1 / 29

Algorithmic Aspects of Dynamic Intelligent Systems Part 1: Static networks, routing

Algorithmic Aspects of Dynamic Intelligent Systems Part 1: Static networks, routing. Friedhelm Meyer auf der Heide. Aim of the lectures. I will present the algorithmic part of the CS-view of DIS I will introduce - static and dynamic network models

danil
Download Presentation

Algorithmic Aspects of Dynamic Intelligent Systems Part 1: Static networks, routing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algorithmic Aspects of Dynamic Intelligent SystemsPart 1: Static networks, routing Friedhelm Meyer auf der Heide

  2. Aim of the lectures • I will present the algorithmic part of the • CS-view of DIS • I will introduce • - static and dynamic network models • - communication strategies • - scheduling strategies • - data management strategies • Important feature: strategies of choice are local, distributed, often online

  3. Overview • Part 1: Static networks, routing • Part 2: Introduction to online algorithms, with • applications in data management • Part 3: Page migration in dynamic networks • Part 4: Communication among mobile robots

  4. Part 1: Static networks, routing

  5. How to realize communication among processors? • via shared memory • via busses • via a network

  6. Networks • … consist of nodes and links. • nodes: processing units, for example PC‘s • edges: connections between PC‘s, • forming, for example, a LAN, a WAN, the Internet. • nodes: webpages • links: hyperlinks, • forming the Web-Graph

  7. Some designed topologies • A monolithic parallel computer with many nodes can benefit from a well designed communication network. • Demands: • Good routing quality: • - Small diameter • - High bandwidh • - Robustness • Good layout quality • - Embedable in a small 2D-area with short links

  8. A central problem • Routing: Given a Routing problem , send a packet from v to w, for each (v, w) • - path selection (which path from v to w to use?) • - packet switching (how to resolve conflicts?)

  9. Introduction to Routing • Given a network G = (P, E), and a routing problem . • Path selection: yields a path system • W of paths from v to w in G, • for each (v, w)2R • Dilation: D: • the length of longest path in W • CongestionC : • C(e) = number of paths from W passing through e • C = max {C(e), e2E } • (Note: max (C, D)≤ Routing time ≤ C · D)

  10. An introductory example: Routing on a ring Routing time: Dilation: Congestion:≤ m Ring: - nodes [n]:={0,…n-1}, - edges {i, i+1}, i= 0,..n-2, and {n-1,0} - path system: shortest paths - routing problem R, arbitrary, #R = m Example:

  11. Routing on a ring: a nice proof technique Theorem: Every R, |R| = m, can be routed in time • Conflict resolution: If several packets contend to use the • same edge (v,w) then • the packet with largest distance to its destination is chosen. • If two packets have the same distance to go, the one with larger index wins. • furthest-to-go (FTG) rule Formal: rankv (pi) = (number of edges to be traversed by p from v to destination of p) + i/(m+1) Packet p is prefered against q at v , iff rankv(p) > rankv(q)

  12. Routing on a ring: a stronger result • Theorem: Every R with congestion C and Dilation D can be routed in time D+C-1. There are such R so that this bound is tight. • Proof: • Tight: C messages have same start and destination.

  13. Routing on a ring: a stronger result • The routing protocol: • If several packets contend to use the same edge (v,w) then • the packet with largest distance from its source is chosen. • If two packets have gone the same distance, the one with • larger index wins. (LIS=longest in system) • Properties: • If a packet has started moving, it will never be delayed. • A packet starts moving as soon as there is no other packet in the buffer of its source, except for those with the same source and smaller index. • 1. and 2. imply the result.

  14. The (m,d)-mesh M(m,d)

  15. Some meshes

  16. Permutation network

  17. Recursive definition of permutation networks

  18. Explicit definition of permutation networks

  19. Permutation routing in (n,d)-PN • A permutation routing problem  is defines by a set • R ½ Sources X Sinks • such that every source/sink is start/destination of exactly one packet. • Theorem: For every permutation  there is a vertex disjoint path system in (n,d)-PN.

  20. Permutation routing on (n,d)-PN and M(n,d) • Theorem: For every permutation  there is a vertex disjoint path system in (n,d)-PN. • Corollary: Every permutation in (n,d)-PN can be routed in time 2d-1with buffer size 1, if a preprocessing is allowed for finding the routing paths. (offline routing; routing with preprocessing). • Theorem: Every permutation in M(n,d) can be routed in time (4d-2)(n-1), if preprocessing is allowed. (offline routing; routing with preprocessing). • (Note: d(n-1) is a lower bound, the diameter of (M(n,d).)

  21. Online routing in the Butterfly Network n=2dsinks n=2d sources Can every permutation connecting sources and sinks be routed in time O(log(n)) by an online routing algorithm?

  22. Path system in the Butterfly network There is a unique path from every source to every sink.

  23. Routing in the Butterfly We route from sources to sinks, along the unique paths. What is the role of a routing protocol? Each node has to decide which packet from its buffer to move. FIFO, LIFO, LIS, FTG,…. Random Rank !

  24. The random rank protocol for the Butterfly network • Initially, each packed chooses a random rank. • Rule for each node: • packet with smallest rank is moved. • Theorem 1: If R has congestion C, than routing R in the Butterfly network needs time O(log(n) + C), w.h.p.

  25. About congestion in the Butterfly network • We consider permutation routing. • Worst case congestion: pn • Example : Matrix Transposition, Bit rerversal • Average case congestion: log(n)

  26. Routing times in the Butterfly network • For random permutations: expected O(log(n)) • For worst case permutations: expected O(pn) • What to do with Matrix Transposition?

  27. Valiant‘s trick in the Butterfly network • Route to random sink (expected time (O(log(n))) • Route upwards to source (exp. time (O(log(n))) • Route to final destination (exp. time (O(log(n))) • Result: We can route every permutation in expected time (exp. time (O(log(n)).

  28. Some dynamic networks • mobile ad hoc networks • nodes move, communicateby wireless devices, e.g. WLAN, • nodes can enter and leave the network • goal: guarantee efficient communication among nodes • Peer-to-Peer systems • nodes can enter and leave the system • goal: maintain efficient access to data items in the systems • Grid computing • Processors of a network (LAN, Internet)offer varying computing • resources • goal: Use these resources to execute a parallel program efficiently

  29. Thank you for your attention! Friedhelm Meyer auf der Heide Heinz Nixdorf Institute & Computer Science Department University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 80 Fax: +49 (0) 52 51/60 64 82 Mailto: fmadh@upb.de http://wwwhni.upb.de/alg

More Related