1. Cache Placement in Sensor Networks Under Update Cost Constraint Bin Tang, Samir Das and Himanshu Gupta Department of Computer Science Stony Brook University NY 11790, USA

2. Outline Problem Statement Motivation Algorithm Design Performance Evaluation Conclusion and Future Work

3. Problem Statement Sensor Network Model A data item is sensed and stored in one node (server) and updated at certain frequency. A set of sensor nodes access the data item at certain frequency. Problem Statement Selection of nodes to cache a data item s.t. the total access cost is minimized under a given update cost constraint.

4. Motivation Why caching ? Caching can save communication cost, thus achieve energy efficiency. Why update constraint ? Server node and its surrounding nodes bear most of the communication cost incurred in updating.

5. Problem Formulation Given : A sensor network graph G=(V,E) A data item stored in a server and its update frequency Access frequency of each client node Update cost constraint Δ Goal : Select cache nodes to minimize the total access cost Total update cost is less than Δ

6. Algorithm Design Outline Tree network: optimal dynamic programming algorithm General graph network: Multiple-unicast update cost model: constant-factor algorithm Steiner tree update cost model: heuristic and distributed

7. Tree Network

8. Subtree notation

9. Dynamic Programming Algorithm for Tv under update cost constraint δ path (v,x) on leftmost branch are cache nodes Ov is the optimal set of caches (not including path(v,x)) in Tv u is the leftmost deepest node of Ov in Tv path(v,u) are all cache nodes Claim: Ov can be computed recursively using the optimal set of caches in Rv,u with the update cost constraint (δ – the length between u and path(v,x))

10. DP recursive equation for Tv Let Γ(Tv, x, δ) denote the optimal access cost for all the nodes in Tv under additional update cost constraint δ, which does not include cost of updating already selected caches on path(v,x). Γ(Tv, x, δ) = min u є Tv ( access cost of Lv,u + access cost of Tu + Γ(Rv,u, p(u), δ – the length between u and path(v,x) ) where p(u) is the parent node of u.

11. Time complexity for original Tr under update cost constraint Δ Original problem is to find Γ(Tr, r, Δ) Time complexity is O(n 4 +n 3 Δ)

12. General Graph Network Two Update Cost Models Multiple-Unicast Optimal Steiner Tree

13. Multiple-Unicast Update Model Update cost: the sum of the individual shortest path lengths from the server to each cache node Benefit of node A: the decrease in total access cost resulting due to the selection of A as a cache node Benefit per unit update cost:

14. Greedy Algorithm Consists of rounds In each round, the node with highest benefit per update cost is selected as a new cache Iterates until the update cost of cache nodes reaches allowed constraint Theorem: the Greedy Algorithm returns a solution whose benefit is at least 63% of the optimal benefit

15. Steiner Tree Update Cost Model Steiner update cost: the cost of a 2-approximation Steiner tree over cache nodes Incremental Steiner update cost of node A: the increase in the Steiner update cost due to the selection of A as cache node Greedy-Steiner Algorithm: In each round, the node with highest benefit per incremental Steiner update cost is selected as a new cache

16. Distributed Greedy-Steiner Algorithm In each round, each non-cache node estimates its benefit per incremental update cost If the estimate is the maximum among all its non-cache neighbors, then it decides to cache At the end of each round, the server computes the Steiner tree cost involving all the cache nodes The remaining update cost is broadcast to the network and a new round is initiated

17. Performance Evaluation (i) network-related -- the number of nodes and network density, (ii) application-related -- the number of clients accessing each data item. a randomly generated sensor network of 2,000 to 5,000 nodes in a square region of 30 x 30.

18. Compared Caching Schemes Centralized Greedy Algorithm Centralized Greedy-Steiner Algorithm Distributed Greedy-Steiner Algorithm Dynamic Programming on Shortest Path Tree of Clients Dynamic Programming on Steiner Tree over Clients and Server

19. Varying Network Size – Transmission radius Tr=2, number of clients = 50% of the number of nodes, update cost = 25% of the Steiner tree cost

20. Varying Transmission Radius - Network size = 4000, number of clients = 2000, update cost = 25% of the Steiner tree cost

21. Varying number of clients – Tr=2, update cost = 50% of the Steiner tree cost, network size = 3000

22. Conclusion and Future Work Data caching problem under update cost constraint. Optimal algorithm for tree; a bounded algorithm for general graph. Efficient distributed implementations. More general cache placement problem: (a) under memory constraint; (b) multiple data items.