graph algorithms have received considerable attention lately because massive graphs
arise naturally in many applications. Recent web crawls, for example, produce
graphs of on the order of 200 million nodes and 2 billion edges. Recent work in
web modeling uses depth-first search, breadth-first search, shortest paths, and
connected components as primitive operations for investigating the structure of
Massive graphs are
also often manipulated in Geographic Information Systems (GIS), where many fundamental
problems can be formulated as basic graph problems. The graphs arising in GIS
applications are often planar. Yet another example of massive graphs is AT&T's
20TB phone call graph . When working with such large data sets, the transfer
of data between internal and external memory, and not the internal memory computation,
is often the bottleneck. Thus, I/O-efficient algorithms can lead to considerable
search (BFS) and depth-first search (DFS) are the two most fundamental graph-searching
strategies. They are extensively used in internal memory algorithms, as they are
easy to perform in linear time; yet they provide valuable information about the
structure of the given graph. Unfortunately, no I/O-efficient algorithms for BFS
and DFS in arbitrary sparse graphs are known, while existing algorithms perform
reasonably well on dense graphs. Together with recent results on single-source
shortest paths (SSSP) and DFS, our algorithm leads to I/O-efficient algorithms
for SSSP, BFS, and DFS on undirected embedded planar graphs.
The algorithms in this paper are designed and analyzed
in the Parallel Disk Model (PDM). In thismodel, D identical disks of unlimited
size are attached to a machine with an internal memory capable of holding M data
items. These disks constitute the external memory of the machine. Initially, all
data is stored on disk. Each disk is partitioned into blocks of B data items each.
An I/O-operation is the transfer of up to D blocks, at most one per disk, to or
from internal memory from or to external memory.
The complexity of an algorithm
in the PDM is the number of I/O-operations it performs. Sorting, permuting, and
scanning an array of N consecutive data items are primitive operations often used
in external memory algorithms. Their I/O-complexities are sort(N) = Q((N=DB) logM=B(N=B)),
perm(N) = Q(min(N; sort(N))), and scan(N) = O(N=DB), respectively.
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