Approximating Sensor Network Queries Using In-Network Summaries Alexandra Meliou Carlos Guestrin Joseph Hellerstein

Approximate Answer Queries Approximate representation of the world: Discrete locations Lossy communication Noisy measurements Applications do not expect accurate values (tolerance to noise) Example: Return the temperature at all locations ±1C, with 95% confidence Query Satisfaction: On expectation the requested portion of sensor values lies within the error range

In-network Decisions Query Use in-network models to make routing decisions No centralized planning

In-network Summaries Spanning tree T(V,E’) + Models M v for all nodes v M v represents the whole subtree rooted at v.

Model Complexity Need for compression Gaussian distributions at the leaves: good for modeling individual node measurements Gaussian distributions at the leaves: good for modeling individual node measurements

Talk “outline” Compression Traversal Construction In-network summaries

Collapsing Gaussian Mixtures Compress an m-size mixture to a k-size mixture. Look at simple case (k=1) Minimize KL- divergence? “Fake” mass

Quality of Compression Depends on query workload Query with acceptable error window W Query with acceptable error window W’<W

Compression Accurate mass inside interval No guarantee on the tails

Talk “outline” Compression Traversal Construction In-network summaries

Query Satisfaction A response R={r 1 …r n } satisfies query Q(w,δ) if: In expectation the values of at least δn nodes lie within [r i -w,r i +w] In-network summary Q R [r 1, r 2, r 3, r 4, r 5, r 6, r 7, r 8, r 9, r 10 ] Within error bounds

Optimal Traversal Given: tree and models Find: subtree such that Can be computed with Dynamic Programming response [μ leaves ]

Greedy Traversal If local model satisfies Return μ Else descend to child node More conservative solution: enforces query satisfiability on every subtree instead of the whole tree More conservative solution: enforces query satisfiability on every subtree instead of the whole tree

Traversal Evaluation

Talk “outline” Compression Traversal Construction In-network summaries

Optimal Tree Construction Given a structure, we know how to build the models But how do we pick the structure?

Traversal = cut Theorem: In a fixed fanout tree, the cost of the traversal is where |C| is the size of the cut, and F the fanout Theorem: In a fixed fanout tree, the cost of the traversal is where |C| is the size of the cut, and F the fanout Intuition: minimize cut size Group nodes into a minimum number of groups which satisfy the query constraints Clustering problem

Optimal Clustering Given a query Q(w,δ), optimal clustering is NP-hard Related to the Group Steiner Tree Problem Greedy algorithm with factor log(n) approximation Greedily pick max size cluster Issue: does not enforce connectivity of clusters

Greedy Clustering Include extra nodes to enforce connectivity Augment clusters only with accessible nodes (losing the logn guarantee)

Clustering comparison 2 distributed clustering algorithms are compared to the centralized greedy clustering

Talk “outline” Compression Traversal Construction In-network summaries Enriched models

Support more complex models k-mixtures Compress to a k-size mixture instead of a SGM Virtual nodes Every component of the k-size mixture is stored as a separate “virtual node” SGMs on multiple windows Maintain additional SGMs for different window sizes More space, more expensive model updates (SGM = Single Gaussian Model)

Evaluation of enriched models SGM surprisingly effective in representing the underlying data

Sensitivity analysis Talk “outline” Compression Traversal Construction In-network summaries

Tree Construction Parameters and Effect on Performance Confidence Performance for workloads of different confidence than the hierarchy design Error window Broader vs narrower ranges of window sizes Assignment of windows across tree levels Temporal changes How often should the models be updated

Confidence Workload of 0.95 confidence Design confidence does not have a big impact on performance

Error windows A wide range is not always better, because it forces the traversal of more levels

Model Updates

Sensitivity analysis Conclusions Analyzed compression schemes for in-network summaries Evaluated summary traversal Studied optimal hierarchy construction Studied increased complexity models Showed that simple SGM are sufficient Analyzed the effect on efficiency of various parameters Compression Traversal Construction In-network summaries Enriched models