Interpolating the Air for Optimizing Wireless Data Broadcast Fotis Tsakiridis, Panagiotis Bozanis, Dimitrios Katsaros Dept. of Computer & Communication Engineering University of Thessaly, Volos, Greece MobiWAC 2007, October 22, Chania, Crete Island, Greece

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 2/21 Introduction Rapid advent of wireless technology + popularity of smart mobiles = pervasive information services Examples

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 3/21 Preliminaries (1/2) Servers: – cyclically broadcast totally ordered data Mobile clients: – tune in and actively find the needed data downlink broadcast channel A BC D A BC D

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 4/21 Preliminaries (2/2) An aid to the search process: interleaving of auxiliary index items within the data items Querying process Tuning time Latency Probe wait Bcast wait

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 5/21 Previous Work (state-of-the-art) (1/2) The Exponential Index – simulates the exponential searching technique over the periodical broadcast When exponent=2  binary search OR skip-list – uses aux information each bucket consists of logarithmic number of pairs (pntr; maxKey)

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 6/21 Previous Work (state-of-the-art) (2/2) In an error-free broadcasting environment – The average access latency is: I/2 + N/2, N the size of bcast – The worst case tuning time is: where t(l) is the tuning time for a data chunk that is l chunks away from the current chunk – the index space overhead per chunk is: where C is the number of data chunks – The average tuning time is:

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 7/21 Interpolation: Basic Notions (1/2) How can we achieve log-log behaviour in main memory? – Suppose that there is an array of totally ordered keys x 1 < x 2 < … < x n, drawn independently from a uniform distribution over the range (x 0, x n+1 ) – Searching by interpolation for the item y in the array proceeds as follows: – Let be the percentage of the keys expected to be less than y – Then, compare y to, and, in case of inequality, we search recursively either: subarray or subarray The above procedure gives loglogn average access time and linear worst case time

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 8/21 Interpolation: Basic Notions (2/2) A further improvement to this scheme can be achieved: – If y is successively compared with to locate the smallest i such that – If y is successively compared with – In either case, the located pertinent subarray of size is recursively searched It can be shown that: – the average access time is bounded by: – its worst case time complexity is:

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 9/21 Interpolation and Broadcasting Exploit interpolation search to design an air index Broadcast Index set up – Every bucket is a hybrid one, containing: The minimum (m 1 ) and the maximum (M n ) keys Pairs (pntr, maxKey), where maxkey is the maximum data entry of the bucket pntr slots ahead, such that: – the 0-th entry refers to the immediately upcoming bucket, – the i-th entry holds i.e., the maximum data entry slots ahead Space consumption – loglogn+1 entries per bucket

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 10/21 The access protocol (1/3) Let – k : be the bucket the client currently tuned into – y : the search item – l : the level of recursion – j : the probing position

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 11/21 The access protocol (2/3) When client tunes for the first time a)Both y and j lie ahead of k: If, c d, the client dozes until bucket 1 arrives, and then he conducts linear search with sqrt(n) long jumps. Else, the client switches to doze mode until bucket arrives, c a being an interpolation adjustment parameter. In case of successful prediction, client conducts linear search from this point, else starts the linear search from bucket 1. b)j precedes k, and y succeeds k: The client starts switching between doze and active mode, with sqrt(n) intervening slots, until he locates the desired sqrt(n)-sized interval, to which he recurs. c)y precedes k, and j succeeds k: The client missed the broadcast. if, the client dozes until the arrival of bucket 1 and linear searches. Else, he goes back to active mode when bucket arrives. If he is lucky, he starts linear searching. Else, the bcast is useless and starts linear search from bucket 1.

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 12/21 The access protocol (3/3) The General Case (l level of recursion): y is lying between buckets k and d)y seems to belong after bucket, while j lies between k and : The client dozes until arrives and then linear searches to locate the pertinent subinterval. e)y is located between k and : Nothing to do! Client is already in the pertinent subinterval, to which he recurs. f)Both y and j lie after bucket : If, the client linear searches the rest of the bcast from slot. Else, he dozes until bucket j- c a sqrt(n) arrives. If the adjusted prediction is proven to be true, he linear searches; Else, he missed the bcast, and linear searching is performed from “secure” bucket

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 13/21 A Searching Instance c d =1.2, c a =c x =1, and 40 is the key the client searches 1.Initially tunes into eighth bucket. 2.Since 40<60, he interpolates to get 3.Since, he switches into doze mode until the bucket arrives. 4.21<40, so must continue linear searching in upcoming buckets. 5.Since level 1 pointer (52) bounds 40, recurs in level 2. 6.Applies once again interpolation, giving 7.Because 40 is located after, he sleeps until bucket 4 is transmitted, where he figures out that 40 belongs to the following bucket 5.

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 14/21 Performance The average tuning time is O(loglog n) Heuristic tuning of parameter C

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 15/21 Tuning Time

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 16/21 Access Latency

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 17/21 Space Overhead (Same Tuning Time)

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 18/21 Tuning Time (Same Space Overhead)

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 19/21 Conclusions Interpolation Air Index – efficient air index – exhibits a linear structure, suitable for the broadcast environment – the index space overhead is log-logarithmic per transmitted bucket – tuning time is log-logarithmically proportional to the broadcast size – very easily configurable the access latency and the tuning can be simply adjusted by a single parameter – the experimental results attest that our index outperforms the exponential index (state-of-the-art air-index) both in tuning time and space overhead while achieving the same access latency

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 20/21 Future Work How skewed data can be accommodated in our scheme? What is its performance in multi-channel data broadcast environments?

MobiWAC’07, October 22, Chania, Crete Island, Greece Dept. of Computer & Communication Engineering, Univ. of Thessaly, Greece 21/21 Any questions? Thank you for your attention!