TEDI: Efficient Shortest Path Query Answering on Graphs Author: Fang Wei SIGMOD 2010 Presentation: Dr. Greg Speegle

Shortest Path Problem Graph G=(V,E) V = set of vertices E = set of edges (u,v) in E, u, v in V Path from u to v Sequence of edges Shortest Path Fewest edges in path Unweighted

Issues of Shortest Path Solutions Algorithms require O(|V| 2 ) time Transitive closure requires |V| 2 space Challenge: Do better!

TEDI TreE Decomposition based Indexing Construct representation (tree decomposition) Algorithms on representation Equivalent results More efficient Index creation flexible

Tree Decomposition Convert graph G into tree T Nodes in T Labeled (integers in paper) Subset of V Properties: All vertices in some node All edges in some node Connectedness condition

Connectedness Condition Consider a vertex v Consider nodes with v Such nodes must form a subtree (i.e., be connected)

Tree Paths Tree vertex If v is in Tree node i, {v,i}. Inner Edges (u,v) in E Both u and v in i (some such i must exist) ({u,i},{v,i}) is inner edge ({v,i},{u,i}) also inner edge Multiple i possible

Tree Paths (cont'd) Inter Edges For v in V, exists set of nodes with v If edge between nodes i and j ({v,i},{v,j}) is inter edge Tree Paths Inner and Inter Edges “Up and down” tree Represent as sequence of tree edges

Path Equivalence There exists a path in G from u to v, iff there exists a tree path from {u,i} to {v,j} Note: No restriction on {u,i} and {v,j} Tree path to path easy  Inner Edges are path  Exists tree node with every edge Path to tree path is inductive proof (see paper)

Shortest Path Assume sdist(u,w) computed for all w seen so far Assume sdist(t,z) computed for all t,z in parent Compute sdist(u,z) as min(sdist(u,t)+sdist(t,z)) for all t in both parent and child, for all z only in parent Union results with previous computations

Algorithm Shortest Path Lookup Let u.r (v.r) be root of tree for u (v) Find lca(u.r,v.r) Compute sdist for all points between u.r and lca and v.r and lca Compute sdist(u,v) at lca Shortest path is concat of shortest paths to lca

Experimentation

Conclusion Faster than existing algorithms Smaller index creation Scales to large graphs Best current solution