1. On-line Routing of Real-Time Messages on Computer Networks

2. Messages routing over a network
- simultaneous transmission of messages from a source to a destination.
- can go multiple routes
Real-Time Routing
- message routing with release time and deadline constraints

3. Reference:

4. Our Focus
Difference in our study, is the network is not a complete Graph
Must go through intermediate nodes
Focus on the complexity of the message routing problem under various restrictions of four parameters
origin node
destination node
release time
deadline

5. Notation/Terminology
Represent a network by a directed graph

6. Edges (communication links)
A node can simultaneously send and receive several messages transmitted on different communication links 1 3 2 4

7. Set of n messages origin node destination node length release time
deadline

8. Preemptive v.s Non-preemptive
Non-preemptive transmission- a message once transmitted on a communication link (u, v) must continue until the entire message is received by node v.
Preemptive transmission- transmission can be interrupted and resumed later

9. Message-Routing Network System

10. Configuration ”q”

11. Message-Routing Network System (Example)

12. G =
M =
feasible!
non-preemptive transmission

13. G =
M =
Not feasible!
nonpreemptive transmission

14. Theorem
Given MRNS=(G,M), where G is an arbitrary directed graph and M is a set of messages with identical:
origin nodes
destination nodes
release times
deadlines
The problem of determining whether MRNS is feasible with respect to nonpreemtivea and preemptive transmission is NP-complete.

15. Proof to Theorem
Show that if the network is an arbitrary directed graph, determining whether there is a feasible transmission (preemptive or nonpreemptive) for a message-routing network system is NP-complete, even when all four parameters are fixed. Our reduction is from the 3-Partition problem, which is known to be strongly NP- complete.

16. Recall: 3-partition
NOT the Partition Problem (2-partition) with 3 parts.
Definition:

17. Definition in ENGLISH!
The 3-partition problem is to decide whether a given multiset of integers can be partitioned into triples that all have the same sum.

18. Unidirectional Ring
In our example with m=4 nodes...

19. Example: Unidirectional Ring
fixed

20. Example: Non-preemtive vs Preemptive non-preemptive
chopped
chopped

21. OFFLINE ROUTING OF VARIABLE-LENGTH MESSAGES

22. Algorithms Algorithm A :Earliest Available Message Strategy.
Algorithm B : Earliest Available Message and Farthest Destination Strategy
Algorithm C : Earliest AvailableMessage and EarliestDeadline Strategy.

23. Theorem
A set of messages with identical origin nodes, release times, and deadlines is feasible with respect to nonpreemptive transmission if and only if the transmission generated by Algorithm B is feasible.

24. Algorithm D: (Earliest Available Message, Earliest Deadline, and Farthest Destination Strategy).
Whenever a node is free for transmission, send that ready message which is available at the node the earliest.
If there is a tie, choose the one with the earliest deadline.
If again there is a tie, choose the one with the farthest destination.
Any further ties can be broken arbitrarily.

25. Theorem
A set of messages with three of the four parameters fixed is feasible with respect to nonpreemptive transmission if and only if the transmission generated by Algorithm D is feasible

26. Online Routing of Variable-Length Messages

27. Online Routing of Variable-Length Messages
In online routing, each node in the network routes messages without any knowledge of the messages in the order nodes or the release times of the messages.
The study of online routing is meaningful only when the release times of the messages are arbitrary.

28. Online Routing of Variable-Length Messages
Algorithm D ( Earliest Available Message, Earliest Deadline, and Farthest Destination Strategy ).
Whenever a node is free for transmission, send that ready message which is available at the node the earliest.
If there is a tie, choose the one with the earliest deadline. If again there is a tie, choose the one with the farthest destination. Any further ties can be broken arbitrarily.

29. Online Routing of Variable-Length Messages
Theorem:
For a set of messages with identical origin nodes, destination nodes, and deadlines, Algorithm D is an optimal nonpreemptive (preemptive) online algorithm.

30. Online Routing of Variable-Length Messages
Theorem: No optimal nonpreemptive (preemptive) online algorithm can exist for a set of messages with identical origin nodes and destination nodes.

31. Online Routing of Variable-Length Messages
Theorem : No optimal nonpreemptive (preemptive) online algorithm can exist for a set of messages with identical origin nodes and deadlines.

32. Online Routing of Variable-Length Messages
Theorem : No optimal preemptive online algorithm can exist for a set of messages with identical destination nodes and deadlines.

33. Online Routing of Variable-Length Messages
Theorem : No optimal nonpreemptive online algorithm can exist for a set of messages with identical destination nodes and deadlines

34. Online Routing of Equal-Length Messages

35. Online Routing of Equal-Length Messages
Previous sections: message routing problem for a set of variable-length messages. This section we consider only equal-length messages.
Ques is:
Is it possible to have an optimal online algorithm for the following networks — unidirectional ring, out-tree, in-tree, bidirectional tree, and bidirectional ring.
Optimal Online Routing Algorithm:
An optimal nonpreemptive (preemptive) online algorithm is one that produces a feasible nonpreemptive (preemptive) transmission, whenever one exists.

36. Online Routing of Equal-Length Messages
Four parameters —
origin node
destination node
release time
and deadline.
Notation used:
quintuple notation, (si , ei , li , ri , di) → quadruple notation, (si , ei , ri , di)
(∵ lengths of messages, li , are equal)

37. Online Routing of Equal-Length Messages
Set of n unit-length messages M = {M1, ... , Mn} needs to be routed through the network.
Mi is represented by the quadruple (si ,ei ,ri , di)
si denotes the origin node (i.e., Mi originates from si).
ei denotes the destination node (i.e., Mi is to be sent to ei),
ri denotes the release time (i.e., Mi originates from si at time ri), and
di denotes the deadline (i.e., Mi must reach ei by time di).
*assume that release times and deadlines are integers, while the length of each message is one unit

38. Unidirectional Networks
The three online algorithms are given below.
Earliest Deadline (ED) Algorithm. Whenever a communication link is free for transmission, send that ready message with the earliest deadline. Ties can be broken arbitrarily.
Farthest Away (FA) Algorithm. Whenever a communication link is free for transmission, send that ready message whose destination node is the farthest away. Ties can be broken arbitrarily.
Smallest Slack Time (SST) Algorithm. Whenever a communication link is free for transmission, send that ready message with the smallest slack time. Ties can be broken arbitrarily.

39. Unidirectional Networks
Theorems below - show that the ED, FA, and SST algorithms are optimal when one of the four parameters is fixed.
Thm: The ED algorithm is optimal for a set of messages with identical destination nodes.
Thm: The FA algorithm is optimal for a set of messages with identical deadlines.
Thm: The SST algorithm is optimal for a set of messages with identical origin nodes.
Thm: The SST algorithm is optimal for a set of messages with identical release times.
Thm - shows that no optimal algorithm can exist if all four parameters are arbitrary.
Thm: No optimal algorithm can exist for a set of messages with all four parameters arbitrary.

40. Kinesthetic Learning Activity
modeling of routing real-time messages on networks

41. Traditional Learning Shortcomings… Hard to concentrate
Easily distracted by the computers in front of you
Passive rather than an active role
One way communication without feedback
Ineffective… learn nothing!

42. I don’t feel awkward being on stage anymore….

43. Modeling Message-Routing Network System (MRNS) with……… KitKat Bars!

44. Materials Required: 1.5 Ounce KIT KAT Bars Plastic Knifes
Edge/Time Table Worksheet

45. KitKat KLA Challenge
Provided a with the following arbitrary directed graph and a set of messages
Using the network and message constraints table, construct a feasible non-preemptive transmission.

46. KitKat KLA Challenge
Provided a with the following arbitrary directed graph and a set of messages
Using the network and message constraints table, construct a feasible non-preemptive transmission.

47. KitKat KLA Challenge
Simulate each message-span with the provided KitKat bar by crafting into the appropriate length. Place the crafted message/KitKat over the correct edge/time slot on the worksheet. (Label each message/KitKat segment)
WARNING! You lose if you chop the message length too short!
Hint: Plan in advance before cutting (no room for error).

48. KitKat KLA Challenge ?

49. KitKat KLA Challenge Solution: