Efficient Verification of Shortest Path Search via Authenticated Hints. Presented by: Nuttiiya Seekhao

Transcription

1 Efficient Verification of Shortest Path Search via Authenticated Hints Presented by: Nuttiiya Seekhao

2 Overview Motivation Architectural Framework Goals Solutions: Proof Integrity Shortest Path Performance Summary

3 Motivation for Shortest Path No one like to take the long route!

4 Motivation for Shortest Path

5 Motivation for Shortest Path

6 Motivation for Shortest Path Verification Model: Data Owner: trusted road network data owner,, etc. Service Provider (SP): Third Party..,, etc. Clients Malicious purposely returning you sub-optimal paths Save computational resources Profit purposes Good getting infiltrated by

7 Architectural Framework

8 Goals The proof size should be as small as possible The offline construction cost and storage overhead of authenticated hints should be low

9 Integrity Proofs Γ T Assuming shortest path proof, Γ S, is given Case 1: Γ S is a subgraph proof Γ S SG Γ S G, a subgraph which contains enough information to proof path optimality Need to prove: G contains only existing nodes Full adjacency information for each of the node is accurately reported Case 2: Γ S is a distance proof Γ S DT Need to prove: Integrity of nodes in P rslt (shortest path returned)

10 Integrity Proofs Γ T Merkel Hash Tree on Graph Nodes Extended tuple, Attributes of v, contains Adjacency list (with weight info) of Φ v Let Digest of a node Φ( v) ( ) = v. id, v. x, v.y, " H ( ) { v,w ( v, v ") ( v, v ") E} be a secure hash function v V is H ( Φ( v) ) v

11 Integrity Proofs Γ T ( ) Data Owner,, builds a Merkle tree on Φ v with respect to some graph-node ordering

12 Integrity Proofs Γ T Service Provider,, generates Γ T from Γ S A hash entry h i is inserted into the integrity proof Γ T if i. The sub tree of h i contains no tuple Φ(v) in Γ S and ii. The parent hash entry of h i doesn t satisfy condition (i) Example: wants to send subgraph G! contains nodes v 32, v 33, v 42 to client. It builds a set Γ S = { Φ( v 32 ), Φ( v 33 ), Φ( v 42 )} Then examine the Merkel tree to generate integrity proof Γ T = H ( Φ( v 31 )), H ( Φ( v 41 )), H ( Φ( v 43 )), h 1, h 2, h 5, h 6, h 18 { }

13 Shortest Path Proofs Γ S Basic Solutions Dijkstra Subgraph Verification (DIJ) Fully Materialized Distances (FULL) More Practical Authenticated Hints Landmark-based Verification Method (LDM) Hyper-graph Verification Method (HYP)

14 Basic: DIJ Γ S SG Γ S contains the Φ v of each vertex that is within distance dist v s, v t from Shortest path verification, client performs: Dijkstra s algorithm on the subgraph defined by proof Γ S and compute If Γ S contains all nodes required by Dijkstra s algorithm then it is said to be valid Path is deemed correct if i. Γ S is valid ii. ( ) v ( ) v s ( ) dist v s, v t The shortest path distance on Γ S is the same as distance report by the service provider

15 Basic: FULL Γ S DT Data Owner,, applies the Floyd-Warshall algorithm to compute the shortest path distance for each pair of nodes, v j V v i dist ( v i, v j ) Use MHT to store and check integrity of shortest distances Client checks integrity of: Shortest distance returned from Shortest path returned from Pre-computation complexity is O V 3! Total number of distances to store is Not scalable! ( ) ( ) O V 2

16 What s wrong with DIJ and FULL? DIJ: Very big proof! -> High communication overhead FULL: High pre-computation cost and storage overhead O( V 3 ) and O( V 2 ) respectively Let s move towards more practical approach!

17 LDM Γ S SG Landmark Distance Vector Ψ ( v) = dist s 1,v Lower bound distance between v and v dist LB v, v! Extended Tuple ( ),dist ( s 2,v),...,dist ( s c,v) ( ) = max i! " 1,c # $ dist s i,v Φ ( v) = v.id,v.x,v.y,ψ ( v),! ( ) dist ( s i, v& ) { v,w ( v, v!) ( v, v!) E}

18 LDM Γ S SG The service provider generates proof { Γ S SG = Φ v ( ),Φ ( v! ) ( v, v! ) E,v V,dist ( v s,v) + dist LB ( v,v t ) dist ( v s,v t )} The client performs A* search using the landmarkbased lower bound distance dist LB (.) How is this better than DIJ? Smaller search space (touches less # of vertices)

19 HYP Γ S SG, Γ S DT HiTi Graph Partitioned into cells Shortest paths of every pair or border cells are precomputed and stored as hyper edges

20 HYP Γ S SG, Γ S DT Build graph-node MHT, just like earlier except: Φ( v) = v. id, v. x, v.y, { v ",W ( v, v ") ( v, v ") E}, v. c, v. is _ border Build distance MHT similar to FULL method

21 HYP Γ S SG Subgraph Proof on Concise Coarse Graph Γ SG S contains i. Φ( v i ) for all v i in C s and C t ii. Hyper-edge E * ( v, v bsi bt where is j ) v bsi any border node in C s and vbt j is any border node in C t Server generates integrity proof for i. Using graph-node MHT ii. Using distance MHT

22 HYP Γ S DT Distance Proof on the Original Fine Graph Γ S DT is essentially integrity proof of all nodes in the shortest path The client checks whether the path distance of the fine proof (Γ S DT ) is the same as the shortest distance from the coarse proof (Γ S SG )

23 Experimental Results DIJ: No pre-computation, high communication overhead FULL: lowest comm. overhead, highest pre-computation LDM, HYP: Decent tradeoffs between the two

24 References Goldberg, Andrew V., and Chris Harrelson. "Computing the shortest path: A search meets graph theory." Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, Yiu, Man Lung, Yimin Lin, and Kyriakos Mouratidis. "Efficient verification of shortest path search via authenticated hints." Data Engineering (ICDE), 2010 IEEE 26th International Conference on. IEEE, 2010.

25 Questions?