Showing papers by "Ming-Yang Kao published in 1997"

Reducing randomness via irrational numbers

Abstract: We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently com- putable 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 algo- rithms 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.

Algorithms for Informed Cows

Abstract: We extend the classic on-line search problem known as the cow-path problem to the case in which goal locations are selected according to one of a set of possible known probability distributions. We present a polynomial-time linear-programming algorithm for this problem.

All-Cavity Maximum Matchings

Abstract: Let G = (X, Y, E) be a bipartite graph with integer weights on the edges Let n, m, and N denote the vertex count, the edge count, and an upper bound on the absolute values of edge weights of G, respectively For a vertex u in G, let Gu denote the graph formed by deleting u from G The all-cavity maximum matching problem asks for a maximum weight matching in Gu for all u in G This problem finds applications in optimal tree algorithms for computational biology We show that the problem is solvable in O(√nmlog(nN)) time, matching the currently best time complexity for merely computing a single maximum weight matching in G We also give an algorithm for a generalization of the problem where both a vertex from X and one from Y can be deleted The running time is O(n21og n + nm) Our algorithms are based on novel linear-time reductions among problems of computing shortest paths and all-cavity maximum matchings

General techniques for comparing unrooted evolutionary trees

Abstract: This paper presents two sets of techniques for comparing unrooted evolutionary trees, namely, label compression and four-way dvnamic programming. The technique of four-way dynamic programming transforms existing algorithms for computing rooted maximum agree ment subtrees into new ones for unrooted trees. Let n be the size of the two input trees. This technique leads to an O(n log n)-time algorithm for unrooted trees whose degrees are bounded by a constant, matching the best known complexity for the rooted binary case. The technique of label compression is not based on dynamic programming. With this technique, we obtain an O(nl”5 log n)-time algorithm for unrooted trees with arbitrary degrees, also matching the best algorithm for the rooted unbounded degree case.

Total Protection of Analytic-Invariant Information in Cross-Tabulated Tables

Abstract: 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.

Efficient Detection and Protection of Information in Cross Tabulated Tables II: Minimal Linear Invariants

Abstract: To protect sensitive information in a cross tabulated table, it is acommon practice to suppress some of the cells. A linear combination of thesuppressed cells is called a linear invariant if it has a unique feasible value.Intuitively, the information contained in a linear invariant is not protectedbecause its value can be uniquely determined. Using a decomposition approach,this paper establishes a fundamental correspondence between linear invariantsof a table and edge cuts of a graph induced from the table. Thiscorrespondence is employed to give a linear-time algorithm for finding animportant class of linear invariants called therow or column linear invariants. In subsequent papers, thiscorrespondence is used to solve via graph theoretic techniques a wide varietyof problems for protecting information in a table.

Security problems for statistical databases with general cell suppressions

Abstract: Studies statistical database problems for 2D tables whose regular cells, row sums, column sums and table sums may be suppressed. Using graph-theoretical techniques, we give optimal or efficient algorithms for the query system problem, the adversary problem and the minimum complementary suppression problem. These three problems are considered for a variety of data security requirements such as those of protecting linear invariants, analytic invariants, k rows (or columns) as a whole, and a table as a whole.

Multiple-size divide-and-conquer recurrences

Abstract: 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.

Tree Contractions and Evolutionary Trees

Abstract: 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 T1 and T2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log2n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd2 log d log2n) time or alternatively in O(nd√d log3n) time.

Optimal Bidding Algorithms Against Cheating in Multiple-Object Auctions

Abstract: 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.

On-line difference maximization

Abstract: In this paper we examine problems motivated by on-line nancial 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 niO(1), and so diers from the best possible o-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 =8i (n logn) An analysis shows that the optimal expected o-line gain is n 2 =6 + (1), so the performance of our on-line algorithm is within a factor of 3=4 of the best o-line strategy

Optimal bidding algorithms against cheating in multiple-object auctions

Abstract: 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.