Optimal Reachability and a Space-Time Tradeoff for Distance Queries in Constant-Treewidth Graphs

We consider data-structures for answering reachability and distance queries on constant-treewidth graphs with n nodes, on the standard RAM computational model with wordsize W=Theta(log n). Our first contribution is a data-structure that after O(n) preprocessing time, allows (1) pair reachability queries in O(1) time; and (2) single-source reachability queries in O(n/log n) time. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. The data-structure uses at all times O(n) space. Our second contribution is a space-time tradeoff data-structure for distance queries. For any epsilon in [1/2,1], we provide a data-structure with polynomial preprocessing time that allows pair queries in O(n^{1-\epsilon} alpha(n)) time, where alpha is the inverse of the Ackermann function, and at all times uses O(n^epsilon) space. The input graph G is not considered in the space complexity.

[1]  Torben Hagerup.  Dynamic Algorithms for Graphs of Bounded Treewidth , 1997, ICALP.

[2]  Torben Hagerup,et al.  Parallel Algorithms with Optimal Speedup for Bounded Treewidth , 1995, ICALP.

[3]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[4]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[5]  Krishnendu Chatterjee,et al.  Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs , 2015, CAV.

[6]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[7]  L. R. Ford,et al.  NETWORK FLOW THEORY , 1956 .

[8]  Andreas Jakoby,et al.  Logspace Versions of the Theorems of Bodlaender and Courcelle , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[9]  Richard Bellman,et al.  ON A ROUTING PROBLEM , 1958 .

[10]  Christos D. Zaroliagis,et al.  Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms , 2000, Algorithmica.

[11]  Uzi Vishkin,et al.  On Finding Lowest Common Ancestors: Simplification and Parallelization , 1988, AWOC.

[12]  Takuya Akiba,et al.  Shortest-path queries for complex networks: exploiting low tree-width outside the core , 2012, EDBT '12.

[13]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[14]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[15]  Michael J. Fischer,et al.  Boolean Matrix Multiplication and Transitive Closure , 1971, SWAT.

[16]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[17]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[18]  Atsuko Yamaguchi,et al.  Graph Complexity of Chemical Compounds in Biological Pathways , 2003 .

[19]  Krishnendu Chatterjee,et al.  Algorithms for algebraic path properties in concurrent systems of constant treewidth components , 2016, POPL.

[20]  Paul D. Seymour,et al.  Graph minors. III. Planar tree-width , 1984, J. Comb. Theory B.

[21]  Ignaz Rutter,et al.  Search-space size in contraction hierarchies , 2013, Theor. Comput. Sci..

[22]  Nils J. Nilsson,et al.  A Formal Basis for the Heuristic Determination of Minimum Cost Paths , 1968, IEEE Trans. Syst. Sci. Cybern..

[23]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[24]  Takuya Akiba,et al.  Fast and scalable reachability queries on graphs by pruned labeling with landmarks and paths , 2013, CIKM.

[25]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[26]  R. Halin.  S-functions for graphs , 1976 .

[27]  Hans L. Bodlaender,et al.  Dynamic Programming on Graphs with Bounded Treewidth , 1988, ICALP.

[28]  Krishnendu Chatterjee,et al.  Faster Algorithms for Algebraic Path Properties in Recursive State Machines with Constant Treewidth , 2015, POPL.

[29]  Stephen Warshall,et al.  A Theorem on Boolean Matrices , 1962, JACM.

[30]  Norbert Zeh,et al.  I/O-Efficient Algorithms for Graphs of Bounded Treewidth , 2001, SODA '01.

[31]  Hans L. Bodlaender,et al.  Discovering Treewidth , 2005, SOFSEM.

[32]  Mathijs de Weerdt,et al.  Computing All-Pairs Shortest Paths by Leveraging Low Treewidth , 2011, ICAPS.

[33]  Mikkel Thorup,et al.  All Structured Programs have Small Tree-Width and Good Register Allocation , 1998, Inf. Comput..

[34]  Michael A. Bender,et al.  The LCA Problem Revisited , 2000, LATIN.

[35]  Ignaz Rutter,et al.  Search-space size in contraction hierarchies , 2016, Theor. Comput. Sci..

[36]  Takuya Akiba,et al.  Fast exact shortest-path distance queries on large networks by pruned landmark labeling , 2013, SIGMOD '13.