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1. On energy-delay tradeoff in uncoordinated MAC

   Yuming Han, Zixiang Xiong (October 2023, Proc. 2023 Allterton conference)

   Polyanskiy [1] proposed a framework for the MAC problem with a large number of users, where users employ a common codebook in the finite blocklength regime. In this work, we extend [1] to the case when the number of active users is random and there is also a delay constraint. We first define a random-access channel and derive the general converse bound. Our bound captures the basic tradeoff between the required energy and the delay constraint. Then we propose an achievable bound for block transmission. In this case, all packets are transmitted in the second half of the block to avoid interference. We then study treating interference as noise (TIN) with both single user and multiple users. Last, we derive an achievable bound for the packet splitting model, which allows users to split each packet into two parts with different blocklengths. Our numerical results indicate that, when the delay is large, TIN is effective; on the other hand, packet splitting outperforms as the delay decreases. 

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2. Balanced Allocation Through Random Walk

   Frieze, A.; Petti, S (December 2018, Information processing letters)

   We consider the allocation problem in which $$m \leq (1-\epsilon) dn $$ items are to be allocated to $$n$$ bins with capacity $$d$$. The items $$x_1,x_2,\ldots,x_m$$ arrive sequentially and when item $$x_i$$ arrives it is given two possible bin locations $$p_i=h_1(x_i),q_i=h_2(x_i)$$ via hash functions $$h_1,h_2$$. We consider a random walk procedure for inserting items and show that the expected time insertion time is constant provided $$\epsilon = \Omega\left(\sqrt{ \frac{ \log d}{d}} \right).$$ 

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3. USING SIMULATION TO STUDY THE LAST TO ENTER SERVICE DELAY ANNOUNCEMENT IN MULTISERVER QUEUES WITH ABANDONMENT

   https://doi.org/10.1109/WSC40007.2019.9004709  

   Shah, Aditya; Wikum, Anders; Pender, Jamol (December 2019, Proceedings of the Winter Simulation Conference)

   The Last to Enter Service (LES) delay announcement is one of the most commonly used delay announcements in queueing theory because it is quite simple to implement. Recent research has shown that using a convex combination of LES and the conditional mean delay are optimal under the mean squared error and the optimal value depends on the correlation between LES and the virtual waiting time. To this end, we show using simulation that it is important to be careful when using finite queue sizes, especially in a heavy traffic setting. Using simulation we demonstrate that the correlation between LES and the virtual waiting time can differ from heavy traffic results and can therefore have a large impact on the optimal announcement. Finally, we use simulation to assess the value of giving future information in computing correlations with virtual waiting times and show that future information is helpful in some settings. 

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4. Testing k-Monotonicity

   Canonne, C; Grigorescu, E.; Guo, S; Kumar, A; Wimmer, K. (January 2017, ITCS)

   A Boolean {\em $$k$$-monotone} function defined over a finite poset domain $${\cal D}$$ alternates between the values $$0$$ and $$1$$ at most $$k$$ times on any ascending chain in $${\cal D}$$. Therefore, $$k$$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $$1$$-monotone} functions. Motivated by the recent interest in $$k$$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $$k$$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $$k$$-monotone (or are close to being $$k$$-monotone) from functions that are far from being $$k$$-monotone. Our results include the following: \begin{enumerate} \item We demonstrate a separation between testing $$k$$-monotonicity and testing monotonicity, on the hypercube domain $$\{0,1\}^d$$, for $$k\geq 3$$; \item We demonstrate a separation between testing and learning on $$\{0,1\}^d$$, for $$k=\omega(\log d)$$: testing $$k$$-monotonicity can be performed with $$2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$$ queries, while learning $$k$$-monotone functions requires $$2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$$ queries (Blais et al. (RANDOM 2015)). \item We present a tolerant test for functions $$f\colon[n]^d\to \{0,1\}$$ with complexity independent of $$n$$, which makes progress on a problem left open by Berman et al. (STOC 2014). \end{enumerate} Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $$[n]^d$, and draw connections to distribution testing techniques. 

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5. Directed Isoperimetric Theorems for Boolean Functions on the Hypergrid and an $$~O(n\sqrt{d})$$ Monotonicity Tester

   Hadley Black, Deeparnab Chakrabarty (June 2023, Proceedings of the annual ACM Symposium on Theory of Computing)

   The problem of testing monotonicity for Boolean functions on the hypergrid, $$f:[n]^d \to \{0,1\}$$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making $$\otilde(\eps^{-2}\sqrt{d})$$ queries. Up to polylog $$d$$ and $$\eps$$ factors, this bound matches the $$\widetilde{\Omega}(\sqrt{d})$$-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n > 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $$\otilde(d^{5/6})$$-query upper bound (SODA 2020), quite far from the $$\sqrt{d}$$ bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant $$n$$, up to $$\poly(\eps^{-1}\log d)$$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $$\otilde(\eps^{-2}n\sqrt{d})$$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube. 

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