Efficient Maximum Flow Algorithms

Though maximum flow algorithms have a long history, revolutionary progress is still being made.

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# Robert E. Tarjan

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The quest for efficiency in computational methods yields not only fast algorithms, but also insights that lead to elegant, simple, and general problem-solving methods.
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Shape the Future of Computing

Efficient Maximum Flow Algorithms

Though maximum flow algorithms have a long history, revolutionary progress is still being made.

Relaxed heaps: an alternative to Fibonacci heaps with applications to parallel computation

The relaxed heap is a priority queue data structure that achieves the same amortized time bounds as the Fibonacci heap—a sequence of m decrease_key and n delete_min operations takes time O(m + n log n). A variant of relaxed heaps achieves similar bounds in the worst case—O(1) time for decrease_key and O(log n) for delete_min. Relaxed heaps give a processor-efficient parallel implementation of Dijkstra's shortest path algorithm, and hence other algorithms in network optimization. A relaxed heap is a type of binomial queue that allows heap order to be violated.

Planar point location using persistent search trees

A classical problem in computational geometry is the planar point location problem. This problem calls for preprocessing a polygonal subdivision of the plane defined by n line segments so that, given a sequence of points, the polygon containing each point can be determined quickly on-line. Several ways of solving this problem in O(log n) query time and O(n) space are known, but they are all rather complicated. We propose a simple O(log n)-query-time, O(n)-space solution, using persistent search trees. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be accessed. We develop a persistent form of binary search tree that supports insertions and deletions in the present and queries in the past. The time per query or update is O(log m), where m is the total number of updates, and the space needed is O(1) per update. Our planar point location algorithm is an immediate application of this data structure. The structure also provides an alternative to Chazelle's "hive graph" structure, which has a variety of applications in geometric retrieval.

A locally adaptive data compression scheme

A data compression scheme that exploits locality of reference, such as occurs when words are used frequently over short intervals and then fall into long periods of disuse, is described. The scheme is based on a simple heuristic for self-organizing sequential search and on variable-length encodings of integers. We prove that it never performs much worse than Huffman coding and can perform substantially better; experiments on real files show that its performance is usually quite close to that of Huffman coding. Our scheme has many implementation advantages: it is simple, allows fast encoding and decoding, and requires only one pass over the data to be compressed (static Huffman coding takes two passes).

Amortized efficiency of list update and paging rules

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 (Belady's MIN algorithm) by a factor that depends on the size of fast memory. No on-line paging algorithm has better amortized performance.

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