CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

1
CPSC 689 Discrete Algorithms for Mobile and
Wireless Systems

• Spring 2009
• Prof. Jennifer Welch

2
Lecture 15

• Topic
• Link Reversal
• Sources
• Gafni Bertsekas
• Busch et al.
• Charron-Bost et al.
• Park Corson
• MIT 6.885 Fall 2008 slides

3
Motivation for Link Reversal

• Assume communication channels in the system can
  be modeled as an undirected graph (i.e.,
  bidirectional links)
• Link reversal algorithms impose logical
  directions on the edges (or links) of the graph
  at certain times the directions on some edges are
  reversed to achieve some goal.
• Example Suppose goal is to send info to a
  distinguished node D.

4
Outline

• Gafni Bertsekas (1981) introduced link
  reversal algorithms for constructing and
  maintaining paths to a destination node and
  proved their correctness.
• Busch, Surapaneni Tirthapura (2003) analyzed
  the complexity of the GB link reversal algorithms
  --- worst-case time and work.
• Charron-Bost, Gaillard, Welch Widder (2008)
  gave a more fine-grained analysis of work
  complexity for a more general algorithm.
• Park Corson (1997) modified the GB partial
  reversal algorithm to detect when nodes are
  partitioned from the destination --- result is
  the Temporally Ordered Routing Algorithm (TORA)
  for MANETs.

5
Motivation GB

• Network contains a distinguished central
  station.
• Central station collects information from other
  nodes via contingency routes.
• Construct contingency routes that
• provide redundant routes to station
• are loop-free
• dont require flooding
• rely only on local information

6
Abstract Graph Problem
Definition a directed acyclic graph (DAG) with a
distinguished node D is destination-oriented if D
is the one and only sink in the graph. Problem
statement Given a DAG with distinguished node D
that is not destination-oriented, convert it into
a destination-oriented DAG by reversing the
directions of some of the links.
7
Full Reversal Algorithm

• Whenever a node is a sink (but not the
  destination), reverse all its incident links,
  i.e., all its incident links become outgoing
• Assuming perfect knowledge of state of the link
  and ability to instantaneously change it
• some other algorithms relax this assumption

8
Full Reversal Algorithm Example
9
Implementation of Full Reversal Agorithm The
Pair Algorithm

• Each node i keeps a pair (a,i), where a is an
  integer pair is called height
• Link between two nodes is considered to be
  directed from node with higher height to node
  with lower height
• At each iteration, if a node i has no outgoing
  links, then set a to 1 greater than the maximum
  a-value of all of is neighbors at the previous
  iteration.

10
Pair Algorithm Example
11
Partial Reversal Algorithm

• Try to reduce the number of link reversals by
  having a sink node reverse only some of its
  incident links.
• Whenever a node is a sink (but not the
  destination)
• reverse all incoming links that have not been
  reversed since last time this node did a reversal

12
Partial Reversal Algorithm Example
13
"Implementation" of Partial Reversal Algorithm
The Triple Algorithm

• Use a triple (a,b,i) for the height.
• At each iteration, if a node i has no outgoing
  links, then, using the neighbors heights from
  the previous iteration,
• set a to 1 greater than the minimum a-value of
  any neighbor
• set b to 1 less than the smallest b-value of all
  neighbors with the new a-value (if no such
  neighbor then dont change b)
• Why does this implement partial reversal?

14
Triple Algorithm Example
15
Analysis of Link Reversal Algorithms

• GB proved correctness of a general class of
  algorithms that includes the full and partial
  reversal algorithms Every algorithm in the
  class terminates in a finite number of iterations
  with a destination-oriented DAG.

16
GB Correctness Proof

• Define an abstraction of the heights taken from
  a set A that is
• countable infinite
• totally ordered
• partitioned into n disjoint subsets, one per node
  (other than D), s.t.
• each subset is countably infinite and unbounded
• there is an operator on each subset under which
  it is an Abelian group
• Link (i,j) is considered to be directed from node
  i to node j if ai is larger than aj in the total
  order

17
General Algorithm GB

• Given a vector v (v1, v2, , vn), where vi is
  the (generalized) height of node i, algorithm M
  has these properties
• M(v) is a set of vectors (nondeterministic)
• if no sinks in v, then M(v) is empty
• for each v' in M(v)
• either v'i vi
• or v'i gi(vi) (only allowed if node i is a sink
  in v)
• gi(vi) gt vi and depends only on heights in v of
  node i and i's neighbors
• some unbounded increase condition for gi (A3)

18
Why Nondeterminism?

• Execution of algorithm is a sequence of vectors,
  starting with some initial assignments of
  heights, where each vector v' in the sequence is
  contained in M(v), where v is the preceding
  vector
• If there is one or more sinks in the graph, then
  at each step, one or more of the sinks can take a
  step
• timing and order of steps is immaterial
• captures asynchronous execution

19
Correctness of General Algorithm

• Acyclic follows because the heights are totally
  ordered
• Proposition 1 algorithm terminates in a finite
  number of iterations (no sinks, other than the
  destination D)
• since acyclic and no sinks, every node has a
  directed path to D
• Proposition 2 once a node has a directed path
  to D, it is never is a sink

20
Complexity of Link Reversal Algorithms

• GB proved correctness of a general class of
  algorithms that includes the full and partial
  reversal algorithms Every algorithm in the
  class terminates in a finite number of iterations
  with a destination-oriented DAG.
• But how many iterations? I.e., what is the
  complexity?

21
BST, BT Complexity Results
Worst case bounds
algorithm time work
pair algorithm Q(n2) Q(n2)
triple algorithm Q(n a n2) Q(n a n2)
better

• time number of iterations with maximum
  concurrency
• work number of link reversals
• n number of nodes with no path to destination
• a max a min a in initial state

22
Counting Reversals

• Both examples had all sink nodes reversing in
  parallel at each iteration.
• GB, and BST,BT, show that the order in which
  reversals are done is immaterial -- resulting DAG
  is always the same.
• BST,BT proof uses notion of a dependency graph
  of reversals
• number of node reversals is number of vertices in
  dependency graph
• time is at least the depth of the dependency graph

23
Upper Bounds

• Pair algorithm BST,BT prove that each node
  reverses at most n times, implying
• O(n2) reversals
• taking O(n2) time (if sequential)
• Triple algorithm BST,BT prove that each node
  reverses at most a n times, so
• O(n a n2) reversals
• taking O(n a n2) time (if sequential)

24
Lower Bounds for Pair Algorithm
Bad graph for work
node i does i reversals gt W(n2) reversals
Bad graph for time
BST,BT prove each node in right half of chain
reverses n/2 times. These reversals must be
sequential since neighbors cannot be sinks
simultaneously gt W(n2) time.
25
Lower Bounds for Triple Algorithm
Bad graph for work (links alternate direction)
BST prove node i reverses Q(a i) times gt
W(n a n2) reversals
Bad graph for time
BST prove each node in the red oval reverses
W(a n) times. These reversals must be
sequential since neighbors cannot be sinks
simultaneously gt W(n a n2) time.
26
Some Additional Questions

• What about the original, abstract, full and
  partial reversal algorithms?
• not hard to show that the pair algorithm
  implements full reversal
• not at all straightforward to see that the triple
  algorithm implements partial reversal
• Are unbounded counters necessary for doing link
  reversal?
• Is the generalized height approach useful for
  developing alternative algorithms, or would some
  other formalism be better?
• Can we find the exact expression for the time and
  work complexity of link reversal algorithms for
  any node and in any graph?

27
CGWW

• New formulation of FR and PR using only binary
  labels on the links
• Simple distributed algorithm for finding routes
  in acyclic graphs
• Identify sufficient conditions on initial
  labeling for correctness
• FR and PR are special cases
• Easy to state new algorithms (use different
  initial labelings)
• Much simpler proof of correctness
• first known proof that PR preserves acyclicity

28
LR Algorithm Schema

• Input is a directed acyclic graph G with
• distinguished node D
• each link labeled with 0 ("unmarked) or 1
  ("marked")
• while there exists a sink v ? D do
• if v has an incident unmarked link then
• reverse all incident unmarked links
• flip the labels on all incident links
• else // all of v's incident links are marked
• reverse all incident links // leave them marked

LR1
LR2
29
LR Example
30
Special Cases of LR Algorithm

• Full Reversal Initially all labels are 1
• Only ever execute LR2 step
• All labels are always 1
• Partial Reversal Initially all labels are 0
• Execute both LR1 and LR2
• Labels change

31
PR Example with LR
32
Basic Definitions

• Given undirected graph G
• G is a directed version of G
• a chain in G corresponds to an (undirected) path
  in G might not have all links directed the same
  way
• a circuit in G corresponds to an (undirected)
  cycle in G might not form a directed cycle

33
LR Terminates

• Theorem LR terminates.
• Proof At least one node takes a finite number
  of steps (D). Suppose in contradiction at least
  one node takes an infinite number of steps.

finite number of steps
infinite number of steps
Case 1 After u's last step, link is directed
toward u.
v cannot become a sink again, contradiction.
34
LR Terminates
Case 2 After u's last step, link is directed
toward v.
Case 2.1 After v's next step, link is directed
toward u.
v cannot become a sink again, contradiction.
35
LR Terminates
Case 2 After u's last step, link is directed
toward v.
Case 2.2 After v's next step, link is still
directed toward v. Then it must be labeled 0.
After v's next step, link is directed toward u
(labeled 1).
v cannot become a sink again, contradiction.
36
LR Preserves Acyclicity

• Theorem If LR input is a DAG in which every
  circuit satisfies a certain condition AC, then
  every step maintains acyclicity.
• Theorem If every node of a DAG has the same
  label on all its incoming links (a "uniform"
  labeling), then every circuit satisfies condition
  AC.

37
Comparing Initializations
38
What about Complexity of LR?

• No systematic study until work of Busch,
  Surapaneni, and Tirthapura (2003)
• They studied work
• total number of reversals done by all nodes
• and time
• total number of rounds, assuming maximum
  concurrency for sinks reversing
• of GB's pair and triple algorithms
• Results were of this form for every n, there
  exists a graph with n nodes in which ?(f(n))
  reversals are done / ?(f(n)) rounds are executed.

39
BST,BT Bounds Again
Worst case bounds
algorithm time work
pair Q(n2) Q(n2)
triple Q(n a n2) Q(n a n2)

• time number of iterations
• work number of node reversals
• n number of nodes with no path to destination
• a max a min a in initial state

40
CGWW Contributions

• Expression for the exact number of steps (work)
  taken by any node in any graph during the generic
  algorithm
• Expression depends only on the input graph
• Has simple formulas when specialized to FR and PR
• Exact formula helps in finding best and worst
  topologies

41
Work Complexity Overview

• For any node v, define a nonnegative quantity ?
• Every time v takes a step, ? decreases by either
  1 or 2
• When other nodes take steps, ? is unchanged
• Once v has a directed path to D, ? 0

42
Work Complexity Pattern for FR
Number of reversals by node v equals number of
links directed away from D in the chain to v.
Quantity decreases by 1 when v takes a step.
43
Work Complexity Overview

• Break ? into 2 pieces ? ? - ?
• Show ? never changes
• Show ? increases by either 1 or 2 when v takes a
  step
• Increment depends on a classification of v into
  one of 3 groups (S, N, or O)
• Thus ? decreases by either 1 or 2 whenever v
  takes a step

44
Work Complexity Overview

• Consider any execution of LR starting with a
  graph whose circuits all satisfy
• condition AC.
• Let k be initial value of ? for node v.

initial group of v number of reversals by v
S (k1)/2
N k/2
O k
45
Work Complexity of FR

• Corollary Number of steps taken by v in FR is
  min, over all chains between v and D, of number
  of links directed away from D.
• Worst case graph is chain with all edges directed
  away from D total work complexity is n(n1)/2

46
Work Complexity of PR

• Corollary Number of steps taken by v in PR is
• min, over all chains b/w v and D, of number of
  nodes on the chain with two incoming unmarked
  chain links, if v is a sink or a source
• (plus 1 in some cases)
• If v is neither a sink nor a source, then the
  number of steps is twice the above quantity
• Worst-case graph is chain with links that
  alternate direction total work complexity is
  n2/41 (for even n)

47
Discussion of CGWW

• Fresh approach to specify and analyze link
  reversal algorithms of GB
• Developed a generalization of the algorithms
• bounded space complexity
• Exact expression for work complexity of any node
  in any graph for general algorithm
• BST,BT work was exact for pair algorithm but
  not exact for triple algorithm

48
Open Question

• Time complexity can we come up with a formula
  for the last round at which a given node takes a
  step (assuming all sinks take steps
  simultaneously at each round)?

49
Effect of Partitions

• The full and partial reversal algorithms work if
  nodes are not partitioned from the destination.
• What happens if they are?
• Full reversal

50
Park Corson, 1997 (TORA)

• Temporally Ordered Routing Algorithm
• Modify GB partial reversal algorithm for routing
  messages in mobile ad hoc networks
• Use n copies of the link reversal algorithm, one
  for each possible destination node.
• Detect when a node has been partitioned from a
  destination and stop trying to sending messages
  to it
• Height is a 5-tuple.

51
TORA System Assumptions

• Undirected communication graph G
• Nodes have unique ids.
• Bidirectional communication on all edges.
• Set of links changes with time.
• Nodes may also fail.
• Nodes know their neighbors (because of some lower
  level protocol).
• Nodes communicate through message broadcast with
  all neighbors.
• Transmitted packets are received correctly and in
  order of transmission, but with some delay
• Different than the previous papers

52
TORA Overview

• Creating routes
• A node initiates establishment of a route to
  destination only if all its adjacent links are
  undirected.
• Route creation essentially assigns directions to
  undirected links.
• Maintaining routes
• When directed portions of the graph deviate from
  being destination oriented, perform reversals to
  try to restore this property.
• Erasing routes
• Detect partitions, and accommodate, by clearing"
  all the links in the portion disconnected from
  the destination, that is, making them undirected
  again.

53
Heights in TORA
reference level
delta
t , oid , r , d , i
54
Maintaining Routes in TORA

• If node i loses last outgoing link
• due to a link failure (Case Generate)
• set ref level to (current time, i, 0), a full
  reversal
• due to a height change and
• nbrs dont have same ref level (Case Propagate)
• adopt max ref level and set d to effect a partial
  reversal
• nbrs have same ref level with r 0 (Case
  Reflect)
• adopt new ref level, set r to 1, set d to 0 (full
  reversal)
• nbrs have same ref level with r 1 and oid i
  (Case Detect)
• Partition! Start process of erasing routes.

55
TORA Example Partition
56
TORA Example in Detail
(0,0,0,0,6)
(0,0,0,2,5)
Link (3,6) fails at time 8
5
6
4
1
2
3
(0,0,0,2,4)
(0,0,0,1,3)
(0,0,0,2,2)
(0,0,0,3,1)
(8,3,0,0,3)
Node 3 generates new ref level (8,3,0), bcasts
new height to all neighbors
57
TORA Example in Detail
(0,0,0,0,6)
(0,0,0,2,5)
5
6
4
1
2
3
(0,0,0,2,4)
(8,3,0,0,3)
(0,0,0,2,2)
(0,0,0,3,1)
(8,3,0,1,4)
(8,3,0,1,2)
Nodes 2 and 4 propagate ref level (8,3,0), bcast
new height to all neighbors
58
TORA Example in Detail
(8,3,1,0,5)
(0,0,0,0,6)
(0,0,0,2,5)
5
6
4
1
2
3
(8,3,0,0,3)
(8,3,0,1,2)
(0,0,0,3,1)
(8,3,0,1,4)
(8,3,1,0,1)
Nodes 1 and 5 reflect ref level (8,3,1), bcast
new height to all neighbors
59
TORA Example in Detail
(0,0,0,0,6)
(8,3,1,0,5)
5
6
4
1
2
3
(8,3,0,0,3)
(8,3,0,1,2)
(8,3,1,0,1)
(8,3,0,1,4)
(8,3,1,1,4)
(8,3,1,1,2)
Nodes 2 and 4 propagate ref level (8,3,1), bcast
new height to all neighbors
60
TORA Example in Detail
(0,0,0,0,6)
(8,3,1,0,5)
5
6
4
1
2
3
(8,3,1,1,4)
(8,3,0,0,3)
(8,3,1,1,2)
(8,3,1,0,1)
Node 3 detects partition from destination 6
61
Effects of Time Errors

• Clocks are used to establish temporal order of
  the link loss events.
• Logical clocks could also be used to establish
  this order.
• What happens if the logical time order isn't the
  same as the real time order?
• They claim that the resulting algorithm is still
  in the BG general class.
• They conclude that it's good to use clocks that
  are sufficiently synchronized so that the time
  between failures is much greater than the clock
  error.

62
Conclusion

• No simulation results. But, they do have a table
  of claimed worst-case complexity of many
  algorithms.
• Possible improvements
• Over time, the DAG may become less optimally
  directed than it was initially.
• They claim that initially, the ?s give the
  minimum distance (in hops) to d, but this can
  degrade with changes.
• We could enhance the protocol by periodically
  propagating refresh packets outwards from d, to
  reset the reference level to 0 (?) and restore
  the meaning of the ?s to be the shortest
  distances
• Such a strategy could also support correction of
  corrupted state information.
• Refreshes should occur at a low rate, in the
  background.