Search results for "A* algorithm"

On the number of prime divisors of the order of elliptic curves modulo p

2005

Almost Tight Bound for the Union of Fat Tetrahedra in Three Dimensions

2007

For any AND-OR formula of size N, there exists a bounded-error N1/2+o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or "approximately balanced," formulas can be evaluated in O(radicN) queries, which is optimal. It follows that the (2-o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.

Span programs for functions with constant-sized 1-certificates

2012

Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the element distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O(n13/10) of the best p…

Irredundant tandem motifs

2014

Eliminating the possible redundancy from a set of candidate motifs occurring in an input string is fundamental in many applications. The existing techniques proposed to extract irredundant motifs are not suitable when the motifs to search for are structured, i.e., they are made of two (or several) subwords that co-occur in a text string s of length n. The main effort of this work is studying and characterizing a compact class of tandem motifs, that is, pairs of substrings {m1, m2} occurring in tandem within a maximum distance of d symbols in s, where d is an integer constant given in input. To this aim, we first introduce the concept of maximality, related to four specific conditions that h…

Complexity of decision trees for boolean functions

2004

For every positive integer k we present an example of a Boolean function f/sub k/ of n = (/sub k//sup 2k/) + 2k variables, an optimal deterministic tree T/sub k/' for f/sub k/ of complexity 2k + 1 as well as a nondeterministic decision tree T/sub k/ computing f/sub k/. with complexity k + 2; thus of complexity about 1/2 of the optimal deterministic decision tree. Certain leaves of T/sub k/ are called priority leaves. For every input a /spl isin/ {0, 1}/sup n/ if any of the parallel computation reaches a priority leaves then its label is f/sub k/ (a). If the priority leaves are not reached at all then the label on any of the remaining leaves reached by the computation is f/sub k/. (a).

Quantum Queries on Permutations with a Promise

2009

This paper studies quantum query complexities for deciding (exactly or with probability 1.0) the parity of permutations of n numbers, 0 through n *** 1. Our results show quantum mechanism is quite strong for this non-Boolean problem as it is for several Boolean problems: (i) For n = 3, we need a single query in the quantum case whereas we obviously need two queries deterministically. (ii) For even n , n /2 quantum queries are sufficient whereas we need n *** 1 queries deterministically. (iii) Our third result is for the problem deciding whether the given permutation is the identical one. For this problem, we show that there is a nontrivial promise such that if we impose that promise to the …

Enlarging the gap between quantum and classical query complexity of multifunctions

2013

Quantum computing aims to use quantum mechanical effects for the efficient performance of computational tasks. A popular research direction is enlarging the gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design for multivalued functions that allow to achieve a large quantum versus classical complexity separation. To compute a basic finite multifunction in a quantum model only one query is enough while classically three queries are required. Then, we present two generalizations and a modification of the original algorithm, and obtain the following complexity gaps: Q UD (M′) ≤ N versus C UD (M′) ≥ 3N,…

On the Low-Dimensional Steiner Minimum Tree Problem in Hamming Metric

2011

It is known that the d-dimensional Steiner Minimum Tree Problem in Hamming metric is NP-complete if d is considered to be a part of the input. On the other hand, it was an open question whether the problem is also NP-complete in fixed dimensions. In this paper we answer this question by showing that the problem is NP-complete for any dimension strictly greater than 2. We also show that the Steiner ratio is 2 - 2/d for d ≥ 2. Using this result, we tailor the analysis of the so-called k-LCA approximation algorithm and show improved approximation guarantees for the special cases d = 3 and d = 4.

The node-depth encoding

2008

The node-depth encoding has elements from direct and indirect encoding for trees which encodes trees by storing the depth of nodes in a list. Node-depth encoding applies specific search operators that is a typical characteristic for direct encodings. An investigation into the bias of the initialization process and the mutation operators of the node-depth encoding shows that the initialization process has a bias to solutions with small depths and diameters, and a bias towards stars. This investigation, also, shows that the mutation operators are unbiased. The performance of node-depth encoding is investigated for the bounded-diameter minimum spanning tree problem. The results are presented f…

A Star-Variety With Almost Polynomial Growth

2000

Abstract Let F be a field of characteristic zero. In this paper we construct a finite dimensional F -algebra with involution M and we study its ∗ -polynomial identities; on one hand we determine a generator of the corresponding T -ideal of the free algebra with involution and on the other we give a complete description of the multilinear ∗ -identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the ∗ -variety generated by M , var( M , ∗ ) has almost polynomial growth, i.e., the sequence of ∗ -codimensions of M cannot be bounded by any polynomial function but any proper ∗ -subvariety of var( M , ∗ ) has polynomial growth. If G 2 is…