Application of Computational Geometry in Wireless Networks Ad hoc wireless networking – a survey By: xiang-yang li Mohammed alali spring ‘16 CG – Dr. dragan

Content Definitions Topology Control Bounded Degree Spanners Planar Spanner Bounded Degree Planar Spanner

Definitions Ad Hoc Wireless Networks: No wired infrastructures or cellular networks. Each mobile node has a transmission range. Node v can receive the signal from node u : if node v is within the transmission range of the sender u. else relay messages by intermediate nodes (multi-hop wireless links) Each node is acting as a router (message forwarding)

Definitions (Cont.) Ad Hoc Wireless Networks have two main classes: Static : Position of the wireless node does not change (or change slowly) Example: Sensor networks. Mobile: Wireless nodes move arbitrarily. Their Topology changes frequently and often without any regular pattern. Example: smartphone ad hoc networks.

Challenges Wireless nodes are often powered by batteries only. They have a limited transmission range. They often have limited memories\computational power. They can adjust their transmission power.

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Bounded Degree Planar Spanner

Topology Control Definitions Power Attenuation Model Unit Disk Graph (UDG) Planar Graph Sparse Graph Vertex Degree Spanners λ-Precision (Civilized Graphs) Centralized, Localized and distributed algorithms

Topology Control Definitions Power Attenuation Model: To transmit a signal from node v to u, the consumed power consists of 3 parts: The power needed (path loss) to support the link uv is ∥𝑢𝑣∥ 𝛽 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑡ℎ𝑒 𝐸𝑢𝑐𝑙𝑖𝑑𝑒𝑎𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑢 𝑎𝑛𝑑 𝑣 𝑎𝑛𝑑 2≤𝛽≤5 ( 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑒𝑛𝑣𝑖𝑟𝑜𝑛𝑒𝑚𝑛𝑡) V U Preparing signal Transmitting signal (distance dependent) Receiving, storing, and processing signal 𝑝 𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑘 𝑒=𝑢𝑣

Topology Control Definitions UDG (V) Unit Disk Graph (UDG) Geometrically: it is the intersection graph of a family of unit disks in the Euclidean plane. In wireless ad hoc networks: UDG is a set 𝑉 of n wireless nodes distributed in a 2D plane, where : There is an edge between 𝑢 and 𝑣 iff ∥𝑢𝑣∥ ≤1 Therefore, all nodes have a maximum transmission range of 1 unit. Problems: Too dense → up to 𝑂( 𝑛 2 ) Not planar

Topology Control Definitions Unit Disk Graph (UDG) : Examples

Topology Control Definitions Unit Disk Graph (UDG) : Examples

Topology Control Definitions Planar graph A graph that can be embedded in the plane. Its edges only intersect at their endpoints (no crossing).  Has at most 3𝑛−6 edges → sparse Planarity would reduce signal interference. Guaranteed delivery.

Topology Control Definitions Sparse graph A graph in which the number of edges is much less than the possible number of edges 𝑛 𝑛−1 /2 . Sparse but : Not too sparse. Connected

Topology Control Definitions Vertex degree The number of edges incident to a vertex. Bounded degree : In- and out-degree are bounded by a constant. High (unbounded) degree → overhead, collision, interference … Low degree → low fault tolerance, may increase energy consumption …

Topology Control Definitions Spanners A spanner is a subgraph with constant stretch factor. A spanner is called a sparse spanner if it has only a linear number of links. Stretch Factor Let 𝐺=(𝑉, 𝐸) be an n-vertex connected weighted graph, and 𝑑 𝐺 (𝑢,𝑣) = the total weight (length) of the shortest path between 𝑢 and 𝑣 A subgraph 𝐻=(𝑉,𝐸`), where 𝐸`⊆𝐸, is a t-spanner of 𝐺 if for every 𝑢,𝑣∈𝑉, 𝑑 𝐻 𝑢,𝑣 ≤𝑡. 𝑑 𝐺 (𝑢,𝑣) The value of t is called the Stretch Factor (length or power). 𝑝 𝐻 𝑢,𝑣 = ∥𝑢𝑣∥ 𝛽 ⇒ 𝑝( ) = 𝑖=1 ℎ ∥ 𝑣 𝑖−1 𝑣 𝑖 ∥ 𝛽 (Total transmission power) 𝑝 𝐻 𝐺 = max 𝑢,𝑣 ∈ 𝑉 𝑝 𝐻 𝑢,𝑣 𝑝 𝐺 𝑢,𝑣 (Power Stretch Factor) 𝐺=(𝑉, 𝐸) u 𝑑 𝐺 (𝑢,𝑣) 𝑑 𝐻 (𝑢,𝑣) v

Topology Control Definitions Spanners (cont.) A graph with a constant bounded length stretch factor must also have a constant bounded power stretch factor, but the reverse is not true.

Topology Control Definitions λ-Precision (Civilized Graphs) A unit disk graph is civilized graph if the distance between any two nodes is ≥ a positive constant λ. λ > 0

Topology Control Definitions Centralized, Localized and distributed algorithms Centralized: construction is done in a central node. Distributed Localized algorithm (preferred): Part of the topology is constructed locally within each node to build the underlying topology in a distributive manner. Every node u can exactly decide all edges incident on u based only on the information of all nodes within a constant hops of u. Examples: YG(V), RNG(V), GG(V) can be constructed locally. EMST(V) and Del(V) can not be constructed by any localized algorithm.

Topology Control Problem formulation (the question) Is it possible (if possible, then how) to design a network, which is a subgraph of the unit disk graph, such that it ensures both attractive network features: bounded node degree low-stretch factor, and linear number of links and attractive routing schemes such as localized routing with guaranteed performances.

Topology Control Goal The primary goals of topology control in wireless networks is to Maintain network connectivity => Planar Optimize network lifetime and throughput, and => Sparse Make it possible to design power-efficient routing. => Spanner Make it fault tolerance. => Degree bi-connected : there are at least two disjoint paths for any pair of wireless nodes.

Topology Control Assumptions Each wireless node has omnidirectional antenna. Each wireless node knows its position (GPS OR signal strength Est.) Wireless nodes are assumed to be quasi-static during the short period of topology reconstruction. the original unit disk graph UDG is bi-connected. Size of network is restricted to reduce routing information.

Content Topology Control Definitions Bounded Degree Spanners Relative Neighborhood Graph RNG Gabriel Graph GG Yao Graph YG Planar Spanner Bounded Degree Planar Spanner

Topology Control Relative Neighborhood Graph RNG(V) u v An edge uv is included iff ∥𝑢𝑣∥ ≤1 , and The disks’ intersection contains no other nodes. i.e. there does not exist a third point that is closer to both points than they are to each other. Maintains connectivity used for efficient broadcasting minimizing the number of retransmission

Topology Control Relative Neighborhood Graph RNG(V) Properties: Planar YES Length Stretch Factor O(n) Power Stretch Factor Spanner NO Degree Unbounded

Topology Control Relative Neighborhood Graph RNG(V) EMST(V) ⊂ RNG(V) if UDG(V) is connected

Topology Control Gabriel Graph GG(V) An edge uv is included iff ∥𝑢𝑣∥ ≤1 , and The disks(u,v) contains no other nodes. i.e. there does not exist a third point that is closer to both points than they are to each other. Used in face routing protocol and GPSR routing protocol which guarantee delivery. u v

Topology Control Gabriel Graph GG(V) Properties: Planar YES Length Stretch Factor O( 𝑛 1/2 /2) Power Stretch Factor 1 Optimal Spanner Degree O(n) Unbounded

Topology Control Gabriel Graph GG(V) RNG(V) ⊂ GG(V)

Topology Control Yao Graph YGk(V) At each node u, any k equally-separated rays originated at u define k cones. k is an integer parameter ≥ 6 In each cone, choose the shortest edge uv among all edges ≤ 1 from u, if there is any, and add a directed link uv. K=8 equal cons

Topology Control Yao Graph YGk(V) Properties: It is strongly connected if UDG(V) is connected. Planar NO Length Stretch Factor O( 1 1−2𝑠𝑖𝑛 𝜋 𝑘 ) Power Stretch Factor O( 1 1− (2𝑠𝑖𝑛 𝜋 𝑘 ) 𝛽 ) Spanner YES Degree O(n) Unbounded Out-degree k In-degree n

Topology Control Summary Planar Spanner Bounded Degree RNG ✔ X GG (Optimal power stretch factor = 1) Yao (but out-degree = k) Topology construction is completely localized Unbounded degree

Content Topology Control Definitions Bounded Degree Spanners Relative Neighborhood Graph RNG Gabriel Graph GG Yao Graph YG Enhanced Yao Structures Sink Structure YaoYao Structure Symmetric Yao Structure Planar Spanner Bounded Degree Planar Spanner

Topology Control Enhanced Yao Structures YGGk(V) Apply Yao structure on top of the GG structure GYGk(V) Apply GG structure on top of the Yao structure Properties: Sparser than YGk Connected → by showing RNG is a subgraph of both Constant power stretch factor Planar Spanner Out-Degree In-Degree Power Stretch Factors YES k n Same as YGk

Topology Control Enhanced Yao Structures (Cont.) Problems: Unbounded in-degree Some nodes may have a very large in-degree → overhead Results in more complex routing algorithms Therefore, it is important to construct a sparse network topology such that both the in-degree and the out-degree are bounded by a constant while it is still power-efficient.

Topology Control | Yao Structures: Sink Structure 𝑌𝐺 𝐾 ∗ (𝑉) Bounded Degree Spanners Replace the directed star consisting of all links toward a node u in YGk(V) by a directed tree T(u) Node u constructs the tree T(u) and then broadcasts the structure of T(u) to all nodes in T(u) The union of all trees T(u) is called the sink structure 𝑌𝐺 𝐾 ∗ (𝑉) Star formed by links toward to u. Directed tree T(u) sinked at u.

Topology Control | Yao Structures: Sink Structure 𝑌𝐺 𝐾 ∗ (𝑉) Bounded Degree Spanners Properties: Planar NO Length Stretch Factor O( 1 1−2𝑠𝑖𝑛 𝜋 𝑘 ) Power Stretch Factor O( 1 1− (2𝑠𝑖𝑛 𝜋 𝑘 ) 𝛽 ) Spanner YES Degree Bounded sparse Out-degree k In-degree (𝑘+1) 2 -1

Topology Control | Yao Structures: YaoYao Structure 𝑌𝑌 𝐾 (𝑉) Bounded Degree Spanners Each node vi is assigned a unique identification ID(vi)=i Each edge uv in YGk(V) is assigned an ID = (∥𝑢𝑣∥ , ID(u), ID(v)) Select the edges with smallest ID( vu ) among all directed links vu in each cone of YGk(V) The union of all chosen directed links is called the YaoYao structure 𝑌𝑌 𝐾 (𝑉)

Topology Control | Yao Structures: YaoYao Structure 𝑌𝑌 𝐾 (𝑉) Bounded Degree Spanners Properties: * Li et al. conjectured a constant bound in any UDG Planar NO Power Stretch Factor - Bounded by a constant in civilized graph * - Small (in practice) Spanner YES Degree Bounded sparse

Topology Control | Yao Structures: Symmetric Yao Structure 𝑌𝑆 𝐾 (𝑉) Bounded Degree Spanners An edge is selected iff: Both directed edges uv and vu are in the Yao graph YGk(V) Maximum node degree = k Strongly connected if UDG(v) is connected and K>=6

Topology Control | Yao Structures: Symmetric Yao Structure 𝑌𝑆 𝐾 (𝑉) Bounded Degree Spanners Properties: Planar NO Power Stretch Factor - Small (in practice) - Theoretically proven, cannot be bounded (same for length) Spanner (proven Theoretically) Degree Bounded (at most k) sparse

Topology Control | Summary Bounded Degree Spanners Planar Spanner Bounded Degree SINK 𝑌𝐺 𝐾 ∗ (𝑉) X ✔ YaoYao 𝑌𝑌 𝐾 (𝑉) Symmetric Yao 𝑌𝑆 𝐾 (𝑉) (small power stretch factor in practice) at most K

Topology Control | Summary Bounded Degree Spanners The node degrees of different topologies

Topology Control | Summary Bounded Degree Spanners

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Localized Delaunay triangulation Partial Delaunay Triangulation Restricted Delaunay Triangulation Bounded Degree Planar Spanner

Topology Control | Delaunay Triangulation Del(V) Planar Spanner Assume that there are no four vertices of V that are co-circular : ∆ (uvw) is constructed if disk(u,v,w) is empty. Properties: Planar YES Length Stretch Factor ≈ 2.42 Spanner Degree Unbounded Sparse

Topology Control | Delaunay Triangulation Del(V) Planar Spanner Problems: It is expensive to construct the Delaunay triangulation in a distributed manner because of the possible massive communication it requires. It contains edges longer than the unit length.

Topology Control | Unit Delaunay Triangulation UDel(V) Planar Spanner Delete edges longer than the unit length (the possible transmission range) from Del(V). 𝑈𝐷𝑒𝑙 𝑉 =𝐷𝑒𝑙 𝑉 ∩𝑈𝐷𝐺(𝑉) Properties: Planar YES Length Stretch Factor ≈ 2.42 Spanner Degree Unbounded Sparse

Topology Control | Unit Delaunay Triangulation UDel(V) Planar Spanner We solved the long edges problem but still it is unknown how to construct UDel(V) locally. One solution: Li et al. gave a localized algorithm that constructs sequence graphs, called localized Delaunay LDel(k) (V), which are supergraphs of UDel(V).

Topology Control | Localized Delaunay Triangulation 𝐿𝐷𝑒𝑙 𝑘 (𝑉) Planar Spanner u v Related definitions: Gabriel edge : ∥𝑢𝑣∥ ≤1 The open disk using uv as a diameter does not contain vertices from V. k-localized Delaunay triangle : Disk(u,v,w) doesn't contain any vertex of V that is a k-neighbor (k-hop) of u, v or w all edges of ∆uvw have length no more than one unit

Topology Control | Localized Delaunay Triangulation 𝐿𝐷𝑒𝑙 𝑘 (𝑉) Planar Spanner Related definitions: k-Localized Delaunay graph 𝐿𝐷𝑒𝑙 𝑘 (𝑉): Has exactly all Gabriel edges and all edges of K-localized Delaunay triangles. Properties: Planar YES Only If k ≥ 2 Length Stretch Factor ≈ 2.42 Spanner Degree Unbounded Sparse NOTE: disk(u, v, w) is not necessarily covered by unit disks centered at u and v. But it is empty of other vertices from N1 (u) U N1 (v) U N1(w).

Topology Control | Localized Delaunay Triangulation: 𝐿𝐷𝑒𝑙 𝑘 (𝑉) Planar Spanner 𝐿𝐷𝑒𝑙 1 (𝑉) 𝐿𝐷𝑒𝑙 2 (𝑉)

Topology Control | Localized Delaunay Triangulation: 𝐿𝐷𝑒𝑙 1 (𝑉) Planar Spanner Construct disk(u,v,w) if it is empty from all nodes in 𝑁1(u) U 𝑁1(v) U 𝑁1(w) Where 𝑁1(i) = 1-hop neighbors of i Localized construction: O(n log n) bits cost Properties: Planar NO Li et al. Spanner YES Degree Unbounded Sparse

Topology Control | Localized Delaunay Triangulation: 𝐿𝐷𝑒𝑙 2 (𝑉) Planar Spanner Construct disk(u,v,w) if it empty from all nodes in 2-hop neighbors. Properties: Problem: Total communication: O(m log n) bits cost M = number of edges in UDG(V) could be as large as O(𝑛2) This is more complicated than non-planar t-spanner. Planar YES Spanner Degree Unbounded

Topology Control | Localized Delaunay Triangulation: 𝐿𝐷𝑒𝑙 𝑘 (𝑉) Planar Spanner What we do to reduce commination cost to O(n log n) .. (Li et al.) Do not construct 𝐿𝐷𝑒𝑙 2 (𝑉), instead Extract a planar graph 𝑃𝐿𝐷𝑒𝑙 (𝑉) out o 𝐿𝐷𝑒𝑙 1 𝑉 . Why? Remember 𝐿𝐷𝑒𝑙 1 𝑉 : Has localized construction = O(n log n) It is guaranteed to be sparse (proven)

Topology Control | Localized Delaunay Triangulation: 𝑃𝐿𝐷𝑒𝑙 (𝑉) Planar Spanner Li et al. gave 2 novel algorithms that: Constructs 𝐿𝐷𝑒𝑙 1 (𝑉) in O(n log n) time. Palanarize 𝐿𝐷𝑒𝑙 1 (𝑉) in O(n log n) time. Algorithms guarantee planarity (proven). The final graph still contains a subgraph UDel(V) Properties: Planar YES Spanner Degree Unbounded Sparse

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Localized Delaunay triangulation Partial Delaunay Triangulation Restricted Delaunay Triangulation Bounded Degree Planar Spanner

Topology Control | Localized Delaunay Triangulation: Partial Delaunay Triangulation PDT: 𝑃𝐷𝑒𝑙 (𝑉) Planar Spanner Localized and contains GG as a subgraph. Construction: For any two vertices u,v : If Disk(u,v) contains no other vertices uv is an edge of GG therefore belongs to PDT Else if Disk(u,v) contains vertices on both sides of uv uv is not an edge of PDT Else Disk(u,v) contains vertices on one side of uv Further checks needed …

Topology Control | Localized Delaunay Triangulation: Partial Delaunay Triangulation PDT: 𝑃𝐷𝑒𝑙 (𝑉) Planar Spanner Construction (cont.): case #1 (Only N1(u) is known to u) Else Disk(u,v) contains vertices on one side of uv Find node w in Disk(uv) which maximizes angle(uwv) where : w is in N1(u) Draw the Disk(u,v,w) For all other nodes in N1(u) : If disk(u,v,w) is empty of neighbors of u, AND disk(u,v,w) is covered by transmission range of u Then: add uv to PDT

Topology Control | Localized Delaunay Triangulation: Partial Delaunay Triangulation PDT: 𝑃𝐷𝑒𝑙 (𝑉) Planar Spanner Construction (cont.): case #2 (u knows N1(u) and v knows N1(v)) Else Disk(u,v) contains vertices on one side of uv Find node w in Disk(uv) which maximizes angle(uwv) where : w is in N1(u) U N1(v) → a common neighbor Draw the Disk(u,v,w) For all other nodes in N1(u) U N1(v) : If disk(u,v,w) is empty of neighbors of u, AND disk(u,v,w) is covered by transmission range of u∪𝑣. Then: add uv to PDT

Neumann et al. has proven it to be ≤ 1+ 5 4 𝜋 2 [6] Topology Control | Localized Delaunay Triangulation: Partial Delaunay Triangulation PDT: 𝑃𝐷𝑒𝑙 (𝑉) Planar Spanner Properties: Planar YES Length Stretch Factor Could be very Large Neumann et al. has proven it to be ≤ 1+ 5 4 𝜋 2 [6] Spanner Degree Unbounded Sparse

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Localized Delaunay triangulation Partial Delaunay Triangulation Restricted Delaunay Triangulation Bounded Degree Planar Spanner

Topology Control | Localized Delaunay Triangulation: Restricted Delaunay Graph RDG Planar Spanner Gao et al. : any planar graph containing UDel(V) is an RDG. Each node u maintains a set of edges E(u) incident to u Edges in E(u) satisfy: Each edge has length at most 1 unit. Edge uv ϵ E(u) iff uv ϵ E(v) → consistent Obtained graph is planar. The graph UDel(V) is in the union of all edges E(u) Problem: Computation and communication cost of each node obtaining E(u) is not optimal. Properties: Planar YES Spanner Good spanning ratio Degree Unbounded Sparse

Topology Control | Summary Planar Spanner Planar Spanner Bounded Degree Sparse Delaunay Del(V) ✔ X Unit Delaunay UDel(V) Localized Delaunay LDel k (V) ✔ k ≥ 2 LDel 1 (V) LDel 2 (V) Planar Localized Delaunay P LDel k (V) Partial Delaunay PDT(V) Restricted Delaunay RDG(V) None of previous graphs satisfy bounded degree, planer and spanner together

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Bounded Degree Planar Spanner Centralized Construction for UDG Localized Construction

Topology Control | Centralized Construction for UDG Bounded Degree Planar Spanner Bose et al. algorithm → runs O(n log n) Properties: Impossible to have a localized or even distributed version Due to the use of BFS and other operations on polygons. Planar YES Length Stretch Factor 2π(π+1)/((3cos π/6)(1 +ε)) Spanner Degree bounded At most 27

Topology Control | Centralized Construction for UDG : BPS1(UDG(V)) Bounded Degree Planar Spanner Li et al. Borrows some ideas from Bose’s algorithm → also O(n log n) The basic idea of his method is to combine (localized) Delaunay triangulation and ordered Yao structure Properties: Smaller bounded node degree than Bose’s. It can be localized. Planar YES Length Stretch Factor A constant Spanner Degree bounded At most 19 + 2𝜋 𝛼

Content Topology Control Definitions Bounded Degree Spanners Planar Spanner Bounded Degree Planar Spanner Centralized Construction for UDG Localized Construction

Topology Control | Localized Construction for UDG : BPS2(UDG(V)) Bounded Degree Planar Spanner The algorithm is based on a planar spanner LDel(2) (V) for UDG proposed by Li et al. Using an approach by Calinescu to collect 2-hop neighbors in O(n) messages. Properties: LDel(2) (V) could have degree as large as O(n) ← gave an efficient algorithm to bound it Communication cost: O(n log n) → n messages each cost log n Planar YES Length Stretch Factor A constant Spanner Degree bounded At most 19 + 2𝜋 𝛼

References X.-Y. Li ,Application of Computational Geometry in Wireless Networks. Department of Computer Science, Illinois Institute of Technology, Chicago, IL Xiang-Yang Li, Peng-Jun Wan, Yu Wang, and Ophir Frieder,“ Sparse power efficient topology for wireless networks," in IEEE Hawaii Int. Conf. on System Sciences (HICSS), 2002. Encyclopedia of Algorithms. 1st ed. US: Springer, 2008. Springer. Springer-Verlag US. Web. 5 Apr. 2016. X.-Y. Li, P.-J. Wan, and Y. Wang, Power Efficient and Sparse Spanner for Wireless Ad Hoc Networks Proc. IEEE Int'l Conf. Combbputer Comm. and Networks (ICCCN01), pp. 564-567, 2001 Yu Wang and Xiang-Yang Li, Distributed Spanner with Bounded Degree for Wireless Ad Hoc Networks F. Neumann and H. Frey, “On the Spanning Ratio of Partial Delaunay Triangulation,” in Proceedings of the 9th IEEE International Conference on Mobile Ad-hoc and Sensor Systems (MASS 2012), Las Vegas, NV, USA, Oct. 2012, pp. 434–442. Proximity Structures for Wireless Communication http://www.csc.villanova.edu/~mdamian/Research/CCF0728909.html https://en.wikipedia.org/wiki/Dense_graph https://xlinux.nist.gov/dads//HTML/sparsegraph.html http://www.cs.kent.edu/~dfuhry/presentations/simple_topologies.pdf Omar Meqdadi’s and Nahla Abid’s previous years presentations.