Michael L. Fredman

Fibonacci heaps and their uses in improved network optimization algorithms

In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, <italic>Fibonacci heaps</italic> (abbreviated <italic>F-heaps</italic>), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an <italic>n</italic>-item heap in <italic>O</italic>(log <italic>n</italic>) amortized time and all other standard heap operations in <italic>O</italic>(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where <italic>n</italic> is the number of vertices and <italic>m</italic> the number of edges in the problem graph:<list><item><italic>O</italic>(<italic>n</italic> log <italic>n</italic> + <italic>m</italic>) for the single-source shortest path problem with nonnegative edge lengths, improved from <italic>O</italic>(<italic>m</italic>log<subscrpt>(<italic>m/n</italic>+2)</subscrpt><italic>n</italic>); </item><item><italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>log <italic>n</italic> + <italic>nm</italic>) for the all-pairs shortest path problem, improved from <italic>O</italic>(<italic>nm</italic> log<subscrpt>(<italic>m/n</italic>+2)</subscrpt><italic>n</italic>); </item><item><italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>log <italic>n</italic> + <italic>nm</italic>) for the assignment problem (weighted bipartite matching), improved from <italic>O</italic>(<italic>nm</italic>log<subscrpt>(<italic>m/n</italic>+2)</subscrpt><italic>n</italic>); </item><item><italic>O</italic>(<italic>mβ</italic>(<italic>m, n</italic>)) for the minimum spanning tree problem, improved from <italic>O</italic>(<italic>m</italic>log log<subscrpt>(<italic>m/n</italic>+2)</subscrpt><italic>n</italic>); where <italic>β</italic>(<italic>m, n</italic>) = min {<italic>i</italic> ↿ log<supscrpt>(<italic>i</italic>)</supscrpt><italic>n</italic> ≤ <italic>m/n</italic>}. Note that <italic>β</italic>(<italic>m, n</italic>) ≤ log<supscrpt>*</supscrpt><italic>n</italic> if <italic>m</italic> ≥ <italic>n</italic>. </item></list>Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.

Storing a Sparse Table with 0 (1) Worst Case Access Time

We describe a data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time. Both the data structure and the query algorithm are easy to implement.

The pairing heap: a new form of self-adjusting heap

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue) called theFibonacci heap. Although theoretically efficient, Fibonacci heaps are complicated to implement and not as fast in practice as other kinds of heaps. In this paper we describe a new form of heap, called thepairing heap, intended to be competitive with the Fibonacci heap in theory and easy to implement and fast in practice. We provide a partial complexity analysis of pairing heaps. Complete analysis remains an open problem.

Fibonacci Heaps And Their Uses In Improved Network Optimization Algorithms

In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in 0(log n) amortized time and all other standard heap operations in 0(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms.

On the Size of Separating Systems and Families of Perfect Hash Functions

This paper presents two applications of an interesting information theoretic theorem about graphs. The first application concerns the derivation of good bounds for the function

A Lower Bound on the Complexity of Orthogonal Range Queries

Y(b,k,n)

Data structures for traveling salesmen

, which is defined to be the minimum size of a family of functions such that for every subset of size k from an n element universe, there exists a perfect hash function in the family mapping the subset into a table of size b. The second application concerns the derivation of good bounds for the function

On the complexity of computing the measure of ∪[a i ,b i ]

M(i,j,n)

Hash functions for priority queues

, which is defined to be the minimum size of an

Storing a sparse table with O(1) worst case access time

(i,j)