Maintaining minimum spanning forests in dynamic graphs

Monika Henzinger
2001
Journal paper

We present the first fully dynamic algorithm for maintaining a minimum spanning forest in time o(sqrt(n)) per operation. To be precise, the algorithm uses O(n1/3 log n) amortized time per update operation. The algorithm is fairly simple and deterministic. An immediate consequence is the first fully dynamic deterministic algorithm for maintaining connectivity and bipartiteness in amortized time O(n1/3 log n) per update, with O(1) worst case time per query.

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Monika Henzinger
2001
Journal paper

Abstract

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https://infoscience.epfl.ch/record/99364?ln=en

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Average-case analysis of dynamic graph algorithms

Monika Henzinger

We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., not random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2- edge connectivity, k-edge connectivity, k-vertex connectivity, and bipartiteness. Given a random graph G with m0 edges and n vertices and a sequence of l update operations such that the graph contains mi edges after operation i, the expected time for performing the updates for any l is O(l log(n) + sum(i=1 to l) n/sqrt(m_i)) in the case of minimum spanning forests, connectivity, 2-edge connectivity, and bipartiteness. The expected time per update operation is O(n) in the case of maximum matching. We also give improved bounds for k-edge and k-vertex connectivity. Additionally we give an insertions-only algorithm for maximum cardinality matching with worst- case O(n) amortized time per insertion.

1998

Randomized fully dynamic graph algorithms with polylogarithmic time per operation

Monika Henzinger

This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of 0(m0) operations, where m0 is the number of edges in the initial graph, the expected time for p updates is O(p log3 n) (Throughout the paper the logarithms are base 2.) for connectivity and bipartiteness. The worst-case time for one query is O(log n/log log n). For the k-edge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p.pow(log(n),3)) and the expected time for q queries is O(p_k.pow(log(n),3)). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O(p_k.pow(log(n),3)). This implies an algorithm to maintain a 1 + e- approximation of the minimum spanning tree in expected time O((p.pow(log(n),3).log U)/e) for p updates, where the weights of the edges are between 1 and U.

1999

Fully dynamic cycle-equivalence in graphs

Monika Henzinger

Two edges e_1 and e_2 of an undirected graph are cycle-equivalent iff all cycles that contain e_1 also contain e_2, i.e., iff e_1 and e_2 are a cut-edge pair. The cycle-equivalence classes of the control-flow graph are used in optimizing compilers to speed up existing control-flow and data-flow algorithms. While the cycle-equivalence classes can be computed in linear time, we present the first fully dynamic algorithm for maintaining the cycle-equivalence relation. In an n-node graph our data structure executes an edge insertion or deletion in O(sqrt(n.log n)) time and answers the query whether two given edges are cycle-equivalent in O(pow2(log(n))) time. We also present an algorithm for plane graphs with O(log n) update and query time and for planar graphs with O(log n) insertion time and O(log2 n) query and deletion time. Additionally, we show a lower bound of Ω(log n/log log n) for the amortized time per operation for the dynamic cycle-equivalence problem in the cell probe model

1994

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Dynamic programming

Dynamic programming is both a mathematical optimization method and an algorithmic paradigm. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, fr

Minimum spanning tree

A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and w