1. The Model & Basic Computations
Paola Flocchini
The Model & Basic Computations
Chapter 1 and 2
The Model
Broadcast
Spanning Tree Construction
Traversal
Wake-up
Paola Flocchini

2. Distributed Environment
Paola Flocchini
Distributed Environment
Multiplicity
Autonomy
clock
memory
1+2 =3
computing capabilities
Interaction
typically by
exchange of messages
Paola Flocchini

3. The Model Collection of entities that communicate
Paola Flocchini
The Model
Collection of entities that communicate
by exchanging messages
communication
link
entity, node, site ...
(processor, process,
object ….)
message-passing
Paola Flocchini

4. Entity x registers In memory: state(x) value(x) Possible operations:
Paola Flocchini
Entity x
registers
In memory:
state(x) value(x)
Possible operations:
- local storage and processing
- transmission of messages
- (re)setting the clock
- changing the value of the registers
Paola Flocchini

5. Entity x state(x) Finite set of possible system states
Paola Flocchini
Entity x
state(x)
Finite set of possible system states
(ex: {idle, computing, waiting, processing ….})
Always defined
At any time an entity is in one of these states
Paola Flocchini

6. External Events The behavior of an entity is reactive:
Paola Flocchini
External Events
The behavior of an entity is reactive:
triggered by events
Possible events:
clock tick
receiving a message
spontaneous impulse
Paola Flocchini

7. Paola Flocchini
clock ring
Paola Flocchini

8. Paola Flocchini
message arrival
Paola Flocchini

9. Paola Flocchini
spontaneous impulse
Paola Flocchini

10. The reaction of an entity depends on the event and on its state
Paola Flocchini
The reaction of an entity
depends on the event and on its state
State x Event
Action
Paola Flocchini

11. Actions Action: sequence of activities, e.g., computing
Paola Flocchini
Actions
Action: sequence of activities, e.g.,
computing
sending message
change state
an action is atomic
the activities cannot be interrupted
an action is terminating
the activities must terminate within finite time
Paola Flocchini

12. Entity Behavior Rule State x Event Action
Paola Flocchini
Entity Behavior
Rule
State x Event
Action
Behavior B(x) = set of rules of entity x for all possible events and all possible states
The algorithm
The protocol
DETERMINISTIC
(state, event) --> only ONE action
COMPLETE
( (state, event)  an action)
Paola Flocchini

13. System Behavior B = { B(x) : x  E}
Paola Flocchini
System Behavior
B = { B(x) : x  E}
A system is SYMMETRIC (or homogeneous)
if all the entities have the same behavior
B(x) = B(y),  x,y  E
Property: Every system can be made symmetric
Paola Flocchini

14. role =(workstation/server )
Paola Flocchini
Workstation
Very different rules, states ….
Workstation
Workstation
Server
Workstation
If role = workstation
ActionW(s,e)
else
ActionS(s,e)
s x e
role =(workstation/server )
Paola Flocchini

15. Communication Message: finite sequence of bits Communication Network:
Paola Flocchini
Communication
Message:
finite sequence of bits
Communication Network:
Paola Flocchini

16. Communication Point-to-point Model No(x) = out-neighbors of entity x
Paola Flocchini
Communication
Point-to-point Model y x w
No(x) = out-neighbors of entity x
Ni(x) = in-neighbors of entity x
N(x) = No(x)  Ni(x) z
Graph describing the
COMMUNICATION TOPOLOGY
G = (V, A)
V: Entities
A: Arcs defined by N
Paola Flocchini

17. An entity x can send a message only to its out-neighbors No(x)
Paola Flocchini
An entity x can send a message only to its
out-neighbors No(x)
and receive from the in-neighbours Ni(x)
Paola Flocchini

18. Axioms Finite Transmission Delays
Paola Flocchini
Axioms
Finite Transmission Delays
In absence of faults a message from x to
its out-neighbour y reaches y in finite time x y
Local orientation
Each entity distinguishes
among its neighbors
Paola Flocchini

19. Local orientation: more precisely
Paola Flocchini
Local orientation: more precisely
Each entity distinguishes among its out-neighbors 1 2 3
Send Message to 3
distinct labels =
(out) port numbers
Paola Flocchini

20. Local orientation: more precisely
Paola Flocchini
Local orientation: more precisely
Each entity distinguishes among its in-neighbors a 2 5 b
distinct labels =
(in) port numbers
When a message arrives, the entity can detect from which port
Paola Flocchini

21. Local orientation: more precisely
Paola Flocchini
Local orientation: more precisely
for an edge (x,y) there are two labels: y x a 31
x(x,y)
y(x,y)
Topology = labeled graph (G, )
Paola Flocchini

22. Restrictions of the model: examples
Paola Flocchini
Restrictions of the model: examples
Communication Restrictions
queue
Message Ordering
In absence of failures, msgs transmitted
along the same link arrive in the same order.
(FIFO)
Paola Flocchini

23. Restrictions of the model: examples
Paola Flocchini
Restrictions of the model: examples
Communication Restrictions
Bidirectional Links
 x, Ni(x) = No(x) =N(x) and
 y  N(x): x(x,y) = x(y,x) d a x y a c b d d a b b b b c a c z a
Paola Flocchini

24. Restrictions of the model: examples
Paola Flocchini
Restrictions of the model: examples
Reliability Restrictions:
1. Guaranteed delivery:
Any message that is sent will be received uncorrupted
2. Partial Reliability:
There will be no failures
3. Total Reliability:
No failures have occurred nor will occur
…..
Paola Flocchini

25. Restrictions of the model: examples
Paola Flocchini
Restrictions of the model: examples
Topological restriction:
The graph G is strongly connected
…..
Knowledge Restrictions
Knowledge of number of nodes
Knowledge of number of links
Knowledge of diameter ….
…..
Paola Flocchini

26. Restrictions of the model: examples
Paola Flocchini
Restrictions of the model: examples
Time restriction:
Bounded Communication Delay:
There exists a constant  such that, in absence of failures,
the communication delay of any message on any link is
at most 
Synchronized clocks:
All local clocks are incremented by one unit simultaneously
and interval are constant
…..
Paola Flocchini

27. Complexity measures - Performance
Paola Flocchini
Complexity measures - Performance
1. Amount of communication
point of view
of SYSTEM
number of messages exchanged
(finer granularity: number of bits)
2. Time
point of view
of USER
Communication delays are in general
unpredictable !!!
Ideal time:
1 unit of time to transmit 1 message
Paola Flocchini

28. Assumptions = Restrictions
Paola Flocchini
Example - Broadcast
Assumptions = Restrictions
Unique Initiator
Total reliability
Bidirectional links
G is connected
By definition of problem
Simplifying assumptions
Otherwise unsolvable
Paola Flocchini

29. Paola Flocchini
Algorithm FLOOD
The idea: If an entity knows something, it sends the info to its neighbours
One entity is INITIATOR, the others are SLEEPING
INITIATOR
spontaneously
send(I) to N(x)
SLEEPING
receiving(I)
send(I) to N(x)
Paola Flocchini

30. The idea: If an entity knows something, it sends
Paola Flocchini
The idea: If an entity knows something, it sends
it to its neighbours except the sender
INITIATOR
spontaneously
send(I) to N(x)
SLEEPING
receiving(I)
send(I) to N(x) - {sender}
It does not terminate
Paola Flocchini

31. S = {initiator, sleeping, done}
Paola Flocchini
S = {initiator, sleeping, done}
Algorithm for node x:
INITIATOR
spontaneously
send(I) to N(x)
become(DONE)
SLEEPING
receiving(I)
send(I) to N(x) – {sender}
become(DONE)
Paola Flocchini

32. Algorithm FLOOD Algorithm for node x: INITIATOR spontaneously
Paola Flocchini
Algorithm FLOOD
Algorithm for node x:
INITIATOR
spontaneously
send(I) to N(x)
become(DONE)
SLEEPING
receiving(I)
send(I) to N(x) – {sender}
become(DONE)
DONE
Paola Flocchini

33. Algorithm FLOOD Algorithm for node x: State If INITIATOR spontaneously
Paola Flocchini
Algorithm FLOOD
Algorithm for node x:
State
If INITIATOR
spontaneously
send(I) to N(x)
become(DONE)
If SLEEPING
receiving(I)
send(I) to N(x) – {sender}
become(DONE)
If DONE
do-nothing
Paola Flocchini

34. Algorithm for node x: Event If INITIATOR spontaneously send(I) to N(x)
Paola Flocchini
Algorithm for node x:
Event
If INITIATOR
spontaneously
send(I) to N(x)
become(DONE)
If SLEEPING
receiving(I)
send(I) to N(x) – {sender}
become(DONE)
If DONE
do-nothing
Paola Flocchini

35. Algorithm for node x: Action If INITIATOR spontaneously
Paola Flocchini
Algorithm for node x:
If INITIATOR
spontaneously
send(I) to N(x)
become(DONE)
Action
If SLEEPING
receiving(I)
send(I) to N(x) – {sender}
become(DONE)
If DONE
do-nothing
Paola Flocchini

36. Paola Flocchini
Example
Paola Flocchini

37. Correctness Termination The Algorithm terminates in finite time
Paola Flocchini
Correctness
The Algorithm terminates in finite time
It follows from: G connected and total reliability
Termination
Local Termination: when DONE
Global Termination: when?
Paola Flocchini

38. Message complexity Worst Case  |N(x)| = 2m m = number of links
Paola Flocchini
Message complexity
Worst Case
Worst for all possible initiators
and for all possible executions
m = number of links
Messages:  2 on each link
 2m
O(m)
More precisely:
Let s be the initiator
|N(s)| +  (|N(x)|-1)
x s
=  |N(x)| -  1 x
x s
 |N(x)| = 2m x
= 2m - (n-1)
Paola Flocchini

39. Time Complexity - Ideal Time Worst Case
Paola Flocchini
Time Complexity - Ideal Time
Worst Case
Worst for all possible initiators
and for all possible executions
Time: (ideal time)
Max{d(x,s)} = eccentricity of s
O(n) x
 Diameter(G)  n-1
Paola Flocchini

40. Time and Events External events: spontaneously receiving when (clock)
Paola Flocchini
Time and Events
External events:
spontaneously
receiving
when (clock)
Actions may generate events
send generates receiving
set-clock generates when
Generated events might not occur (in case of faults).
If they occur, they occur later.
In the case of receiving with some unpredictable delay.
Paola Flocchini

41. Different delays ---> different executions
Paola Flocchini
Different delays ---> different executions
Different executions could have different outcomes
(Spontaneous events are considered generated
before execution starts: initial events)
An executions is fully described by the sequence of events
that have occurred
Paola Flocchini

42. Time x Event diagram Spontaneous event receiving event x y time
Paola Flocchini
Time x Event diagram
Spontaneous event
receiving event
time x
as seen by an observer y
Paola Flocchini

43. Example: Time x Event diagram of Flooding
Paola Flocchini
Example: Time x Event diagram of Flooding
One possible execution x y z x y
Paola Flocchini z

44. Example: Time x Event diagram of Flooding
Paola Flocchini
Example: Time x Event diagram of Flooding
Another execution x y z x y
Paola Flocchini z

45. Knowledge P = fact; x = entity; S = set of entities.
Paola Flocchini
Knowledge
P = fact; x = entity; S = set of entities.
Local knowledge LK
P  LK(x).
Implicit knowledge IK
P  IK(S) if x  S: P  LK(x).
Explicit knowledge EK P  EK(S) if x  S: P  LK(x). x P S x P P P S P P P P
Paola Flocchini

46. x  S (x  S, P  LK(x))  LK(x) 
Paola Flocchini
Common knowledge CK
P  CK(S) if
x  S, P  LK(x) 
x  S (x  S, P  LK(x))  LK(x) 
x  S ((x  S, P  LK(x))  LK(x))  LK(x)  ...
Paola Flocchini

47. Examples Implicit knowledge Explicit knowledge I am in a ring
Paola Flocchini
Examples
Implicit knowledge
I am in a ring
Explicit knowledge
I am in a ring
I am in a ring
I am in a ring
I am in a ring
Paola Flocchini

48. Common knowledge I know I’m in a ring, everyone knows it,
Paola Flocchini
Common knowledge
I know I’m in a ring,
everyone knows it,
everyone knows that
everyone knows it, ... ….
Paola Flocchini

49. muddy forehead t=1 t=2 t=3 t=4
Paola Flocchini
How to reach common knowledge in FINITE TIME ?
P  CK(S) if
x  S, P  LK(x) 
x  S (x  S, P  LK(x))  LK(x) 
x  S ((x  S, P  LK(x))  LK(x))  LK(x)  ...
muddy forehead
I can see you disobeyed
t=1
t=2
t=3
t=4
Paola Flocchini

50. Paola Flocchini
Paola Flocchini

51. I see one dirty forehead
Paola Flocchini ??
I see one dirty forehead
Paola Flocchini

52. I see everybody else is clean
Paola Flocchini
I see everybody else is clean
I must be dirty !!!!!
Paola Flocchini

53. Paola Flocchini
t=1
Paola Flocchini

54. Paola Flocchini ?
t=1
Paola Flocchini

55. Paola Flocchini
The other didn’t go forward … I must be dirty too !!!! ? ?
t=2 ? ? ? ? ? ? ? ? ?
The other didn’t go forward …
I must be dirty too !!!!
Paola Flocchini

56. Paola Flocchini
t=2
Paola Flocchini

57. Paola Flocchini
In general ….
If I see k dirty foreheads and they do not go forward all together at time k, I go forward at time k+1
Paola Flocchini

58. Some types of knowledge
Paola Flocchini
Some types of knowledge
Topological knowledge
Graph type (“G is a ring”...), adjacency matrix of G ...
Metric knowlege
Number of nodes, diameter, eccentricity...
Sense of direction
Information on link labels
Information on node labels
As the available knowledge grows, the algorithm becomes less portable (rigid). Generic algorithms do not use any knowledge.
Paola Flocchini

59. Example: impact of knowledge
Paola Flocchini
Example: impact of knowledge
In specific topologies flooding can be avoided
and broadcast can be much more efficient
(if the topology is known).
What is the complexity of flooding in a complete graph ?
How can it be done more efficiently ?
What is the complexity of flooding in a tree ?
Can it be done more efficiently ?
Paola Flocchini

60. Example: The labeled hypercube
Paola Flocchini
Example: The labeled hypercube 01 00 1 11 10 2 1
010
011
101
000
001
111
110
100 1 3 2
Each link between two nodes is labeled by the dimension of the bit by which the nodes‘ name differ.
Paola Flocchini

61. x2x1x0 X= xkxk-1 ….. x1x0 first bit K-bit name 010 011 101 000 001 111
Paola Flocchini
010
011
101
000
001
111
110
100 1 3 2
x2x1x0 X=
xkxk-1 ….. x1x0
K-bit name
first bit
Paola Flocchini

62. Paola Flocchini 1 3 2
0000
0001 4 4
0010
0011 4
0100
0101 4
0110
0111 4 4 4 1 1 2
1000
1001 4 2 3 2 3
1010 1 1
1011 2 3 3 4 3 4 3 1 1 2
1100
1101 3 3 2 4 2 4
1111 2
1110 1 1
Paola Flocchini

63. A hypercube of dimension k has n = 2k nodes Each node has k links
Paola Flocchini
A hypercube of dimension k has n = 2k nodes
Each node has k links
 m = n k/2 = O(n log n)
Flooding would cost O(n log n)
Paola Flocchini

64. HyperFlood - Efficient Broadcast
Paola Flocchini
HyperFlood - Efficient Broadcast
The initiator sends the message to all its
neighbours
2) A node receiving the message from link l,
sends it only to links with label l’ < l
Paola Flocchini

65. For each pair of nodes x and y there exists a
Paola Flocchini
Correctness
Every node is touched
Based on the lemma:
For each pair of nodes x and y there exists a
unique path of decreasing labels X=
xk,xk-1 ….. x1,x0
yk,yk-1 ….. y1,y0 Y=
Paola Flocchini

66. For each pair of nodes x and y there exists a
Paola Flocchini
Correctness
Every node is touched
Based on the lemma:
For each pair of nodes x and y there exists a
unique path of decreasing labels X=
xk,xk-1 ….. x1,x0
yk,yk-1 ….. y1,y0 Y=
Consider positions where they differ in decreasing order …
Paola Flocchini

67. Paola Flocchini
Example: 5 8 4 x 3 8 X= 5 Y= 4 3 x y y
Paola Flocchini

68. ---> the messages create a spanning tree ….
Paola Flocchini
---> the messages create a spanning tree ….
and every node is touched 1 1 2 4 4 2 3 2 3 2 1 1 3 3 4 3 3 1 1 2 3 3 2 4 2 2 1 1 4 4 4 1 1 2 4 2 3 2 3 2 1 1 3 3 3 4 4 3 1 1 2 3 3 2 4 2 4 2 1 1
Paola Flocchini

69. Complexity: n-1 (OPTIMAL) Because every entity receives the info
Paola Flocchini
Complexity: n-1 (OPTIMAL)
Because every entity receives the info
only ONCE.
Paola Flocchini

70. General Flooding: 2m - (n-1)
Paola Flocchini
In Special Topologies
General Flooding: 2m - (n-1)
Ad-hoc algorithm in hypercube: (n-1)
Ad-hoc algorithm in complete network: (n-1)
In the tree Flooding is optimal: (n-1)
Paola Flocchini

71. State x Event ---> Action
Paola Flocchini
Important Facts
State x Event ---> Action
(x,t) = internal state of entity x
content of memory (registers, clock, …) at time t
Paola Flocchini

72. If the same event happens to x at time t in two different
Paola Flocchini
E1, E2: different environments a a
1 = 2 E1 E2
If the same event happens to x at time t in two different
executions and if the internal states 1 and 2 of x in
the two executions at that time are equal, then
the new internal state of x will be the same in both
executions
Paola Flocchini

73. 2) If the same event happens to x and y at time t in the
Paola Flocchini a a x y
(x) = (y)
2) If the same event happens to x and y at time t in the
same execution and if the internal states (x) and (y)
are equal, then the new internal states of x and y will be
the same.
Paola Flocchini

74. Theorem: Under the set of assumptions:
Paola Flocchini
An example
Back to Broadcast …
Theorem: Under the set of assumptions:
unique Initiator
G is connected
no failures
bidirectional links
Every generic broadcast protocol requires, in the worst
case, m messages.
Paola Flocchini

75. Lower Bounds for Broadcast
Paola Flocchini
Lower Bounds for Broadcast
Proof.
m(G) = n. of edges in G
By contradiction.
Let A be an algorithm that broadcasts exchanging
less than m(G) messages (in all executions, and
for any graph G) under those assumption.
Then there is at least a link in G where no messages
are sent.
Paola Flocchini

76. Let e = (x,y) be such a link. G
Paola Flocchini
G=(V,E)
Let e = (x,y) be such a link. G y x y x G'
Construct a new graph G’ z
G' =(V {z}, E-e  {(x,z),(y,z)}
(remember: n is unknown)
Execute the same algorithm on G’ with the same
time delays, same initial internal states for all
nodes except for z which is sleeping
Paola Flocchini

77. Two executions in two environments
Paola Flocchini
Two executions in two environments E1 E2 G G' y y x x z
For all nodes, except z, the two executions are identical
x and y never send to each other in E1
--->
x and y never send to z in E2
Paola Flocchini

78. Within finite time the protocol terminates
Paola Flocchini E1 E2 G G' y y x x z
Within finite time the protocol terminates
but in E2 node z will never be reached.
Paola Flocchini

79. 1) Dense networks = more messages
Paola Flocchini
Observations:
1) Dense networks = more messages
(ex. in complete networks m = n (n-1) …)
2) It is optimum in acyclic graphs
Idea: to solve broadcast.
1. Build a spanning tree of G
2. Execute flooding
Spanning Tree construction Problem
Paola Flocchini