1 Tight Bounds for Delay-Sensitive Aggregation Yvonne Anne Oswald Stefan Schmid Roger Wattenhofer Distributed Computing Group LEA

2Yvonne Anne PODC 2008 Introduction Distributed Computing Trade-off: examples: gossiping data gathering organization theory time complexitymessage complexityvs Dijkstra Prize 2008

3Yvonne Anne PODC 2008 Introduction Particularly in sensor networks limited energy (battery): transmission/reception expensive goal: be up-to-date without much delay time complexitymessage complexityvs Distributed Computing Trade-off:

4Yvonne Anne PODC 2008 Model communication network: rooted spanning tree root

5Yvonne Anne PODC 2008 Model communication network: rooted spanning tree transmission cost c nodes synchronized, time slotted events occur at nodes (online, worst case) Goal: forward events to root Root, be fast and energy-efficient!

6Yvonne Anne PODC 2008 Model communication network: rooted spanning tree transmission cost c nodes synchronized, time slotted events occur at nodes (online, worst case) Goal: forward events to root, be fast and energy-efficient! minimize (c¢ # transmissions + delay cost) messages can be merged Root e.g.,1 per event per time slot until arrival at root => reduce # transmissions

7Yvonne Anne PODC 2008 Oblivious Algorithm DEFINITION: OBLIVIOUS ALGORITHM decision (transmit/wait) of node v depends on # events currently at node v when events arrived at node v decision of node v does NOT depend on history (messages forwarded earlier) v ’s location in the aggregation network perfect for sensor nodes! I’ve got a memory like a sieve and I don’t know where I am..

8Yvonne Anne PODC 2008 Related Work and our Contributions Trees O(min(c,h)) oblivious  (min(c,h)) Chains O(min(c,h 1/2 )) oblivious  (min(c, h 1/2 )) WSN model log(  e c(e)) improvement higher lower bound Link: Dooly et al.[JACM01]: TCP, offline OPT online O(1) Karlin et al.[STOC01] online randomized e/(e-1) Tree: Khanna et al.[ICALP02] model: edge e -> cost c(e) distributed bounds O(h log(  e c(e))  (h 1/2 )

9Yvonne Anne PODC 2008 Algorithm AGG ([DGS01],[KNR01]) AGG: node v forwards msg m as soon as delay(m,t) ¸ c Balance delay cost and total communication cost ski rental on trees Details m : message at v, containing |m| events delay(m,t) : delay associated with m at time t no transmission: delay(m,t+1) = delay (m,t) + |m| transmission: delay(m,t+1) = delay (m,t) + |m| - c

10Yvonne Anne PODC 2008 cost = 17+9 events v 1 v 2 t = t = delay at v 1 v 2 v 3 t = t = t = t = t = V1V1 v2v2 V3V3 |m|=1 delay = 1 |m| =2 delay = 3 |m|=2 delay=2 |m| =2 delay = 0 |m| =2 delay = 2 |m|=2 delay=4 |m|=2 delay=1 |m|=4 delay=7 Example (4 nodes, 4 events, c=3) root

11Yvonne Anne PODC 2008 Related Work and our Contributions Trees O(min(c,h)) oblivious  (min(c,h)) Chains O(min(c,h 1/2 )) oblivious  (min(c, h 1/2 )) WSN model log(  e c(e)) improvement higher lower bound Link: Dooly et al.[JACM01]: TCP, offline OPT online O(1) Karlin et al.[STOC01] online randomized e/(e-1) Tree: Khanna et al.[ICALP02] model: edge e -> cost c(e) distributed bounds O(h log(  e c(e))  (h 1/2 )

12Yvonne Anne PODC 2008 Lower Bound on Trees Thm: any oblivious deterministic online algorithm ALG has a competitive ratio of at least  (min(c,h)) on the tree. … root t=1events at nodes v 1..v n/2-1 t=w messages leave v i t=w+1 messages at nodes v n/2..v n-1 … ALG: cost 2  (c+w)h 2

13Yvonne Anne PODC 2008 Lower Bound on Trees … root ALG: cost 2  (c+w)h 2 OPT: cost 2 O(ch+h 2 ) => ratio  (min(c,h)) Thm: any oblivious deterministic online algorithm ALG has a competitive ratio of at least  (min(c,h)) on the tree.

14Yvonne Anne PODC 2008 Upper Bound on Chains … root Thm: AGG has a competitive ratio of at most O(min(c,h 1/2 )) on chains. proof sketch assume no messages merged: ratio O(h 1/2 ) include u merges at depth i: cost reduction AGG  (uci) cost reduction OPT O(uci) generalize for many merges at any depth: ratio O(h 1/2 ) combine with result from trees: ratio O(min(c,h 1/2 ) assume : #msg OPT = x h 1/2 #msg AGG, x 2  (1) => cost AGG 2 O(x h 1/2 hc) find sequence keeping cost opt minimal => msg size increases with t yet no merges for AGG => cost OPT 2  (xhc) time difference ensures no merges before i => bound for reduction cost

15Yvonne Anne PODC 2008 Teaser on Value-Sensitive Aggregation What if urgency of delivery depends on value? root knows v r (t), value at leaf v l (t) cost := transmissions +  t |v r (t) –v l (t)| Results (2 nodes): offline: dynamic programming O(#changes 3 ) online: competitive ratio 2 O(c/  ), where  smallest difference between values Online AGG: forward at (t+1) if  last sent |v r (t) –v l (t)| ¸ c consider intervals between consecutive transmissions

16Yvonne Anne PODC 2008 Summary event aggregation Tree: O(min(c,h)) oblivious  (min(c,h)) Chain: O(min(c,h 1/2 )) oblivious  (min(c, h 1/2 )) value-sensitive event aggregation model optimal algorithm for link O(# changes 3 ) online algorithm for link O(c/min.change) ski rental on trees

17Yvonne Anne PODC 2008 The End! Thank you! Questions? Comments?