Robert Endre Tarjan

Depth-First Search and Linear Graph Algorithms

The value of depth-first search or “backtracking” as a technique for solving problems is illustrated by two examples. An improved version of an algorithm for finding the strongly connected components of a directed graph and at algorithm for finding the biconnected components of an undirect graph are presented. The space and time requirements of both algorithms are bounded by

Fibonacci heaps and their uses in improved network optimization algorithms

k_1 V + k_2 E + k_3

Amortized efficiency of list update and paging rules

for some constants

Efficiency of a Good But Not Linear Set Union Algorithm

k_1 ,k_2

Self-adjusting binary search trees

, and

Time bounds for selection

k_3

Three partition refinement algorithms

, where V is the number of vertices and E is the number of edges of the graph being examined.

Fast algorithms for finding nearest common ancestors

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.

Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs

In this article we study the amortized efficiency of the “move-to-front” and similar rules for dynamically maintaining a linear list. Under the assumption that accessing the ith element from the front of the list takes &thgr;(i) time, we show that move-to-front is within a constant factor of optimum among a wide class of list maintenance rules. Other natural heuristics, such as the transpose and frequency count rules, do not share this property. We generalize our results to show that move-to-front is within a constant factor of optimum as long as the access cost is a convex function. We also study paging, a setting in which the access cost is not convex. The paging rule corresponding to move-to-front is the “least recently used” (LRU) replacement rule. We analyze the amortized complexity of LRU, showing that its efficiency differs from that of the off-line paging rule (Beladys MIN algorithm) by a factor that depends on the size of fast memory. No on-line paging algorithm has better amortized performance.

Efficient Planarity Testing

TWO types of instructmns for mampulating a family of disjoint sets which part i tmn a umverse of n elements are considered FIND(x) computes the name of the (unique) set containing element x UNION(A, B, C) combines sets A and B into a new set named C. A known algorithm for implementing sequences of these mstructmns is examined I t is shown that , if t(m, n) as the maximum time reqmred by a sequence of m > n FINDs and n -1 intermixed UNIONs, then kima(m, n) _~ t(m, n) < k:ma(m, n) for some positive constants ki and k2, where a(m, n) is related to a functional inverse of Ackermanns functmn and as very slow-growing.