cpsc 689: discrete algorithms for mobile and wireless systems

CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

Spring 2009

Prof. Jennifer Welch

Discrete Algs for Mobile Wireless Sys 2

Lecture 11 Topic:

Yet More Localization Sources:

Moore, Leonard, Rus & Teller Aspnes, Eren, Goldenberg, Morse, Whiteley, Yang,

Anderson, & Belhumeur MIT 6.885 Fall 2008 slides


Robust Quadrilaterals[Moore, Leonard, Rus, Teller]

2D setting, claim ideas extend to 3D. Problem:

Given graph with “enough” edges, distances on all edges, recover 2D coordinates of all nodes.

No anchors, so results can be unique only up to translation, rotation, reflection.

Issues: Relative coordinates only. Distance info may not be enough to yield global rigidity. Distance measurements may be noisy (approximate). Scalability.


Noisy Distance Estimates Can introduce anomalies, for some graphs and distance

assignments. Exact measurements on edges can yield a unique, exact solution

(up to T, R, R), But when a little noise is introduced:

We might no longer get exact solutions, but may have some error (difference between given and computed distances). Tolerable, unavoidable.

Solutions may no longer be unique, even allowing for small errors: varying the measurements a tiny amount could yield drastically different best solutions.

See Figure 2: A (complicated) point formation for which varying the measurements a little leads to an entirely different-looking best solution.

Errors caused by noisy measurements can become compounded through an iterative coordinate assignment procedure (GDOP).


Overview Robust quadrilateral:

Robust with respect to a bound e on error in distance calculations.

4 nodes in 2D, edges (and distances) between all pairs. Nodes spaced so, even with errors, only one realization is

possible. Algorithm:

Parameterized by bound e on measurement error. Start with a robust quad, “localize” it (assign it consistent

coordinates). Iterate, at each stage localizing one node in a new robust

quad that shares 3 nodes with a previously-localized robust quad, using atomic trilateration.

Algorithm is distributed, supports node insertion/deletion, mobility.


Overview

Similar to [Savvides] iterative multilateration, but with redundancy.

Algorithm properties: Avoids certain kinds of ambiguities (“flip”,

“discontinuous flex”). Each node localized correctly (with high probability) or

not at all. Drawback: Under conditions of low connectivity or high

noise, may not localize many nodes. Linear time in size of graph.


What is a Robust Quadrilateral? A completely-connected quadrilateral:

know distances between all four nodes A quadrilateral is globally rigid: Unique up to T,R,R Any 2 quads that share 3 nodes form a 5-vertex graph that

is also globally rigid. But global rigidity isn't enough to guarantee unique graph

realization when distance measurements are noisy. So, include additional constraints, on lengths of sides and

sizes of angles.


Flip Ambiguity Suppose we are trying to localize node D based on locations of A, B and

C We have measured distances between these nodes Distance from D to B and from D to C give two possible positions for D Use distance from D to A to disambiguate:

B

C

Ad'd

If distance from D to A is d, then D goes on the left

If distance from D to A is d', then D goes on the right

What if difference between d and d' is within the error of the measurement??

Flip ambiguity


Robust Triangle Cause of the flip ambiguity is that A, B

and C are almost collinear Avoid triangles that have very small

angles or very short sides Let b be length of shortest side and be

smallest angle in a triangle Require b sin2 > dmin for some threshold

dmin Analysis relates choice of dmin to

probability of error, assuming a distribution on measurement error

B

C

Ad'd

b


What is a Robust Quadrilateral?

Definition: A quadrilateral with all 6 edges, and distances on the edges, is robust provided that each of the four triangles that appear in the quad is robust.


Algorithm Overview

Assume graph is dense, nodes have many neighbors. Nodes have ranging capability, to determine distances. Cluster = a node (the root) + all its neighbors. Phase I: Cluster localization

Each node identifies the cluster with itself as the root. Determines distances to all its neighbors. Estimates locations for its neighbors that can be unambiguously

localized: Identifies all robust quads within the cluster. Finds the largest subgraph composed of “sufficiently

overlapping” (3 nodes) robust quads. Computes coordinates using a chain of quads and trilaterating

at each stage. Each node gets a local coordinate system with itself at the origin

(0,0).


Algorithm Overview Phase II: Cluster optimization

Refines position estimates using numerical optimization methods: reduces and redistributes error.

Phase III: Cluster Transformation “Stitch together” local coordinate systems:

Compute transformations (combinations of translations, rotations, reflections) between local coordinate systems that best align the clusters.


More on Phase III Now we have a lot of clusters localized separately, from

different root nodes. Must make all of these localizations consistent, by rotating,

translating, and reflecting. Can reduce the problem (?) to that of merging two separate

coordinate systems. Q: Does this assume that there is a single, generally-known,

order in which to merge? How is this determined (by a distributed algorithm)?

For the two-coordinate-system case, they refer to a “stitching” technique by Horn. LTTR.


One kind of severe ambiguity that could arise in localization: Flip ambiguity, as before:

Avoiding Flip Ambiguities

12

34

1

2

34


Discontinuous flex: Remove edge, then deform continuously to a different configuration, then reinsert removed edge with the same distance: Here, remove (5,3).

Flex Ambiguities

2

3

1

45

1

45

2

3


Assessment Requires known error model: probability distribution on the

measurement errors Not a theory paper---could use more theoretical analysis. Requires high graph density: degree 10 or more! Computational complexity:

In general, finding a realization of a weighted graph that is known to have a realization is NP-hard. (E.g., [Aspnes, Eren,…])

This algorithm is polytime. But it doesn't always find a realization if one is possible. Refuses to localize nodes that have position ambiguities---the

cases that typically cause algorithms to behave badly.


Assessment Experiments:

Small network of Crickets. Simulations, to study scalability:

183 nodes, placed uniformly at random. Connectivity only when not obstructed by (simulated) walls.

Fraction of nodes successfully localized: Measured by average percentage localized per cluster, and by fraction

of nodes in entire network that are localized into a single coordinate system.

Ability to localize decreases as measurement error increases. Excellent with small amounts of error (as good as with no error). Hardly ever localizes everything, because of obstructions (walls).


Assessment Accuracy of algorithm output:

Compared (relative) positions produced by algorithm to manual measurements.

Algorithm's errors are only slightly greater than basic measurement errors.

Error propagation reduced significantly over approaches based on basic trilateration.

Localizing mobile nodes: Algorithm can recompute cluster localizations (Phases I and II)

periodically. Fast, works well, accommodates mobile nodes. Don’t perform global Phase III with mobile nodes. Practical issues with mobile nodes:

Ranging estimates can be inaccurate. Noise can be misconstrued as observed motion. Trilateration inaccurate for a moving device.


Theory of Network Localization[Aspnes, Eren, Goldenberg,…]

Describe foundations for network localization in terms of graph rigidity theory.

Inspired by [Savvides,…], claim to provide foundations for the mobile-assisted localization and robust quad papers.

Focus on: Characterizations for unique localization. Computational complexity of localization.

Consider d dimensions, focusing on d = 2, 3


Network Localization Problem

Connected undirected graph G = (V, E), where V = {1,2,…,n}, and where {1,…,m} are “beacons” (anchors).

Network localization problem: Given graph G = (V, E), distances on all edges, and

positions p1, …, pm of the anchors, consistent with the given edge distances,

Determine positions of the non-anchors pm+1,…,pn, consistent with the given edge distances.

Problem is (uniquely) solvable if there is exactly one vector of positions (p1,…,pn) consistent with the given graph, edge distances, and anchor positions.


Network Localization Problem A variant: Uniqueness of localization:

Given graph G = (V, E) and a position vector (p1,…,pn) for all the nodes,

Determine whether there is a different position vector (p1,…,pm,qm+1,…,qn) having all the same edge distances.

Formulate this problem in terms of “global rigidity” of “point formations”…


Point Formations

d-dimensional point formation F = ((p1,…,pn), L) is a vector of n points in d-space, plus a set L of pairs (i, j) of point indices, representing the edges. Distance for link (i, j) is the Euclidean distance between

positions pi and pj. Determines a graph (V, L), where V = {1,…,n}, plus a

distance function defined by distances for links.

Point formation F is globally rigid if every other point formation F’ with the same graph and same distance function is congruent to F. Thus, allows translations, rotations, reflections.


Characterization Theorem 1:

Given graph G = (V, E) and positions p1,…,pn, for all the nodes.

Assume some simple non-collinearity,… conditions for nodes 1,…,m.

Let F be the point formation ((p1,…,pn), L), where L = E {(i, j) : 1 i, j m}. Adds in edges between all the anchors, to preserve

their relative positions. Then localization is unique for the given graph and

positions if and only if F is globally rigid.

Proof: LTTR.


Other Notions of Rigidity

Generically rigid graph G: A strong notion. Limits the types of “infinitesimal flexing” that G

may allow. Theorem: Characterization for generically rigid

graphs in 2D [Laman 02]. Generically globally rigid graph G: Some open

dense set of point formations are globally rigid. Redundantly rigid graph G: The removal of any

single edge results in a graph that is generically rigid.


Another Characterization

Theorem: Characterization for generically globally rigid graphs, in 2D: A graph G is generically globally rigid in 2D if and only if it is 3-connected and redundantly rigid.

More sources: Tina Nolte’s summary of Aspnes et al. paper and overview

of rigidity theory, Lecture 5 of 2006 version of MIT couse 6.885.

Connelly, R. Manuscript on rigidity theory, Cornell. Basic Concepts.


Results [Aspnes, et al.] Multilateration construction to construct generically globally

rigid point formations. Construction and proof based on [Savvides].

Computational complexity results: Testing whether localization is possible is NP-hard [Saxe]. Assigning coordinates is NP-hard, even if it is known that

localization is possible. True even for restricted (unit-disk) graphs, and even for

approximate solutions. But localization is polytime for multilateration graphs.

Study rigidity properties of random graphs.


Questions Theory still needs clarification?

Point formations vs. graphs, graphs with anchors vs. without,…

Many notions of rigidity: local vs. global, generic (applies not just at one point formation, but in a small region), redundant,…

How do these notions explain the properties of practical localization algorithms? In particular: How can the theory precisely justify the “mobile assisted”

strategy? How can the theory precisely justify the “robust

quadrilaterals” strategy?

cpsc 689: discrete algorithms for mobile and wireless systems

Documents

distance measurements

mobile wireless syslecture

mobile wireless syswhat

node d

wireless systemsspring

noisy measurements

exact measurements

d coordinates