AN OPTIMAL ALGORITHM FOR CONFLICT-FREE COLORING FOR TREE OF RINGS

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1 AN OPTIMAL ALGORITHM FOR CONFLICT-FREE COLORING FOR TREE OF RINGS Enollah Pra The Busness Tranng Center of TabrzIran ABSTRACT An optmal algorthm s presented about Conflct-Free Colorng for connected subgraphs of tree of rngs. Suppose the number of the rngs n the tree s T and the maxmum length of rngs s R. A presented algorthm n [] for a Tree of rngs used O(log T.log R ) colors but ths algorthm uses O(log T +log R ) colors. The colorng earned by ths algorthm has the unque-mn property that s the unque color s also mnmum. KEYWORDS Conflct-Free Colorng Tree Tree of Rngs. INTRODUCTION A vertex colorng of graph G=(VE) s an assgnment of colors to the vertces such that two adjacent vertces are assgned dfferent colors. A hypergraph H = (VE) s a generalzaton of a graph for whch hyperedges can be arbtrary-szed non-empty subsets of V. A vertex colorng C of hypergraph H s called conflct-free f n every hyperedge there s a vertex whose color s unque among all other colors n the hyperedge. Suppose the hypergraph H=(VD) of a graph G=(VE) be defned as follows: The set of vertces V of H s the same as that of G and the set of hyperedges D conssts of all possble subsets of V that nduce connected subgraphs of G. Another possble generalzaton [5] s the followng one: Defnton. A vertex colorng of a hypergraph H=(VD) s called conflct-free f n every hyperedge e there exsts at least one vertex whch has a unque color among all other colors used for vertces n that hyperedge. A vertex colorng of a hypergraph such that the mnmum (maxmum) color of any vertex of a hyperedge s unque (assgned to only one vertex n ths hyperedge) s conflct-free and s called unque-mn (resp. unque-max) (confct-free) colorng. The problems of computng a unquemn colorng s equvalent to computng a unque-max colorng snce we can replace every color by c max where cmax s the maxmum color among all vertces []. In ths paper frst study unque-mn (confct-free) colorng n chan rng and tree second present a new algorthm for a tree of rngs. Conflct-free colorng have varous applcatons. For Example n [2] consder the followng scenaro: vertces represent base statons of a cellular network nterconnected through a backbone. Moble clent connect to the network by rado lnks and the recepton range of each agent s a connected subgraph of the base statons graph. Then t may be desrable that n each agent's range there s a base staton transmttng n a unque frequency n order to avod nterference. The problem of mnmzng the number of necessary frequences s equvalent to Connected Subgraphs Conflct-Free Colorng.

2 Related work. The study of conflct-free colorng was ntated n [2] as a geometrc problem wth applcatons to cellular networks. Some of the problems proposed n that paper can be defned as hypergraph conflct-free colorng problems. The algorthm that uses O(log 2 n) colors (where n s the number of vertces) s gven n [] about CF-colorng for trees and trees of rngs. Some of the problems presented n [2] can be defned as hypergraph conflct-free colorng problems. In [34] the conflct-free colorng was studed for grds. In [6] the conflct-free colorng of n ponts wth respect to (closed) dsks were studed and were proved a lower bound of (log n) colors. In [7] the conflct-free colorng of n ponts wth respect to axs-parallel rectangles were studed. Varous other conflct-free colorng problems have been consdered n very recent papers [ ]. The problem becomes more nterestng when the vertces are gven onlne by an adversary. For example at every gven tme step a new vertex v s gven and the algorthm must assgn v a color such that the colorng s a conflct-free colorng of the hypergraph that s nduced by the vertces V { v v2... v}. Once v s assgned a color that color cannot be changed n the future. Ths s an onlne settng so the algorthm has no knowledge of how vertces wll be gven n the future. In [5] there s the onlne verson of conflct-free colorng of a hypergraph. The onlne verson of Connected Subgraphs Conflct-Free Colorng n chans was presented n [8]. Also n the case of ntervals there are several algorthms []. Ther randomzed algorthm uses O (logn loglog n) colors wth hgh probablty. Ther determnstc algorthm uses O(log 2 n) colors n the worst case. Recently randomzed algorthms that use O (logn) colors have been found n [90]. 2. PRELIMINARIES The topologes study durng ths paper are chan rng tree and tree of rngs. A graph s a rng when all ts vertces V are connected n such a way that they form a cycle of length V. A tree of rngs can be defned recursvely n the followng manner [8]: t s ether a sngle rng or a rng R attached to a tree of rngs T by dentfyng exactly one vertex of R to one vertex of T. An Example of a tree of rngs s dsplayed n Fgure. Algorthm for unque-mnmum conflct-free colorng n a chan: n [2] there exsts an algorthm that uses log n colors for chans. The algorthm for a chan {2 n} as follows: n step : Color vertex 2 wth color n n n step 2: Color vertces wth color 2 n n n n n n n n step 3: Color vertces wth color 3. n step : Color verces 2... n n n n... 2 wth color n Color s used only f 2 so n fact log n colors are used by the algorthm. For example f n=8 the colorng s It s clearly to see that the colorng s unquemnmum conflct-free colorng.

3 The above algorthm wth a small change can be used to solve the unque-mnmum conflctfree colorng n a rng. Pck an arbtrary vertex v and color t wth a color (not to be reused anywhere else n the colorng). The remanng vertces form a chan that color wth the algorthm descrbed above. Ths algorthm colors a rng of n vertces wth log ( n ) 2 colors. For example f n=8 the colorng s where `' s the frst unque color used for v. It s not dffcult to see that the colorng s conflct-free: All paths that nclude v are conflctfree colored and the remanng graph G v s a chan of n vertces so paths of G v are also conflct-free colored. R 2 R 3 R 4 R 9 R 0 R R 2 R 5 R R 6 R 7 R 8 G v 2 v 23 v 34 v 459 v 90 v 0 v 56 v 67 v 78 v 2 T(G) Fgure. A tree of rngs G and the correspondng tree representaton T(G). An mportant noton for my algorthm s α-separator. Defnton 2. An α-separator (α<) of a graph G=(VE) s a vertex u the removal of whch parttons G to connected components of sze at most α V. It s obvous from the above defnton that on a general graph an α-separator does not always exst. It s a folklore result that n trees a (/2)-separator always exsts; moreover t can be found n polynomal tme [9]. In my algorthm wll often make use of (/2)-separators. Algorthm for unque-mnmum conflct-free colorng n a tree: n [] there exsts an algorthm that uses log n colors for trees. The algorthm for a tree s dsplayed n Fgure 2.

4 Algorthm : Unque-Mn Colorng for a Tree Input: a tree T Output: a colorng of vertces of T. : Set T :=T :=. 2: whle T do 3: Fnd (/2)-separators on all connected components of forest T. 4: Add these separators to set V. 5: Color vertces n V wth color. 6: Construct forest T + by removng vertces V from T. 7: Set :=+. 8: end whle Fgure 2. Algorthm 3. AN ALGORITHM FOR TREES OF RINGS In order to present my algorthm for a tree of rngs wll use the noton of tree representaton of a tree of rngs. Assume a tree of rngs G s R R2... R T. Let me frst descrbe how to construct such a representaton T(G) of a tree of rngs G: Connect all vertces together that led n ntersecton of rngs. An Example of a tree of rngs and ts tree representaton s dsplayed n Fgure. The algorthm for a tree of rngs s dsplayed n Fgure 3. Algorthm 2: Unque-Mn Colorng for a Tree of Rngs Input: a tree of rngs G by names R R2... R T Output: a colorng of vertces of G : Construct the tree representaton T(G) of the tree of rngs G. 2: Color the tree T(G) wth algorthm. 3: for := to T - do Color the vertex n ntersecton (f exsts and before ddn t colour) of the rngs R R by color vertex v ( ) n tree T(G). end for 4: for := to T do 5: set cm:=a max color of the colored vertces of rng R. 6: Delete the colored vertces of rng R and connect the neghbors of them. 7: Let R denote the resultng cycle. 8: Color cycle R wth sad algorthm n secton 2 by usng colors from { cm... cm log R 2}. 9: end for Fgure 3. Algorthm Analyss of the algorthm Lemma. The colorng obtaned by Algorthm 2 conflct-free colorng. s a connected-subgraphs unque-mn

5 Proof. Assume that C s a path n G. There are two cases for C. Case : C s part of a rng or a rng tself. f C does not contan the common vertces of the rngs C wll be colored n a unquemn way because C colored n lne 8 from algorthm 2. f C contans the common vertces of the rngs C wll be colored n a unque-mn way because the colorng of t start from the max of the colors of the common vertces of the rngs (see lnes 58 from algorthm 2). Case 2: C les on a connected subset of rngs say R... R ; the correspondng vertces of these rngs n T(G) say v ( )... v( j) j j. Snce these vertces of T(G) n lne 2 from algorthm 2 are colored n a unquemn way and each rng R k n C les between vertces v ( k) k vk( k) that colored n lne 8 from algorthm 2 therefore C has been colored n a unque-mn way. Lemma 2. The Algorthm 2 uses O(log T +log R ) colors. Proof. The number of colors for colorng T(G) equal log T. For colorng the rngs n lne 5 from algorthm 2 the maxmum of cm s s log T therefore the maxmum color s used n lne 8 are log T +2+log R. Thus the Algorthm 2 uses O(log T +log R ) colors. 4. CONCLUSIONS I have presented an optmal algorthm for colorng a tree of rngs such that each connected subgraph has a vertex wth a unque mnmum color. Also have proved ths algorthm uses O(log T +log R ) colors. ACKNOWLEDGEMENTS Ths research has been supported by the busness tranng center of tabrziran. REFERENCES [] Georga K. & Ars P. & Katerna P. (2007) Conflct-free Colorng for Connected Subgraphs of Trees and Trees of Rngs In Proc. th Panhellenc Conference n Informatcs. [2] Even G. & Lotker Z. & Ron D. & Smorodnsky S. (2002) Conflct-free colorngs of smple geometrc regons wth applcatons to frequency assgnment n cellular networks SIAM Journal on Computng 33: Also n Proceedngs of the 43rd Annual IEEE Symposum on Foundatons of Compj nu8v uter Scence (FOCS). [3] Bar-Noy A. & Chelars P. & Lamps M. & Zachos S. (2006) Conflct-free colorng graphs and other related problems Manuscrpt. [4] Chelars P. & Specker E. & Zachos S. (2006) Neochromatca. Manuscrpt. [5] Bar-Noy A. & Chelars P. & Smorodnsky S. (2006) Conflct-free colorng for ntervals: from offlne to onlne In Proceedngs of the 8th Annual ACM Symposum on Parallel Algorthms and Archtectures (SPAA 2006) Cambrdge Massachusetts USA July 30 August 2 pages [6] Har-Peled S. & Smorodnsky S. (2005) Conflct-free colorng of ponts and smple regons n the plane Dscrete and Computatonal Geometry 34: [7] Pach J. & Toth G.(2003) Conflct free colorngs. In Dscrete and Computatonal Geometry The Goodman-Pollack Festschrft pages Sprnger Verlag. [8] Alon N. & Smorodnsky S. (2006) Conflct-free colorngs of shallow dsks n Proceedngs of the 22nd annual Symposum on Computatonal Geometry (SCG 06) pages 4-43 New York NY USA ACM Press. [9] Bar-Noy A. & Chelars P. & Smorodnsky S. (2006) Randomzed onlne conflct-free colorng for hypergraphs Manuscrpt.

6 [0] Chen K. (2006) How to play a colorng game aganst a color-blnd adversary In Proceedngs of the 22nd Annual ACM Symposum on Computatonal Geometry (SoCG) pages [] Fat A. & Lev M. & Matousek J. & Mossel E. & Pach J. & Sharr M. & Smorodnsky S. & Wagner U. & Welzl E. (2005) Onlne conflct-free colorng for ntervals In Proceedngs of the 6th Annual ACM-SIAM Symposum on Dscrete Algorthms (SODA) pages [2] Ajwan D. & Elbasson K. & Govndarajan S. & Ray S. (2007) Conflct-free colorng for. 382 rectangle ranges usng O ( n ) colors In Proc. 9th ACM Symp. on Parallelsm n algorthms and Archtectures (SPAA) pages [3] Bar-Noy A. & Chelars P. & Olonetsky S. & Smorodnsky S. (200) Onlne conflct-free colorng for hypergraphs Combn. Probab. Comput. 9: [4] Chen K. & Kaplan H. & Sharr M. (2009) Onlne conflct-free colorng for halfplans congruent dsks and axs-parallel rectangles ACM Transactons on Algorthms5(2):6:- 6:24. [5] Lev-Tov N. & Peleg D. (2009) Conflct-free colorng of unt dsks Dscrete Appl. Math. 57(7): [6] Pach J. & Tardos G. (2009) Conflct-free colorngs of graphs and hypergraphs. Combn. Probab. Comput. 8(5): [7] Smorodnsky S. (2009) Improved conflct-free colorngs of shallow dscs manuscrpt. [8] Erlebach T. (200) Approxmaton Algorthms and complexty results for path problems n trees of rngs In Proceedngs of the 26th Internatonal Symposum on Mathematcal Foundatons of Computer Scence (MFCS 200) Maranske Lazne Czech Republc August pages [9] Erlebach T. & Pagourtzs A. & Potka K. & Stefanakos S (2003) Resource allocaton problems n multfber WDM tree networks In Graph-Theoretc Concepts n Computer Scence 29th Internatonal Workshop WG 2003 Elspeet The Netherlands June Revsed Papers LNCS 2880 pages BIOGRAPHY M.S. n Computer Engneerng Sharf Unversty of Technology Tehran Iran [ ] B.S. n Computer Engneerng Tarbat Moallam Unversty of Technology Tehran Iran [ ] Dploma n Math. and Physcs Raz Hgh School ajabshr Iran [ ]