Network Optimization

Network Optimization Lecture 16 - 2

Range Assignment Problems (RA) • Problem Definition of RA • Approximation for Two-dimension RA • Symmetric Versions of RA

Problem Definition of RA (1/3) • Range Assignment Problem • Nodes can change the transmit power level in many scenarios • Choosing the nodes’ transmit power levels in such a way that the network topology satisfies certain properties becomes relevant • Definition 7.1.1 (RA problem) • Let N be a set of nodes in d-dimensional space, with d = 1, 2, 3. • Determine a range assignment function RA such that • corresponding communication graph is strongly connected, • c(RA)=u∈N(RA(u))α is minimum,α is distance-power gradient.

Problem Definition of RA (2/3)  = 2 2 3 2 3 1 Not strongly connected Strongly connected c(RA)= 22 + 32+ 12 = 14 c(RA)= 22 + 32+ 32 = 22

Problem Definition of RA (3/3) • RA problem can be seen as a generalization of the problem of determining the CTR for connectivity. • Computation Complexity • CTR in 1D networks: O(nlogn) • CTR in 2D networks : O(n2) • RA in 1D networks: O(n4) • RA in 2D networks: NP-hard

Approximation for Two-dimension RA (1/4) • Give a set of nodes N = {u1, . . . , un} on 2-dimension space • Construct a weighted completed graph G = (N, E) • Weight of edge (ui, uj)∈ E is δ(ui, uj)α • Find a minimum weight spanning tree T of G RA*: Optimal RA Theorem 7.3.2 (Kirousis et al. 2000)

c(T’) < c(RA*) Approximation for Two-dimension RA (2/4) • Proof: • Choosing any node u in N • Construct a shortest path destination tree rooted at u • Change all directed edges to undirected edges to build a spanning tree T ’ of G • RA*: optimal range assignment of G RA*(v)  RA*(w)  v w u u • N T’

Approximation for Two-dimension RA (3/4) • Proof (cont.): • By definition of MST, c(T) c(T’) c(T) = 7 c(T’) = 8 5 4 u u 3 3 MST T Shortest path rooted tree T ’ c(T) c(T’) < c(RA*)

Approximation for Two-dimension RA (4/4) • Proof (cont.): • During the construction of RAT, each edge of T can be chosen as the longest edge (as the transmitting rage) at most by two nodes (endpoints of the edge) 4 4 u MST T u 3 3 c(RAT) < 2c(T) c(T) = 4 + 3 = 7 c(RAT) = 4 +4 +3 = 11 c(RAT) < 2c(T) < 2c(RA*)

Symmetric Versions of RA (1/5) • Implementing of unidirectional wireless links is technically feasible, but theadvantageis questionable • For routing: high overhead needed to handle unidirectional links (e.g. AODV, DSR) • For MAC: considerable modification of the current implementation (e.g. RTS/CST in IEEE 802.11) • Certain symmetry constraints should be imposed on the communication graph • Weakly Symmetric Range Assignment (WSRA) • Symmetric Range Assignment (SRA)

Symmetric Versions of RA (1/5) • WSRA • Unidirectional links (dashed edges) are allowed, but they are not essential for connectivity

Symmetric Versions of RA (1/5) • SRA • All the links in the communication graph must be bidirectional

u5 u6 u4 u3 u2 u1 SRA in One-dimension Networks (1/3) • Step 1: Order the nodes according to their spatial coordinate {u1, . . ., un} • Step 2: Assign to node • RA(u1) = δ(u1, u2), • RA(un) = δ(un−1, un), • RA(ui) = max{δ(ui−1, ui ), δ(ui, ui+1)}.

u5 u6 u4 u3 u2 u1 SRA in One-dimension Networks (2/3) • Step 3: Augment the transmitting range of some of the nodes in order to preserve symmetry, repeated until all the edges in the graph are bidirectional

SRA in One-dimension Networks (3/3) • Proof of the optimality : • To achieve connectivity, every node must be connected at least to its left and right closest neighbor • To achieve symmetry, augmentation procedure at step 3 increases a node’s transmitting range of the minimal amount of the range assignment • Computational complexity • Symmetric eases the task of finding the optimal solution • SRA is a restricted version of WSRA

Network Optimization