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Independent Sets and Graph Coloring with Applications to the Frequency Allocation Problem in Wireless Networks Evi Papaioannou PhD Thesis Department of Computer Engineering and Informatics University of Patras

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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

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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

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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

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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]

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Interference graph Irregular networks

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Interference graph Cellular networks –Reuse distance (k): the min distance between two cells where the same frequency can be used

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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

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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

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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

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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

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Competitive analysis Frequency allocation Cost: Number of frequencies used Competitive ratio: Call Control Benefit: Number of users served Competitive ratio:

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Frequency allocation

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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]

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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

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Proof – Upper bound

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D

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...α 0...α 1...α 2...α 3...α 4...α 5...α 6 D

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Proof – Upper bound...α 0...α 1...α 2...α 3...α 4...α 5...α 6 D a 0 2.5D

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Proof – Lower bound

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Call control

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Previous results Greedy algorithm, networks of maximum degree Δ that support one frequency [PPS97] Greedy algorithm Benefit = 1 Optimal algorithm Benefit = Δ

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

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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 Δ Chromatic number Δ+1 Δ+1 times worse

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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]

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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

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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 = 1Benefit = 3

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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

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Idea Accept the call with probability p

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(1-p) t 0: w.h.p. one of the calls is accepted Marking TechniqueIdea Accept the call with probability p

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Marking TechniqueIdea Accept the call with probability p (1-p) t 0: w.h.p. one of the calls is accepted

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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

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

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

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

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

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

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Upper bounds Best upper bound: 2.651 Study in detail all neighborhoods containing an optimal call and express the competitive ratio as a function of p

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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

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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

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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 0101010101 1010101010 0101010101 1010101010 232323232 323232323 232323232

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Algorithm CRS-A 0101010101 1010101010 0101010101 1010101010 232323232 323232323 232323232 3333 33333 3333 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

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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

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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

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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

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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

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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 2 t mod7 out of 2 t 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

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Algorithm CRS-k Use λ=3k 2 -3k +1 colors 0,1,… 3k 2 -3k and 3k 2 -3k +1 for coloring the interference graph and define color classes appropriately so that each color belongs to –v=3k 2 /4 color classes, if k even –v=(3k 2 +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

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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

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Networks of autonomous transmitters

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Disk graphs

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Unit disk graphs

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Disk graphs σ-bounded disk graphs Unit disk graphs

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.54,41 No 35

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.54,41 No 35

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R/4 R/2 R 2R 4R Algorithm Classify

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R/4 R/2 R 2R 4R Algorithm Classify

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R/4 R/2 R 2R 4R Algorithm Classify R/4 R/2 R 2R 4R

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R/4 R/2 R 2R 4R Algorithm Classify R/4 R/2 R 2R 4R

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R/4 R/2 R 2R 4R Algorithm Classify R/4 R/2 R 2R 4R

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σ = 2 R/4 R/2 R 2R 4R Algorithm Classify R/4 R/2 R 2R 4R

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.54,41 No 35

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R/4 R/2 R 2R 4R Algorithm Guess Tries to guess σ

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R/4 R/2 R 2R 4R Algorithm Guess Tries to guess σ

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R/4 R/2 R 2R 4R Algorithm Guess Tries to guess σ

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.54,41 No 35

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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

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Benefit = log σ

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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 Ε[Benefit] = 2

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 }) O(min{n, σ 2 }) Unit disk graphs Yes 2.54,41 No 35

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Adversary Ε1

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Adversary Ε1 Ε2

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Adversary Ε1 Ε2 Ε3

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Adversary Ε1 Ε2 Ε3 Εκ-1

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Adversary Ε1 Ε2 Ε3 Εκ-1 Εκ

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Optimal Algorithm Benefit = κ+1 Adversary Ε1 Ε2 Ε3 Εκ-1 Εκ

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Optimal Algorithm Benefit = κ+1 Adversary Ε1 Ε2 Ε3 Εκ-1 Εκ Randomized Algorithm Ε[Benefit] 2

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Disk graph for κ=Ω(min{n, σ 2 })

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.5 4,41 No 35

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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’) 4

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123456789101112 3 3 4 3 3 3 2 3 - 3 -2 3 -3 3 -4

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123456789101112 3 3 4 3 3 3 2 3 - 3 -2 3 -3 3 -4 π 1 2 4x234x23 4.41 1

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Independent Sets Representation Lower boundUpper bound σ-bounded disk graphs Yes Ω(min{n, logσ}) O(min{n, logσ}) O(min{n, Π j=1 log*σ log (j) σ}) No Ω(min{n, σ 2 })O(min{n, σ 2 }) Unit disk graphs Yes 2.5 4,41 No 3 5

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Minimum coloring σ-bounded disk graphs Yes Ω(min{logn, loglogσ})O(min{logn, logσ}) No O(min{logn, logσ }) Unit disk graphsNo 25

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Algorithm Layered At most logσ+1 layers

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Algorithm Layered At most logσ+1 layers

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First Fit Layered Combination of Layered and First Fit

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First Fit Layered D1 Combination of Layered and First Fit

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First Fit Layered D1d1 Combination of Layered and First Fit

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First Fit Layered D1d1 Di di-1 Combination of Layered and First Fit

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First Fit Layered D1d1 Di di-1 logn OPT logσ OPT Combination of Layered and First Fit

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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

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Publications I. Caragiannis, C. Kaklamanis, E. Papaioannou On-line Call Control in Cellular Networks. Foundations of Mobile Computing (satellite workshop of FST&TCS 99), 1999. I. Caragiannis, C. Kaklamanis, E. Papaioannou Efficient 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. Papaioannou Competitive 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.

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Publications I. Caragiannis, C. Kaklamanis, E. Papaioannou Efficient 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.

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