Planar graphs, negative weight edges, shortest paths, and near linear time

Fakcharoenphol, Jittat; Rao, Satish

doi:10.1109/sfcs.2001.959897

Cited by 93 publications

(194 citation statements)

References 17 publications

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“…While the query times are very good (less than 20 µs for ǫ = 0.01), the preprocessing time and space consumption are quite high (2.5 hours and 2 GB, respectively). Using O(n log 3 n) space and preprocessing time, query time O( √ n log n) can be achieved [16] for directed planar graphs without negative cycles.…”

Section: Related Workmentioning

confidence: 99%

Engineering highway hierarchies

Sanders

Schultes

2012

ACM J. Exp. Algorithmics

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Highway hierarchies exploit hierarchical properties inherent in real-world road networks to allow fast and exact point-to-point shortest-path queries. A fast preprocessing routine iteratively performs two steps: first, it removes edges that only appear on shortest paths close to source or target; second, it identifies low-degree nodes and bypasses them by introducing shortcut edges. The resulting hierarchy of highway networks is then used in a Dijkstra-like bidirectional query algorithm to considerably reduce the search space size without losing exactness. The crucial fact is that 'far away' from source and target it is sufficient to consider only high-level edges. Various experiments with real-world road networks confirm the performance of our approach. On a 2.0 GHz machine, preprocessing the network of Western Europe, which consists of about 18 million nodes, takes 13 minutes and yields 48 bytes of additional data per node. Then, random queries take 0.61 ms on average. If we are willing to accept slower query times (1.10 ms), the memory usage can be decreased to 17 bytes per node. We can guarantee that at most 0.014% of all nodes are visited during any query. Results for US road networks are similar. Highway hierarchies can be combined with goal-directed search, they can be extended to answer manyto-many queries, and they are a crucial ingredient for other speedup techniques, namely for transit-node routing and highway-node routing.

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Section: Related Workmentioning

confidence: 99%

Engineering highway hierarchies

Sanders

Schultes

2012

ACM J. Exp. Algorithmics

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“…While no nontrivial theoretical results are known for the general P2P problem, there has been work on the special case of undirected planar graphs with slightly super-linear preprocessing space. The best bound in this context is c 0000 (copyright holder) due to Fakcharoenphol and Rao [14]. Algorithms for approximate shortest paths that use preprocessing have also been studied; see e.g., [4,28,48].…”

Section: Introductionmentioning

confidence: 99%

Reach for A*: shortest path algorithms with preprocessing

Goldberg

Kaplan²,

Werneck³

2009

DIMACS Series in Discrete Mathematics And Theoretical Computer Science

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We study the point-to-point shortest path problem with preprocessing. Given an input graph, we preprocess it so as to be able to answer a series of source-to-destination queries efficiently. Our work is motivated by an algorithm of Gutman [ALENEX'04], based on the notion of reach, which measures how important each vertex is with respect to shortest paths. We present a simplified version of his algorithm that does not require explicit lower bounds during queries. We also show how the addition of shortcuts to the graph greatly improves the performance of both preprocessing and queries. Finally, we combine a reach-based algorithm with landmark-based A * search to obtain a wide range of space-time trade-offs. For our motivating application, driving directions for road networks, the resulting algorithm is very efficient and practical. The road networks of the USA and Western Europe have roughly 20 million vertices, but on average our algorithm must visit fewer than a thousand to find the distance between two points. Our algorithm also works reasonably well on 2-dimensional grid graphs with random arc weights.

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“…Thereafter, a few solutions [3,32,35,40] were proposed whose running times are better than a total re-computation. However, these solutions only work for integer weights.…”

Section: Discussionmentioning

confidence: 99%

Adaptive algorithms for routing and traffic engineering in stochastic networks

Misra

Oommen

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Planar graphs, negative weight edges, shortest paths, and near linear time

Cited by 93 publications

References 17 publications

Engineering highway hierarchies

Engineering highway hierarchies

Reach for A*: shortest path algorithms with preprocessing

Adaptive algorithms for routing and traffic engineering in stochastic networks

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