We address the question of theoretical vs. practical behavior of algorithms for the minimum spanning tree problem. We review the factors that influence the actual running time of an algorithm, from choice of language, machine, and compiler, through low-level implementation choices, to purely algorithmic issues. We discuss how to design a careful experimental comparison between various alternatives. Finally, we present the results from a study in which we used: multiple languages, compilers, and machines; all the major variants of the comparison-based algorithms; and eight varieties of graphs in five families, with sizes of up to 0.5 million vertices (in sparse graphs) or 1.3 million edges (in dense graphs).

Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine. Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T ∗ are T ∗ (m, n) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edge-exposure martingale for Gn,p.

We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum span-ning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε−2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dε−2 log d ε) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε−2 log 1 ε) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connected-components algorithm picks O(1/ε2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. 1

The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a linear-time algorithm that obtains an orderly pair (H

We present a simple shortest path algorithm. If the input lengths are positive and uniformly distributed, the algorithm runs in linear time. The worst-case running time of the algorithm is O(m + n log C), where n and m are the number of vertices and arcs of the input graph, respectively, and C is the ratio of the largest and the smallest nonzero arc length.

The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1 Introduction 1.1 The 'Were-you-last?' game A Dream Team, consisting of m players, is held captive in the dungeon of their adversary, Hannibal. He now makes them play his favourite game, Were-you-last?. Before the game starts the players of the Team are allowed to meet to discuss a strategy (obviously, Hannibal has the room bugged and is listening in). After the discussion they are led to separate waiting rooms. Then Hannibal leads each of the players of the team, one by one, to the playing field. The players do not know the order in which they are led to the field and they spend their time there alone. The playing field is a room, containing an infinite number of boxes, labelled 0, 1, 2, 3, . . . . Inside each box is a switch that can be ...

We present two new data structure tools---disjoint set union with bottom-up linking, and pointer-based radix sort---and combine them with bottom-level microtrees to devise the first linear-time pointer-machine algorithms for off-line least common ancestors, minimum spanning tree (MST) verification, randomized MST construction, and computing dominators in a flowgraph.

We develop succinct data structures to represent (i) a sequence of values to support partial sum and select queries and update (changing values) and (ii) a dynamic array consisting of a sequence of elements which supports insertion, deletion and access of an element at any given index. For the partial sums problem...

Given a set of obstacles and two distinguished points in the plane the problem of finding a collision free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research. 1 Introduction Given a set of obstacles and two distinguished points in the plane, the problem of finding a collision free path subject to a certain optimization function is a fundamental probl...

A linear time algorithm is presented for finding dominators in control flow graphs.