On External-Memory Planar Depth-First Search

Transcription

1 On External-Memory Planar Depth-First Search Laura Toma Duke University Lars Arge Ulrich Meyer Norbert Zeh Duke University Max-Plank Institute Carlton University August 2

2 External Graph Problems Applications Geographic Information Systems (GIS): Terrain analysis: flow modeling, topographic indices Routing (e.g. find optimal routes given US road network) Web modeling Web crawl of 2M nodes, 2M links: shortest paths, (strongly) connected components, breadth/depth first search, diameter [BK] Search engines I/O bottleneck Data reside on disk Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 2

3 Disk Model [Aggarwal & Vitter 88] Block I/O D V = # of vertices E = # of edges M = # of vertices/edgesthatfitinmemory B = # of vertices/edges per disk block M I/O complexity Basic bounds P scan(e) = E B E sort(e) =Θ( E B log M/B E B ) E Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh

4 Depth-First Search (DFS) DFS(u) mark u for each v in Adj(u) if v not marked then DFS(v) Internal memory: O(V + E) External memory: O(V + E) =O(E) I/Os I/O per vertex to load adjacency list I/O per edge to check if v is marked External memory DFS and BFS are hard! Lower bounds: Ω(sort(V )) Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 4

5 I/O-Efficient DFS and BFS Previous Work General undirected graphs: DFS O ( ) V + V E M B O ( (V + E B ) log ) V B [CGG95] [KS96] BFS O ( V + E V sort(v )) = O(V +sort(e)) [MR99] Planar undirected graphs: O(N) I/Os, O(N) space N O( γlog B +sort(nbγ )) I/Os, O(NB γ ) space [M] Grid graphs: DFS O( N B ) [M], BFS O(sort(N)) [AT] Outerplanar graphs: O(sort(N)) [MZ99] Trees: O(sort(N)) [CGG95] Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 5

6 Planar undirected graphs Our Contributions An O(sort(N) log N) DFS algorithm o(n) iflog M/B N B log N<B An O(sort(N)) DFS to BFS reduction DFS O(sort(N)) BFS [ABT] O(sort(N)) External multi way separation [ABT] O(sort(N)) SSSP Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 6

7 Known O(sort(N)) solutions List ranking [CGG95] I/O-Toolbox List given as (u, succ(u)) pairs. Compute for each vertex the number of edges on its path to the end of the list. Trees BFS, DFS, expression evaluation, Euler tour [CGG95,BGV] Planar graphs Minimum spanning tree (MST), spanning tree (ST), connected components (CC), biconnected components (BCC) [CGG95] Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 7

8 A Divide-and-Conquer DFS Algorithm on Planar Graphs Based on DFS PRAM algorithm [Smith86] s v w C Compute a simple cycle-separator C Compute a path from s to C: s v w Compute DFS recursively in the CC G i of G \ C Attach the DFS trees of G i onto the path Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 8

9 A Divide-and-Conquer DFS Algorithm on Planar Graphs Compute a simple cycle-separator C A simple 5 6-cycle-separator of a biconnected plane graph can be computed in O(sort(N)) I/Os. Compute a path from s to C O(sort(N)) using spanning tree, list ranking etc. Attach the DFS trees of G i onto the path O(sort(N)) connected components, list ranking, sorting etc. O(log N) recursion steps, O(sort(N) I/Os per step = O(sort(N) log N) Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 9

10 Computing a Cycle-separator of a Planar Graph Use a spanning tree T of the dual graph Subtree rooted at v T defines subgraph G(v) Boundary of G(v) is a simple cycle r r v G(v) for each node v T can be computed bottom-up in O(sort(N)) I/Os. Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh

11 Computing a Cycle-separator of a Planar Graph Based on PRAM algorithm [JK88] Case : Search for face in G with boundary 6 E Case 2: Search for v T such that 6 E G(v) 5 6 E r v Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh

12 Computing a Cycle-separator of a Planar Graph Case : Find v T such that G(v) > 5 6 E and G(w i) < 6 E for all children w i of v Find a subset {w i } such that 6 E v i G(w i ) 5 6 E v Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 2

13 Computing a Cycle-separator of a Planar Graph Problem: Boundary must be a simple cycle! Example: boundary of v G(w ) not a simple cycle G(w ) v G(w ) G(w 2 ) Compute a permutation σ(), σ(2),... such that the boundary of v i k G(w σ(i) ) is a simple cycle, for any k =,2,... Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh

14 A DFS to BFS Reduction on Planar Graphs Idea: Partition the faces of G into levels around a source face containing s and grow DFS level-by-level. s s Computing DFS of each level and gluing them together is simpler! Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 4

15 Partitioning the Faces into Levels BFS in the vertex-on-face graph Level of a face = BFS depth in vertex-on-face graph ss Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 5

16 A DFS to BFS Reduction on Planar Graphs G i = union of the boundaries of faces at level i T i =DFStreeofG i H i =G i \G i s H H H 2 Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 6

17 A DFS to BFS Reduction on Planar Graphs Algorithm: Compute a DFS-forest of H i and attach it onto T i. A spanning tree is a DFS tree if and only if it does not have cross edges. Compute CC H i of H i For each H i find deepest node in T i which connects to a node r i in H i. If root DFS tree of H i at r i = gluing it onto T i does not introduce cross-edges. Lemma: The bicomps of H i are the boundary cycles of G i. Compute bicomps and bicomp-cut-point tree for each H i Compute the DFS of H i rooted at r i Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 7

18 A DFS to BFS Reduction on Planar Graphs I/O-analysis Graphs H i can be computed in O(sort(N)) I/Os For each level i CC H i of H i Deepest node in T i which connects to H i Bicomps, bicomp-cut-point tree and DFS tree in H i can be computed in O(sort(H i )) I/Os = total O(sort(N)) I/Os Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 8

19 Conclusion Our results: DFS on planar graphs O(sort(N) log N) I/Os A reduction to BFS in O(sort(N)) I/Os DFS O(sort(N)) BFS [ABT] O(sort(N)) External multi way separation [ABT] O(sort(N)) SSSP External planar multi-way separation: O(sort(N)) I/Os [MZ] = BFS, DFS, SSSP in O(sort(N)) I/Os! Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 9

20 Open problems BFS, DFS on general graphs Reductions on general graphs Practical O(sort(N)) DFS for special classes of graphs encountered in practice (triangulations, grid graphs) Lars Arge, Ulrich Meyer, Laura Toma, Norbert Zeh 2