Home  About us  Contact
 

Space Complexity (space + complexity)

Distribution by Scientific Domains
Mathematics and Statistics75%

Selected Abstracts

Space complexity of random formulae in resolution

RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2003
Eli Ben-Sasson
We study the space complexity of refuting unsatisfiable random k -CNFs in the Resolution proof system. We prove that for , , 1 and any , > 0, with high probability a random k -CNF over n variables and ,n clauses requires resolution clause space of ,(n/,1+,). For constant ,, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density , , n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with ,n clauses requires treelike refutation size of exp(,(n/,1+,)), for any , > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92,109, 2003 [source]

An improved direct labeling method for the max,flow min,cut computation in large hypergraphs and applications

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 1 2003
Joachim Pistorius
Algorithms described so far to solve the maximum flow problem on hypergraphs first necessitate the transformation of these hypergraphs into directed graphs. The resulting maximum flow problem is then solved by standard algorithms. This paper describes a new method that solves the maximum flow problem directly on hypergraphs, leading to both reduced run time and lower memory requirements. We compare our approach with a state,of,the,art algorithm that uses a transformation of the hypergraph into a directed graph and an augmenting path algorithm to compute the maximum flow on this directed graph: the run,time complexity as well as the memory space complexity are reduced by a constant factor. Experimental results on large hypergraphs from VLSI applications show that the run time is reduced, on average, by a factor approximately 2, while memory occupation is reduced, on average, by a factor of 10. This improvement is particularly interesting for very large instances, to be solved in practical applications. [source]

Near-shortest and K-shortest simple paths

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005
W. Matthew Carlyle
Abstract We present a new algorithm for enumerating all near-shortest simple (loopless) s - t paths in a graph G = (V, E) with nonnegative edge lengths. Letting n = |V| and m = |E|, the time per path enumerated is O(nS(n, m)) given a user-selected shortest-path subroutine with complexity O(S(n, m)). When coupled with binary search, this algorithm solves the corresponding K -shortest paths problem (KSPR) in O(KnS(n, m)(log n+ log cmax)) time, where cmax is the largest edge length. This time complexity is inferior to some other algorithms, but the space complexity is the best available at O(m). Both algorithms are easy to describe, to implement and to extend to more general classes of graphs. In computational tests on grid and road networks, our best polynomial-time algorithm for KSPR appears to be at least an order of magnitude faster than the best algorithm from the literature. However, we devise a simpler algorithm, with exponential worst-case complexity, that is several orders of magnitude faster yet on those test problems. A minor variant on this algorithm also solves "KSPU," which is analogous to KSPR but with loops allowed. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 98,109 2005 [source]

Space complexity of random formulae in resolution

RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2003
Eli Ben-Sasson
We study the space complexity of refuting unsatisfiable random k -CNFs in the Resolution proof system. We prove that for , , 1 and any , > 0, with high probability a random k -CNF over n variables and ,n clauses requires resolution clause space of ,(n/,1+,). For constant ,, this gives us linear, optimal, lower bounds on the clause space. One consequence of this lower bound is the first lower bound for size of treelike resolution refutations of random 3-CNFs with clause density , , n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with ,n clauses requires treelike refutation size of exp(,(n/,1+,)), for any , > 0. Our space lower bound is the consequence of three main contributions: (1) We introduce a 2-player Matching Game on bipartite graphs G to prove that there are no perfect matchings in G. (2) We reduce lower bounds for the clause space of a formula F in Resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. (3) We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the Pigeonhole principle based on bipartite graphs G. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 92,109, 2003 [source]