Randomized algorithms for approximating a Connected Dominating Set in Wireless Sensor Networks
Summary (3 min read)
Introduction
- A Connected Dominating Set (CDS) of a graph representing a Wireless Sensor Network can be used as a virtual backbone for routing through the network.
- In this paper the authors present three randomized algorithms for constructing a CDS.
- The construction of a CDS provides the network with a virtual backbone over which routing, multicast and broadcast can be performed since every node is either in the backbone or has a neighbor in the backbone.
- The authors simulations also showed that for networks of a given density, the randomized algorithm could be repeated at each sensor a certain number of times and would result in a CDS with a very high probability.
- In Section II the authors present related work on this problem and introduce the K2 and GreedyConnect algorithm in some detail since they use ideas from both algorithms in their algorithms.
A. K2 Marking and Pruning
- The K2 [4] algorithm has two phases, first the construction of a connected dominating set, then the reduction of unnecessary vertices from the CDS.
- This phase has a time complexity of O(∆2) where ∆ is the maximum degree in the graph.
- This set is likely to contain many more nodes than necessary since the marking process was very simple.
- In order to reduce the size of the set, the authors use a kreduction (where k is the number of hops the algorithm is looking at) to remove unnecessary sensors from the set.
- For each sensor in the dominating set the authors consider every k-hop group of neighbors where each member of the group is in the dominating set.
B. Centralized Random Selection
- The authors first algorithm takes a simple approach to creating a CDS.
- This algorithm randomly selects sensors and adds them to the future CDS.
- This is the primary reason that the algorithm is centralized - an outside source is needed to manage the construction of the CDS and determine that a CDS has been formed.
- The base station for the network would typically serve in this role.
- This algorithm requires the CDS constructor to be aware of two sets: C which represents the CDS under construction and R with represents vertices to be potentially selected.
C. Deriving the probability for each sensor to join the DS
- Since the process described above adds all nodes not in R to D to give a dominating set, their assumption of a d-regular graph does not impact correctness.
- For the purpose of their algorithms, each sensor works with its local degree instead of d.
D. Probability with Rounds
- Like most randomized algorithms, the premise behind their first randomized algorithm was simple: randomly add sensors until a CDS has been formed.
- In their experiments the authors found that 6 rounds always sufficed for this topology density - in other words every sensor flipping a coin with a probability p six times is enough to form a CDS for this network size.
- The figure shows the variation in the number of rounds needed based on the density (as measured by the number of neighbors) of the graph.
- Each sensor provides its ID to its neighborhood and then tosses a coin a constant number of times.
- For analyzing the message complexity, the authors note that in phase one each sensor broadcasts its ID so a total of n messages are passed.
E. Random Distributed Dominating Set
- Since the distributed with rounds algorithm could be implemented in a centralized or localized manner (centralized when an observer told sensors when to stop adding themselves, localized when k tosses were used) the authors next looked at a purely localized algorithm.
- Each sensor tosses a coin weighted with the probability p as derived above.
- Any sensor which adds itself to the DS then alerts its neighbors to update their color to grey.
- This ensures a dominating set because any nondominated sensor will add itself and also dominate any white neighbors.
- This also limits the algorithm to two rounds.
IV. Simulation Results
- The authors compared the size of the CDS created by their three randomized algorithms with that of K2 [4] and GreedyConnect [5].
- For each of their algorithms, their simulation takes in a randomly generated graph with a specified number of sensors distributed randomly across a 100 by 100 region and returns the size of the CDS after running a specific algorithm.
- For the remainder of this section the authors discuss the results of running the simulation on each algorithm with the following conditions:.
- The algorithms the authors simulated would run asynchronously on actual sensor networks.
A. Dominating Set Construction
- Set algorithm the authors discussed in Section III-E created a dominating set (not connected).
- In the following sections of this chapter the authors use the connection algorithm from Greedy Connect to connect the components and create a CDS.
- The authors see that the greedy DS is fairly consistent in size and that the randomized algorithm varies depending on the topology of the graph.
- Figure 3 contains the average DS size for a particular range.
- The authors note the greedy algorithm creates a DS 50% to 60% smaller than the size of the randomized algorithm for every range.
B. Connected Dominating Set Construction
- In this section the authors compare the size of the CDS created by each algorithm.
- From Figure 4 the authors see that the size of the centralized randomized algorithm is not consistent based on range.
- The other four algorithms fairly consistently make the same size CDS based on sensing range.
- As can be seen from the figure, the Random Distributed algorithm presented in Section III-E performs the best and is pretty close to K2 even without any modifications, particularly at higher densities.
- Greedy consistently out-performs every other algorithm and out-performs their reference algorithm, K2, by roughly 15%.
C. Trimming the CDS
- The randomized algorithms were out-performed by both GreedyConnect and K2.
- For this reason, the authors ran the simulation again with the only change being the addition of the trimming algorithm from Phase 2 of K2 as explained in Section II-A. Comparing Figure 6 to Figure 4 the reduction in size for all three randomized algorithms is evident.
- GreedyConnect still holds the lead for creating the smallest CDS by almost 25% for every range.
- Next the authors summarize the time and message complexity of all the algorithms.
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"Randomized algorithms for approxima..." refers methods in this paper
...A Connected Dominating Set (CDS) of a graph representing a Wireless Sensor Network can be used as a virtual backbone for routing through the network....
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3,269 citations
"Randomized algorithms for approxima..." refers methods in this paper
...A Connected Dominating Set (CDS) of a graph representing a Wireless Sensor Network can be used as a virtual backbone for routing through the network....
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1,525 citations
"Randomized algorithms for approxima..." refers background in this paper
...A variation to this problem is that of the Connected Dominating Set (CDS) which can be defined as a set that is dominating and induces a connected subgraph....
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"Randomized algorithms for approxima..." refers background in this paper
...The applications of a connected dominating set to routing in ad hoc networks were first outlined in [7] where they presented the idea of constructing a connected dominating set and using it as a backbone in the network for routing....
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Frequently Asked Questions (12)
Q2. What have the authors stated for future works in "Randomized algorithms for approximating a connected dominating set in wireless sensor networks" ?
Further studies using implementation on a test bed are planned in order to quantify this gain.
Q3. How does the algorithm reduce the size of the set?
In order to reduce the size of the set, the authors use a kreduction (where k is the number of hops the algorithm is looking at) to remove unnecessary sensors from the set.
Q4. What is the phase of the K2 algorithm?
The K2 [4] algorithm has two phases, first the construction of a connected dominating set, then the reduction of unnecessary vertices from the CDS.
Q5. How many rounds does a CDS take to form?
Although a CDS was often created after 4 rounds, in their experiments the authors found that 6 rounds always sufficed for this topology density - in other words every sensor flipping a coin with a probability p six times is enough to form a CDS for this network size.
Q6. How do the authors use rounds to distinguish phases?
Using rounds to distinguish phases is a common technique in WSNs and these are usually implemented by using a fixed interval of time after a message signaled by the base station.
Q7. How many times do the sensors broadcast their ID?
For analyzing the message complexity, the authors note that in phase one each sensor broadcasts its ID so a total of n messages are passed.
Q8. What is the message complexity for this?
The message complexity for this is minimal, since the central manager is doing all calculations, the only messages passed is at the end of the algorithm to inform sensors if they are in the CDS.
Q9. Why is the CDS not being formed?
Due to the randomized nature of the previous algorithm, k tosses would not necessarily create a CDS every time, although a CDS not being formed would be statistically unlikely.
Q10. What is the benefit of the K2 algorithm?
For the purposes of this paper the authors let k = 2 as larger values of k have a message and time complexity which would drain the network more it would benefit it by creating a path for routing.
Q11. How many iterations of random selection have been enough to create a CDS?
Through their simulations the authors have experimentally found that there is a small value k based on the density of the network such that k iterations of randomly selecting nodes with the derived probability were sufficient every time to create a CDS.
Q12. What is the difference between the distributed with rounds algorithm and the GreedyConnect algorithm?
COLOR ← BLACK C = C ⋃ vend if end forflips++ end whileSince the distributed with rounds algorithm could be implemented in a centralized or localized manner (centralized when an observer told sensors when to stop adding themselves, localized when k tosses were used) the authors next looked at a purely localized algorithm.