arXiv subjects

Michael Sipser

Publications and source records attributed to Michael Sipser.

3 recordsLinked to original sources

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↗

arXiv2000

Quantum Computation by Adiabatic Evolution

We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.

quant-ph↗

arXiv1999

Invariant Quantum Algorithms for Insertion into an Ordered List

We consider the problem of inserting one item into a list of N-1 ordered items. We previously showed that no quantum algorithm could solve this problem in fewer than log N/(2 log log N) queries, for N large. We transform the problem into a "translationally invariant" problem and restrict attention to invariant algorithms. We construct the "greedy" invariant algorithm and show numerically that it outperforms the best classical algorithm for various N. We also find invariant algorithms that succeed exactly in fewer queries than is classically possible, and iterating one of them shows that the insertion problem can be solved in fewer than 0.53 log N quantum queries for large N (where log N is the classical lower bound). We don't know whether a o(log N) algorithm exists.

quant-ph↗