Dimitrios Katsaros 1,2 Nikos Dimokas 1 Yannis Manolopoulos 1

1. Generalized Indexing forEnergy-Efficient Access to Partially Ordered Broadcast Data in Wireless Networks Dimitrios Katsaros1,2 Nikos Dimokas1 Yannis Manolopoulos1 1Informatics Dept., Aristotle University, Thessaloniki, Greece 2Computer & Comm. Engineering Dept., University of Thessaly, Volos, Greece 10th IEEE IDEAS Symposium, New Delhi, India,11-13/12/2006

2. Data broadcasting in WSN Heterogeneous sensor net: resource-rich & ordinary sensor nodes • Resource-rich nodes (proxies, base stations) serving instructions to ordinary nodes • Ordinary nodes “carry data forward”

3. Data broadcasting in Cellular nets • Data “on air” • General interest data: e.g. stock market • Local interest data: e.g. restaurants, hotels

4. Data broadcasting in MANETs Automated battlefield • intelligence • tactical information

5. Data broadcasting characteristics • Contradictory requirements: • Small access latency, i.e., the time between when a node needs some data and the moment the node gets these data • Small tuning time, i.e., the time a node spends monitoring the communication channel (to save energy) • To achieve energy savings: • Mobile hosts support active or doze mode • Ordinary sensor nodes support active (transmit, receive, idle) or sleeping mode • Characteristics: • Not all data are of interest to all clients (skewed access pattern) • Not necessary global ordering among data (only partial ordering)

6. Low energy consumption  Indexing • Common in the database world • B+-trees, R-trees, Hashing …. etc. • Complication • magnetic disk: random access medium • broadcast channel: “one-dimensional” medium • Broadcast indexing (so far) only for global data ordering • Variations of B-trees, skip-lists, hashing, signatures • Good for uniform access pattern • Variations of Huffman and Alphabetic trees • Unbalanced structures (not binary but k-ary) • Good for skewed access pattern • Our proposal : the POBI index supports • Skewed access pattern • Partial data ordering • Generalizes Huffman trees and Alphabetic trees

7. Terminology and assumptions • resource-rich host (server) broadcasting n equi-sized items through a singlebroadcast channel, each item denoted as Ri • resource-starving ordinary hosts tune into the channel • flat broadcast : each item Ri appears exactly once in the broadcast cycle; neither client caching nor prefetching • server is aware of the item popularitiesPr(Ri) • Ipb(Ri) : number of index probes to reach Ri • d(αi) : fanout of an index node αi • Path(Ri) : set of index nodes from tree root to Ri • we adopt a generic model for the average cost

8. Relevant work – Broadcast Indexing Uniform access pattern (unrealistic) • (1,m) indexing : interleave m copies of the broadcast index, alike a B-tree IEEE TKDE’97 • Distributed index : improve upon (1,m)-indexing • Exponential index : distributed structure, alike skip-lists IEEE TKDE’06

9. Relevant work – Broadcast Indexing Skewed access pattern (realistic) • Variant Fanout tree (VF): k-ary version of the classic binary Huffman tree • pairs of [Ri, Pr(Ri)] : record and access probability of the record • assumes no ordering at all among Ri, thus it is not a search tree, i.e, internal nodes (a1,a2,a3,…) can not guide the searching IEEE TKDE’03

10. Relevant work – Broadcast Indexing Skewed access pattern (realistic) • k-ary Alphabetic tree (kAT) : k-ary version of the classic binary Alphabetic tree • pairs of [Ri, Pr(Ri)] : record and access probability of the record • assumes global ordering among Ri, thus it is a search tree, i.e, internal nodes (1,2,3,…) can guide the searching ACM MONET’96

11. Less relevant – WSN Indexing • index is NOT BROADCASTED over the channel, but STORED in distributed fashion among nodes • examples • GHT : distributed geographic hashing scheme • DIM : based on the k-d quadtree structure: divides network into zones; each node mapped to one zone; maps m-d space to zones; zones organized into a virtual binary tree • DIFS : based on the quadtree structure: every node (except the root) has more than one parent for relieving hot-spots • DIST : based on the quadtree structure: different spatial resolutions • TSAR : based on Skip Graphs

12. Generalizing VF and kAT • Suppose the existence of bins (groups) Bi.Bin items are not ordered, items in different bins are ordered • Case 1: Only one bin B1 and all items in it R1 VF tree Indexing ? R2 Rn • Case 2: As many bins as the items; exactly one item in each bin R1 R2 Indexing ? kAT tree Rn

13. POBI: Generalizing VF and kAT • A bin may contain more than one item • If only one bin, then previous Case 1 • If as many bins as items, then previous Case 2 • Practical problem instances • Sensor measurements : temperature vs. humidity • Battlefields : enemy movements vs. friendly losses • Cellular : different projections of relations

14. POBI: Problem definition • Problem definition • n data items and their access probabilities • m number of bins and a membership function • construct the index with minimal cost by respecting the partial order, i.e., in an inorder tree traversal x precedes y, if xBi and yBj and i<j

15. POBI design – First attempts • Brute force  exponentially many permutations • generate all possible permutation of the n items obeying group membership and inter-group ordering • build an alphabetic tree for the groups • Random ordering inside each group and build k-ary alphabetic tree for the grpoups: kATr • Sort the items of each group in non-descending (non-increasing) order and build an alphabetic tree for the groups: kATi (kATd)

16. POBI design – Final attempt • Objective: push the less popular items of each group deeper into the resulting broadcast tree • Method • create subtrees; each subtree corresponds to one group (bin) • treat each subtree as a node; the subtree’s cost is its root’s weight • apply alphabetic tree construction method to all subtrees • Challenge: devise a subtree creation method

17. POBI design – Variations • MostPop: place the most popular item at the tree root, then proceed similarly wrt the branches of the root • EqWeig: choose a root that equalizes the weight of the branches • POBI: construct a Huffman tree with variant fanout over the items of each group • Create a father node x with children all the items n1, …, ny • Sort n1, …, ny in non-ascending popularity • Find a node z such that: • Create a new node nx as child of x, father of nodes nz+1, …, ny • Recurse wrt both nodes x and nx until no change

18. Evaluation setting • Since no prior similar work exists, we compare: • Straightforward extension of VF, with random ordering inside each group • Straightforward extension of kAT, with random ordering inside each group • kATi and kATd • MostPop and EqWeig and POBI • Evaluation wrt: • number of items, default 500 • number of groups, default 10 • relative group size, default 0.1 (Zipf skew theta) • relative group popularity, default 0.1 (Zipf skew theta) • Performance metric • Index access cost

19. Impact of the #nodes (1/2)

20. Impact of the #nodes (2/2)

21. Impact of the #groups (1/2)

22. Impact of the #groups (2/2)

23. Impact of the relative group size (1/2)

24. Impact of the relative group size (2/2)

25. Impact of the relative group popularity

26. Summary and contributions • Defined and investigated for the first time indexing broadcast information of partially ordered data • Proved that it naturally generalizes two problems proposed earlier in the literature • Proposed approximate algorithms to generate the broadcast search trees; optimal algorithms require solving exponential number of subproblems • Simulated an environment to evaluate the performance • POBI – Partial Ordering Broadcast Index has been proven to prevail