Min-Zheng Shieh

Decoding Frequency Permutation Arrays Under Chebyshev Distance

A frequency permutation array (FPA) of length n = mλ and distance d is a set of permutations on a multiset over m symbols, where each symbol appears exactly λ times and the distance between any two elements in the array is at least d. FPA generalizes the notion of permutation array. In this paper, under the Chebyshev distance, we first prove lower and upper bounds on the size of FPA. Then we give several constructions of FPAs, and some of them come with efficient encoding and decoding capabilities. Moreover, we show one of our designs is locally decodable, i.e., we can decode a message bit by reading at most λ+1 symbols, which has an interesting application to private information retrieval.

On the inapproximability of maximum intersection problems

Given u sets, we want to choose exactly k sets such that the cardinality of their intersection is maximized. This is the so-called MAX-k-INTERSECT problem. We prove that MAX-k-INTERSECT cannot be approximated within an absolute error of 12n^1^-^2^@e+O(n^1^-^3^@e) unless P=NP. This answers an open question about its hardness. We also give a correct proof of an inapproximable result by Clifford and Popa (2011) [3] by proving that MAX-INTERSECT problem is equivalent to the MAX-CLIQUE problem.

Computing the ball size of frequency permutations under chebyshev distance

Let Sλn be the set of all permutations over the multiset equation where n = mλ. A frequency permutation array (FPA) of minimum distance d is a subset of Sλn in which every two elements have distance at least d. FPAs have many applications related to error correcting codes. In coding theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived from the size of balls of certain radii. We propose two efficient algorithms that compute the ball size of frequency permutations under Chebyshev distance. Both methods extend previous known results. The first one runs in equation time and space. The second one runs in equation time and equation space. For small constants λ and d, both are efficient in time and use constant storage space.

More on the Magnus-Derek game

We consider the so called Magnus-Derek game, which is a two-person game played on a round table with n positions. The two players are called Magnus and Derek. Initially there is a token placed at position 0. In each round Magnus chooses a positive integer m@?n/2 as the distance of the targeted position from his current position for the token to move, and Derek decides a direction, clockwise or counterclockwise, to move the token. The goal of Magnus is to maximize the total number of positions visited, while Dereks is to minimize this number. If both players play optimally, we prove that Magnus, the maximizer, can achieve his goal in O(n) rounds, which improves a previous result with O(nlogn) rounds. Then we consider a modified version of the Magnus-Derek game, where one of the players reveals his moves in advance and the other player plays optimally. In this case we prove that it is NP-hard for Derek to achieve his goal if Magnus reveals his moves in advance. On the other hand, Magnus has an advantage to occupy all positions. We also consider the circumstance that both players play randomly, and we show that the expected time to visit all positions is O(nlogn).

Decoding frequency permutation arrays under infinite norm

A frequency permutation array (FPA) of length n = m⋋ and distance d is a set of permutations on a multiset over m symbols, where each symbol appears exactly ⋋ times and the distance between any two elements in the array is at least d. FPA generalizes the notion of permutation array. In this paper, under the distance metric ℓ∞-norm, we first prove lower and upper bounds on the size of FPA. Then we give a construction of FPA with efficient encoding and decoding capabilities. Moreover, we show our design is locally decodable, i.e., we can decode a message bit by reading at most ⋋+1 symbols, which has an interesting application for private information retrieval.

Decoding permutation arrays with ternary vectors

We give an explicit decoding scheme for the permutation arrays under Hamming distance metric, where the encoding is constructed via a distance-preserving mapping from ternary vectors to permutations (3-DPM).

Jug measuring: Algorithms and complexity

We study the hardness of the optimal jug measuring problem. By proving tight lower and upper bounds on the minimum number of measuring steps required, we reduce an inapproximable NP-hard problem (i.e., the shortest GCD multiplier problem [G. Havas, J.-P. Seifert, The Complexity of the Extended GCD Problem, in: LNCS, vol. 1672, Springer, 1999]) to it. It follows that the optimal jug measuring problem is NP-hard and so is the problem of approximating the minimum number of measuring steps within a constant factor. Along the way, we give a polynomial-time approximation algorithm with an exponential error based on the well-known LLL basis reduction algorithm.

MapTalk: mosaicking physical objects into the cyber world

ABSTRACT Digital map is considered as a cyber world, which maps visual representations (cyber objects) to the physical objects in the real world and allows the user to interact with these physical objects through their cyber representations. However, it typically requires significant programming effort to create a map application. This paper proposes MapTalk, a web-based visual map platform that allows the user to interact with the physical objects through their cyber representations in a visual map. We show how the administrator can add applications to the map without any programming effort. The novel idea in our approach is to utilise the IoT concept. Specifically, we implement the map as an output IoT device and all physical objects to be mosaicked in the map as input IoT devices. We show how to automatically create the device features of IoT devices when the administrator creates applications in the Map web page. We have deployed over 20 services in MapTalk including smart home, smart farm, tracking (bus, person, dog, etc), video monitoring, travel route planning, sensing of CO2, PM2.5, Internet of Things (IoT). temperature, humidity, and so on. Abbreviation: Internet of Things (IoT)

A Local Customizable Gateway in General-Purpose IoT Framework

In the emergence of the Internet of Things, the local gateway plays an important role because it connects devices over heterogeneous networks and provides local intelligence. In this paper, we propose a structure of a general-purpose gateway which provides customizable connectivity for a wide range of devices and services such as home appliances, environment sensors, and webpage parsers. In addition, the gateway has an interface to communicate with the IoTtalk platform which has a graphical user interface for device management and enables users create new network applications by programming in Python. The gateway is also implemented on Android platform.

Inapproximability Results for the Weight Problems of Subgroup Permutation Codes

A subgroup permutation code is a set of permutations on <i>n</i> symbols with the property that its elements are closed under the operation of composition. In this paper, we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well-known metrics. Based on previous works, we prove that under Hamming, Lee, Cayley, Kendalls tau, Ulams, and <i>lp</i> distance metrics, 1) there is no polynomial-time 2^log1-ε<i>n</i>-approximation algorithm for the minimum weight problem for any constant ε >; 0 unless NP ⊆ DTIME(2^polylog(<i>n</i>)) (quasi-polynomial time), and 2) there is no polynomial-time <i>r</i>-approximation algorithm for the minimum weight problem for any constant <i>r</i> >; 1 unless P = NP. Under <i>l</i>_∞-metric, we prove that it is NP-hard to approximate the minimum weight problem within factor 2-ε for any constant ε >; 0. We also prove that for any constant ε >; 0, it is NP-hard to approximate the maximum weight within <i>p</i> √{[ 3/ 2]}-ε under ℓ_p distance metric, and within [ 3/ 2]-ε under Hamming, Lee, Cayley, Kendalls tau, and Ulams distance metrics.