Topology Control Dr. WenZhan Song Professor, Computer Science

Topology Control Dr. WenZhan Song Professor, Computer Science

Goals of this chapter Networks can be too dense – too many nodes in close (radio) vicinity This chapter looks at methods to deal with such networks by Reducing/controlling transmission power Deciding which links to use Turning some nodes off Focus is on basic ideas, some algorithms Complexity results are only very superficially covered Experimental studies

Overview Motivation, basics Flat control Hierarchical Control Experimental Study

Motivation: Dense networks In a very dense networks, too many nodes might be in range for an efficient operation Too many collisions/too complex operation for a MAC protocol, too many paths to chose from for a routing protocol, … Idea: Make topology less complex Topology: Which node is able/allowed to communicate with which other nodes Topology control needs to maintain invariants, e.g., connectivity

Options for topology control Topology control Control link activity – deliberately use/not use certain links Control node activity – deliberately turn on/off nodes Topology control Hierarchical network – assign different roles to nodes; exploit that to control node/link activity Flat network – all nodes have essentially same role Power control Backbones Clustering

Flat networks Main option: Control transmission power Do not always use maximum power Selectively for some links or for a node as a whole Topology looks “thinner” Less interference, … Alternative: Selectively discard some links Usually done by introducing hierarchies

Hierarchical networks – backbone Construct a backbone network Some nodes “control” their neighbors – they form a (minimal) dominating set Each node should have a controlling neighbor • Controlling nodes have to be connected (backbone) • Only links within backbone and from backbone to controlled neighbors are used • Formally: Given graph G=(V,E), construct D  V such that

Hierarchical network – clustering Construct clusters Partition nodes into groups (“clusters”) Each node in exactly one group Except for nodes “bridging” between two or more groups Groups can have clusterheads • Typically: all nodes in a cluster are direct neighbors of their clusterhead • Clusterheads are also a dominating set, but should be separated from each other – they form an independent set • Formally: Given graph G=(V,E), construct C  V such that

Aspects of topology-control algorithms Connectivity – If two nodes connected in G, they have to be connected in G0 resulting from topology control Stretch factor – should be small Hop stretch factor: how much longer are paths in G0 than in G? Energy stretch factor: how much more energy does the most energy-efficient path need? Throughput – removing nodes/links can reduce throughput, by how much? Robustness to mobility Algorithm overhead

Power control – magic numbers? Question: What is a good power level for a node to ensure “nice” properties of the resulting graph? Idea: Controlling transmission power corresponds to controlling the number of neighbors for a given node Is there an “optimal” number of neighbors a node should have? Is there a “magic number” that is good irrespective of the actual graph/network under consideration? Historically, k=6 or k=8 had been suggested as such “magic numbers” However, they optimize progress per hop – they do not guarantee connectivity of the graph!! ! Needs deeper analysis

Controlling transmission range Assume all nodes have identical transmission range r=r(|V|), network covers area size A, V nodes, uniformly distribution. Fact: Probability of connectivity goes to zero if: Fact: Probability of connectivity goes to 1 for if and only if as a function of |V| Fact (uniform node distribution, density ):

Controlling number of neighbors Knowledge about range also tells about number of neighbors Assuming node distribution (and density) is known, e.g., uniform Alternative: directly analyze number of neighbors Assumption: Nodes randomly, uniformly placed, only transmission range is controlled, identical for all nodes, only symmetric links are considered Result: For connected network, required number of neighbors per node is  (log |V|) It is not a constant, but depends on the number of nodes! For a larger network, nodes need to have more neighbors & larger transmission range! – Rather inconvenient Constants can be bounded

Centralized power control algorithm E E E F F F 2) Connect A-B 1) Connect A-C and B-D 4 4 3 C C C D D D 1 1 1 1 1 1 B B B A A A 2 2 E E F F 3) Connect C-D 5) Remove edge A-B 4) Connect C-E and D-F 4 4 E F 3 3 C C D D 4 4 1 1 1 1 3 C B B A A 2 1 1 B A 2 Topology D

What is it? Maintain network connectivity and efficiency with less links Why need it? Conserve energy Do not send with maximum transmission power if not necessary Sending over near neighbors is more efficient than sending directly to a far target Improve network performance Less interference More spectrum reuse Geometric Topology Control v u w 1 1 1.8 p(u,v) ~ |uv| (2<5) is the energy cost for direct communication between u and v.

Preferred Properties Unicast Power Spanner: keep an energy efficient path between any pair of nodes. Degree Bounded: limit the workload of each node. Planar: support existing position-based routing protocols. Broadcast Any broadcast protocol in G can be interpreted as flood-based broadcast in a subgraph H of G. Low-weighted structure is asymptotically optimal for broadcast among all subgraphs constructed by localized algorithm. u v H G u v Given any pair of nodes (u,v), let the most energy efficient path in original graph G=(V,E) is PG(u,v) when all nodes use maximum transmission range. In the selected subgraph H=(V,E’), there must keep a path PH(u,v) satisfying that p(PH(u,v))  t· p(PG(u,v)) Here t is called power spanning ratio.

Preferred Properties Unicast Power Spanner: keep an energy efficient path between any pair of nodes. Degree Bounded: limit the workload of each node. Planar: support existing position-based routing protocols. Broadcast Any broadcast protocol in G can be interpreted as flood-based broadcast in a subgraph H of G. Low-weighted structure is asymptotically optimal for broadcast among all subgraphs constructed by localized algorithm. u • Limit the workload of each node • Other impact • routing • packet scheduling • load balancing • Bluetooth scatternet

Preferred Properties Unicast Power Spanner: keep an energy efficient path between any pair of nodes. Degree Bounded: limit the workload of each node. Planar: support existing position-based routing protocols. Broadcast Any broadcast protocol in G can be interpreted as flood-based broadcast in a subgraph H of G. Low-weighted structure is asymptotically optimal for broadcast among all subgraphs constructed by localized algorithm. t u s Localized position-based routing is widely used in ad hoc networks. Greedy routing from s to t will fail in the node u, where local minimum occurs. Face-based routings which guarantee delivery demands planar topology.

Preferred Properties Unicast Power Spanner: keep an energy efficient path between any pair of nodes. Degree Bounded: limit the workload of each node. Planar: support existing position-based routing protocols. Broadcast Any broadcast protocol in G can be interpreted as flood-based broadcast in a subgraph H of G. Low-weighted structure is asymptotically optimal for broadcast among all subgraphs constructed by localized algorithm. Broadcasting based on any locally constructed structures may consume energy more than (n-1) times optimum, unless the structure has low-weight. Low weight:if the total edge length of a structure H is proportional to the weight of MST, then H is called low weighted structure.

Example 1: Relative Neighborhood Graph (RNG) Edge between nodes u and v if and only if there is no other node w that is closer to either u or v Formally: RNG maintains connectivity of the original graph Easy to compute locally Planar But: Worst-case spanning ratio is  (|V|) Average degree is 2.6

Example 2: Gabriel graph Gabriel graph (GG) similar to RNG Difference: Smallest circle with nodes u and v on its circumference must only contain node u and v for u and v to be connected Formally: Properties: Maintains connectivity and planar, Worst-case spanning ratio (|V|1/2), energy stretch O(1) (depending on consumption model!), worst-case degree  (|V|)

Example 3: Delaunay triangulation Assign, to each node, all points in the plane for which it is the closest node !Voronoi diagram Constructed in O(|V| log |V|) time Connect any two nodes for which the Voronoi regions touch !Delaunay triangulation Problem: Might produce very long links; not well suited for power control – can not be constructed locally -> Localized Delaunay research [COCOON 2003] Edges of Delaunay triangulation Voronoi region for upper left node

Example 4: Cone-based topology control Assumption: Distance and angle information between nodes is available Two-phase algorithm Phase 1 Every node starts with a small transmission power Increase it until a node has sufficiently many neighbors What is “sufficient”? – When there is at least one neighbor in each cone of angle   = 5/6 is necessary and sufficient condition for connectivity! Phase 2 Remove redundant edges: Drop a neighbor w of u if there is a node v of w and u such that sending from u to w directly is less efficient than sending from u via v to w Essentially, a local Gabriel graph construction

Example 4: Cone-based topology control (2) Properties: simple, local construction Extensions for k-connectivity (Yao graph) /2 /2 /2 /2 /2 /2 /2 /2 w u v x Yao Graph Out-Degree-Bounded Spanner

Example 5: LowSGG – Low-weighted, Degree-bounded, Planar, Power Spanner Step 1: Bound Degree of Planar Spanner Once a node u has smallest ID among unprocessed neighbors: Node u creates a list A by sorting its processed neighbors according to distance increasing, then similarly sorts its unprocessed neighbors and appends to the list A. Node u scans the list A from left to right: keep the current Valid neighbor v, and mark the others dominated by v as Invalid. Nodes u notifies its neighbors of the status update. After every node finishes, the graph SGG is formed. One of Three Best Papers ACM MobiCom 2005 k d b e j n i h u a c l f g m o

Example 5: LowSGG – Low-weighted, Degree-bounded, Planar, Power Spanner (2) Step 2: Make Spanner Low-Weighted The previous algorithm [INFOCOM 2004] to generate low-weighted structure: blindly delete uivi, as long as there exist another link ukvk such that ||uivi||>max(||ukvk||,||uiuk||,||vivk||). Notice ukvk could be deleted later it can not keep spanner property. This new algorithm: keep independent set of links based on localized ordering delete uivi if exist another processed link ukvk such that ||uivi||>max(||ukvk||,3||uiuk||,3||vivk||). un-1 u1 u2 un u3 u4 v2 vn vn-1 v3 v4 v1 un-1 u1 u2 un u3 u4 v2 vn v3 v4 v1 vn-1

Hierarchical networks – backbones Idea: Select some nodes from the network/graph to form a backbone A connected, minimal, dominating set (MDS or MCDS) Dominating nodes control their neighbors Protocols like routing are confronted with a simple topology – from a simple node, route to the backbone, routing in backbone is simple (few nodes) Problem: MDS is an NP-hard problem Hard to approximate, and even approximations need quite a few messages

Hierarchical networks – clustering Partition nodes into groups of nodes – clusters Many options for details Are there clusterheads? – One controller/representative node per cluster May clusterheads be neighbors? If no: clusterheads form an independent set C:Typically: clusterheads form a maximum independent set May clusters overlap? Do they have nodes in common?

Hierarchical networks – clustering Further options How do clusters communicate? Some nodes need to act as gateways between clustersIf clusters may not overlap, two nodes need to jointly act as a distributed gateway How many gateways exist between clusters? Are all active, or some standby? What is the maximal diameter of a cluster? If more than 2, then clusterheads are not necessarily a maximum independent set Is there a hierarchy of clusters?

Maximum independent set Computing a maximum independent set is NP-complete Can be approximate within ( +3)/5 for small , within O( log log  / log ) else;  bounded degree A maximum independent set is also a dominating set, but a dominating set is not necessarily an independent set and an independent set is also not necessarily a dominating set. How to prove? Maximum independent set not necessarily intuitively desired solution Example: Radial graph, with only

A basic construction idea for independent sets Use some attribute of nodes to break local symmetries Node identifiers, energy reserve, mobility, weighted combinations… - matters not for the idea as such (all types of variations have been looked at) Make each node a clusterhead that locally has the largest attribute value Once a node is dominated by a clusterhead, it abstains from local competition, giving other nodes a chance 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 7 7 7 7 7 6 6 6 6 6 5 5 5 5 5 4 4 4 4 4 Step 2: Step 1: Step 3: Init: Step 4:

Guha and Khuller’s algorithm 2 Centralized CDS Algorithms PR = 3 + ln(Δ) C C G Iteratively select the node with the max number of white neighbors as a dominating node. E D D A A F B B C G G The first phase terminates when there are no white nodes. E E D A F F B C G Color some grey nodes black to connect all the black nodes. E D D A F B B

Efficient Distributed Method -1 Tree Based Approach for UDG model Using O(n log n) messages Approximation bound: 8 Wan et al, MobiHoc 2002 Approach Find a leader, say s, which takes O(n log n) messages Starting from s, node s builds a spanning tree T rooted at s Node s marks itself Dominator, and informs its neighbors When a node u gets a message from its neighbor v saying that v is dominator, node u marks itself Dominatee, and informs its neighbors If node u has the smallest rank among all its neighboring nodes that are not dominatee nor dominator, then it elects itself as dominator Rank of a node is (its level in the tree, ID)

CDS from a tree structure Build a tree rooted at s Such as BFS s

CDS from a tree structure s dominator dominatee 5 3 4 2 1

Connect dominators A dominator u (not s) starts to find a connector ‘upward’ There must have another dominator v that are at most 2-hops away from u and v has a level smaller than u Let w be the node connecting u and v Node u then chooses w as connector to connect u to v All above operations are done by node u Exactly DS-1 connectors are selected Messages are O(n) in this steps

CDS from a tree structure s dominator dominatee connector 5 3 4 2 1

Performance Guarantee It is 8-approximation in UDG networks Found dominators form an Independent Set Root OPT as a tree, and root dominates at most 5 nodes from DS found by previous method Each other node u of OPT dominates at most 4 new nodes from DS found previously Each node in DS found is dominated by a node in OPT DS OPT DS ≤ 4 OPT+1 Connectors ≤ DS-1≤ 4 OPT Thus, DS+ Connectors ≤ 8 OPT+1 Size is small, but, it is not a spanner

Efficient Distributed Method -2 Distributed method O(n) messages for UDG Remove the leader election step: let many nodes start together. A node starts if it has the smallest rank among all its neighboring nodes whose status are not decided yet Rank: node ID, or node degree, or node position Status: dominator, dominatee, white (not decided)

Construct a DS Same as maximal IS Each node u broadcasts its ID to N1(u) if node u has smallest ID among all unprocessed nodes of N1(u) Node u be dominator and broadcasts to N1(u) A node v becomes a dominatee if v got a message from its neighbor in N1(v) saying that u is a dominator Node v broadcast to their neighbors---it is dominatee and its dominator

Finding Dominators Illustration 30 s dominator 25 24 23 dominatee 5 3 4 2 1 6 9 10 11 13 12 7 8 20 18 21 22 15 17 16 19 14

Performance Communication cost O(n): each node need broadcast twice One for broadcasting its ID One for broadcasting its role: dominator/ee Size: within 5 times of smallest DS OPT

Performance There are constant number of IS within constant hop

Make it connected Connect two dominators u and v If they are within two hops, node u knows v already (through w1,w2,…) Node u picks the node with smallest ID Assume ID(u)<ID(V) w1 w2 u v w3

Make it connected If they are within three hops Each wi, xi knows all dominators within 2-hop Node wi, xi broadcast their 2-hop dominators Node u knows its 3-hop dominators--constant Node u picks the node with smallest ID x1 w1 x2 w2 u w3 x3 x4 v

Finding Connectors Illustration 30 s dominator 25 24 23 dominatee connector 5 3 4 2 1 6 9 10 11 13 12 7 8 20 18 21 22 15 17 16 19 14

Performance Communication cost O(n): each node need broadcast constant times There are constant number of IS within constant hop The backbone plus the links between dominator/ee is sparse spanner Both length and hop spanner With O(n) edges Length spanner means that for any two nodes s and t, the shortest path using backbone has length at most a constant (6) factor of the optimum in the UDG Hop spanner means that for any two nodes s and t, the path with smallest hops using backbone use links at most a constant (3) times of the optimum in the UDG

Spanner Property wn w1 un u1 Assume a shortest path u1u2…un, then we have an approximate path from w1 to wn Its hop is < 3n, its length is <3n/2 Notice length of u1u2…un is at least n/2.

Another approach - start big, make lean Idea: start with some, possibly large, connected dominating set, reduce it by removing unnecessary nodes Initial construction for dominating set All nodes are initially white Mark any node black that has two neighbors that are not neighbors of each other (they might need to be dominated) ! Black nodes form a connected dominating set (proof by contradiction); shortest path between ANY two nodes only contains black nodes Needed: Pruning heuristics

Topology Control Dr. WenZhan Song Professor, Computer Science