Tight Bounds for Delay-Sensitive Aggregation

1. Tight Bounds for Delay-Sensitive Aggregation Yvonne Anne Oswald Stefan Schmid Roger Wattenhofer LEA DistributedComputingGroup

2. Introduction Distributed Computing Trade-off: time complexity vs message complexity Dijkstra Prize 2008 examples: • gossiping • data gathering • organization theory Yvonne Anne Oswald @ PODC 2008

3. Introduction Distributed Computing Trade-off: time complexity vs message complexity Particularly in sensor networks • limited energy (battery): transmission/reception expensive • goal: be up-to-date without much delay Yvonne Anne Oswald @ PODC 2008

4. root Model • communication network: rooted spanning tree Yvonne Anne Oswald @ PODC 2008

5. Root 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! Yvonne Anne Oswald @ PODC 2008

6. 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) e.g.,1 per event per time slot until arrival at root • messages can be merged Root => reduce # transmissions Yvonne Anne Oswald @ PODC 2008

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

8. Related Work and our Contributions • 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)) • (h1/2) • Trees • O(min(c,h)) • oblivious (min(c,h)) • Chains • O(min(c,h1/2)) • oblivious (min(c, h1/2)) • WSN model • log(e c(e)) improvement • higher lower bound Yvonne Anne Oswald @ PODC 2008

9. Algorithm AGG ([DGS01],[KNR01]) Balance delay cost and total communication cost AGG: node v forwards msg m as soon as delay(m,t) ¸ c 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 Yvonne Anne Oswald @ PODC 2008

10. V3 v2 V1 Example (4 nodes, 4 events, c=3) events v1 v2 t = 1 1 0 t = 2 1 2 delay at v1 v2 v3 t = 1 1 0 0 t = 2 3 2 0 t = 3 0 4 2 t = 4 0 0 7 t = 5 0 0 0 root |m|=4 delay=7 |m| =2 delay = 2 |m| =2 delay = 0 |m|=2 delay=1 |m|=2 delay=2 |m|=2 delay=4 |m|=1 delay = 1 |m| =2 delay = 3 cost = 17+9 Yvonne Anne Oswald @ PODC 2008

11. Related Work and our Contributions • 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)) • (h1/2) • WSN model • log(e c(e)) improvement • higher lower bound • Trees • O(min(c,h)) • oblivious (min(c,h)) • Chains • O(min(c,h1/2)) • oblivious (min(c, h1/2)) Yvonne Anne Oswald @ PODC 2008

12. 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 ALG: t=1 events at nodes v1..vn/2-1 t=w messages leave vi t=w+1 messages at nodes vn/2..vn-1 … cost 2(c+w)h2 Yvonne Anne Oswald @ PODC 2008

13. 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 ALG: cost 2(c+w)h2 OPT: cost 2O(ch+h2) => ratio (min(c,h)) Yvonne Anne Oswald @ PODC 2008

14. assume : #msgOPT = x h1/2 #msgAGG, x 2(1) => costAGG2O(x h1/2 hc) find sequence keeping costopt minimal => msg size increases with t yet no merges for AGG => costOPT 2 (xhc) Upper Bound on Chains Thm: AGG has a competitive ratio of at most O(min(c,h1/2))on chains. … root time difference ensures no merges before i => bound for reduction cost • proof sketch • assume no messages merged: ratio O(h1/2) • include u merges at depth i: • cost reduction AGG (uci) • cost reduction OPT O(uci) • generalize for many merges at any depth: ratio O(h1/2) • combine with result from trees: ratio O(min(c,h1/2) Yvonne Anne Oswald @ PODC 2008

15. Teaser on Value-Sensitive Aggregation What if urgency of delivery depends on value? root knows vr(t), value at leaf vl(t) cost := transmissions + t |vr(t) –vl(t)| • Results (2 nodes): • offline: dynamic programming O(#changes3) • online: competitive ratio 2 O(c/), • where  smallest difference between values Online AGG: forward at (t+1) if lastsent |vr(t) –vl(t)| ¸ c consider intervals between consecutive transmissions Yvonne Anne Oswald @ PODC 2008

16. Summary • event aggregation • Tree: O(min(c,h)) • oblivious (min(c,h)) • Chain: O(min(c,h1/2)) • oblivious (min(c, h1/2)) • value-sensitive event aggregation • model • optimal algorithm for link O(# changes3) • online algorithm for link O(c/min.change) ski rental on trees Yvonne Anne Oswald @ PODC 2008

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