Welcome to the course Algorithm Design

Transcription

1 Welcome to the course Algorithm Design Summer Term 2011 Friedhelm Meyer auf der Heide Lecture 12, Friedhelm Meyer auf der Heide 1

2 Randomised Algorithms Friedhelm Meyer auf der Heide 2

3 Topics - Divide & conquer - Dynamic programming - Greedy Algorithms - Randomized Algorithms - Approximation Algorithms - Online Algorithms Friedhelm Meyer auf der Heide 3

4 Online Algorithms An online algorithm is one that can process its input piece-by-piece, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand. Input: a sequence of requests. Task: process the requests as efficiently as possible Online: i th request has to be processed before future requests are known Offline: All requests are known in advance Friedhelm Meyer auf der Heide 4

5 How to measure quality of online algorithms? 1. Assume some a priori knowledge about request sequence, e.g., requests are chosen randomly 2. Assume worst case measure, compare online cost to offline cost Online : standard competitive analysis competitive ratio Online randomized: Friedhelm Meyer auf der Heide 5

6 Paging Friedhelm Meyer auf der Heide 6

7 Paging: A basic problem for an operating system Paging: given a main memory that can hold k pages. Main memory Size k=6 Input: request sequence of pages to be used by the processor Goal: Need as few page faults ( requests for pages that have to be moved from disk to main memory) The algorithm has to store the requested page in main memory in case of a page fault, and has to choose a page to be removed from main memory. Disk Friedhelm Meyer auf der Heide 7

8 Paging Optimal offline: Friedhelm Meyer auf der Heide 8

9 Paging Two online strategies: Theorem: LRU and FIFO are k-competitive. This competitive ratio is best possible. Friedhelm Meyer auf der Heide 9

10 Paging In practise, both algorithms are much better, the observed competitive ratio decreases with increasing memory size k. Furthermore, LRU turns out to be better than FIFO. Reasons: In practise, request sequences exhibit locality, i.e., they tend to use the same pages more often, and have dependencies among pages. ( If page A is accessed, then it is likely that page B will be accessed shortly afterwards ) Way out: Model restrictions to the adversary, i.e. the bad guy that generates the worst case sequences. This is done using access graphs. Friedhelm Meyer auf der Heide 10

11 Page Migration Friedhelm Meyer auf der Heide 11

12 Page Migration Model (1) Page migration Classical online problem processors connected by a network v 3 v 2 v 4 v 5 v 1 v 6 v 7 There are costs of communication associated with each edge. Cost of communication between pair of nodes = cost of the cheapest path between these nodes. Costs of communication fulfill the triangle inequality. Friedhelm Meyer auf der Heide 12

13 Page Migration Model (2) Alternative view: processors in a metric space v 3 v 2 v 4 v 5 v 1 v 6 v 7 Indivisible memory page of size one processor (initially at ) in the local memory of Friedhelm Meyer auf der Heide 13

14 Page Migration Model (3) Input: sequence of processors, dictated by a request adversary - processor which wants to access (read or write) one unit of data from the memory page. v 3 v 2 v 4 v 5 v 1 v 7 v 6 After serving a request an algorithm may move the page to a new processor. Friedhelm Meyer auf der Heide 14

15 Page Migration (cost model) Cost model: The page is at node. Serving a request issued at costs. Moving the page to node costs. Friedhelm Meyer auf der Heide 15

16 A randomized algorithm Memoryless coin-flipping algorithm CF [Westbrook 92] In each step, after serving a request issued at, move page to with probability. Theorem: CF is 3-competitive against an adaptive-online adversary (may see the outcomes of the coinflips). Remark: This ratio is optimal against adaptive-online adversary. Friedhelm Meyer auf der Heide 16

17 Proof of competitiveness of CF We run CF and OPT in parallel on the input sequence We define potential function There are two events to consider in each step Request occurs at a node, CF and OPT serve the requests, part 1 CF optionally moves the page OPT optionally moves the page } part 2 For each part separately, we prove that Friedhelm Meyer auf der Heide 17

18 Proof of competitiveness of CF Note: This is a telescopic sum. Thus the cancel out and we get the competitive ratio 3. Friedhelm Meyer auf der Heide 18

19 Competitiveness of CF, a step Page in and resp. Request occurs at CF and OPT serve the requests CF optionally moves the page to part 1 OPT optionally moves the page part 2 to Friedhelm Meyer auf der Heide 19

20 Competitiveness of CF part 1 Request occurs at Cost of serving requests: in CF : a, in OPT : b Expected cost of moving the page: Potential before: Exp. potential after: Exp. change of the potential: Friedhelm Meyer auf der Heide 20

21 Competitiveness of CF part 2 OPT moves to Friedhelm Meyer auf der Heide 21

22 Deterministic algorithm Algorithm Move-To-Min (MTM) [Awerbuch, Bartal, Fiat 93] After each steps, choose to be the node which minimizes, and move to. ( is the best place for the page in the last steps) Theorem: MTM is 7-competitive Remark: The currently best deterministic algorithm achieves competitive ratio of Friedhelm Meyer auf der Heide 22

23 Results on page migration The best known bounds: Deterministic Randomized: Oblivious adversary Algorithm [Bartal, Charikar, Indyk 96] [Westbrook 91] Lower bound [Chrobak, Larmore, Reingold, Westbrook 94] [Chrobak, Larmore, Reingold, Westbrook 94] Randomized: Adaptive-online adversary [Westbrook 91] [Westbrook 91] Friedhelm Meyer auf der Heide 23

24 Data management in networks Friedhelm Meyer auf der Heide 24

25 Scenario Networks have low bandwidth, global objects are small, access is fine grained. typical for parallel processor networks, partially also for the internet. bottleneck: link-congestion task: distribute global objects (maybe dynamically) among processors such that an application (sequence of read/write access to global variables) can be executed using small link-congestion storage overhead is small. - Exploit Locality - Friedhelm Meyer auf der Heide 25

26 Basic Strategy Design strategy for trees Produce strategy for target-network by tree embedding Friedhelm Meyer auf der Heide 26

27 Dynamic Model Application: Sequence of read / write requests from processors to objects. Each processor decides solely based on its local knowledge. distributed online-strategy Goal: Develop strategy that produces only by a factor c more congestion than an optimal offline strategy. c-competitive strategy (and by a factor m more storage per processor (m, c) competitive strategy ) Friedhelm Meyer auf der Heide 27

28 Dynamic strategy for trees v writes to x : v creates (or updates) copy of x in v, and invalidates all other copies (consistency!) v reads x: v reads the closest copy of x and creates copies in every processor on the path back to v. (Remark: Data Tracking in trees is easy!) Friedhelm Meyer auf der Heide 28

29 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 0 Friedhelm Meyer auf der Heide 29

30 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 Friedhelm Meyer auf der Heide 30

31 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 Friedhelm Meyer auf der Heide 31

32 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 Friedhelm Meyer auf der Heide 32

33 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 Friedhelm Meyer auf der Heide 33

34 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v 0 v 2 v 3 Friedhelm Meyer auf der Heide 34

35 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v k v 0 v 2 v 3 Friedhelm Meyer auf der Heide 35

36 Example and Analysis Consider phase write (v 0 ), read (v 1 ), read (v 2 ),..., read (v k-1 ), write (v k ) v 1 v k v 0 v 2 3 Each strategy has to use each link of the red subtree at least once. Our strategy uses each of these links at most three times. Strategy is 3-competitive for trees v Friedhelm Meyer auf der Heide 36

37 Thank you for your attention! Friedhelm Meyer auf der Heide Heinz Nixdorf Institute & Computer Science Department Fürstenallee Paderborn, Germany Tel.: +49 (0) 52 51/ Fax: +49 (0) 52 51/ Friedhelm Meyer auf der Heide 37