Deterministic fully dynamic graph algorithms are presented for connectivity and minimum spanning forest. For connectivity, starting with no edges, the amortized cost for maintaining a spanning forest is O(log² n) per update, i.e. per edge insertion or deletion. Deciding connectivity between any two given vertices is done in O(log n= log log n) time. This matches the previous best randomized bounds. The previous best deterministic bound was O( 3 p n log n) amortized time per update but constant time for connectivity queries. For minimum spanning trees, first a deletions-only algorithm is presented supporting deletes in amortized time O(log² n). Applying a general reduction from Henzinger and King, we then get a fully dynamic algorithm such that starting with no edges, the amortized cost for maintaining a minimum spanning forest is O(log^4 n) per update. The previous best deterministic bound was O( 3 p n log n) amortized time per update, and no better randomized bounds were ...

This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b. For approximate shortest paths with an error factor of (2 + ffl), for any positive constant ffl, the amortized update time is O(n 2 log 2 n= log log n); for an error factor of (1 + ffl) the amortized update time is O(n 2 log 3 (bn)=ffl 2 ). For exact shortest paths the amortized update time is O(n 2:5 p b log n). Query time for exact and approximate shortest distances is O(1); exact and approximate paths can be generated in time proportional to their lengths. Also presented is a fully dynamic transitive closure algorithm with update time O(n 2 log n) and query time O(1). The previously known fully dynamic transitive closure algorithm with fast query time has one-sided error and update time O(n 2:28 ). The algorithms use simple data structures, and are deterministic.

Introduction In many applications of graph algorithms, including communication networks, graphics, assembly planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of edges or vertices. In the last decade there has been a growing interest in such dynamically changing graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered. This chapter is intended as an overview of this field. In a typical dynamic graph problem one would like to answer queries on graphs that are undergoing a sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a dynamic graph algorithm is to update efficiently the solution of a problem after dynamic changes, rather than having to recompute it from scratch each time. Given their powerful versatility, it is not surprising that dynamic algorithms and dynamic data structures are often more difficult to design and analyze than their static c

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(n 3/2) in the matrix model and in Θ ( √ nm) in the array model, while the complexity of Connectivity is also in Θ(n 3/2) in the matrix model, but in Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

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 ...

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA. We give a near-optimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with word-size b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.

We consider Las Vegas randomized dynamic algorithms for on-line connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O(minfn 2 ; m log ng+ p nm log 2:5 n) expected total time. This is amortized constant time per operation if we start with a complete graph. The deletions may be interspersed with connectivity queries, each of which is answered in constant time. The previous best bound was O(m log 2 n) by Henzinger and Thorup (1996), which covered both insertions and deletions. Our bound is stronger for m=n = !(log n). The result is based on a general randomized reduction of many deletions-only queries to few deletions and insertions queries. Similar results are thus derived for 2-edge-connectivity, bipartiteness, and q-weights minimum spanning tree. For the decremental dynamic k-edge-connectivity problem of deleting the edges of a graph starting with m edges ...

We give a number of new lower bounds in the cell probe model with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations.

We prove an Ω(lg n) cell-probe lower bound on maintaining connectivity in dynamic graphs, as well as a more general trade-off between updates and queries. Our bound holds even if the graph is formed by disjoint paths, and thus also applies to trees and plane graphs. The bound is known to be tight for these restricted cases, proving optimality of these data structures (e.g., Sleator and Tarjan’s dynamic trees). Our trade-off is known to be tight for trees, and the best two data structures for dynamic connectivity in general graphs are points on our trade-off curve. In this sense these two data structures are optimal, and this tightness serves as strong evidence that our lower bounds are the best possible. From a more theoretical perspective, our result is the first logarithmic cell-probe lower bound for any problem in the natural class of dynamic language membership problems, breaking the long standing record of Ω(lg n / lg lg n). In this sense, our result is the first data-structure lower bound that is “truly ” logarithmic, i.e., logarithmic in the problem size counted in bits. Obtaining such a bound is listed as one of three major challenges for future research by Miltersen [13] (the other two challenges remain unsolved). Our techniques form a general framework for proving cell-probe lower bounds on dynamic data structures. We show how our framework also applies to the partial-sums problem to obtain a nearly complete understanding of the problem in cell-probe and algebraic models, solving several previously posed open problems.

. We state a new sampling lemma and use it to improve the running time of dynamic graph algorithms. For the dynamic connectivity problem the previously best randomized algorithm takes expected time O(log 3 n) per update, amortized over\Omega (m) updates. Using the new sampling lemma, we improve its running time to O(log 2 n). There exists a lower bound in the cell probe model for the time per operation of\Omega (log n= log log n) for this problem. Similarly improved running times are achieved for 2-edge connectivity, k-weight minimum spanning tree, and bipartiteness. 1 Introduction In this paper we present a new sampling lemma, and use it to improve the running times of various dynamic graph algorithms. We consider the following type of problem: Let S be a set with a subset R ` S. Membership in R may be efficiently tested. For a given parameter r ? 1, either (i) find an element of R, or (ii) guarantee with high probability that the ratio jRj=jSj is at most 1=r, i.e. that rjRj j...