Facility Location: Distributed Approximation

Transcription

1 Faility Loation: Distributed Approximation Thomas Mosibroda Roger Wattenhofer Distributed Computing Group PODC 2005

2 Where to plae ahes in the Internet? A distributed appliation that has to dynamially plae ahes. Thomas Mosibroda, ETH PODC

3 Where to plae ahes in the Internet? A distributed appliation that has to dynamially plae ahes. Where should data be ahed? On all/one servers...? On servers with plenty of storage...? On servers in proximity to lients..?... Cahing at a host inurs osts (storage, traffi, maintenane,..) Clients want to aess data from lose-by hosts Classi trade-off aptured by faility loation problems Thomas Mosibroda, ETH PODC

4 Faility Loation Problems Given: Set of lients C (ities, demands,...) Set of failities F (servers,..) Connet every lient to an open faility Opening a faility i F inurs opening osts f i Conneting lient j C to an open faility i inurs onnetion osts ij F f i ij C Open a subset of the failities and onnet all lients to an open faility suh that the sum of opening and onnetion osts is minimized! Thomas Mosibroda, ETH PODC

5 Faility Loation Problems This is the basi non-metri, unapaitated faility loation Important in many appliations: Best geographi loation for warehouses, industrial failities Cahing in distributed systems Energy-effiient lustering in wireless networks Inherently... distributed! Different appliations lead to many important variants Connetion osts ij form metri Capaitated failities Fault-tolerant faility loation (lients must onnet to several failities) Hierarhial faility loation (failities onnet to higher level failities) et... Thomas Mosibroda, ETH PODC

6 Previous work on Faility Loation Faility Loation well studied in (entralized) approximation theory! Non-metri ase: Greedy Algorithm yields O(log n) approximation [Hohbaum, 82] Corresponding Ω(log n) lower bound follows from redution to set over [Lund, Yannakakis, 94] [Feige 98] Metri ase: Various algorithms with onstant approximation ratio based on primal-dual algorithms [Jain, Vazirani, 99]... based on greedy-like strategies [Jain et al., 03],..... based on randomized rounding [Shmoys, Tardos, Aardal, 97]... based on loal searh [Korupolu, Plaxton, Rajaraman, SODA 98] Inherently distributed! These approximation algorithms are entralized! How about distributed approximation? Thomas Mosibroda, ETH PODC

7 Distributed Approximation Distributed Approximation: Distributed Faility Loation Approximation Nodes must ome up with a good global solution! Problems: Loal knowledge, bounded messages,... Many new results on distributed approximation in reent years [see survey by Elkin 2004] Hardness of approximation of distributed MST [Elkin, STOC 04] New upper bounds on MST [Elkin, SODA 04], [Lotker et al., SPAA 03] Hardness of approximation of distributed minimum vertex over, maximum mathing, minimum dominating set,... [Kuhn, Mosibroda, Wattenhofer, PODC 04] Effiient (loal) algorithms for minimum dominating set [Jia, Rajaraman, Suel, PODC 01], [Kuhn, Wattenhofer, PODC 03] Approximation in restrited graphs (unit disk graph, graphs with bounded growth, et...) and on problem variants [Grandoni et al..., PODC 05] Thomas Mosibroda, ETH PODC

8 Outline Motivation of distributed faility loation Related work and distributed approximation Model Algorithm & analysis Conlusions & Open Problems Thomas Mosibroda, ETH PODC

9 Model Complete bipartite graph G=(F C, E). F denotes set of failities, F =m F f i C denotes set of lients, C =n For eah i F, opening ost f i 0 C ij For eah i F, j C, onnetions ost ij 0 We study the lassi synhronous, bounded message size model: We have k ommuniation rounds. In eah round, a lient an send a message to eah faility Fixed Time- Complexity! a faility an send a message to eah lient. Eah message is bounded by O(log n) bits Our algorithm: Clients/Failities send equal message to all. Thomas Mosibroda, ETH PODC

10 Is it diffiult after all? Communiation graph G is omplete. Why not sending all information to a leader-node who an then omputes the solution loally? Communiation Bottlenek! Problem: Every lient has m links to failities with different ij! But message size is bounded by O(log (n+m)) bits A lient needs Ω(m) rounds to tell all its osts to a faility Thomas Mosibroda, ETH PODC

11 Is it diffiult after all? Why not distribute existing entralized algorithms...? 1) Simple greedy algorithm Always open the faility with the best ost-effiieny Problem: Failities are piked one by one Too slow...! 2) Improved greedy algorithm In parallel open many failities with good ost-effiieny Problem: A lient may ontribute to many of these failities! Bad Approximation...! Thomas Mosibroda, ETH PODC

12 Improved Greedy Approah Example 1 f f f Assume all f i, ij are equal All failities have same ost-effiieny Algorithm should open exatly one faility! Example 2 f f f j: ij is for one i, and for all other i Eah faility has exatly one lose lient All failities have same ost-effiieny Algorithm must open all failities in k rounds! Whether a faility should be opened depends on all ij Communiation bottlenek renders distintion diffiult! Thomas Mosibroda, ETH PODC

13 Outline Motivation of distributed faility loation Related work and distributed approximation Model Algorithm & analysis Conlusions & Open Problems Thomas Mosibroda, ETH PODC

14 Algorithm Overview Input: Bipartite Graph Frational Faility Loation Open Failities Connet Cities Phase A: Distributed approximation of linear program Phase B: Distributed randomized rounding Phase C: Connet lients to losest open faility. In 2k+1 rounds In 2 rounds Thomas Mosibroda, ETH PODC

15 The linear program and... Compute an approximate frational solution in O(k) rounds y i denotes whether faility i is opened or not. x ij denotes whether lient j is onneted to faility i or not The following ILP aptures the faility loation problem: Every lient must be onneted A lient must be onneted to an open faility. In the frational first phase, we relax this to x ij, y i 0. Thomas Mosibroda, ETH PODC

16 The linear program and its dual LP The dual linear program is given by: Thomas Mosibroda, ETH PODC

17 Algorithm and Analysis Overview The algorithm adopts a distributed primal-dual approah Faility i stores y i, and lient j stores x ij for all i F Initially, all y i, x ij, α j, and β ij are 0. Primal solution (P) is infeasible Dual solution (D) is feasible, but suboptimal How...?? In the algorithm, nodes gradually inrease y i, x ij, α j, β ij, suh that always... until (P) beomes feasible! (D) may no longer be feasible! By LP duality: If α/ρ 1 and β/ρ 2 is a feasible dual solution The algorithm has approximation ratio ρ 1. Thomas Mosibroda, ETH PODC

18 The algorithm Basi struture: for do failities i with good ost-effiieny inrease y i A faility s ost effiieny: In iteration s, faility i has good ost-effiieny if : : Ative lients are not yet frationally onneted, i.e. ρ is a parameter that depends on the ost values of the given instane. This does not inrease primal feasibility! What happens to the lient s x ij? These failities i inrease their y i Thomas Mosibroda, ETH PODC

19 The algorithm Basi struture: for do if then faility i inreases y i by i lose lients j inrease x ij by i A lient j is tight to faility i if: lose lients Intuitively, a lient j inreases the x ij to failities i if - j is lose to i - and i has a good ost-effiieny! Gradually, the solution will beome more and more feasible! Thomas Mosibroda, ETH PODC

20 The algorithm Problem: a lient may be tight to several failities! If a lient inreases x ij to all failities, solution deteriorates! Solution: a seond loop is required! Gradually redue the number of failities to whih a lient an be tight! Eah outer-loop iteration onsists of inner-loop iterations. Eah inner-loop iteartion onsists of ommuniation rounds Total running time is thus: Initialization Thomas Mosibroda, ETH PODC

21 Results In O(k) ommuniation rounds, the algorithm omputes an approximation to the frational faility loation problem. In 2 ommuniation rounds, the frational solution an be rounded to an integer faility loation solution whose approximation ratio is with high probability. The dependeny on ρ an be avoided using a saling tehnique [Bartal, Byers, Raz, FOCS 97] For any k>0, the distributed algorithm ahieves an approximation ratio of in O(k) ommuniation rounds. Thomas Mosibroda, ETH PODC

22 Conlusions / Open Problems Numerous diretions for future researh... Improve approximation ratio Lower bounds Doubling metris, Eulidean metris? Better (entralized) approximations in metris Distributed metri approximations? Many important variants remain unstudied Capaitated, Hierarhial,... Pratial onsiderations: joining & leaving nodes, adversial nodes Thomas Mosibroda, ETH PODC

23 Questions? Comments? Questions? Comments? Distributed Computing Group Thomas Mosibroda Roger Wattenhofer Thomas Mosibroda, ETH PODC 2005