arXiv subjects

Ming-Yang Kao

Publications and source records attributed to Ming-Yang Kao.

At least 37 records · Page 2Linked to original sources

arXiv2001

Fast Pricing of European Asian Options with Provable Accuracy: Single-stock and Basket Options

This paper develops three polynomial-time pricing techniques for European Asian options with provably small errors, where the stock prices follow binomial trees or trees of higher-degree. The first technique is the first known Monte Carlo algorithm with analytical error bounds suitable for pricing single-stock options with meaningful confidence and speed. The second technique is a general recursive bucketing-based scheme that can use the Aingworth-Motwani-Oldham aggregation algorithm, Monte-Carlo simulation and possibly others as the base-case subroutine. This scheme enables robust trade-offs between accuracy and time over subtrees of different sizes. For long-term options or high frequency price averaging, it can price single-stock options with smaller errors in less time than the base-case algorithms themselves. The third technique combines Fast Fourier Transform with bucketing-based schemes for pricing basket options. This technique takes polynomial time in the number of days and the number of stocks, and does not add any errors to those already incurred in the companion bucketing scheme. This technique assumes that the price of each underlying stock moves independently.

cs.CE↗

arXiv2001

Cavity Matchings, Label Compressions, and Unrooted Evolutionary Trees

We present an algorithm for computing a maximum agreement subtree of two unrooted evolutionary trees. It takes O(n^{1.5} log n) time for trees with unbounded degrees, matching the best known time complexity for the rooted case. Our algorithm allows the input trees to be mixed trees, i.e., trees that may contain directed and undirected edges at the same time. Our algorithm adopts a recursive strategy exploiting a technique called label compression. The backbone of this technique is an algorithm that computes the maximum weight matchings over many subgraphs of a bipartite graph as fast as it takes to compute a single matching.

cs.CE↗

arXiv2001

Total Protection of Analytic Invariant Information in Cross Tabulated Tables

To protect sensitive information in a cross tabulated table, it is a common practice to suppress some of the cells in the table. An analytic invariant is a power series in terms of the suppressed cells that has a unique feasible value and a convergence radius equal to +\infty. Intuitively, the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed. This paper gives an optimal linear-time algorithm for testing whether there exist nontrivial analytic invariants in terms of the suppressed cells in a given set of suppressed cells. This paper also presents NP-completeness results and an almost linear-time algorithm for the problem of suppressing the minimum number of cells in addition to the sensitive ones so that the resulting table does not leak analytic invariant information about a given set of suppressed cells.

cs.CR↗

arXiv2001

Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings

Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using {4/3}m-1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5+2\log{3})\min\{n,f\}-7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.

cs.DS↗

arXiv2001

Data Security Equals Graph Connectivity

To protect sensitive information in a cross tabulated table, it is a common practice to suppress some of the cells in the table. This paper investigates four levels of data security of a two-dimensional table concerning the effectiveness of this practice. These four levels of data security protect the information contained in, respectively, individual cells, individual rows and columns, several rows or columns as a whole, and a table as a whole. The paper presents efficient algorithms and NP-completeness results for testing and achieving these four levels of data security. All these complexity results are obtained by means of fundamental equivalences between the four levels of data security of a table and four types of connectivity of a graph constructed from that table.

cs.CR↗

arXiv2001

Optimal Constructions of Hybrid Algorithms

We study on-line strategies for solving problems with hybrid algorithms. There is a problem Q and w basic algorithms for solving Q. For some lambda <= w, we have a computer with lambda disjoint memory areas, each of which can be used to run a basic algorithm and store its intermediate results. In the worst case, only one basic algorithm can solve Q in finite time, and all the other basic algorithms run forever without solving Q. To solve Q with a hybrid algorithm constructed from the basic algorithms, we run a basic algorithm for some time, then switch to another, and continue this process until Q is solved. The goal is to solve Q in the least amount of time. Using competitive ratios to measure the efficiency of a hybrid algorithm, we construct an optimal deterministic hybrid algorithm and an efficient randomized hybrid algorithm. This resolves an open question on searching with multiple robots posed by Baeza-Yates, Culberson and Rawlins. We also prove that our randomized algorithm is optimal for lambda = 1, settling a conjecture of Kao, Reif and Tate.

cs.DM↗

arXiv2001

Tree Contractions and Evolutionary Trees

An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T_1 and T_2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log^2(n)) and is within a log(n) factor of optimal. For general d, this algorithm runs in O(n d^2 log(d) log^2(n)) time or alternatively in O(n d sqrt(d) log^3(n)) time.

cs.CE↗

arXiv2001

On-Line Difference Maximization

In this paper we examine problems motivated by on-line financial problems and stochastic games. In particular, we consider a sequence of entirely arbitrary distinct values arriving in random order, and must devise strategies for selecting low values followed by high values in such a way as to maximize the expected gain in rank from low values to high values. First, we consider a scenario in which only one low value and one high value may be selected. We give an optimal on-line algorithm for this scenario, and analyze it to show that, surprisingly, the expected gain is n-O(1), and so differs from the best possible off-line gain by only a constant additive term (which is, in fact, fairly small -- at most 15). In a second scenario, we allow multiple nonoverlapping low/high selections, where the total gain for our algorithm is the sum of the individual pair gains. We also give an optimal on-line algorithm for this problem, where the expected gain is n^2/8-Θ(n\log n). An analysis shows that the optimal expected off-line gain is n^2/6+Θ(1), so the performance of our on-line algorithm is within a factor of 3/4 of the best off-line strategy.

cs.DS↗

arXiv2001

A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs

We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property pi is called a pi-graph. If pi satisfies certain properties, then an n-node m-edge pi-graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most beta(n)+o(beta(n)) bits for any continuous super-additive function beta(n) so that there are at most 2^{beta(n)+o(beta(n))} distinct n-node pi-graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such pi include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from {1, ..., ell_1} for ell_1 <= n; (5) the edges of G are labeled with labels from {1, ..., ell_2} for ell_2 <= m; and (6) each node (respectively, edge) of G has at most ell_3 = O(1) self-loops (respectively, ell_4 = O(1) multiple edges). Moreover, ell_3 and ell_4 are not required to be O(1) for the cases of pi being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes.

cs.DS↗

arXiv2001

An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm.

cs.CV↗

arXiv2001

A Dynamic Programming Approach to De Novo Peptide Sequencing via Tandem Mass Spectrometry

The tandem mass spectrometry fragments a large number of molecules of the same peptide sequence into charged prefix and suffix subsequences, and then measures mass/charge ratios of these ions. The de novo peptide sequencing problem is to reconstruct the peptide sequence from a given tandem mass spectral data of k ions. By implicitly transforming the spectral data into an NC-spectrum graph G=(V,E) where |V|=2k+2, we can solve this problem in O(|V|+|E|) time and O(|V|) space using dynamic programming. Our approach can be further used to discover a modified amino acid in O(|V||E|) time and to analyze data with other types of noise in O(|V||E|) time. Our algorithms have been implemented and tested on actual experimental data.

cs.CE↗

arXiv2001

Combinatorial Toolbox for Protein Sequence Design and Landscape Analysis in the Grand Canonical Model

In modern biology, one of the most important research problems is to understand how protein sequences fold into their native 3D structures. To investigate this problem at a high level, one wishes to analyze the protein landscapes, i.e., the structures of the space of all protein sequences and their native 3D structures. Perhaps the most basic computational problem at this level is to take a target 3D structure as input and design a fittest protein sequence with respect to one or more fitness functions of the target 3D structure. We develop a toolbox of combinatorial techniques for protein landscape analysis in the Grand Canonical model of Sun, Brem, Chan, and Dill. The toolbox is based on linear programming, network flow, and a linear-size representation of all minimum cuts of a network. It not only substantially expands the network flow technique for protein sequence design in Kleinberg's seminal work but also is applicable to a considerably broader collection of computational problems than those considered by Kleinberg. We have used this toolbox to obtain a number of efficient algorithms and hardness results. We have further used the algorithms to analyze 3D structures drawn from the Protein Data Bank and have discovered some novel relationships between such native 3D structures and the Grand Canonical model.

cs.CE↗

arXiv2001

Multiple-Size Divide-and-Conquer Recurrences

This short note reports a master theorem on tight asymptotic solutions to divide-and-conquer recurrences with more than one recursive term: for example, T(n) = 1/4 T(n/16) + 1/3 T(3n/5) + 4 T(n/100) + 10 T(n/300) + n^2.

cs.GL↗

arXiv2000

Provably Fast and Accurate Recovery of Evolutionary Trees through Harmonic Greedy Triplets

We give a greedy learning algorithm for reconstructing an evolutionary tree based on a certain harmonic average on triplets of terminal taxa. After the pairwise distances between terminal taxa are estimated from sequence data, the algorithm runs in O(n^2) time using O(n) work space, where n is the number of terminal taxa. These time and space complexities are optimal in the sense that the size of an input distance matrix is n^2 and the size of an output tree is n. Moreover, in the Jukes-Cantor model of evolution, the algorithm recovers the correct tree topology with high probability using sample sequences of length polynomial in (1) n, (2) the logarithm of the error probability, and (3) the inverses of two small parameters.

cs.DS↗

arXiv2000

Optimal Bidding Algorithms Against Cheating in Multiple-Object Auctions

This paper studies some basic problems in a multiple-object auction model using methodologies from theoretical computer science. We are especially concerned with situations where an adversary bidder knows the bidding algorithms of all the other bidders. In the two-bidder case, we derive an optimal randomized bidding algorithm, by which the disadvantaged bidder can procure at least half of the auction objects despite the adversary's a priori knowledge of his algorithm. In the general $k$-bidder case, if the number of objects is a multiple of $k$, an optimal randomized bidding algorithm is found. If the $k-1$ disadvantaged bidders employ that same algorithm, each of them can obtain at least $1/k$ of the objects regardless of the bidding algorithm the adversary uses. These two algorithms are based on closed-form solutions to certain multivariate probability distributions. In situations where a closed-form solution cannot be obtained, we study a restricted class of bidding algorithms as an approximation to desired optimal algorithms.

cs.CE↗

arXiv2000

Reducing Randomness via Irrational Numbers

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.

cs.DS↗

arXiv2000

Optimal Buy-and-Hold Strategies for Financial Markets with Bounded Daily Returns

In the context of investment analysis, we formulate an abstract online computing problem called a planning game and develop general tools for solving such a game. We then use the tools to investigate a practical buy-and-hold trading problem faced by long-term investors in stocks. We obtain the unique optimal static online algorithm for the problem and determine its exact competitive ratio. We also compare this algorithm with the popular dollar averaging strategy using actual market data.

cs.CE↗

arXiv2000

Linear-Time Approximation Algorithms for Computing Numerical Summation with Provably Small Errors

Given a multiset $X=\{x_1,..., x_n\}$ of real numbers, the {\it floating-point set summation} problem asks for $S_n=x_1+...+x_n$. Let $E^*_n$ denote the minimum worst-case error over all possible orderings of evaluating $S_n$. We prove that if $X$ has both positive and negative numbers, it is NP-hard to compute $S_n$ with the worst-case error equal to $E^*_n$. We then give the first known polynomial-time approximation algorithm that has a provably small error for arbitrary $X$. Our algorithm incurs a worst-case error at most $2(\mix)E^*_n$.\footnote{All logarithms $\log$ in this paper are base 2.} After $X$ is sorted, it runs in O(n) time. For the case where $X$ is either all positive or all negative, we give another approximation algorithm with a worst-case error at most $\lceil\log\log n\rceil E^*_n$. Even for unsorted $X$, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most $\lceil\log n\rceil E^*_n$, while $E^*_n$ was known to be attainable in $O(n \log n)$ time using Huffman coding.

cs.DS↗