1 Maximizing Information in Unreliable Sensor Networks under Deadline and Energy Constraints Srikanth Hariharan*, Zizhan Zheng and Ness B. Shroff Abstract—We study the problem of maximizing the infor- mation in a wireless sensor network with unreliable links. We consider a sensor network with a tree topology, where the root corresponds to the sink, and the rest of the network detects an event and transmits data to the sink. We formulate a combi- natorial optimization problem that maximizes the information that reaches the sink under deadline, energy, and interference constraints. This framework allows using a variety of error recovery schemes to tackle link unreliability. We show that this optimization problem is NP-hard in the strong sense when the input is the maximum node degree of the tree. We then propose a dynamic programming framework for solving the problem exactly, which involves solving a special case of the Job Interval Selection Problem (JISP) at each node. Our solution has a polynomial time complexity when the maximum node degree is O(log N ) in a tree with N nodes. For trees with higher node degrees, we further develop a sub-optimal solution, which has low complexity and allows distributed implementation. We investigate tree structures for which this solution is optimal to the original problem. The efficiency of the sub-optimal solution is further demonstrated through numerical results on general trees. I. I NTRODUCTION A wireless sensor network is a wireless network consisting of a number of sensors that sense a desired aspect of the region in which they are deployed. These networks are used in a number of military and civilian applications, such as target tracking and environment monitoring. Sensor measurements are prone to errors due to environmental factors and resource constraints. Therefore, sinks cannot rely on the data sensed by a single sensor. In many applications, the sinks only desire a certain function of the data sensed by different sensor nodes (e.g., average temperature, maximum pressure, detect a signal, etc.). When sinks require certain classes of functions of the sensed data, performing in-network computation (intermediate This work was supported in part by ARO MURI Awards W911NF-07- 10376 (SA08-03) and W911NF-08-1-0238, and NSF Awards 0626703-CNS, 0635202-CCF, and 0721236-CNS. S. Hariharan is with the Department of Electrical and Computer Engineer- ing, The Ohio State University, 2015 Neil Ave., Columbus, OH 43210, USA srikanth.hariharan@gmail.com Z. Zheng is with the Department of Electrical and Computer Engineering, The Ohio State University, 2015 Neil Ave., Columbus, OH 43210, USA zhengz@ece.osu.edu N. B. Shroff is with the Department of Electrical and Computer En- gineering and the Department of Computer Science and Engineering, The Ohio State University, 2015 Neil Ave., Columbus, OH 43210, USA shroff@ece.osu.edu * Corresponding author. A preliminary version of this paper by S. Hariharan and N. B. Shroff titled “Deadline Constrained Scheduling for Data Aggregation in Unreliable Sensor Networks” appeared in the proceedings of the 9 th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WIOPT), 2011 [1]. nodes in the network aggregate data from all their predeces- sors, and only transmit the aggregated data) greatly reduces the communication overhead [2]. A tree structure is commonly used for data aggregation in wireless sensor networks [3], [4]. In this paper, we consider a tree topology with the sink as the root of the tree. An event is observed by a subset of nodes in the tree called the source nodes. All source nodes transmit their data about the event to the sink. Our goal is to maximize the information obtained by the sink. The information obtained by the sink is a representation of the quality of the data that reaches the sink. For example, it could be the sum of the inverses of the error variances of the data from various sources that reaches the sink [5]. It could also represent other relevant metrics such as the Log-Likelihood Ratio if detection is being performed by the network, distortion, etc. Much of the existing work in data aggregation does not take channel errors, and interference into account. However, wireless channels are inherently prone to errors due to fading and environmental factors. Also, interference is a critical component of the wireless environment. We consider a one- hop interference model where two nodes that are one hop away from each other cannot transmit simultaneously. We also consider unreliable links where the errors across different links are independent of each other, and allow for the usage of various error-recovery schemes including retransmissions, coding, etc. Delay is also an important parameter in a wireless sensor network. While most works focus only on energy, minimizing the delay can help save a huge amount of energy. For instance, suppose that a sensor network is tracking a target. In order to ensure good tracking quality, the sink must obtain previous measurements in a timely manner so that the best subset of sensors for the next measurement is chosen. If the sink does not get the measurements in a timely manner, the target might have moved too far resulting in a poor measurement quality during future measurements. On the other hand, if the sink decides to ensure good tracking quality by activating a large number of sensors at all times, a large amount of energy could be wasted. Therefore, it is critical that the sink obtains sensor measurements in a delay efficient manner. With this model, we provide an optimization framework for maximizing the information received at the sink under a dead- line constraint at the sink, and per-sensor energy constraints. The output of this framework is a schedule of time slots for each node within the deadline, and the amount of energy to be expended by each node on transmissions and receptions. The main contributions of this work are summarized as