1 Delay-efficient Data Gathering in Sensor Networks Bin Tang, Xianjin Zhu and Deng Pan

2 Outline Introduction Problem Statement Data Gathering Algorithms Conclusions

3 Introduction Data gathering in sensor networks To explore energy efficiency To explore data correlation and redundancy Time critical applications Patient Monitoring Surveillance sensor networks Fine-grained environmental monitoring Delay Efficiency is the goal of our work.

4 Our assumptions One half duplex interface Collision free – each node can not receive more than one transmissions Each node can HEAR more than one transmissions Each node (except sink) has one message

5 Problem Statement Disk graph G=(V,E) sink node S Each node (except S) has one message Delay-efficient Data Gathering problem: Schedule the collision-free message transmission for each node, s.t. the delay when all messages are received by S is minimized. Minimum delay is 7 time slots S

6 Our Solution Linear Topology Star Topology  NP-completeness Tree Topology General Topology

7 Linear Topology Divide the nodes into two sets, so that all the nodes in each set can transmit simultaneously 4321 sink N=4 Odd={1,3}, Even={2,4}

8 Linear Topology: Proof of Optimum Step 1: Delay of above algorithm is 2N-1, where N is the number of nodes (excluding sink) Step 2: 2N-1 is the lower bound of delay in linear topology  For messages from node 2, 3, …, N: 2 slots are needed to pass through node 1, for each message  For message from node 1: 1 slot is needed to send to sink  Total delay≥2(N-1)+1=2N-1 Thus, our algorithm is optimal sink1N-1N2

9 Star Topology In order to achieve the minimum delay, different branches need to be pipelined sink

10 Star Topology More branches? Divide the branches into two groups, so that each group has equal number of nodes How to divide branches into 2 equal-set? NP-hard

11 Star Topology: NP-complete Prove that our problem is NP-complete by reduction from the Integer Partition problem Integer Partition:  Give a set of integers X={x1,x2,…,xn} and a target y. Is there a subset X’ of X, such that the sum of all the elements in X’ is equal to y?

12 Star Topology: NP Reduction For each xi, create a branch with xi nodes Create another branch with (∑X-2y) nodes Connect all the branches with the sink For the created star topology graph, is its minimum gathering delay 2(∑X-y)?

13 Star Topology: NP Reduction

14 Star Topology: NP Complete If the integer partition problem has a solution, say X=X’ ∪ X’’, X’ ∩ X’’={}, and ∑X’=y. Then, the minimum gathering delay of the created graph is 2(∑X-y) Since ∑X’+ (∑X-2y) =(∑X+ (∑X-2y) )/2 Similarly, vice versa

15 Tree Topology For the tree topology, if the root has only one child, it can be viewed as the conjunction of several lines. Similarly, minimum delay = (2N-1), where k is the number of nodes in the sub-tree

16 Tree Topology How about if the root has more than one child? Similar to the star topology with multiple branches

17 General Topology Generate a BFS tree, the general graph can be processed as a tree topology. Heuristics to divide the BFS tree into two parts with similar number of nodes. The maximal gathering delay has an upper bound of 2N-1, and the optimal delay is at least N. So our solution is a 2-approximation algorithm

18 Conclusions Current status  Formulate the collision-free data gathering problem  There is optimal algorithm in linear topology  In tree/star/general graph, our problem is NP- complete  2 approximation algorithm in general graph Future work  Distributed algorithms  Simulation