1. Presented by: Dardan Xhymshiti Spring 2016:

2. Authors: Publication:  ICDM 2015 Type:  Research Paper 2 Michael ShekelyamGregor JosseMatthias Schubert Institute of Informatics, Ludwig-Maximilians-University Munich

3.  In many application areas, data is organized as a network of graph.  Important task: compute a cost-optimal path between a start node and a target one.  Example:  Road networks (Cost criteria: travel time, travel distance, energy consumption etc.)  Computer networks (Cost criteria: bandwidth and the latency between routers.)  Cost vector: when considering more than one criterion at a time, the cost of complete path is called cost vector. 3 Cost criteria 1Cost criteria 2Cost criteria 3

4. 4  How to define if a path is an optimal path?  1. Map the cost vector to a value by employing a monotonic combination function, and then sort the paths. The top n paths are the optimal ones. Problems: a) Hard to find a suitable function, b) Different types of cost might have different levels of scale.  2. Compute the pareto optimal (mathematical definition of Skyline) cost vectors. (This is also known as conventional path skyline). Problems: a) The number of pareto optimal paths might increase exponentially as function of distance and the amount of considered cost criteria. b) Showing to the user a large amount of results is not very helpful.

5.  There actually exist solutions for computing linear path skylines, but they are restricted to the specific case of having just two cost criteria. Problem: cannot be generalized to more criteria.  The number of skyline paths increases exponentially with the distance between the locations and the number of cost criteria. Thus the result set might be too big. 5

6.  Come up with a new approach of computing the results set of path skylines, which provides better and faster results. 6

7.  Recall: Conventional path skyline computes all potential optimal paths, but the result is too big.  Idea: reduce the result set, to only show the paths which are optimal under a weighted sum function or linear combination of cost criteria.  Intuitively saying: The user weights each type of cost with a percentage describing its importance.  How to compute the linear path skyline?  Naive approach: compute the conventional path skyline and then compute the convex hull on the resulting cost vectors. (Inefficient). 7

8.  What is Convex Hull?  Definition: In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X. 8

9.  The authors come up with an algorithm called LSCH (Linear Skyline Convex Hull) which constructs a linear path skyline successively while only adding new paths which are members of the result set.  Implementation overview:  1. To add a new cost vector, a single search is performed which combine the cost vectors based on the normal vectors of the hyper planes currently limiting the linear path skyline.  2. To identify the areas on the linear skyline where additional results might still exits, the algorithms applies multidimensional convex hull computation. 9

10.  Experiments are run in two different types of networks:  1. Munich road network with five cost criteria.  2. Artificial lattice graphs that allow to simulate different problem instances and parameter settings. 10

11.  Computing route Skylines algorithms.  Convex hull algorithm. 11