This paper concerns lower bounds on the straight line complexity of multi-variate polynomials. We obtain a conditional result showing that certain explicit linear operators must greatly increase the complexity of some polynomials. We do so by showing that if these operators roughly preserve the complexity of all polynomials then, co-NP is in AM. We show that certain explicit operators must vastly increase the straight line complexity of certain polynomials.
Automata on infinite words are used for specification and verification of nonterminating programs. Different types of automata induce different levels of expressive power, of succinctness, and of complexity. Alternating automata have both existential and universal branching modes and are particularly suitable for specification of programs. In a weak alternating automaton, the state space is partitioned into partially ordered sets, and the automaton can proceed from a certain set only to smaller sets. Reasoning about weak alternating automata is easier than reasoning about alternating automata with no restricted structure. Known translations of alternating automata to weak alternating automata involve determinization, and therefore involve a double-exponential blow-up. In this paper we describe a quadratic translation, which circumvents the need for determinization, of Buchi (1962) and co-Buchi alternating automata to weak alternating automata. Beyond the independent interest of such a translation, it gives rise to a simple complementation algorithm for nondeterministic Buchi automata
A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. We apply it to modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of (Ma and Hsu, 1991) for modular decomposition of chordal graphs and (Habib et al., 1995) for inheritance graph decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.
The number of molecular strands used by a molecular algorithm is an important measure of the algorithm's complexity. This measure is also called the *space* used by the algorithm. We prove that three important polynomial-time models of molecular computation with bounded space are equivalent to models of polynomial-time Turing machine computation with bounded nondeterminism. Without any assumption, we show that the Split operation does not increase the power of polynomial-time molecular computation. Assuming a plausible separation between Turing machine complexity classes, the Amplify operation does increase the power of polynomial-time molecular computation.
We consider the broadcasting operation in point-to-point packet-switched parallel and distributed networks of processors. We develop a general technique for the design of optimal broadcast algorithms on a wide range of such systems. Our technique makes it easier to design such algorithms and, furthermore, provides tools that can be used to derive precise analyses of their running times. As direct applications of this method we give an exact analysis of a known algorithm for the POSTAL model, and design and analyze an optimal broadcast algorithm for the MULTI PORT MULTI-MEDIA model. We then show how our method can be applied to networks with different underlying topologies, by designing and giving an exact analysis of an optimal broadcast algorithm for the OPTICAL RING.
We study the subtree isomorphism problem: Given trees H and G, find a subtree of G which is isomorphic to H or decide that there is no such subtree. We give an O(~k/sup 1.8//log k\ n) time algorithm for this problem, where k and n are the number of vertices in H and G respectively. This improves over the O(k/sup 1.5/n) algorithms of Chung (1987) and Matula (1978). We also give a randomized (Las Vegas) O(min(k/sup 1.45/n, kn/sup 1.43/))-time algorithm for the decision problem.
Kearns introduced the "statistical query" (SQ) model as a general method for producing learning algorithms which are robust against classification noise. We extend this approach in several ways, in order to tackle algorithms that use "membership queries": focusing on the more stringent model of "persistent noise". The main ingredients in the general analysis are: (1) Smallness of dimension of both the targets' class and the queries' class. (2) Independence of the noise variables. Persistence restricts independence forcing repeated invocation of the same point x to give the same label. We apply the general analysis and ad-hoc considerations to get noise-robust version of Jackson's Harmonic Sieve (1995), which learns DNF under the uniform distribution. This corrects an error in his earlier analysis of noise tolerant DNF learning.
We consider polynomial-time Turing machines that have access to two oracles and investigate when the order of oracle queries is significant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries does not matter. This improves upon the previous result of Hemaspaandra, Hemaspaandra and Hempel, who showed that the order of the queries does not matter if the base machine asks only one query to each oracle. On the other hand, we prove that, for computing functions, the order of oracle queries does matter unless PH collapses.
A local stabilizer protocol that takes any on-line or of-line distributed algorithm and converts it into a synchronous self-stabilizing algorithm with local monitoring and repairing properties is presented. Whenever the self-stabilizing version enters an inconsistent state, the inconsistency is detected, in O(1) time, and the system state is repaired in a local manner. The expected computation time that is lost during the repair process is proportional to the largest diameter of a faulty region.
We study the Private Simultaneous Messages (PSM)} model which is a variant of the model proposed by Feige, Kilian and Naor (STOC '94). In the PSM model there are n players P_1,\ldots,P_n, each player P_i holding a secret input x_i (say, a bit), and all having access to a common random string. Each player sends a single message to a special player, Carol, depending on its own input and the random string (and independently of all other messages). Based on these messages, Carol should be able to compute f(x_1,\ldots,x_n) (for some predetermined function f) but should learn no additional information on the values of x_1,\ldots,x_n.Our results go in two directions. First, we present efficient PSM protocols, which improve the efficiency of previous solutions, and extend their scope to several function classes for which no such solutions were known before. These classes include most of the important linear algebraic functions; as a result, we get efficient constant-round private protocols (in the standard model) for these classes. Second, we present reductions that allow transforming PSM protocols into solutions for some other problems, thereby demonstrating the power of this model. An interesting reduction of this sort shows how to construct, based on a standard \OT (Oblivious Transfer) primitive, generalized-OT (GOT) primitives that, we believe, might be useful for the design of cryptographic protocols.
We propose a new model for the analysis of data transmission protocols in lossy communication networks. The overall goal of a data transmission protocol is to successfully transmit a message from the sender to the receiver. We study the performance of protocols in an adversarial setting where the loss pattern and latencies of packets are determined by an adversary.We advocate the modular decomposition of data transmission protocols into a time scheduling policy, which determines when packets are to be sent, and a data selection policy, which determines what data is to be placed in each sent packet. We concentrate on the data selection policy and require that the protocol will achieve high bandwidth utilization in transmitting any prefix of the transmitted message. The simple and universal data selection policy we introduce is provably close to optimal in the following sense: For any time scheduling policy and any network behavior, in the worst case prefix measure our data selection policy performs as well as any other data selection policy up to a constant additive term.Our explicit modular decomposition of a transmission protocol into two policies should be contrasted with existing network protocols such as TCP/IP. Our result shows that the performance of the overall transmission protocol would not degrade in performance (and could improve dramatically) if it used our universal data selection policy in place of its own. We therefore reduce the problem of designing a data transmission protocol to the task of designing a time scheduling policy.
We study connection networks in which certain pairs of nodes have to be connected by k edge-disjoint paths, and study bounds for the minimal sum of lengths of such k paths. We define the related notions of total/sub k/-distance for a pair of nodes and total/sub k/-diameter of a connection network, and study the value TD/sub k/(d) which is the maximal such total/sub k/-diameter of a network with diameter d. These notions have applications in fault-tolerant routing problems, in ATM networks, and in compact routing in networks. We prove an upper bound on TD/sub k/(d) and a lower bound on the growth of TD/sub k/(d) as functions of k and d; those bounds are tight, /spl theta/(d/sup k/), when k is fired. Specifically, we prove that TD/sub k/(d)/spl les/2/sup k-1/d/sup k/, with the exceptions TD/sub 2/(1)=3, TD/sub 3/(1)=5, and that for every k, d/sub 0/0, there exists (a) an integer d/spl ges/d/sub 0/ such that TD/sub k/(d)/spl ges/d/sup k/k/sup k/; and (b) a k-connected simple graph G with diameter d such that d/spl ges/d/sub 0/, and td/sub k/(G)/spl ges/(d-2)/sup k//k/sup k/.
In this paper we present structural and algorithmic results for problems involving the packing of T-joins. We explore minimax relations that relate the size of a packing of T-joins with the size of a minimum T-cut in a graph. We present a new conjecture stating that if all T-cuts have the same parity then the maximum size of a family of T-joins that uses each edge at most twice equals the double of the size of a minimum T-cut. We show that this conjecture is equivalent to a famous conjecture for perfect matchings. We also prove a theorem for the case |T|/spl les/8 and describe a polynomial time algorithm for the maximization problem.
This paper studies four combinatorial search models of reconstructing a fixed unknown Hamiltonian cycle in the complete graph by means of queries about subgraphs. For each model, an efficient algorithm is proposed that matches asymptotically the information-theoretic lower bound. The problem is motivated by an application to genome physical mapping.
A stereoscopic family of permutations maps an m-dimensional mesh into several one-dimensional lines, in a way that jointly preserves distance information. Specifically, consider any two points and denote their distance on the m-dimensional mesh by d. Then the distance between their images, on the line on which these images are closest together, is O(d^m).We initiate a systematic study of stereoscopic families of permutations. We show a construction of these families that involves the use of m+1 images. We also show that under some additional restrictions (namely, adjacent points on the image lines originate at points which are not too far away on the mesh), three images are necessary in order to construct such a family for the two-dimensional mesh.We present two applications for stereoscopic families of permutations. One application is an algorithm for routing on the mesh that guarantees delivery of each packet within a number of steps that depends upon the distance between this packet's source and destination, but is independent of the size of the mesh. Our algorithm is exceptionally simple, involves no queues, and can be used in dynamic settings in which packets are continuously generated. Another application is an extension of the construction of non-expansive hash functions of Linial and Sasson (STOC 96) from the case of one dimensional metrics to arbitrary dimensions.
MAX SAT (the maximum satisfiability problem) is stated as follows: given a set of clauses with weights, find a truth assignment that maximizes the sum of the weights of the satisfied clauses. In this paper, we consider approximation algorithms for MAX SAT proposed by Yannnkakis and Goemans-Williamson and present an approximation algorithm which is an improvement of Yannakakis' algorithm. Although Yannakakis' original algorithm has no better performance guarantee than Goemans-Williamson, our improved algorithm has a better performance guarantee and leads to a 0.770 approximation algorithm.
We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upper-bounded by a polynomial in the worst case. We show that a natural generalization of Simon's problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worst-case running time. This is achieved by generalizing both Simon's and Grover's algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box.
We consider the k-payment problem: given a total budget of N units, the problem is to represent this budget as a set of coins, so that any k exact payments of total value at most N can be made using k disjoint subsets of the coins. The goal is to minimize the number of coins for any given N and k, while allowing the actual payments to be made on-line, namely without the need to know all payment requests in advance. The problem is motivated by the electronic cash model, where each coin is a long bit sequence, and typical electronic wallets have only limited storage capacity. The k-payment problem has additional applications in other resource-sharing scenarios. Our results include a complete characterization of the k-payment problem as follows. First, we prove a necessary and sufficient condition for a given set of coins to solve the problem. Using this characterization, we prove that the number of coins in any solution to the k-payment problem is at least k H_{N/k}, where H_n denotes the nth element in the harmonic series. This condition can also be used to efficiently determine k (the maximal number of exact payments) which a given set of coins allows in the worst case. Secondly, we give an algorithm which produces, for any N and k, a solution with minimal number of coins. In the case that all denominations are available, the algorithm finds a coin allocation with at most (k+1)H_{N/(k+1)} coins. (Both upper and lower bounds are the best possible.) Finally, we show how to generalize the algorithm to the case where some of the denominations are not available.
We classify the relativizable closure properties of GapP, solving a problem posed by (Hertrampf et al., 1995). We also give a simple proof of their classification of functions that are relativizable closure properties of #P.
The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter.
