Information Dissemination in Highly Dynamic Graphs Regina O’Dell Roger Wattenhofer

DIALM-POMC 2005Flooding in Highly Dynamic Graphs2 Motivation Highly Dynamic Networks –Mobility zebra herd, flock of birds, cars,... –Stationary nodes BUT unstable links Roof-top network Sensor network in changing environment Flooding –High mobility  reactive protocol  flooding element –Guaranteed information delivery –Fundamental ingredient Routing Service discovery Sensor network management …

DIALM-POMC 2005Flooding in Highly Dynamic Graphs3 Motivation – Previous Work Simulations –Lots of mobility models –Algorithm correctness depends on model? “Coarse-grained” mobility –Periods of stability –Link failures between completed route requests Worst-case analysis [Awerbuch et al, 2005] –Assumptions on path stabilities Graph restrictions –Unit Disk Graph (UDG) –Quasi-UDG –“bounded-growth” graphs

DIALM-POMC 2005Flooding in Highly Dynamic Graphs4 Our View  “fine-grained”, theoretical worst-case mobility analysis We ask: -how mobile can the network, -how limited can the system be such that flooding is still possible? anything?

DIALM-POMC 2005Flooding in Highly Dynamic Graphs5 Outline Model –Network, mobility, algorithm Flooding –Knowing |V| –Storing IDs –Finding max ID Impossibility Conjecture Routing Outlook

DIALM-POMC 2005Flooding in Highly Dynamic Graphs6 Model – “Environmental Challenges” Graph G t = (V,E t ), V same, –G t connected for all times t –Time between N(v) changes ¸ T for all nodes v –T = max message transmission time Broadcast medium Negligible local processing time Asynchronous message transmissions –x 2 N(v) during entire transmission Events: –Message receipt –Neighborhood change (  N(v)  0) Obs: Transmission at v will reach some node after at most 2T time. arbitrary changes Node w enters N(v)  w should receive message from v “lost messages” possible simultaneously

DIALM-POMC 2005Flooding in Highly Dynamic Graphs7 terminatecorrect Goals – “Engineering Constraints” do not know |V| O(log |V|) space task dependent no more messages sent no upper bound! – storage, header – unique small IDs N(v)N(v)

DIALM-POMC 2005Flooding in Highly Dynamic Graphs8 “3 out of 4” Flooding terminatecorrect do not know |V| O(log |V|) space send nothing send forever

DIALM-POMC 2005Flooding in Highly Dynamic Graphs9 C OUNTER F LOODING Assume: know n ¸ |V| (polynomial upper bound) Algorithm  counter k v = 0  retransmit message when  N(v)  ;, inc k v  while k v < 2n Proof idea –Border intact  normal flooding –Otherwise   N(v) at border node v Comments –Reaches n nodes in time O(n) –Explicit termination? Synchronous: easy! NN

DIALM-POMC 2005Flooding in Highly Dynamic Graphs10 “3 out of 4” Flooding terminatecorrect do not know |V| O(log |V|) space estimate |V|

DIALM-POMC 2005Flooding in Highly Dynamic Graphs11 L IST F LOODING Assume: store & send O(n) IDs Algorithm  list l v of known nodes, set n v = |l v |  receive l w  merge: l v = l v Å l w  if |l v | > n v n v = |l v | C OUNTER F LOODING ( f(n v ),l v ) Proof idea –Set of flooding nodes increases –Or: l max increases Comments –Correct in time O(n 2 ) for f(n) = n + 1 probably in O(n) for f(n) = 2n f(n) = –n + 1 –2n –… l max

DIALM-POMC 2005Flooding in Highly Dynamic Graphs12 Flooding terminatecorrect do not know |V| O(log |V|) space

DIALM-POMC 2005Flooding in Highly Dynamic Graphs13 IDF LOODING Idea: find max ID  upper bound on |V| Algorithm  store n v = max ID seen  receive ID w  if w > n v n v = w C OUNTER F LOODING ( f(n v ),n v ) Proof idea –Same principle as L IST F LOODING –Max ID of flooding nodes will grow Comments –Needs unique polynomial IDs –Intuition: IDs encode information about |V| IDs  strong assumption!

DIALM-POMC 2005Flooding in Highly Dynamic Graphs14 Flooding – No IDs General idea –Receive new information about graph (|l v | > n v, w > n v ) –Update estimate n v –Restart C OUNTER F LOODING with f(n v ) Counter example (idea) General argument? Guessing ID? –dynamic naming/initialization problem –randomized

DIALM-POMC 2005Flooding in Highly Dynamic Graphs15 Routing What about routing?  possible! –Destination: send ACK –Initiates “termination” phase Idea: 2 modes  counter n v  FLOOD : inc n v, send message every  N(v) if received n’ > n v  update n v  TERM : C OUNTER F LOODING (n v ) if received n’ > n v  update n v restart C OUNTER F LOODING (n v ) Correct in time O(n) –Actually: “optimal path”

DIALM-POMC 2005Flooding in Highly Dynamic Graphs16 Outlook First step –Theoretical analysis possible –More general mobility model Lots of open questions –Impossibility of flooding? –Does randomization help? –Explicit termination? Local dynamic synchronizer –Even more mobility Nodes join and leave –Even less mobility Restricted link changes Timing assumptions

DIALM-POMC 2005Flooding in Highly Dynamic Graphs Distributed Computing Group Questions? Comments? Thank You!

DIALM-POMC 2005Flooding in Highly Dynamic Graphs18 Goals – “Engineering Constraints” Requirements: –Correctness (task dependent) –Termination(no messages sent) Conditions: –n = |V| unknown, nor any upper bound –O(log n) overhead (storage, message header) Unique O(log n)-bit IDs Neighborhood table  prohibitively expensive Idea: –Separate conditions  analyze effect must haves design, cost, environment, …