Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees
Abstract
We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman’s equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the levelwise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability.
TheoremTheorem
I Introduction
Consider a distributed detection network consisting of a set of sensors and fusion nodes. The objective is to collectively solve a binary hypothesis testing problem. The sensors make observations from a common event, and then communicate quantized messages to other fusion nodes, according to the network architecture. Each fusion node fuses the received messages from its child nodes into a new message and then sends it to the fusion node at the next level for further integration. A final decision is eventually made at a central fusion node, usually called the fusion center. A fundamental question is how to fuse messages at each fusion node such that the fusion center makes the best decision, in the sense of optimizing a global objective function. For example, under the NeymanPearson criterion, the objective is to minimize the probability of missed detection with an upper bound constraint on the probability of false alarm; under the Bayesian criterion, the objective is to minimize the total error probability.
The distributed detection problem has been investigated extensively in the context of different network architectures. In the wellstudied parallel network [1]–[18] where sensors communicate with the fusion center directly, with the assumption of independent sensor observations conditioned on either hypothesis, the optimal fusion rule under the Bayesian criterion is simply a likelihoodrate test with a threshold given by the ratio of the prior probabilities.
The tandem network has been considered in [19]–[24], in which each fusion node combines the observation from its own sensor with the message it receives from its child node at one level down, and then transmits the combined message to its parent node at the next level up. We call a collection of fusion rules at all fusion nodes a fusion strategy. Specifically, [20] considers the problem of finding the optimal fusion strategy in such a tandem network under both NeymanPearson and Bayesian criteria.
The boundedheight tree network has been considered in [25]–[33], where the leaves are sensors, the root is the fusion center, and every other node is a fusion node that fuses the messages from its child nodes and sends a new message to its parent node. In general, finding a fusion strategy that minimizes the total error probability at the fusion center in boundedheight trees is computationally intractable even for a network with moderate number of nodes. Therefore, many recent papers focus on the asymptotic decay rate of the total error probability as the number of sensors goes to infinity. In balanced boundedheight trees where all the leaf nodes are at the same distance from the fusion center, a fusion strategy that achieves the optimal decay exponent is studied [26], in which all the fusion nodes at the same level use the same likelihoodratio test as the fusion rule.
The unboundedheight tree network has been considered in [34]–[38]. In particular, [34] considered balanced binary relay trees with the structure shown in Fig. 1. In this configuration, the leaf nodes, depicted as circles, are sensors generating binary messages independently and forward these binary messages to their parent nodes. Each node depicted as a diamond is a fusion (relay) node, which fuses the two binary messages received from its child nodes and forwards the new message upward. Ultimately, the fusion center at the root makes an overall decision. This tree is balanced in the sense that all the leaf nodes are at the same distance from the fusion center, and it is binary in the sense that each nonleaf node has two child nodes. This architecture is of interest because it represents the worstcase scenario in the sense that the minimum distance from the sensors to the fusion center is the largest. Assuming that all nonleaf nodes use the same fusion rule, the unitthreshold likelihoodratio test (ULRT). [34] shows the convergence of detection error probabilities using a Lyaponov method. Under the same assumptions, we further show in [35] that the decay rate of the total error probability is , where is the number of sensors. Under the equallylikely prior probability assumption, ULRT is the locally optimal fusion rule in the sense that the total error probability of each node is minimized after each fusion. However, we do not expect the strategy consisting of repeated ULRT fusion rules (which we call the greedy strategy) to be globally optimal in the sense that the total error probability at the fusion center is minimized.
In this paper we are interested in the following questions:

What is the globally optimal strategy for balanced binary relay trees?

How much difference in terms of the total error probability is there between the globally optimal strategy and the greedy strategy?
We answer the first question by formulating the problem as a dynamic program and characterizing the optimal strategy using Bellman’s equations. We answer the second question by introducing the notion of stringsubmodularity and showing that the reduction in the total error probability is a stringsubmodular function. Subsequently, we show that the reduction in the total error probability achieved by the greedy strategy is at least a factor of that achieved by the globally optimal strategy.
Ii Problem Formulation
We consider the problem of testing binary hypothesis between and in a balanced binary relay tree, with structure shown in Fig. 1. Let be any fusion node (i.e., is a nonleaf node). We denote by the set of child nodes of . Suppose that receives binary messages from every (i.e., from its child nodes), and then summarizes the two received binary messages into a new binary message using a fusion rule :
The new message is then communicated to the parent node (if any) of . Ultimately, the fusion center makes an overall decision.
It turns out that the only meaningful rules to aggregate two binary messages in this case are simply ‘AND’ and ‘OR’ rules defined as follows:

AND rule (denoted by ): a parent node decides if and only if both its child nodes send ;

OR rule (denoted by ): a parent node decides if and only if both its child nodes send .
Henceforth, we only consider the case where each fusion node in the tree choose a fusion rule from .
We assume that all sensors are independent with identical Type I error probability and identical Type II error probability . Moreover, we assume that all the fusion nodes at level use the same fusion rule ; i.e., for each node that lies at the th level of the tree, . In this case, all the output binary messages for nodes at level have the same Type I and Type II error probabilities, which we denote by and respectively. Given a fusion rule , we can show that the error probabilities evolve as follows:
Remark: Note that the evolution of the error probability pair is symmetric with respect to the line . Hence, it suffices to consider the case where the initial pair satisfies . We can derive similar result for the case where (e.g., by only flipping the decision at the fusion center). In the case where , the Type I and II error probabilities add up to one regardless of the fusion rule used. Hence, this case is not of interest.
Notice that the ULRT fusion rule is either the rule or the rule, depending on the values of the Type I and Type II error probabilities at a particular level of the tree. More precisely, we have

If , then the ULRT fusion rule is ;

If , then the ULRT fusion rule is ;

If , then the total error probability remains unchanged after using or . Moreover, the error probability pairs at the next level after using or are symmetric about the line . Therefore, we call both and the ULRT fusion rule in this case.
We define a fusion strategy as a string of fusion rules used at levels , denoted by . Let the collection of all possible fusion strategies with length be :
For a given initial error probability pair at the sensor level, the pair at the fusion center (level ) is a function of and the specific fusion strategy used. We consider the Bayesian criterion in this paper, under which the objective is to minimize the total error probability at the fusion center, where and are the prior probabilities of the two hypotheses, respectively. Equivalently, we can find a strategy that maximizes the reduction of the total error probability between the sensors and the fusion center. We call this optimization problem an optimal problem. Without loss of generality, we assume that the prior probabilities are equal; i.e., , in which case the optimal problem (ignoring a factor of ) can be written as:
(1) 
A fusion strategy that maximizes (1) is called the optimal strategy:
In contrast, the ULRT fusion rule minimizes the stepwise reduction in the total error probability:
Because of the equal prior probability assumption, a maximum a posteriori (MAP) fusion rule is the same as the ULRT fusion rule. In this context, we call a fusion strategy consisting of repeated ULRT fusion rules a ULRT (greedy) strategy.
In the next section, we derive the optimal fusion strategy for balanced binary relay trees with height using Bellman’s equations. We then show that the 2optimal strategy is equivalent to the ULRT strategy. Moreover, we show that the reduction of the total error probability is a stringsubmodular function (as defined in Section IIIC), which implies that the greedy strategy is close to the optimal fusion strategy in terms of the reduction in the total error probability.
Iii Main Results
Iiia Dynamic Programming Formulation
In this section, we formulate the problem of finding the optimal fusion strategy using a deterministic dynamic programming model. First we define the necessary elements of this dynamic model.

Dynamic System: We define the error probability pair at the th level as the system state, denoted by . Notice that and can only take values in the interval . Therefore, the set of all possible states is . Moreover, given the fusion rule, the state transition function is deterministic. If we choose , then
On the other hand, if we choose , then

Rewards: At each level , we define the instantaneous reward to be the reduction of the total error probability after fusing with :
where and are functions of the previous state and the fusion rule .
Let be the cumulative reduction of the total error probability if we start the system at state at level and the strategy is used. Following the above definitions, we have
If we let , that is, we start calculating the reduction from the sensor level, then the above cumulative reward function is the same as the global objective function defined in Section II. Therefore, for given initial state , we have to solve the following optimization problem to find the global optimal strategy over horizon :
The globally optimal strategy is
Notice that depends on the previous state and the fusion rule . Hence we write the state at level to be . The solution of the above optimization problem can be characterized using Bellmam’s equations, which state that
where is the first element of the optimal strategy . Recursively, the solution of the optimization problem is
Moreover, the th element of the optimal strategy is
Remark: The above formulation can easily be generalized to the node and link failure case [36] and even more complicated architectures (e.g., boundedheight trees [26] and ary relay trees [38]) simply by changing the state transition functions and the set of all possible fusion rules. Also, we can generalize the formulation to nonequal prior probability scenarios.
The complexity of the explicit solution to Bellman’s equations grows exponentially with respect to the horizon . Therefore, it is usually intractable to compute the optimal strategy if is sufficiently large. An alternative strategy is the ULRT strategy, which consists of repeating ULRT fusion rule at all levels. We have shown in [35] that the decay rate of the total error probability with this strategy is . Next we study whether the ULRT strategy is the same as the optimal strategy. If not, does the ULRT strategy provide a reasonable approximation of the optimal strategy?
IiiB 2optimal Strategy
In this section, we show that the 2optimal strategy is the same as the ULRT strategy. Moreover, we give an counterexample which shows that the ULRT strategy is not 3optimal.
Consider the 2optimal problem in the balanced binary relay tree with height 2:
where . The 2optimal strategy in this case is
We have the following theorem. (Because of lack of space, many of the proofs here are omitted.)
A strategy is 2optimal if and only if is the ULRT strategy.
This result also applies to any subtree with height 2 within a balanced binary relay tree with arbitrary height . However, the ULRT strategy is not in general optimal for multiple levels; i.e., , as the following counterexample for shows.
Let the initial state be , in which case the ULRT strategy is . As shown in Fig. 2, the solid (red) line denotes the total error probabilities at each level up to 3. However, the 3optimal strategy in this case is . The total error probability curve of this strategy is shown as a dashed (green) line in Fig. 2. Similar counterexamples can be found for cases where . Hence, the ULRT strategy is not in general optimal for . In the next section, we will introduce and employ the notion of stringsubmodularity to quantify the gap in performance between optimal and ULRT strategies for .
IiiC Stringsubmodularity
Submodularity of functions over finite sets plays an important role in combinatorial optimization. It has been shown that the greedy strategy provides at least a constantfactor approximation to the optimal strategy. For example, the celebrated result of Nemhauser et al. [39] states that for maximizing a monotone submodular function over a uniform matroid such that (here denotes the empty set), the value of the greedy strategy is no less than a factor of that of the optimal strategy.
Note that the submodular functions studied in most previous papers are defined on the power set of a given finite set. However, many stochastic optimization problems concern optimizing objective functions over a finite horizon, where we have to choose an action from a given finite set at each iteration. In these cases, the objective function usually depends on the order of actions, and repeating the same action is allowed. Hence, we cannot directly apply the result of Nemhauser et al. [39].
For objective functions defined on strings (finitelength sequences), [40] and [41] provide some sufficient conditions such that the greedy strategy achieves a good approximation for maximizing the string function. In this paper, we improve these results by providing sufficient conditions that are weaker than those in [40] and [41].
Next we formulate the maximization problem using submodular functions defined on strings.

String: Consider a finite set of possible actions. For each step, we choose an action from . Let be a string of actions taken over steps, where for all . Let the set of all strings of actions be
Note that corresponds to the empty string (no action taken), denoted by .

String length: For a given string , we define its string length as , denoted .

String concatenation: Let and be two strings in . We define concatenation as follows:

String dominance: Let and be two strings in . We write if we have
where and for all . In other words, is a prefix of .

Stringsubmodularity: A function from strings to real numbers, , is stringsubmodular if

has the monotone property; i.e.,

has diminishingreturn property; i.e.,
Note that the diminishingreturn property here only requires concatenating one more action.


Globally optimal solution: Consider the problem of finding a string that maximizes under the constraint that the string length is not larger than . Because the function is monotone, it suffices to consider the stronger constraint with fixed length :
(2) 
Greedy solution: A string is called greedy if
For a deterministic dynamic system, the diminishing return property can be simplified as follows.
Lemma 1: For any and , we have
if and only if
Consider a submodular function such that

;

For any greedy strings with a length less than and optimal strings (optimal with respect to (2)), .
Then any greedy string of length satisfies
IiiD Application to Distributed Detection
We consider balanced binary relay trees with even heights. Again we assume that the nodes at the same level use the same fusion rule. Moreover, we assume that two fusion rules of consecutive levels are chosen from the following set . Let be a fusion strategy, where for all . Let be the set of all possible strategies (strings); i.e., . Here we only prove the case where the prior probabilities are equally likely. The following analysis easily generalizes to nonequal prior probabilities. Given the two types of error probability at level 0, the reduction of the total error probability after applying a strategy is
where and represent the Type I and II error probabilities after fusion using .
Next we show that is a stringsubmodular function.
Proposition 2: The function : is stringsubmodular.
For a balanced binary relay trees with height , the global optimization problem is to find a strategy with length such that the above reduction is maximized; that is
(3) 
We have shown that the reduction of the total error probability is a stringsubmodular function. Moreover, we know that the total error probability does not change if there is no fusion; i.e.,
Therefore, we can employ Theorem 2 to the above maximization problem (3).
Consider a balanced binary relay tree with height . We denote by the reduction of the total error probability after using the greedy strategy. We have shown that the ULRT strategy is 2optimal. We have also shown in [35] that the ULRT strategy only allows at most two identical consecutive fusion rules. Hence, we can conclude that a strategy is the ULRT strategy if and only if it is the greedy strategy. We denote by the reduction of the total error probability using the optimal strategy. We have the following theorem.
Consider a balanced binary relay tree with height . We have
Remark: Recall that the fusion strategy is a string of fusion rules chosen from . Thus, the strategies we considered in this section have at most two consecutive repeated fusion rules. For example, the strategy is not considered. It is easy to show that with repeating identical fusion rule, the total error probability goes to 1/2. Therefore, it is reasonable to rule out this situation.
Iv Concluding Remarks
We study the problem of finding a fusion strategy that maximizes the reduction of the total error probability in balanced binary relay trees. We formulate this optimization problem using deterministic dynamic programming and characterize its solution using Bellman’s equations. Moreover, we show that the reduction of the total error probability is a stringsubmodular function. Therefore, the ULRT strategy, which is a string of repeated ULRT fusion rules, is close to the globally optimal strategy in terms of the reduction in the total error probability.
Future work includes studying the overall optimal strategy for other architectures (e.g., boundedheight trees [26] and ary relay trees). We would like to derive the optimal strategy in balanced binary relay trees with node and link failures. In this paper, we assume that all the nodes at the same level use identical fusion rule. What about the case where the nodes at each level are allowed to use different fusion rules? Moreover, what can we say about trees with correlated sensor measurements? These questions are currently been investigated.
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