1. Friedhelm Meyer auf der Heide
Algorithmic Aspects of Dynamic Intelligent Systems Part 2: Introduction to online algorithms
Friedhelm
Meyer auf der Heide

2. Lectures on monday
take place at am
in W0.209

3. 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

4. How to measure quality of online algorithms?
Assume some a priori knowledge about request sequence, e.g., „requests are chosen randomly“
Assume worst case measure, compare online cost to offline cost
Online : standard competitive analysis – competitive ratio
Online randomized:

5. Toy Example: Ski rental problem
I go skiing one year after the other, until I am no longer interested in skiing. I do not know in advance, when I loose interest.
Bying skis costs D € (and I can use them for ever), renting them costs 1€.
Question: When should I buy skis?
Answer: Immediately, if I am interested for at least D years, never otherwise.
This can only be done offline!

6. Toy Example: Ski rental problem
What to do online?
I buy when I go skiing for a D‘th year.
Cost offline : D
Cost online : 2D
This strategy is 2-competitive

7. Paging

8. Paging: A basic problem for an operating system
Main memory
Size k=6
Disk
Paging: given a main memory that can hold k pages.
Input: 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) as possible
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.

9. Paging
Optimal offline:

10. Two online strategies:
Paging
Two online strategies:
Theorem: LRU and FIFO are k-competitive.
Theorem: This competitive ratio is best possible.

11. Furthermore, LRU turns out to be better than FIFO.
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.

12. Page Migration

13. Page Migration Model (1)
Page migration – Classical online problem
processors connected by a network
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. v3 v2 v4 v5 v1 v6 v7

14. Page Migration Model (2)
Alternative view:
processors in a metric space
Indivisible memory page of size in the local memory of
one processor (initially at ) v3 v2 v4 v5 v1 v6 v7

15. 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.
After serving a request an algorithm may move the page
to a new processor. v3 v2 v4 v5 v1 v6 v7

16. Page Migration (cost model)
The page is at node .
Serving a request issued at costs
Moving the page to node costs

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

18. 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

19. Proof of competitiveness of CF
Note:
This is a telescopic sum.
Thus the cancel out
and we get the competitive ratio 3.

20. 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

21. 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:

22. Competitiveness of CF – part 2
OPT moves to

23. Deterministic algorithm
Algorithm Move-To-Min (MTM) [Awerbuch, Bartal, Fiat 93]
Theorem: MTM is 7-competitive
Remark: The currently best deterministic algorithm achieves
competitive ratio of 4.086
After each steps, choose to be the node
which minimizes , and move to .
( is the best place for the page in the last steps)

24. Results on page migration
The best known bounds:
Algorithm
Lower bound
Deterministic
[Bartal, Charikar, Indyk ‘96]
[Chrobak, Larmore, Reingold, Westbrook ‘94]
Randomized:
Oblivious
adversary
[Westbrook ‘91]
Adaptive-online adversary

25. Data management in networks

26. - Exploit Locality - 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 -

27. Basic Strategy
Design strategy for trees
Produce strategy for target-network by tree embedding

28. 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 )

29. 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!)

30. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v0

31. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v1 v0

32. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v1 v0

33. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v1 v0 v2

34. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v1 v0 v2

35. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) v1 v0 v2 v3

36. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) vk v1 v0 v2 v3

37. Example and Analysis
Consider phase write (v0), read (v1), read (v2), ... , read (vk-1), write (vk) vk v1 v0 v2 v3
 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

38. Other networks Goals contradict?!?
Idea: Simulate suitable tree in target-network M.
tree embedding:
Goals: - small dilation
(in order to reduce overall load)
- randomized embedding
(in order to reduce congestion)
Goals contradict?!?

39. Tree embedding Randomized, locality preserving embedding!
Example: nxn-mesh 1 2 3 M‘ v
leaves: nodes of the mesh
link-capacity: # links leaving the submesh

40. Result for meshes The static and dynamic strategies are
O (log(n))-competitive in nxn-meshes, w.h.p.
Finding an optimal static placement for several variables is
NP-hard already on 3x3-meshes.

41. Some Results competitive ratio w.h.p. d-dimensional meshes O(d log n)
Fat Trees
O(log n)
Hypercubes
SE Networks
De Bruijn Networks
Direct Butterflies
Indirect Butterflies
Arbitrary Networks
O(1)
polylog n (Räcke 2002)

42. Conclusions
Provably efficient protocols for data
management in networks
Different models for different
application scenarios
Experimental evaluation (Presto, DIVA)
Some open problems:
- startup times
- combination with load balancing
- randomized, locality preserving
embedding and routing in dynamic
networks

43. Robot Navigation

44. Robot navigation
Theorem: This competitive ratio can be achieved.

45. A simple navigation problem
The playground is a line, no obstacles.
The robot has to find the treaure, but does not know in advance where it is. It only finds it when it touches it.

46. A simple navigation problem
Offline : n
Online : Go left, right, left, right,..
In i‘th step, go distance .
This gives total cost
This algorithm is 9-competitive.
This is best possible!!
Distance n

47. Thank you for your attention!
Heinz Nixdorf Institute
& Computer Science Institute
University of Paderborn
Fürstenallee 11
33102 Paderborn, Germany
Tel.: (0) 52 51/
Fax: (0) 52 51/