arXiv subjects

Amos Korman

Publications and source records attributed to Amos Korman.

At least 19 recordsLinked to original sources

arXiv2026

Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes

Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent L\'evy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (L\'evy exponent $\mu = 2$) uniquely achieves scale-invariant, near-optimal detection across a broad spectrum of target sizes and shapes. In a domain of volume $n$ with boundary conditions, expected detection time for a convex target of surface area $\Delta$ optimally scales as $n/\Delta$. Conversely, L\'evy strategies with $\mu < 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $\mu > 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $\mu = 1$, ceding dominance to surface area as $\mu \to 2$, after which surface area and elongation couple to govern detection. Ultimately, 3D search introduces a pronounced sensitivity to target shape that is absent in lower dimensions. Our work provides a rigorous foundation for the L\'evy flight foraging hypothesis in 3D by establishing the scale-invariant optimality of the Cauchy walk. Furthermore, our results reveal dimensionality-driven shape vulnerabilities and offer testable predictions for biological and engineered systems.

cs.DS↗

arXiv2024

The 2024 Motile Active Matter Roadmap

Activity and autonomous motion are fundamental aspects of many living and engineering systems. Here, the scale of biological agents covers a wide range, from nanomotors, cytoskeleton, and cells, to insects, fish, birds, and people. Inspired by biological active systems, various types of autonomous synthetic nano- and micromachines have been designed, which provide the basis for multifunctional, highly responsive, intelligent active materials. A major challenge for understanding and designing active matter is their inherent non-equilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Furthermore, interactions in ensembles of active agents are often non-additive and non-reciprocal. An important aspect of biological agents is their ability to sense the environment, process this information, and adjust their motion accordingly. It is an important goal for the engineering of micro-robotic systems to achieve similar functionality. With many fundamental properties of motile active matter now reasonably well understood and under control, the ground is prepared for the study of physical aspects and mechanisms of motion in complex environments, of the behavior of systems with new physical features like chirality, of the development of novel micromachines and microbots, of the emergent collective behavior and swarming of intelligent self-propelled particles, and of particular features of microbial systems. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter poses major challenges, which can only be addressed by a truly interdisciplinary effort involving scientists from biology, chemistry, ecology, engineering, mathematics, and physics.

cond-mat.soft↗

arXiv2024

Fast and Robust Information Spreading in the Noisy PULL Model

Understanding how information can efficiently spread in distributed systems under noisy communications is a fundamental question in both biological research and artificial system design. When agents are able to control whom they interact with, noise can often be mitigated through redundancy or other coding techniques, but it may have fundamentally different consequences on well-mixed systems. Specifically, Boczkowski et al. (2018) considered the noisy $\mathcal{PULL}(h)$ model, where each message can be viewed as any other message with probability $\delta$. The authors proved that in this model, the basic task of propagating a bit value from a single source to the whole population requires $\Omega(\frac{n\delta}{h(1-\delta|\Sigma|)^2})$ (parallel) rounds. The current work shows that the aforementioned lower bound is almost tight. In particular, when each agent observes all other agents in each round, which relates to scenarios where each agent senses the system's average tendency, information spreading can reliably be achieved in $\mathcal{O}(\log n)$ time, assuming constant noise. We present two simple and highly efficient protocols, thus suggesting their applicability to real-life scenarios. Notably, they also work in the presence of multiple conflicting sources and efficiently converge to their plurality opinion. The first protocol we present uses 1-bit messages but relies on a simultaneous wake-up assumption. By increasing the message size to 2 bits and removing the speedup in the information spreading time that may result from having multiple sources, we also present a simple and highly efficient self-stabilizing protocol that avoids the simultaneous wake-up requirement. Overall, our results demonstrate how, under stochastic communication, increasing the sample size can compensate for the lack of communication structure by linearly accelerating information spreading time.

cs.DC↗

arXiv2023

Enhanced Food Availability can Deteriorate Fitness through Excessive Scrounging

In group foraging situations, the conventional expectation is that increased food availability would enhance consumption, especially when animals prioritize maximizing their food intake. This paper challenges this conventional wisdom by conducting an in-depth game-theoretic analysis of a basic producer-scrounger model, in which animals must choose between intensive food searching as producers or moderate searching while relying on group members as scroungers. Surprisingly, our study reveals that, under certain circumstances, increasing food availability can amplify the inclination to scrounge to such an extent that it paradoxically leads to a reduction in animals' food consumption compared to scenarios with limited food availability. We further illustrate a similar phenomenon in a model capturing free-riding dynamics among workers in a company. We demonstrate that, under certain reward mechanisms, enhancing workers' production capacities can inadvertently trigger a surge in free-riding behavior, leading to both diminished group productivity and reduced individual payoffs. Our findings underscore the significance of contextual factors when comprehending and predicting the impact of resource availability on individual and collective outcomes.

cs.GT↗

arXiv2023

On the Role of Memory in Robust Opinion Dynamics

We investigate opinion dynamics in a fully-connected system, consisting of $n$ identical and anonymous agents, where one of the opinions (which is called correct) represents a piece of information to disseminate. In more detail, one source agent initially holds the correct opinion and remains with this opinion throughout the execution. The goal for non-source agents is to quickly agree on this correct opinion, and do that robustly, i.e., from any initial configuration. The system evolves in rounds. In each round, one agent chosen uniformly at random is activated: unless it is the source, the agent pulls the opinions of $\ell$ random agents and then updates its opinion according to some rule. We consider a restricted setting, in which agents have no memory and they only revise their opinions on the basis of those of the agents they currently sample. As restricted as it is, this setting encompasses very popular opinion dynamics, such as the voter model and best-of-$k$ majority rules. Qualitatively speaking, we show that lack of memory prevents efficient convergence. Specifically, we prove that no dynamics can achieve correct convergence in an expected number of steps that is sub-quadratic in $n$, even under a strong version of the model in which activated agents have complete access to the current configuration of the entire system, i.e., the case $\ell=n$. Conversely, we prove that the simple voter model (in which $\ell=1$) correctly solves the problem, while almost matching the aforementioned lower bound. These results suggest that, in contrast to symmetric consensus problems (that do not involve a notion of correct opinion), fast convergence on the correct opinion using stochastic opinion dynamics may indeed require the use of memory. This insight may reflect on natural information dissemination processes that rely on a few knowledgeable individuals.

cs.DC↗

arXiv2023

Distributed Alignment Processes with Samples of Group Average

Reaching agreement despite noise in communication is a fundamental problem in multi-agent systems. Here we study this problem under an idealized model, where it is assumed that agents can sense the general tendency in the system. More specifically, we consider $n$ agents, each being associated with a real-valued number. In each round, each agent receives a noisy measurement of the average value, and then updates its value, which is in turn perturbed by random drift. We assume that both noises in measurements and drift follow Gaussian distributions. What should be the updating policy of agents if their goal is to minimize the expected deviation of the agents' values from the average value? We prove that a distributed weighted-average algorithm optimally minimizes this deviation for each agent, and for any round. Interestingly, this optimality holds even in the centralized setting, where a master agent can gather all the agents' measurements and instruct a move to each agent.We find this result surprising since it can be shown that the total measurements obtained by all agents contain strictly more information about the deviation of Agent $i$ from the average value, than the information contained in the measurements obtained by Agent $i$ alone. Although this information is relevant for Agent $i$, it is not processed by it when running a weighted-average algorithm. Nevertheless, the weighted-average algorithm is optimal, since by running it, other agents manage to fully process this information in a way that perfectly benefits Agent $i$.Finally, we also analyze the drift of the center of mass and show that no distributed algorithm can achieve drift that is as small as the one that can be achieved by the best centralized algorithm. In light of this, we also show that the drift associated with our weighted-average algorithm incurs a relatively small overhead over the best possible drift in the centralized setting.

cs.GT↗

arXiv2022

Early Adapting to Trends: Self-Stabilizing Information Spread using Passive Communication

How to efficiently and reliably spread information in a system is one of the most fundamental problems in distributed computing. Recently, inspired by biological scenarios, several works focused on identifying the minimal communication resources necessary to spread information under faulty conditions. Here we study the self-stabilizing bit-dissemination problem, introduced by Boczkowski, Korman, and Natale in [SODA 2017]. The problem considers a fully-connected network of n agents, with a binary world of opinions, one of which is called correct. At any given time, each agent holds an opinion bit as its public output. The population contains a source agent which knows which opinion is correct. This agent adopts the correct opinion and remains with it throughout the execution. We consider the basic PULL model of communication, in which each agent observes relatively few randomly chosen agents in each round. The goal of the non-source agents is to quickly converge on the correct opinion, despite having an arbitrary initial configuration, i.e., in a self-stabilizing manner. Once the population converges on the correct opinion, it should remain with it forever. Motivated by biological scenarios in which animals observe and react to the behavior of others, we focus on the extremely constrained model of passive communication, which assumes that when observing another agent the only information that can be extracted is the opinion bit of that agent. We prove that this problem can be solved in a poly-logarithmic in n number of rounds with high probability, while sampling a logarithmic number of agents at each round. Previous works solved this problem faster and using fewer samples, but they did that by decoupling the messages sent by agents from their output opinion, and hence do not fit the framework of passive communication. Moreover, these works use complex recursive algorithms with refined clocks that are unlikely to be used by biological entities. In contrast, our proposed algorithm has a natural appeal as it is based on letting agents estimate the current tendency direction of the dynamics, and then adapt to the emerging trend.

cs.DC↗

arXiv2022

Public Communication can Facilitate Low-Risk Coordination under Surveillance

Consider a sub-population of rebels that wish to initiate a revolution. In order to avoid initializing a failed revolution, rebels would first strive to estimate their relative "power", which is often correlated with their fraction in the population. However, and especially in non-democratic countries, rebels refrain from disclosing themselves. This poses a significant challenge for rebels: estimating their fraction in the population while minimizing the risk of being identified as rebels. This paper introduces a distributed computing framework aiming to study this question. Our main takeaway message is that the communication pattern has a crucial role in achieving such a task. Specifically, we show that relying on the inherent noise in the communication, "public communication", characterized by the fact that each message announced by an individual can be viewed by all its neighbors, allows rebels to estimate their fraction in the population while keeping a negligible risk of each rebel being identified as such. The suggested estimation protocol, inspired by historical events, is extremely simple and can be executed covertly even under extreme conditions of surveillance. Conversely, we show that under peer-to-peer communication, protocols of similar simplicity are either inefficient or non-covert.

cs.DC↗

arXiv2021

On the Role of Hypocrisy in Escaping the Tragedy of the Commons

We study the emergence of cooperation in large spatial public goods games. Without employing severe social-pressure against "defectors", or alternatively, significantly rewarding "cooperators", theoretical models typically predict a system collapse in a way that is reminiscent of the "tragedy-of-the-commons" metaphor. Drawing on a dynamic network model, this paper demonstrates how cooperation can emerge when the social-pressure is mild. This is achieved with the aid of an additional behavior called "hypocritical", which appears to be cooperative from the external observer's perspective but in fact hardly contributes to the social-welfare. Our model assumes that social-pressure is induced over both defectors and hypocritical players, but the extent of which may differ. Our main result indicates that the emergence of cooperation highly depends on the extent of social-pressure applied against hypocritical players. Setting it to be at some intermediate range below the one employed against defectors allows a system composed almost exclusively of defectors to transform into a fully cooperative one quickly. Conversely, when the social-pressure against hypocritical players is either too low or too high, the system remains locked in a degenerate configuration.

cs.GT↗

arXiv2021

Intermittent Inverse-Square Lévy Walks are Optimal for Finding Targets of All Sizes

Lévy walks are random walk processes whose step-lengths follow a long-tailed power-law distribution. Due to their abundance as movement patterns of biological organisms, significant theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the \emph{Lévy foraging hypothesis}, and are thus expected to impact future experiments on animals performing Lévy walks.

cs.DS↗

arXiv2020

Exploitation of Multiple Replenishing Resources with Uncertainty

We consider an optimization problem in which a (single) bat aims to exploit the nectar in a set of $n$ cacti with the objective of maximizing the expected total amount of nectar it drinks. Each cactus $i \in [n]$ is characterized by a parameter $r_{i} > 0$ that determines the rate in which nectar accumulates in $i$. In every round, the bat can visit one cactus and drink all the nectar accumulated there since its previous visit. Furthermore, competition with other bats, that may also visit some cacti and drink their nectar, is modeled by means of a stochastic process in which cactus $i$ is emptied in each round (independently) with probability $0 < s_i < 1$. Our attention is restricted to purely-stochastic strategies that are characterized by a probability vector $(p_1, \ldots, p_n)$ determining the probability $p_i$ that the bat visits cactus $i$ in each round. We prove that for every $ε> 0$, there exists a purely-stochastic strategy that approximates the optimal purely-stochastic strategy to within a multiplicative factor of $1 + ε$, while exploiting only a small core of cacti. Specifically, we show that it suffices to include at most $\frac{2 (1 - σ)}{ε\cdot σ}$ cacti in the core, where $σ= \min_{i \in [n]} s_{i}$. We also show that this upper bound on core size is asymptotically optimal as a core of a significantly smaller size cannot provide a $(1 + ε)$-approximation of the optimal purely-stochastic strategy. This means that when the competition is more intense (i.e., $σ$ is larger), a strategy based on exploiting smaller cores will be favorable.

cs.DS↗

arXiv2020

Multi-Round Cooperative Search Games with Multiple Players

Assume that a treasure is placed in one of $M$ boxes according to a known distribution and that $k$ searchers are searching for it in parallel during $T$ rounds. We study the question of how to incentivize selfish players so that the success probability, namely, the probability that at least one player finds the treasure, would be maximized. We focus on congestion policies $C(s)$ that specify the reward that a player receives if it is one of $s$ players that (simultaneously) find the treasure for the first time. Our main technical contribution is proving that the exclusive policy, in which $C(1)=1$ and $C(s)=0$ for $s>1$, yields a price of anarchy of $(1-(1-{1}/{k})^{k})^{-1}$, and that this is the best possible price among all symmetric reward mechanisms. For this policy we also have an explicit description of a symmetric equilibrium, which is in some sense unique, and moreover enjoys the best success probability among all symmetric profiles. For general congestion policies, we show how to polynomially find, for any $e>0$, a symmetric multiplicative $(1+e)(1+C(k))$-equilibrium. Together with an appropriate reward policy, a central entity can suggest players to play a particular profile at equilibrium. As our main conceptual contribution, we advocate the use of symmetric equilibria for such purposes. Besides being fair, we argue that in many cases, despite the fact that some small fraction of players crash, symmetric equilibria remain efficient in terms of their group performances and, at the same time, serve as approximate equilibria. We show that this principle holds for a class of games, which we call monotonously scalable games. This applies in particular to our search game, assuming the sharing policy, in which $C(s)=1/s$. For the exclusive policy, this general result does not hold, but we show that the symmetric equilibrium is nevertheless robust under mild assumptions.

cs.GT↗

arXiv2020

Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths

Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from the microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n 1+$Θ$(1/k). While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k > 1. Specifically, up to lower order polylogarithmic factors, the upper bound on the cover time is a polynomial in n of exponent 1+ 1/(2k--1). For k = 2, 3, 4 and 5 the exponent is thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k.

q-bio.QM↗

arXiv2020

Searching Trees with Permanently Noisy Advice: Walking and Query Algorithms

We consider a search problem on trees in which the goal is to find an adversarially placed treasure, while relying on local, partial information. Specifically, each node in the tree holds a pointer to one of its neighbors, termed \emph{advice}. A node is faulty with probability $q$. The advice at a non-faulty node points to the neighbor that is closer to the treasure, and the advice at a faulty node points to a uniformly random neighbor. Crucially, the advice is {\em permanent}, in the sense that querying the same node again would yield the same answer. Let $Δ$ denote the maximal degree. Roughly speaking, when considering the expected number of {\em moves}, i.e., edge traversals, we show that a phase transition occurs when the {\em noise parameter} $q$ is about $1/\sqrtΔ$. Below the threshold, there exists an algorithm with expected move complexity $O(D\sqrtΔ)$, where $D$ is the depth of the treasure, whereas above the threshold, every search algorithm has expected number of moves which is both exponential in $D$ and polynomial in the number of nodes~$n$. In contrast, if we require to find the treasure with probability at least $1-δ$, then for every fixed $\varepsilon > 0$, if $q<1/Δ^{\varepsilon}$ then there exists a search strategy that with probability $1-δ$ finds the treasure using $(δ^{-1}D)^{O(\frac 1 \varepsilon)}$ moves. Moreover, we show that $(δ^{-1}D)^{Ω(\frac 1 \varepsilon)}$ moves are necessary. Besides the number of moves, we also study the number of advice {\em queries} required to find the treasure. Roughly speaking, for this complexity, we show similar threshold results to those previously stated, where the parameter $D$ is replaced by $\log n$.

cs.DS↗

arXiv2018

Intense Competition can Drive Selfish Explorers to Optimize Coverage

We consider a game-theoretic setting in which selfish individuals compete over resources of varying quality. The motivating example is a group of animals that disperse over patches of food of different abundances. In such scenarios, individuals are biased towards selecting the higher quality patches, while, at the same time, aiming to avoid costly collisions or overlaps. Our goal is to investigate the impact of collision costs on the parallel coverage of resources by the whole group. Consider M sites, where a site x has value f(x). We think of f(x) as the reward associated with site x, and assume that if a single individual visits x exclusively, it receives this exact reward. Typically, we assume that if > 1 individuals visit x then each receives at most f(x). In particular, when competition costs are high, each individual might receive an amount strictly less than f(x), which could even be negative. Conversely, modeling cooperation at a site, we also consider cases where each one gets more than f(x). There are k identical players that compete over the rewards. They independently act in parallel, in a one-shot scenario, each specifying a single site to visit, without knowing which sites are explored by others. The group performance is evaluated by the expected coverage, defined as the sum of f(x) over all sites that are explored by at least one player. Since we assume that players cannot coordinate before choosing their site we focus on symmetric strategies. The main takeaway message of this paper is that the optimal symmetric coverage is expected to emerge when collision costs are relatively high, so that the following "Judgment of Solomon" type of rule holds: If a single player explores a site x then it gains its full reward f(x), but if several players explore it, then neither one receives any reward. Under this policy, it turns out that there exists a unique symmetric Nash Equilibrium strategy, which is, in fact, evolutionary stable. Moreover, this strategy yields the best possible coverage among all symmetric strategies. Viewing the coverage measure as the social welfare, this policy thus enjoys a (Symmetric) Price of Anarchy of precisely 1, whereas, in fact, any other congestion policy has a price strictly greater than 1. Our model falls within the scope of mechanism design, and more precisely in the area of incentivizing exploration. It finds relevance in evolutionary ecology, and further connects to studies on Bayesian parallel search algorithms.

cs.GT↗

arXiv2017

Limits for Rumor Spreading in stochastic populations

Biological systems can share and collectively process information to yield emergent effects, despite inherent noise in communication. While man-made systems often employ intricate structural solutions to overcome noise, the structure of many biological systems is more amorphous. It is not well understood how communication noise may affect the computational repertoire of such groups. To approach this question we consider the basic collective task of rumor spreading, in which information from few knowledgeable sources must reliably flow into the rest of the population. In order to study the effect of communication noise on the ability of groups that lack stable structures to efficiently solve this task, we consider a noisy version of the uniform PULL model. We prove a lower bound which implies that, in the presence of even moderate levels of noise that affect all facets of the communication, no scheme can significantly outperform the trivial one in which agents have to wait until directly interacting with the sources. Our results thus show an exponential separation between the uniform PUSH and PULL communication models in the presence of noise. Such separation may be interpreted as suggesting that, in order to achieve efficient rumor spreading, a system must exhibit either some degree of structural stability or, alternatively, some facet of the communication which is immune to noise. We corroborate our theoretical findings with a new analysis of experimental data regarding recruitment in Cataglyphis niger desert ants.

cs.MA↗

arXiv2017

Parallel Search with no Coordination

We consider a parallel version of a classical Bayesian search problem. $k$ agents are looking for a treasure that is placed in one of the boxes indexed by $\mathbb{N}^+$ according to a known distribution $p$. The aim is to minimize the expected time until the first agent finds it. Searchers run in parallel where at each time step each searcher can "peek" into a box. A basic family of algorithms which are inherently robust is \emph{non-coordinating} algorithms. Such algorithms act independently at each searcher, differing only by their probabilistic choices. We are interested in the price incurred by employing such algorithms when compared with the case of full coordination. We first show that there exists a non-coordination algorithm, that knowing only the relative likelihood of boxes according to $p$, has expected running time of at most $10+4(1+\frac{1}{k})^2 T$, where $T$ is the expected running time of the best fully coordinated algorithm. This result is obtained by applying a refined version of the main algorithm suggested by Fraigniaud, Korman and Rodeh in STOC'16, which was designed for the context of linear parallel search.We then describe an optimal non-coordinating algorithm for the case where the distribution $p$ is known. The running time of this algorithm is difficult to analyse in general, but we calculate it for several examples. In the case where $p$ is uniform over a finite set of boxes, then the algorithm just checks boxes uniformly at random among all non-checked boxes and is essentially $2$ times worse than the coordinating algorithm.We also show simple algorithms for Pareto distributions over $M$ boxes. That is, in the case where $p(x) \sim 1/x^b$ for $0< b < 1$, we suggest the following algorithm: at step $t$ choose uniformly from the boxes unchecked in ${1, . . . ,min(M, \lfloor t/σ\rfloor)}$, where $σ= b/(b + k - 1)$. It turns out this algorithm is asymptotically optimal, and runs about $2+b$ times worse than the case of full coordination.

cs.DC↗

arXiv2017

The Dependent Doors Problem: An Investigation into Sequential Decisions without Feedback

We introduce the dependent doors problem as an abstraction for situations in which one must perform a sequence of possibly dependent decisions, without receiving feedback information on the effectiveness of previously made actions. Informally, the problem considers a set of $d$ doors that are initially closed, and the aim is to open all of them as fast as possible. To open a door, the algorithm knocks on it and it might open or not according to some probability distribution. This distribution may depend on which other doors are currently open, as well as on which other doors were open during each of the previous knocks on that door. The algorithm aims to minimize the expected time until all doors open. Crucially, it must act at any time without knowing whether or which other doors have already opened. In this work, we focus on scenarios where dependencies between doors are both positively correlated and acyclic.The fundamental distribution of a door describes the probability it opens in the best of conditions (with respect to other doors being open or closed). We show that if in two configurations of $d$ doors corresponding doors share the same fundamental distribution, then these configurations have the same optimal running time up to a universal constant, no matter what are the dependencies between doors and what are the distributions. We also identify algorithms that are optimal up to a universal constant factor. For the case in which all doors share the same fundamental distribution we additionally provide a simpler algorithm, and a formula to calculate its running time. We furthermore analyse the price of lacking feedback for several configurations governed by standard fundamental distributions. In particular, we show that the price is logarithmic in $d$ for memoryless doors, but can potentially grow to be linear in $d$ for other distributions.We then turn our attention to investigate precise bounds. Even for the case of two doors, identifying the optimal sequence is an intriguing combinatorial question. Here, we study the case of two cascading memoryless doors. That is, the first door opens on each knock independently with probability $p\_1$. The second door can only open if the first door is open, in which case it will open on each knock independently with probability $p\_2$. We solve this problem almost completely by identifying algorithms that are optimal up to an additive term of 1.

cs.AI↗