Showing papers on "Prim's algorithm published in 1980"

An 0 (nm log n) algorithm for maximum network flow

Abstract: This thesis presents a new algorithm for the maximum network flow problem. The problem is this: Given a directed network of m edges and n vertices with two distinguished vertices, a source and a sink, and with a nonnegative capacity associated with each edge, find a way of labeling the edges with nonnegative real numbers representing flow in such a way that: (1) The flow into each vertex equals the flow out (except at the source and sink). (2) The flow at each edge does not exceed the capacity at that edge. (3) The flow out of the source is a maximum over all flows satisfying (1) and (2). Our algorithm finds a maximum flow in O(nmlog n) time, which is a factor of log n faster than the previous fastest algorithm. The central innovation of our algorithm is a sophisticated data structure that is used to represent a certain subset of edges that form a forest. The edges of this forest are further partitioned into two classes: broken and solid. Long paths of solid edges are grouped together and stored in a balanced tree data structure called a biased 2-3 tree. Each leaf of the biased 2-3 tree corresponds to an edge in the path. The most significant property of this data structure is that it allows a variety of operations to be performed in O(log n) time. Some of these are: The traversal from any leaf of the tree to the root while accumulating information about the edges along the path (e.g. finding the minimum capacity edge); the updating of information stored at each edge (e.g. changing the current value of the flow); certain types of tree rearrangements (e.g. removing a subtree and attaching it somewhere else). It appears that this data structure can be used to get fast algorithms for other problems such as the transshipment problem, the deepest common ancestor problem where a cut operation is allowed, and the two color minimum spanning tree problem.