Widest path problem

About: Widest path problem is a research topic. Over the lifetime, 1045 publications have been published within this topic receiving 49159 citations.

A note on two problems in connexion with graphs

Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to

Algorithm 97: Shortest path

Abstract: procedure ari thmetic (a, b, c, op); in t eger a, b, c, op; ¢ o n l m e n t This procedure will perform different order ar i thmetic operations with b and c, put t ing the result in a. The order of the operation is given by op. For op = 1 addit ion is performed. For op = 2 multiplicaLion, repeated addition, is done. Beyond these the operations are non-commutat ive. For op = 3 exponentiat ion, repeated multiplication, is done, raising b to the power c. Beyond these the question of grouping is important . The innermost implied parentheses are at the right. The hyper-exponent is always c. For op = 4 te t ra t ion, repeated exponentiat ion, is done. For op = 5, 6, 7, etc., the procedure performs pentat ion, hexation, heptat ion, etc., respectively. The routine was originally programmed in FORTRAN for the Control Data 160 desk-size computer. The original program was limited to te t ra t ion because subroutine recursiveness in Control Data 160 FORTRAN has been held down to four levels in the interests of economy. The input parameter , b, c, and op, must be positive integers, not zero; b e g i n own i n t e g e r d, e, f, drop; i f o p = 1 t h e n b e g i n a := h-4c; go t o l e n d i f o p = 2 t h e n d := 0; else d := 1; e := c; drop := op 1; for f := I s t e p 1 u n t i l e do b e g i n ari thmetic (a, b, d, drop);

A shortest augmenting path algorithm for dense and sparse linear assignment problems

Abstract: We develop a shortest augmenting path algorithm for the linear assignment problem. It contains new initialization routines and a special implementation of Dijkstra's shortest path method. For both dense and sparse problems computational experiments show this algorithm to be uniformly faster than the best algorithms from the literature. A Pascal implementation is presented.

The shortest path through many points

Abstract: We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

Computing the shortest path: A search meets graph theory

Abstract: We propose shortest path algorithms that use A* search in combination with a new graph-theoretic lower-bounding technique based on landmarks and the triangle inequality. Our algorithms compute optimal shortest paths and work on any directed graph. We give experimental results showing that the most efficient of our new algorithms outperforms previous algorithms, in particular A* search with Euclidean bounds, by a wide margin on road networks and on some synthetic problem families.