Showing papers by "Robert E. Tarjan published in 1972"

Depth-First Search and Linear Graph Algorithms

Abstract: The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples. An improved version of an algorithm for finding the strongly connected componen...

Enumeration of the Elementary Circuits of a Directed Graph

Abstract: An algorithm to enumerate all the elementary circuits of a directed graph is presented. The algorithm uses back-tracking with lookahead to avoid unnecessary work, and it has a time bound of $O ((V+E)(C+1))$ when applied to a graph with $V$ vertices, $E$ edges, and $C$ elementary circuits. Keywords: Algorithm, circuit, cycle, graph

Sorting Using Networks of Queues and Stacks

Abstract: Inspired by Knuth [2, p. 234], we wish to consider the following problem: Suppose we are presented with the layout of a railroad switchyard (Figure 1 ). I f a train is driven into one end of the yard, what rearrangements of the cars may be made before the train comes out the other end? In order to get a handle on the problem, we must introduce some formalization. A switchyard is an acyclic directed graph, with a unique source and a unique sink (Figure 2). Each vertex represents a siding. The vertex/siding is assumed to have indefinite storage space and may be a stack, a queue, or a deque of some sort (see Knuth [2, p. 234]). A stack is a siding which has the property tha t the last element inserted is the first to be removed. A queue has the proper ty tha t the first element inserted is the first to be removed. In the switchyard, the sidings associated with the source and sink are assumed to be queues. Suppose a finite sequence of numbers s = (sl, s2, • • • , sn) is placed in the source queue of a switchyard (Figure 3). We may rearrange s by moving the elements of s through the switchyard. At each step, an element is moved from some siding to another siding along an arc of the switchyard. After a suitable number of such moves, all elements will be in the sink queue. I f they are in order, smallest to largest, we have sorted the sequence s using the switchyard. We wish to analyze the sequences s which may be sorted in a switchyard Y. We lose nothing in our formalism by allowing storage only on the vertices, and not on the arcs of the switchyard. We ignore questions concerning the finite size of the sidings; assuming small sidings complicates the problem considerably. A circuit in the switchyard will allow us to sort any sequence; thus we do not allow circuits. Having established our model, we proceed to discover its properties.

Isomorphism of Planar Graphs (Working Paper)

Abstract: An algorithm is presented for determining whether or not two planar graphs are isomorphic. The algorithm requires O(V log V) time, if V is the number of vertices in each graph.

Linear time bounds for median computations

Abstract: New upper and lower bounds are presented for the maximum number of comparisons, f(i,n), required to select the i-th largest of n numbers. An upper bound is found, by an analysis of a new selection algorithm, to be a linear function of n: f(i,n) ≤ 103n/18 A lower bound is shown deductively to be: f(i,n) ≥ n+min(i,n−i+l) + [log2(n)] − 4, for 2 ≤ i ≤ n−1, or, for the case of computing medians: f([n/2],n) ≥ 3n/2 − 3

Finding a Maximum Clique.

Abstract: : An algorithm for finding a maximum clique in an arbitrary graph is described. The algorithm has a worst-case time bound of k(1.286) sup N for some constant k, where n is the number of vertices in the graph. Within a fixed time, the algorithm can analyze a graph with 2 3/4 as many vertices as the largest graph which the obvious algorithm (examining all subsets of vertices) can analyze. (Author)

Finding the Triconnected Components of a Graph

Abstract: : An algorithm for decomposing a graph into triconnected components is presented. The algorithm requires 0(V + E) time and space when implemented on a random access computer, where V is the number of vertices and E is the number of edges in the graph. The algorithm is both theoretically optimal (to within a constant factor) and efficient in practice.