1. Broadcast & Convergecast Downcast & Upcast
Distributed Algorithms for Multi-Agent Networks
Instructor: K. Sinan YILDIRIM

2. Broadcast from a Single Source
A broadcast operation is initiated by a single processor, the source.
The source wants to send a message to all other nodes in the system.
For any broadcast algorithm
The message complexity of broadcast is at least n – 1.
The source's radius is a lower bound for the time complexity.
PROOF: Every node must receive the message.

3. Tree Broadcast
The message is transmitted from the root to all its children
each internal vertex of the tree getting the message from its parent forwards it to all its children.
any broadcast algorithm can also be used to build a tree in the network.
The message complexity: lower bounded by n - 1
The time complexity: lower bounded by the depth of the tree

4. Flooding Algorithm
The message complexity: lower bounded by the number of edges
The time complexity: lower bounded by the radius of r.
PROOF: If you do not try every edge, you might miss a whole part of the graph behind it.

5. Spanning Tree
Flooding Algorithm implicitly constructs a directed tree T rooted at source r
spanning all vertices of the network
the parent of each non-root vertex is the vertex from which it has received the message for the first time.
synchronous setting - BFS Tree
asynchronous setting

6. Convergecasts
The process of collecting information upwards on a tree is called convergecast
Enables the source to detect the fact that the broadcast operation has completed.
termination detection “This subtree has finished the task.”
collecting back acknowledgements in a leveled fashion on a spanning tree.
The message complexity: is the number of edges
The time complexity: is the depth of the spanning tree

7. Flooding/Echo Algorithm
It is possible to get a broadcast algorithm with acknowledgements
the source can detect the termination of the broadcast
Flooding algorithm
constructs a spanning tree T
this tree can be used for the convergecast
a vertex sends Ack to its parent only after the subtree of T rooted at v has been constructed and all its vertices have received the message.
The message complexity of broadcast is at least n - 1
The source's radius is a lower bound for the time complexity.
PROOF: - Every node sends one message to each of its neighbors and one to its parent.
- The flooding algorithm requires at least a time complexity of the radius of the source to reach all nodes. The Echo phase requires the same amount of time.

8. Flooding/Echo Algorithm State Machine
COLLECT
CHILD
INIT
COLLECT
ECHO
TERMINATE

9. Rule of Thumb In general, considering the line topology
Gives a lower bound for the message complexity
An upper bound for the time complexity
considering fully-connected network
Gives an upper bound for the message complexity
A lower bound for the time complexity

10. Global Function Computation
Suppose that each vertex in the graph holds an input
We would like to compute some global function of these inputs.
Semigroup function
f is well-defined for any subset of the inputs
f is associative and commutative
A semigroup function can be computed efficiently on a tree T by a convergecast process.
the value sent upwards by each vertex in the tree will be the value of the function on the inputs of the vertices in its subtree T 5

11. Converge Algorithm F 4 F T 1 2 T F T 14 12 3 F 4 F T 5 42

12. Downcast Root has m distinct items
each destined to one specific vertex in the tree.
The time complexity of downcast is at least Depth(T) and at most n + Depth(T)
PROOF: (1) If the last message’s destination is the node that is the furthest (2) or the closest

13. Upcast
Data items are initially stored at some of the vertices of the tree T.
The goal is to end up with all the items stored at the root of the tree.
The time complexity of upcast is at most Depth(T)
PROOF: Consider line topology.

14. Applications - I Smallest k-of-m
collect the smallest k elements at the root.
The global function computation scheme
First, find the minimum element and inform all the vertices by broadcasting it throughout the tree.
Now, find the next smallest element by the same method and so on.
This should take O(kDepth(T)) time.
An alternative and faster
Every vertex keeps the elements it knows of in an ordered list.
In each step, each vertex sends to its parent the smallest element that hasn't been sent yet.
Upcasting the k smallest elements on a tree T can be performed in Depth(T) + k time.

15. Applications - II Information gathering and dissemination
Disseminating information among the vertices of a tree.
Suppose that m data items are initially stored at some of the vertices
The goal is to end up with each vertex knowing all the items
collect the items at the root of the tree
then broadcast them one by one.
the broadcast phase can be completed within O(m + Depth(T))
collecting phase can be performed by an upcast operation with a similar complexity.
The entire problem can be solved in time O(m + Depth(T)).

16. Applications - III Route-disjoint matching
Our goal is to find a matching of these vertices into pairs such that the routes connecting them are all edge-disjoint.
"Finding" the matching means that each vertex in W should know the identifier of the vertex with which it is paired

17. Applications - IV The token distribution problem
n tokens are initially distributed among the n vertices of the tree
with no more than K at each processor.
redistribute the tokens
so that each processor will have exactly one token.
The cost of the entire redistribution process
equals the sum of the distances traversed by the tokens in their way to their destinations.