Computer science department university of Southern California Los Angeles. Songhua Xing a PhD candidate at Computer Science Department now at IBM T.J. Watson Research Center Country: China Cyrus Shahabi Director of the Information Laboratory (InfoLAB) Professor of Computer Science and Electrical Engineering at USC Director of the Integrated Media Systems Center Chief Technology Officer and Founder of Geosemble Technologies Ph.D. in Computer Science Bei Pan a PhD candidate at Computer Science Department Microsoft Research Asia, Autodesk, University of Southern California Country:China

New type kNN queries Surface k Nearest Neighbor(skNN) queries Extend to constrained third dimension Two exact methods for highly dynamic environment as arbitrary movement SE(Surface Expansion)-Tree Fat and short - not efficiency an analogous counterpart of the Dijkstra Expansion Tree on land surface ASI(Angular Surface Index)-Tree Thin and tall – low cpu and I/O overhead

Related research Static and dynamic query Problem & Preliminaries kNN methods only on Euclidean and road networks. skNN is for surface distance, represented as triangular meshes. CskNN can be used on continuous queries and the complexities of land surfaces. CskNN algorithm monitor and coordinate among the moving objects. One of CkNN for road map Dijkstra Expansion tree SE-Tree – for Static query More effective – ASI-Tree Details of our surface index(ASI) and its corresponding CskNN Dynamic query Experiments Summary and future work ASI- Tree outperforms SE-Tree

Static(snapshot) query Different constrained environment Road networks Land surface Dynamic query The paper three types of updates Object movements Query movements Fluctuations of edge weights

Assumption and problem definition Moving object Point Of Interest Static query point Three distance metrics Euclidean distance Lower bound of surface distance Network distance Upper bound of surface distance Surface distance

Problem definition query consists of two steps snapshot skNN query continuously monitoring and updating the result sets as the objects move

Shortest surface path computation Chen-Han algorithm costs Unfolding process - expensive

Surface expansion tree Dijkstra Expansion tree, based on the Chen- Han algorithm Definition Surface Expansion Tree is the final result of Chen-Han algorithm and there is only one path from the source to the vertice

Surface expansion tree

Observation 1 makes partitioning these surface shortest paths of an SE-Tree possible. Observation 2 Drawback SE-Tree in general is fat and short

Surface Expansion Tree Initial query processing – two areas

Three categories Within the result boundary Ignore this case Result set remains the same Incoming movement Outgoing movement Two scenarios More Outgoing movement More Incoming movement

Expansion phase, the complexity is In the shrinking phase, there is no surface distance computation Complexity is mlog(m)

Similarity All these methods built an expansion tree rooted at the query point The result boundary and expansion boundary are the same on road networks. Different This naïve approach could be fast during the phase when the SE-Tree shrinks Expansion Two problems: Surface path computation is extremely high Expansion areas of SE-Tree could be large. Overcome by Surface Shortest Path Container store partial results of pre-computation build a novel index schema(Angular Surface Index(ASI))

Angular Surface Index (ASI) Thin and tall Two data structure Surface Shortest Path Container Surface Equidistant Line

Surface Shortest Path Container To pre-compute a complete SE-Tree offline and store its shortest path. Two Steps locate the data object using a spatial index retrieve the shortest path directly from disk Drawbacks a data object lays on the face rather than a vertex, this approach cannot find the exact shortest path and the accurate distance storing all these shortest paths is per site The search time is almost linear. How to speed up take advantage of partial results based on geometric property to speed up the online process.

The advantage is to minimize the search area of Dijkstra algorithm. A new concept of Cover Set and redefine the concept of Shortest Path Container for surface, and then discuss their spatial properties.

According to Observation 1, we can always find a polyline sp from the source s to a point p on the margin of T, which is immediately left to the leftmost shortest path to CS(e) and do not cross any shortest paths, hence sp constitutes the left part of the boundary b.

Container’s boundary consists of the left boundary line, the right boundary line the end boundary line (which only exists if the left and right boundary lines do not converge)

Propose an algorithm to create a surface shortest path container

In Line 6, the end boundary can be NULL if left and right boundaries do not intersect the margin. The time complexity of Algorithm 3 is O(NlogN) due to the sort operation in Line 3. However, since the pre-computation of shortest paths takes, the overall time complexity is.

Designed to partition along the horizontal (latitude) direction These lines are sorted by their increasing distance value to the source point and this order is termed as levels

Based on surface shortest path containers and surface equidistant lines Each partition of is called a surface chunk. With this ASI-Tree, each node represents a container. Compared with SE-Tree, ASI-Tree has the following advantages

Experiment setup Model BH: Bearhead (BH) area in WA, USA which covers an area around 10.7km×14km and 2) EP: Eagle Peak (EP) area in WY, USA with similar size as BH. Create five synthetic surface models with the same size (10km×10km) Device PC with Intel 6420 Dual CPU 2.13G Hz and 3.50 GB RAM The operating system is Windows XP SP2 The parameters 100 CskNN queries, each query is 50 timestamps. The first 6 parameters are tested on both BH and EP (Surface Roughness RA is only for synthetic data sets.)

The Impact of k ASI based algorithm outperforms the naïve algorithm both in query efficiency and I/O operations (least a factor of two for k > 4.) Performance in I/O by an average factor of two because the search is localized to avoid unnecessary access to surface vertices.

Uniform or Gaussian distributions the ASI based algorithm has a slightly better performance for objects with Gaussian distribution than objects with uniform distribution.

The Impact of Object Distribution and DO Both query processing time and I/O cost decrease for both algorithms as DO increases.

The Impact of a and v (a)(b)both query processing time increases slightly as well because the possibility to enlarge the search area is increased. (c)(d)both algorithms are practically unaffected by object speed because the core of both algorithms only concern whether there are object updates rather than how far the objects move.

The Impact of DC the performance is enhanced as more containers are created for both BH and EP. The Impact of RA ASI-based algorithm keeps outperforming the naïve Algorithm rougher terrains could probably generate a larger search area than smooth terrains.

Propose two algorithm naïve algorithm surface index (ASI) based algorithm ASI-based algorithm outperforms the naïve algorithm under all circumstances Simplified problem setting (pre-defined static query points)

Further studying these complex settings, where queries move arbitrarily.