Finding the Median (Obliviously) with Bounded Space

We prove that any oblivious algorithm using space $S$ to find the median of a list of $n$ integers from $\{1,...,2n\}$ requires time $\Omega(n \log\log_S n)$. This bound also applies to the problem of determining whether the median is odd or even. It is nearly optimal since Chan, following Munro and Raman, has shown that there is a (randomized) selection algorithm using only $s$ registers, each of which can store an input value or $O(\log n)$-bit counter, that makes only $O(\log\log_s n)$ passes over the input. The bound also implies a size lower bound for read-once branching programs computing the low order bit of the median and implies the analog of $P \ne NP \cap coNP$ for length $o(n \log\log n)$ oblivious branching programs

Oblivious Median Slope Selection

We study the median slope selection problem in the oblivious RAM model. In this model memory accesses have to be independent of the data processed, i.e., an adversary cannot use observed access patterns to derive additional information about the input. We show how to modify the randomized algorithm of Matou\v{s}ek (1991) to obtain an oblivious version with O(n log^2 n) expected time for n points in R^2. This complexity matches a theoretical upper bound that can be obtained through general oblivious transformation. In addition, results from a proof-of-concept implementation show that our algorithm is also practically efficient.Comment: 14 pages, to appear in Proceedings of CCCG 202

The Bane of Low-Dimensionality Clustering

In this paper, we give a conditional lower bound of $n^{\Omega(k)}$ on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time $n^{O(k^{1-1/d})}$ or $2^{n^{1-1/d}}$ in d dimensions, our work shows that widely-used clustering objectives have a lower bound of $n^{\Omega(k)}$, even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than $n^{o(\sqrt{k})}$, and provide a matching upper bound of $n^{O(\sqrt{k})}$. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity

Turvalisel ühisarvutusel põhinev privaatsust säilitav statistiline analüüs

Väitekirja elektrooniline versioon ei sisalda publikatsioone.Kaasaegses ühiskonnas luuakse inimese kohta digitaalne kirje kohe pärast tema sündi. Sellest hetkest alates jälgitakse tema käitumist ning kogutakse andmeid erinevate eluvaldkondade kohta. Kui kasutate poes kliendikaarti, käite arsti juures, täidate maksudeklaratsiooni või liigute lihtsalt ringi mobiiltelefoni taskus kandes, koguvad ning salvestavad firmad ja riigiasutused teie tundlikke andmeid. Vahel anname selliseks jälitustegevuseks vabatahtlikult loa, et saada mingit kasu. Näiteks võime saada soodustust, kui kasutame kliendikaarti. Teinekord on meil vaja teha keeruline otsus, kas loobuda võimalusest teha mobiiltelefonikõnesid või lubada enda jälgimine mobiilimastide kaudu edastatava info abil. Riigiasutused haldavad infot meie tervise, hariduse ja sissetulekute kohta, et meid paremini ravida, harida ja meilt makse koguda. Me loodame, et meie andmeid kasutatakse mõistlikult, aga samas eeldame, et meie privaatsus on tagatud. Käesolev töö uurib, kuidas teostada statistilist analüüsi nii, et tagada üksikisiku privaatsus. Selle eesmärgi saavutamiseks kasutame turvalist ühisarvutust. See krüptograafiline meetod lubab analüüsida andmeid nii, et üksikuid väärtuseid ei ole kunagi võimalik näha. Hoolimata sellest, et turvalise ühisarvutuse kasutamine on aeganõudev protsess, näitame, et see on piisavalt kiire ja seda on võimalik kasutada isegi väga suurte andmemahtude puhul. Me oleme teinud võimalikuks populaarseimate statistilise analüüsi meetodite kasutamise turvalise ühisarvutuse kontekstis. Me tutvustame privaatsust säilitavat statistilise analüüsi tööriista Rmind, mis sisaldab kõiki töö käigus loodud funktsioone. Rmind sarnaneb tööriistadele, millega statistikud on harjunud. See lubab neil viia läbi uuringuid ilma, et nad peaksid üksikasjalikult tundma allolevaid krüptograafilisi protokolle. Kasutame dissertatsioonis kirjeldatud meetodeid, et valmistada ette statistiline uuring, mis ühendab kaht Eesti riiklikku andmekogu. Uuringu eesmärk on teada saada, kas Eesti tudengid, kes töötavad ülikooliõpingute ajal, lõpetavad nominaalajaga väiksema tõenäosusega kui nende õpingutele keskenduvad kaaslased.In a modern society, from the moment a person is born, a digital record is created. From there on, the person’s behaviour is constantly tracked and data are collected about the different aspects of his or her life. Whether one is swiping a customer loyalty card in a store, going to the doctor, doing taxes or simply moving around with a mobile phone in one’s pocket, sensitive data are being gathered and stored by governments and companies. Sometimes, we give our permission for this kind of surveillance for some benefit. For instance, we could get a discount using a customer loyalty card. Other times we have a difficult choice – either we cannot make phone calls or our movements are tracked based on cellular data. The government tracks information about our health, education and income to cure us, educate us and collect taxes. We hope that the data are used in a meaningful way, however, we also have an expectation of privacy. This work focuses on how to perform statistical analyses in a way that preserves the privacy of the individual. To achieve this goal, we use secure multi-­‐party computation. This cryptographic technique allows data to be analysed without seeing the individual values. Even though using secure multi-­‐party computation is a time-­‐consuming process, we show that it is feasible even for large-­‐scale databases. We have developed ways for using the most popular statistical analysis methods with secure multi-­‐party computation. We introduce a privacy-­‐preserving statistical analysis tool called Rmind that contains all of our resulting implementations. Rmind is similar to tools that statistical analysts are used to. This allows them to carry out studies on the data without having to know the details of the underlying cryptographic protocols. The methods described in the thesis are used in practice to prepare for running a statistical study on large-­‐scale real-­‐life data to find out whether Estonian students who are working during university studies are less likely to graduate in nominal time