Topology Control Andreas Wolf (0325330) 17.01.2007
1 Motivation and Basics 2 Flat networks 3 Hierarchy by dominating sets 4 Hierarchy by clustering 5 Adaptive node activity
Options for topology control Reduce active nodes Reduce active links Order nodes hierarchically
Figure: Dense network Figure: Sparse topology Figure: Using a Backbone Figure: Using Clusters
Aspects of topology-control algorithms Connectivity Stretch factors Graph metrics Throughput Robustness to mobility Algorithm overhead
Connectivity Example 5000 nodes 5000 Maximum component size Probability of connectivity 1 Area of 1000 x 1000 m Transmission range r m known as disc graph model if r = 1 unit disc graph Average size of the target component 4000 3000 2000 1000 0,8 0,6 0,4 0,2 0 0 10 15 20 25 30 35 40 Maximum transmission range
Stretch factors Definition hop stretch factor = max u,v V (u,v) T (u,v) G Definition energy stretch factor = max u,v V E T (u,v) E G (u,v)
1 Motivation and Basics 2 Flat networks 3 Hierarchy by dominating sets 4 Hierarchy by clustering 5 Adaptive node activity
For description of power control problems, a four-tuple (M, P, O, I) is used: Definition M {dir, undir} Graph is either directed or undirected P a property to be guaranteed, such as strongly connected k-node connected (k-nc) k-edge connected (k-ec) O the objective function to be minimized I additional Information, topology control can use
Basic complexity results (undir, 1-NC, total P, -) is NP-hard There exists an approximation algorithm with a performance guarantee of 2. (undir, 1-NC, max P, -) are solvable in polynomial time.
Are there magic numbers? Asumption: A large number of nodes is placed randomly within a given area Questions: how many nodes are required to guarantee n-connectivity when transmission range is limited? Problem is NP-hard, additional heuristics achieve O(k) approximation is there a magic number of minimum neighbours? neighbours < 0.0774log V asymptotically disconnected neighbours > 5.1774log V asymptotically connected
Relative Neigborhood Graph Definition RNG T of graph G = (V, E) is T = (V, E ) u, v V : (u, v) E iff w V : max {d (u, w), d (v, w)} < d (u, v)
Relative Neigborhood Graph Definition RNG T of graph G = (V, E) is T = (V, E ) u, v V : (u, v) E iff w V : max {d (u, w), d (v, w)} < d (u, v) w u v
Algorithm of the RNG for all v N do for all w N do if w == v then continue else if d(u, v) > max[d(u, w), d(v, w)] then eliminate edge (u,v) break end if end for end for
Properties of the RNG always connected if G was connected worst-case spanning ratio is Ω ( V ) energy stretch is polynomal average degree is 2.6
Gabriel Graph Definition GG T of graph G = (V, E) is T = (V, E ) u, v V : (u, v) E iff w V : d 2 (u, w), d 2 (v, w) < d 2 (u, v)
Gabriel Graph Definition GG T of graph G = (V, E) is T = (V, E ) u, v V : (u, v) E iff w V : d 2 (u, w), d 2 (v, w) < d 2 (u, v) w u v
Algorithm of the GG m = midpoint of uv for all v N do for all w N do if w == v then continue else if d(m, w) < d(u, m) then eliminate edge (u,v) break end if end for end for
Properties of the GG always connected if G was connected ( V ) worst-case spanning ratio is Ω energy stretch is O(1) depending on energy-consumption model worst case degree is Ω ( V )
1 Motivation and Basics 2 Flat networks 3 Hierarchy by dominating sets 4 Hierarchy by clustering 5 Adaptive node activity
Motivation and definition Building hierarchical structures to reduce connections simplify routing Using a dominating set as backbone network, which should be minimal: minimum nodes minimum connections Minimum Connected Dominating Set (MCDS)
Centralized algorithms a naïve approach initialize all nodes color to white pick an arbitrary node and color it gray while (there are white nodes) { pick a gray node v that has white neighbors color the gray node v black foreach white neighbor u of v { color u gray add (v,u) to tree T } }
1 2 3 4 needs not to be small or even minimal
Greedy heuristic Picking the next node to turn gray which would turn the most white nodes gray (maximize the yield) seems to enhance the naive algorithm. u d... v
Greedy heuristic Picking the next node to turn gray which would turn the most white nodes gray (maximize the yield) seems to enhance the naive algorithm. u d... v
Greedy heuristic Picking the next node to turn gray which would turn the most white nodes gray (maximize the yield) seems to enhance the naive algorithm. u d... v
Greedy heuristic Picking the next node to turn gray which would turn the most white nodes gray (maximize the yield) seems to enhance the naive algorithm. u d... v
The shortsightedness can be overcome by allowing the algorithm to look ahead one step. u d... v
The shortsightedness can be overcome by allowing the algorithm to look ahead one step. u d... v
The shortsightedness can be overcome by allowing the algorithm to look ahead one step. u d... v
The shortsightedness can be overcome by allowing the algorithm to look ahead one step. u d... v
The shortsightedness can be overcome by allowing the algorithm to look ahead one step. u d... v
Distributed algorithms Centralized algorithms can be adapted Another idea is to first look for connected, possibly large dominating sets and reduce them in size afterwards
Nonoptimal dominating set construction All nodes are initially unmarked each node exchanges neighbor sets with its neighbors Mark any node if it has two neighbors that are not directly connected
Nonoptimal dominating set construction All nodes are initially unmarked each node exchanges neighbor sets with its neighbors Mark any node if it has two neighbors that are not directly connected This leads to the following properties If the original graph is connected, the resulting set of marked nodes is a dominating set The resulting set of marked nodes is connected The shortest path between any two nodes does not include any nonmarked nodes The dominating set is not minimal
Pruning heuristics After constructing a nonminimal dominating set, it can be easily reduced Definition Unmark a node v if its neighborhood is included in the neighborhoods of two marked neighbors u and w and v has the smallest identifier
Pruning heuristics After constructing a nonminimal dominating set, it can be easily reduced Definition Unmark a node v if its neighborhood is included in the neighborhoods of two marked neighbors u and w and v has the smallest identifier u v w a b c d
Performance of distributed algorithm The Algorithm only requires O( 2 ) time to exchange neighborhood sets and constant time to reduce the set.
1 Motivation and Basics 2 Flat networks 3 Hierarchy by dominating sets 4 Hierarchy by clustering 5 Adaptive node activity
Definition Locally mark nodes to have a special role. forming of clusters Clusterheads control neighbor nodes emphasis on local resource arbitration shielding higher layers of dynamics aggregate and compress traffic
Definition Locally mark nodes to have a special role. forming of clusters Clusterheads control neighbor nodes emphasis on local resource arbitration shielding higher layers of dynamics aggregate and compress traffic
Properties of clusters Clusterheads may be neighbors, but often its better to have them separated If Clusters may not overlap, some decision rules are needed to assign nodes also, Gateway nodes are needed for communication
Properties of clusters Clusterheads may be neighbors, but often its better to have them separated If Clusters may not overlap, some decision rules are needed to assign nodes also, Gateway nodes are needed for communication
Properties of clusters If Clusterheads are separated by two nodes, the two nodes can act as distributed gateway Clusters can be k-connected Gateways and clusters form a dominating set Clusters can have variable diameter Clusters can be nested
Properties of clusters If Clusterheads are separated by two nodes, the two nodes can act as distributed gateway Clusters can be k-connected Gateways and clusters form a dominating set Clusters can have variable diameter Clusters can be nested
A basic idea to construct independent sets Use a property that can be locally determined that can be easily exchanged to rank nodes
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 1 2 3 7 6 5 4
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 2 A node can become clusterhead if it has the highest (or lowest) rank among all its undecided neighbors 1 2 3 7 6 5 4
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 2 A node can become clusterhead if it has the highest (or lowest) rank among all its undecided neighbors 3 It changes its state and announces it to all of its neighbors 1 2 3 7 6 5 4
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 2 A node can become clusterhead if it has the highest (or lowest) rank among all its undecided neighbors 3 It changes its state and announces it to all of its neighbors 4 Nodes that hear about a clusterhead next to them switch to cluster member and announce this to their neighbors 1 2 3 7 6 5 4
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 2 A node can become clusterhead if it has the highest (or lowest) rank among all its undecided neighbors 3 It changes its state and announces it to all of its neighbors 4 Nodes that hear about a clusterhead next to them switch to cluster member and announce this to their neighbors 1 2 3 7 6 5 4
a basic idea to construct independent sets For example use a unique node identifier 1 Each node determines its local ranking property and exchanges it with its neighbors 2 A node can become clusterhead if it has the highest (or lowest) rank among all its undecided neighbors 3 It changes its state and announces it to all of its neighbors 4 Nodes that hear about a clusterhead next to them switch to cluster member and announce this to their neighbors 1 2 3 7 6 5 4
Connecting cluster heads As clusterheads are separated by at most three hops, they can connect to all of them, resulting in a backbone network. This may result in more connections than necessary.
Rotation of Clusterheads Clusterheads may have additional tasks and a higher communication amount, resulting in a higher power consumption. The task of being Clusterhead should therefore change over nodes. This could be done by: use of a queue of virtual identifiers consider remaining battery power
A weighted clustering algorithm Definition W v = w 1 d v δ + w 2 u N(v) dist (u, v) + w 3 S (v) + w 4 T (v) w i are the nonnegative weighting factors N (v) are the neighbors of v at maximum power S (v) is the average speed of node v T (v) is the time node v has already served as clusterhead
1 Motivation and Basics 2 Flat networks 3 Hierarchy by dominating sets 4 Hierarchy by clustering 5 Adaptive node activity
Adaptive node activity Nodes can be turned off if they are not active regarding tasks or communication. This may be the case if nodes are redundant, for example in sensor networks.
Geographic Adaptive Fidelity (GAF) Nodes can be turned off if there exists another node which can be uses instead for communication and if they are neither data source nor sink. In the GAF, the area is divided into rectangles small enough to allow each node to communicate with each node of its neighboring rectangles (only at the borders).
Geographic Adaptive Fidelity (GAF) Nodes can be turned off if there exists another node which can be uses instead for communication and if they are neither data source nor sink. In the GAF, the area is divided into rectangles small enough to allow each node to communicate with each node of its neighboring rectangles (only at the borders). r R r
Properties of GAF r R r The node positions needs to be known The distance between critical nodes is It follows that r should fulfill r < R/ 5 r 2 + (2r) 2
The End Bibliography The End Thank you, for your attention!
The End Bibliography Holger Karl and Andreas Willig. Protocols and Architectures for Wireless Sensor Networks. Jon Wiley & Sons, 2005 (reprint July 2006). B. Karp and H. T. Kung. Greedy perimeter stateless routing for wireless networks. Proceedings of the 6th International Conference on Mobile Computing and Networking (ACM Mobicom), 2000.