Piyush P. Kurur

Graph isomorphism is in SPP

We show that graph isomorphism is in the complexity class SPP and hence it is in /spl oplus/P (in fact, it is in Mod/sub k/P for each k/spl ges/2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FP/sup SPP/ algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism (e.g. coset intersection) are in SPP, and thus in Mod/sub k/P for each k>2.

Fast integer multiplication using modular arithmetic

We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an O(N • log N • 2O(log*N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürers algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürers algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.

Computing single source shortest paths using single-objective fitness

Runtime analysis of evolutionary algorithms has become an important part in the theoretical analysis of randomized search heuristics. The first combinatorial problem where rigorous runtime results have been achieved is the well-known single source shortest path (SSSP) problem. Scharnow, Tinnefeld and Wegener [PPSN 2002, J. Math. Model. Alg. 2004] proposed a multi-objective approach which solves the problem in expected polynomial time. They also suggest a related single-objective fitness function. However, it was left open whether this does solve the problem efficiently, and, in a broader context, whether multi-objective fitness functions for problems like the SSSP yield more efficient evolutionary algorithms. In this paper, we show that the single objective approach yields an efficient (1+1) EA with runtime bounds very close to those of the multi-objective approach.

Graph Isomorphism is in SPP

We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, in ModkP for each k ≥ 2). These inclusions for Graph Isomorphism were not known prior to membership in SPP. We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FPSPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FPSPP algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as Graph Isomorphism (e.g., coset intersection) are in SPP, and thus in ModkP for each k ≥ 2.

Fast Integer Multiplication Using Modular Arithmetic

We give an

Bounded color multiplicity graph isomorphism is in the #L hierarchy

N\cdot \log N\cdot 2^{O(\log^*N)}

Can quantum search accelerate evolutionary algorithms

time algorithm to multiply two

Metropolis algorithm for solving shortest lattice vector problem (SVP)

N

On the Complexity of Computing Units in a Number Field

-bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Furers algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Schonhage and Strassen) has complexity

Upper Bounds on the Complexity of Some Galois Theory Problems

O(N \cdot \log N \cdot \log\log N)