Norbert Blum

Computing a maximum cardinality matching in a bipartite graph in time O n 1.5 m/ log n

Abstract We show how to compute a maximum cardinality matching in a bipartite graph of n vertices in time O(n1.5 m log n ). For dense graphs this improves on the O( n m) algorithm of Hopcroft and Karp. The speed-up is obtained by an application of the fast adjacency matrix scanning technique of Cheriyan, Hagerup and Mehlhorn.

A Boolean function requiring 3n network size

Abstract Paul (1977) has proved a 2.5 n -lower bound for the network complexity of an explicit Boolean function. We modify the definition of Pauls function slightly and prove a 3 n -lower bound for the network complexity of that function.

On the Average Number of Rebalancing Operations in Weight-Balanced Trees

Abstract It is shown that the average number of rebalancing operations (rotations and double rotations) in weight-balanced trees is constant.

A new approach to maximum matching in general graphs

We reduce the problem of finding an augmenting path in a general graph to a reachability problem and show that a slight modification of depth-first search leads to an algorithm for finding such paths. As a consequence, we obtain a straightforward algorithm for maximum matching in general graphs of time complexity O(√nm), where n is the number of nodes and m is the number of edges in the graph.

Griebach normal form transformation revisited

We develop a new method for placing a given context-free grammar into Greibach normal form with only polynomial increase of its size. Starting with an arbitrarye-free context-free grammarG, we transformGinto an equivalent context-free grammarHin extended Greibach normal form; i.e., in addition to rules, fulfilling the Greibach normal form properties, the grammar can have chain rules. The size ofHwill beO(|G|3), where |G| is the size ofG. Moreover, in the case thatGis chain rule free,Hwill be already in Greibach normal form. IfHis not chain rule free then we use the standard method for chain rule elimination for the transformation ofHinto Greibach normal form. The size of the constructed grammar isO(|G|4).

Circular convex bipartite graphs: maximum matching and Hamiltonian circuits

Abstract The maximum matching and its related problems in convex bipartite graphs were studied by Glover (1967) and Lipski and Preparata (1981). This paper introduces circular convex bipartite graphs and considers the maximum matching and Hamiltonian circuit problems in these graphs. A bipartite graph G = ( A , B , E ) is circular convex on the vertex set A if A can be ordered on a circle so that for each element b in the vertex set B the elements of A connected to b form a circular arc of A . We present linear time algorithms for finding a maximum matching and a Hamiltonian circuit in circular convex bipartite graphs.

An O( n log n ) implementation of the standard method for minimizing n -state finite automata

Abstract More than 20 years ago, Hopcroft (1971) has given an algorithm for minimizing an n-state finite automaton in O(kn log n) time where k is the size of the alphabet. This contrasts to the usual O(kn2) algorithms presented in text books. Since Hopcrofts algorithm changes the standard method, a nontrivial correctness proof for its method is needed. In lectures given at university, mostly the standard method and its straightforward O(Kn2) implementation is presented. We show that a slight modification of the O(kn2) implementation combined with the use of a simple data structure composed of chained lists and four arrays of pointers (essentially the same as Hopcrofts data structure) leads to an O(kn log n) implementation of the standard method. Thus, it is possible to present in lectures, with a little additional amount of time, an O(kn log n) time algorithm for minimizing n-state finite automata.

An area-maximum edge length trade-off for VSLI layout

We construct an N-node graph G which has (i) a layout with area O(N) and maximum edge length O(N1/2), (ii) a layout with area O(N5/4) and maximum edge length O(N1/4). We prove for 1 ≤ f(N) ≤ (O(N1/8) that any layout for G with area Nf(N) has an edge of length Ω(N1/2/f(N)·log N). Hence G has no layout which is optimal with respect to both measures.

On Negations in Boolean Networks

Although it is well known by a counting argument that relative to the full basis most Boolean functions need exponentially many operations, for explicit Boolean functions only linear lower bounds with small constant factors are known. For monotone networks (i.e., networks without negations) exponential lower bounds for explicit monotone Boolean functions have been proved. We describe the state of the art and give some arguments why techniques developed for the proof of lower bounds for monotone networks cannot easily be extended to Boolean networks with negations.

On the single-operation worst-case time complexity of the disjoint set union problem

We give an algorithm for the disjoint set union problem, within the class of algorithms defined by Tarjan, which has O(log n/loglog n) single-operation time complexity in the worst-case. Also we define a class of algorithms for the disjoint set union problem, which includes the class of algorithms defined by Tarjan. We prove that any algorithm from this class has at least ω(log n/loglog n) single-operation time complexity in the worst-case.