Network Optimization

Network Optimization Lectures 15,16 - 1

Taxonomy of TC

1 1 1 1 Unit Disk Graph • Consisted of a set V of n nodes • Distributed in a two-dimensional plane • Equipped an omnidirectional antenna with the same Pmax • Unit disk graph, UDG(V) V

0.8 0.8 0.4 0.4 0.5 0.5 UDG( ) G’( ) 0.6 Minimum energy for unicasting a to b: 0.8 b to c: 1.0a to c: 0.5 b to d: 0.4a to d: 1.1 c to d: 0.6 Minimum energy for unicastinga to b: 0.8 b to c: 1.3a to c: 0.5 b to d: 0.4a to d: 1.2 c to d: 1.7 0.8 / 0.8 = 1 1.3 / 1.0 = 1.30.5 / 0.5 = 1 0.4 / 0.4 = 11.2 / 1.1 = 1.09 1.7 / 0.6 = 2.83 G’(V)(UDG(V)) = max {1, 1, 1.09, 1.3, 1, 2.83} = 2.83 Power Stretch Factor (1/2) b b a a V V d d c c

Power Stretch Factor (2/2) • Definition: • PG(u, v): least energy path of two nodes u and v in a graph G • Given a set V of n nodes, the power stretch factor of a subgraph S(V) with respect to UDG(V): Upper bound of size n

Proximity Graphs • Proximity graph G(V) of UDG(V) • Sparser, i.e. G(V)  UDG(V) • Can be constructed locally, e.g. 1-hop locations • Well-known proximity graphs • Gabriel graph, GG(V) • Relative neighborhood graph, RNG(V) • Yao graph, YG(V)

Relative Neighborhood Graph • Definition: Given a set V of nodes, RNG(V) consists of all edges uv such that ||uv||  1 and there is no wV such that ||uw|| < ||uv||, and ||wv|| < ||uv|| w w u v u v uv  RNG(V) uv  RNG(V)

Power Stretch Factor – RNG(1/2) • Theorem: RNG(n) = n – 1 • It was proved that EMST(V)  RNG(V) • Any path between u and v in EMST(V) • contain at most (n - 1) edges • each edge has length at most ||uv|| • PRNG(u,v)  PEMST(u,v)  (n - 1)||u,v|| • RNG(n)  n – 1 3 3 2 2 3 3 3 2 3 2 5 5 u v u v

Power Stretch Factor – RNG(2/2) • Theorem: RNG(n) = n – 1 (cont.) •  = /3 +  •  = /3 – 2 • As  0, length of each edge  ||v1v2|| • As  0, RNG(v1,v2)/ UDG(v1,v2)  (n – 1) • RNG(n) > n – 1 –  Asymptotic analysis n is even n is odd

Gabriel Graph • Definition: Given a set V of nodes, GG(V) consists of all edges uv such that ||uv||  1 and the open disk using uv as diameter does not contain any wV. w w u v u v uv  GG(V) uv  GG(V)

Power Stretch Factor – GG • Theorem: GG(n) = 1 ( = 2, c = 0) i.e.GG(V)  EG2,0(V) v u r

Yao Graph • Definition: Given a set V of nodes, and an integer parameter k 6, • At each u, any k equal-separated rays originated at u defined k cones. • In each cone, choose the closest nodevto u with distance at most one, if there is any, and add a directed link uv. • Ties are broken arbitrarily • YGk(V) is the undirected graph by ignoring the direction of each link. v v v u u u w w w

Power Stretch Factor – YGk (1/4) • Theorem: For any integer k 6, • Proof. • Let  = 1/(1-(2sin/k)) • Construct a path u~v in YGk(V) • Prove by induction that P(u~v) < ||uv|| on the number of its edges • Initial: if uv  YGk(V), p(u~v) = uv • Assumption: the claim is true for any path with l edges. • Induction: prove it is also true for any path wit l + 1 edges

Power Stretch Factor – YGk (2/4) • Proof. (cont.) • If uv  YGk(V), p(u~v) = uv • Otherwise, exist a node w in the same cone of v, which is a neighbor of u in YGk(V) such that p(u~v) = p(uw)p(w~v)  uwv is not acute  uwv is acute

The claim Power Stretch Factor – YGk (3/4) • Proof. (Case 1:  uwv is not acute) • ||uw||2 + ||wv||2  ||uv||2 • k  6    /3 • ||uw|| < ||uv||  ||uw||/||uv||  1 • ||wv|| < ||uv||  ||wv||/||uv||  1 • So, ||uw|| + ||wv||  ||uv||, for any   2 • P(u~v) = ||uw|| + P(w~v) = ||uw|| + ||wv||  ||uv||

Power Stretch Factor – YGk (4/4) • Proof. (Case 2:  uwv is acute) • We know ||uw||  ||uv|| • Max length of vw is achieved when ||uw|| = ||uv|| • wuv<  = 2/k • ||wv||  2sin(/2)||uv|| = 2sin(/k)||uv|| • P(u~v)  ||uw|| + ||wv|| • P(u~v)  ||uw|| +  (2sin(/k)||uv||) • P(u~v)  ||uv||

Comparison (1/3)  Node degree of RNG is bound by 6 if no two neighbors having the same distance to a node u v Unbound case of RNG

Comparison (2/3) • YGk has bounded out degree, but some nodes may have a very large in-degree. v w v w u u

Comparison (3/3) • Property (relationships) • EMST(V)  RNG(V) • RNG(V)  GG(V) • RNG(V)  YGk(V), for any k  6 • Property (sparseness) • |RNG(V)|  3n – 10 • |GG(V)|  3n – 8 • |YGk(V)|  nk • Average node degrees are all bounded by a constant

Some Variations • GYGk(V): First Yao then Gabriel graph • YGGk(V): First Gabriel then Yao graph • Second phase can further improve the sparseness • Power stretch factors remains the same of YGk(V) • Out in-degrees are still unbounded

Yao and Sink (YG*k) • In each cone of a node u, recursively construct a spanning towards node u

Yao and Sink (YG*k) w x z v u

Yao and Sink (YG*k) • Node z w x z v u

Yao and Sink (YG*k) w x z v u

Yao and Sink (YG*k) • Replace the directed star consisting of all links towards a node u by a directed tree T(u) as the sink

Yao and Sink (YG*k) YGk YG*k

Ordered Yao Graphs General idea: apply YGk on GG according to some ordering on nodes

OrderYaoGG • Consist of three phases • Construct GG • Compute local ordering of nodes in GG • Construct YGk on GG according to the local ordering

OrderYaoGG • Phase 1: each node self-constructs its neighbors in GG n messages

OrderYaoGG • Phase 2: Each node compute its local ordering in GG Initially, the order is set 0, i.e. unordered.

OrderYaoGG • Phase 2: Each node compute its local ordering in GG

OrderYaoGG • Phase 3: Construct YGk on GG according to the local ordering

OrderYaoGG

SYaoGG

Comparison of Various Yao Graphs

Yao and Sink (YG*k)

Network Optimization