Searching without communicating: tradeoffs between performance and selection complexity

Abstract

We consider the ANTS problem (Feinerman et al.) in which a group of agents collaboratively search for a target in a two-dimensional plane. Because this problem is inspired by the behavior of biological species, we argue that in addition to studying the time complexity of solutions it is also important to study the selection complexity, a measure of how likely a given algorithmic strategy is to arise in nature due to selective pressures. Intuitively, the larger the $\chi$ value, the more complicated the algorithm, and therefore the less likely it is to arise in nature. In more detail, we propose a new selection complexity metric $\chi$, defined for algorithm ${\mathscr {A}}$ such that $\chi ({\mathscr {A}}) = b + \log \ell$, where b is the number of memory bits used by each agent and $\ell$ bounds the fineness of available probabilities (agents use probabilities of at least $1/2^\ell$). In this paper, we study the trade-off between the standard performance metric of speed-up, which measures how the expected time to find the target improves with n, and our new selection metric. Our goal is to determine the thresholds of algorithmic complexity needed to enable efficient search. In particular, consider n agents searching for a treasure located within some distance D from the origin (where n is sub-exponential in D). For this problem, we identify the threshold $\log \log D$ to be crucial for our selection complexity metric. We first prove a new upper bound that achieves a near-optimal speed-up for $\chi ({\mathscr {A}}) \approx \log \log D + \mathscr {O}(1)$. In particular, for $\ell \in \mathscr {O}(1)$, the speed-up is asymptotically optimal. By comparison, the existing results for this problem (Feinerman et al.) that achieve similar speed-up require $\chi ({\mathscr {A}}) = \varOmega (\log D)$. We then show that this threshold is tight by describing a lower bound showing that if $\chi ({\mathscr {A}}) < \log \log D - \omega (1)$, then with high probability the target is not found in $D^{2-o(1)}$ moves per agent. Hence, there is a sizable gap with respect to the straightforward $\varOmega (D^2/n + D)$ lower bound in this setting.

Appendix: Math preliminaries

1.1 Basic probability

In this section, we state a few basic concentration results.

Theorem 7

(Chernoff bound) Let $X_1, \ldots , X_k$ be independent random variables such that for $1 \le i \le k$, $X_i \in \{0,1\}$. Let $X = X_1 + X_2 + \cdots + X_k$ and let $\mu = \mathbb {E}[X]$. Then, for any $0 \le \delta \le 1$, it is true that:

$$\begin{aligned} P[X > (1 + \delta ) \mu ] \le e^{-\delta ^2 \mu /2} \end{aligned}$$

(7)

$$\begin{aligned} P[X < (1 - \delta ) \mu ] \le e^{-\delta ^2 \mu /3} \end{aligned}$$

(8)

Theorem 8

(Two-sided Chernoff bound) Let $X_1, \ldots , X_k$ be independent random variables such that for $1 \le i \le k$, $X_i \in \{0,1\}$. Let $X = X_1 + X_2 + \cdots + X_k$ and let $\mu = \mathbb {E}[X]$. Then, for any $0 \le \delta \le 1$, it is true that:

$$\begin{aligned} P[|X - \mu | > \delta \mu ] \le 2e^{-\delta ^2 \mu /3} \end{aligned}$$

(9)

1.2 Markov chains

In this section, we state some basic results on Markov chains.

Theorem 9

(Feller [15]) In an irreducible Markov chain with period t the states can be divided into t mutually exclusive classes $G_0, \ldots , G_{t-1}$ such that it is true that (1) if $s \in G$ then the probability of being in state s in some round $r \ge 1$ is 0 unless $r = \tau + vt$ for some $v \in \mathbb {N}$, and (2) a one-step transition always leads to a state in the right neighboring class (in particular from $G_{t-1}$ to $G_0$). In the chain with matrix $P^t$ each class G corresponds to an irreducible closed set.

The next theorem establishes a bound on the difference between the stationary distribution of a Markov chain and the distribution resulting after k steps.

Lemma 19

(Rosenthal [26]) Let $P(x,\cdot )$ be the transition probabilities for a time-homogeneous Markov chain on a general state space $\mathscr {X}$. Suppose that for some probability distribution $Q(\cdot )$ on $\mathscr {X}$, some positive integers k and $k_0$, and some $\epsilon > 0$, $\forall x \in \mathscr {X}:\,P^{k_0}(x,\cdot ) \ge \epsilon Q(\cdot )$, where $P^{k_0}$ represents the $k_0$-step transition probabilities. Then for any initial distribution $\pi _0$, the distribution $\pi _k$ of the Markov chain after k steps satisfies $\Vert \pi _k - \pi \Vert \le (1 - \epsilon )^{\lfloor k/k_0 \rfloor }$, where $\Vert \cdot \Vert$ is the $\infty$-norm and $\pi$ is any stationary distribution.

Lemma 20

In any irreducible, aperiodic Markov chain with |S| states, there exists an integer $k \le 2|S|^2$ such that there is a walk of length k between any pair of states in the Markov chain.

Proof

By the definition of periodicity, for each state of the Markov chain, it is true that the greatest common divisor of the lengths of all the cycles that pass through that state is 1. Let the total number of distinct cycles in the Markov chain be m and let $(a_1, \ldots , a_m)$ denote the lengths of these cycles where $a_1 \le \cdots \le a_m$. The Frobenius number $F(a_1, \ldots , a_m)$ of the sequence $(a_1, \ldots , a_m)$ is the largest integer such that it is not possible to express it as a linear combination of $(a_1, \ldots , a_m)$ and non-negative integer coefficients. By a simple bound on the Frobenius number [7], we know that $F(a_1, \ldots , a_m) \le (a_1 - 1)(a_2 - 1) - 1$. Since $a_1$ and $a_2$ refer to cycle lengths in our Markov chain we know that $a_1, a_2 \le |S|$. So, it is true that $F(a_1, \ldots , a_m) \le |S|^2$ and we can express every integer greater than $F(a_1, \ldots , a_m)$ as a non-negative integer linear combination of $(a_1, \ldots , a_m)$.

Let i and j be arbitrary states in the Markov chain and let d(i, j) be the shortest path between i and j. Let $k = 2|S|^2$. By the argument above, we know that there is a walk starting at state i and ending at state i of length $k - d(i,j) \ge |S|^2$. Appending the shortest path between i and j to the end of that walk results in a walk from i to j of length exactly k.