Efficient and Robust Query Processing in Dynamic Environments Using Random Walk Techniques

1
Efficient and Robust Query Processing in
DynamicEnvironments Using Random Walk Techniques

• Chen Avin
• Carlos Brito

2
Outline

• Motivation
• Random Walk and Partial Cover Time
• Efficiency
• Robustness
• Quality
• Load Balancing, Scalability and Latency
• Discussion

3
Motivation

• Sensor Network as large, dense and dynamic
  networks
• Task Query the network
• Common systems depend on state information stored
  in the nodes for proper operation and control
  (i.e. spanning trees, cluster heads)
• Critical points of failure lead to recovery
  mechanism
• Explore the properties of uncontrolled scheme
  like random walk
• Simple process, no critical point of failure, all
  nodes are equally unimportant at all times

4
Random Walk

• Visiting the nodes of the graph in a random order
• At each step, a token moves to a neighbor with
  some distribution(simple uniform)

5
Random Walk for Sensor Nets

• Easily implemented in sensor networks base
  station issues a token with a query
• (almost) Assumption free method, the protocol
  does not require knowledge of
• Location
• Neighbors
• Transmission range
• Symmetric connection
• High density and redundancy are advantage

6
Cover Time

• Cover Time the expected time to visit all the
  nodes in a random walk (starting at the worst
  case node)
• How efficient is the process ?
• hij the expected time to go from node i to j
• hmax max (hij all nodes in the graph)
• Matthews Bound C hmaxlog(n)

7
Cover Time

• Known results
• Worst cases O(n3)
• Lollipop graph
• Line O(n2)
• Best cases O(nlog(n))
• Star
• Complete Graph
• Hypercube
• Grid O(nlog2(n))
• Random sensor networks ?

8
Partial Cover Time (PCT)

• In sensor network we dont need to consult every
  node
• How efficient is to visit 80 of the nodes ?

• LemmaPCT(c) O(hmax )

• O(n) in Hypercube
• O(nlog(n)) in Grid

9
Lemma Proof Sketch

• aV time when node v is first visited
• ? time when more than half of the nodes visited
• c expected time to visit more than half of the
  nodes E?

2k1
?
ai
aj

• (k1) ? ? aV
• E? 1/(k1) ?E aV (2k1)/(k1)hmax
• c lt 2hmax

10
Outline

• Overview of our approach
• Random Walk and Partial Cover Time
• Efficiency
• Robustness
• Quality
• Load Balancing, Scalability and Latency
• Discussion

11
Efficiency Simple Walk
16
14
12
10
8
Number of steps normalize to n
6
4
3.12
2
0
10
20
30
40
50
60
70
80
90
100
of Cover
12
Biased Random Walk

• Can we improve this results?
• Give priority to unvisited nodes
• Define bias parameter 0 bias 1
• Visited neighbor selected with probability
• (1- bias) / d
• Unvisited with
• (1- bias) / d bias / du
• The protocol remain (almost) the same

13
Biased Random Walk
8
Bias 0
7
6
5
4
3.12
Number of steps normalize to n
3
2
1
0
0
10
20
30
40
50
60
70
80
90
100
of Cover
14
Comparison with Clustering

• Analytical result for Cluster Head scheme shows
  that the number of messages for optimal protocol
  on grid require 0.945n7/6
• The efficiency ofboth systems issimilar

15
Outline

• Overview of our approach
• Random Walk and Partial Cover Time
• Efficiency
• Robustness
• Quality
• Load Balancing, Scalability and Latency
• Discussion

16
Robustness to Dynamics

• The probability that a node will fail when it has
  the token is negligible
• No critical point of failure (but do need
  reliable token passing)
• All we require is connectivity in the token
  neighborhood
• Robust to independent and dependent failures
  (disaster areas)

17
Spanning tree in dynamic env.

• Nodes close to the root are more important
• When a node fails all nodes in the sub-tree are
  disconnected from the root and must participate
  in recovery mechanism
• Assuming independent failure (or duty cycle)
  probability p, (q1-p) the expected number of
  nodes to report is O(qh)
• Since R ltlt network area, h is large
• p0.1. h10 ? 65 will not report to the root.

18
Outline

• Overview of our approach
• Random Walk and Partial Cover Time
• Efficiency
• Robustness
• Quality
• Load Balancing, Scalability and Latency
• Discussion

19
Quality of Partial Cover - 1

• How far are the unvisited nodes from visited ones
  ?
• 90 are atmost 2 hops
• Expected random walkwill not leavelarge area
  uncovered

20
Quality of Partial Cover - 2

• How long must a node wait before a walk will
  visit its neighborhood?
• 85 are visitedat most every other run
• At most will need to wait4 runs

21
Application Example

• Find the histogram of the data in the network
• Assume non uniform distribution
• Token report after seeing 80 of the nodes

22
Outline

• Overview of our approach
• Random Walk and Partial Cover Time
• Efficiency
• Robustness
• Quality
• Load Balancing, Scalability and Latency
• Discussion

23
Load Balancing

• The stationary distribution of the Markov chain p
  (p1, , pn) is pidi/2m
• In regular graphsp is uniform,but this only
  afterlong walks
• Here we issue manyshort walks

24
Scalability
25
Latency

• Random walk is sequential process
• The latency is proportional to the number of
  steps to accomplish the task
• Reduce the range of applicability
• Future work combine result from few parallel
  random walks in the network

26
Discussion

• Achieving control in highly dynamic env. is
  problematic, and in many cases not energy
  efficient do to recovery mechanism
• How do we do with uncontrolled process such as
  random walk? Not Bad !
• Not applicable in all cases, but,
• When applicable provides an elegant, simple and
  efficient solution