publications

• I. Caragiannis, C. Kaklamanis, E. PapaioannouOn-line Call Control in Cellular Networks. Foundations of Mobile Computing (satellite workshop of FST&TCS 99), 1999.

• I. Caragiannis, C. Kaklamanis, E. PapaioannouEfficient On-line Communication in Cellular Networks. In Proc. of the 12th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 00), pp. 46-53, 2000.

• I. Caragiannis, C. Kaklamanis, E. PapaioannouCompetitive Analysis of On-line Randomized Call Control in Cellular Networks.In Proc. of the 15th International Parallel and Distributed Processing Symposium (IPDPS 01), IEEE Computer Society Press, 2001.

• I. Caragiannis, C. Kaklamanis, and E. Papaioannou Randomized Call Control in Sparse Wireless Cellular Networks.In Proc. of the 8th International Conference on Advances in Communications and Control (COMCON 01), pp. 73-82, 2001.

Publications

• I. Caragiannis, C. Kaklamanis, E. PapaioannouEfficient On-line Frequency Allocation and Call Control in Cellular Networks. Theory of Computing Systems, Vol. 35 (5), pp. 521-543, 2002.

• I. Caragiannis, C. Kaklamanis, and E. Papaioannou Simple on-line algorithms for call control in cellular networks.In Proc. of the 1st Workshop on Approximation and On-line Algorithms (WAOA 03), LNCS 2909, Springer, pp. 67-80, 2003.

• I. Caragiannis, A. Fishkin, C. Kaklamanis, E. Papaioannou On-line algorithms for disk graphs.In Proc. of the 29th International Symposium on Mathematical Foundations of Computer Science (MFCS 04), LNCS 3153, Springer, pp. 215-226, 2004.

independent sets and graph coloring with applications to the frequency allocation problem in...

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