CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

Discrete Algs for Mobile Wireless Sys2 Lecture 11  Topic: Yet More Localization  Sources: Moore, Leonard, Rus & Teller Aspnes, Eren, Goldenberg, Morse, Whiteley, Yang, Anderson, & Belhumeur MIT Fall 2008 slides

Discrete Algs for Mobile Wireless Sys3 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.

Discrete Algs for Mobile Wireless Sys4 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).

Discrete Algs for Mobile Wireless Sys5 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.

Discrete Algs for Mobile Wireless Sys6 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.

Discrete Algs for Mobile Wireless Sys7 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.

Discrete Algs for Mobile Wireless Sys8 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 A d'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

Discrete Algs for Mobile Wireless Sys9 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 sin 2  > d min for some threshold d min  Analysis relates choice of dmin to probability of error, assuming a distribution on measurement error B C A d'd b 

Discrete Algs for Mobile Wireless Sys10 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.

Discrete Algs for Mobile Wireless Sys11 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).

Discrete Algs for Mobile Wireless Sys12 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.

Discrete Algs for Mobile Wireless Sys13 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.

Discrete Algs for Mobile Wireless Sys14  One kind of severe ambiguity that could arise in localization: Flip ambiguity, as before: Avoiding Flip Ambiguities

Discrete Algs for Mobile Wireless Sys15  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

Discrete Algs for Mobile Wireless Sys16 Discontinuous Flex Ambiguities  Claim that robust quadrilaterals “rule out” the possibility of discontinuous flex ambiguities.  Because removing a single edge leaves a locally rigid structure.  ???  Q: Precisely what does this say about the setting with noisy measurements?  Ruled out by use of robust quads: A flex ambiguity involves removing an edge, thereby obtaining a non- locally-rigid point formation, deforming this formation to a very different one, then putting the edge back (same distance as before). Thus, removing one edge yields a non-locally-rigid point formation. But removing one edge from a robust quad yields a locally rigid point formation. Starting from a locally rigid point formation, perturbing distance measurements slightly leads to correspondingly small changes in the best solution. Thus, small errors in the distance measurements cannot cause large discontinuous flex ambiguities.

Discrete Algs for Mobile Wireless Sys17 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.

Discrete Algs for Mobile Wireless Sys18 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).

Discrete Algs for Mobile Wireless Sys19 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.

Discrete Algs for Mobile Wireless Sys20 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

Discrete Algs for Mobile Wireless Sys21 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 p 1, …, p m of the anchors, consistent with the given edge distances, Determine positions of the non-anchors p m+1,…,p n, consistent with the given edge distances.  Problem is (uniquely) solvable if there is exactly one vector of positions (p 1,…,p n ) consistent with the given graph, edge distances, and anchor positions.

Discrete Algs for Mobile Wireless Sys22 Network Localization Problem  A variant: Uniqueness of localization: Given graph G = (V, E) and a position vector (p 1,…,p n ) for all the nodes, Determine whether there is a different position vector (p 1,…,p m,q m+1,…,q n ) having all the same edge distances.  Formulate this problem in terms of “global rigidity” of “point formations”…

Discrete Algs for Mobile Wireless Sys23 Point Formations  d-dimensional point formation F = ((p 1,…,p n ), 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 p i and p j. 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.

Discrete Algs for Mobile Wireless Sys24 Characterization  Theorem 1: Given graph G = (V, E) and positions p 1,…,p n, for all the nodes. Assume some simple non-collinearity,… conditions for nodes 1,…,m. Let F be the point formation ((p 1,…,p n ), 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.

Discrete Algs for Mobile Wireless Sys25 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.

Discrete Algs for Mobile Wireless Sys26 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 Connelly, R. Manuscript on rigidity theory, Cornell. Basic Concepts.

Discrete Algs for Mobile Wireless Sys27 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.

Discrete Algs for Mobile Wireless Sys28 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?