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.

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