Cache-Oblivious Buffer Heap and Cache-Efficient Computation of Shortest Paths in Graphs

Abstract

We present the buffer heap , a cache-oblivious priority queue that supports Delete-Min , Delete , and a hybrid Insert / Decrease-Key operation in O (1/ B log 2 N / M ) amortized block transfers from main memory, where M and B are the (unknown) cache size and block size, respectively, and N is the number of elements in the queue. We introduce the notion of a slim data structure that captures the situation when only a limited portion of the cache, which we call a slim cache , is available to the data structure to retain data between data structural operations. We show that a buffer heap automatically adapts to such an environment and supports all operations in O (1/λ + 1/ B log 2 N /λ) amortized block transfers each when the size of the slim cache is λ. Our results provide substantial improvements over known trivial cache performance bounds for cache-oblivious priority queues with Decrease-Keys . Using the buffer heap, we present cache-oblivious implementations of Dijkstra’s algorithm for undirected and directed single-source shortest path (SSSP) problems for graphs with non-negative real edge-weights. On a graph with n vertices and m edges, our algorithm for the undirected case performs O ( n + m / B log 2 n / M ) block transfers and for the directed case performs O (( n + m / B ) ċ log 2 n / B ) block transfers. These results give the first non-trivial cache-oblivious bounds for the SSSP problem on general graphs. For the all-pairs shortest path (APSP) problem on weighted undirected graphs, we incorporate slim buffer heaps into multi-buffer-buffer-heaps and use these to improve the cache-aware cache complexity. We also present a simple cache-oblivious APSP algorithm for unweighted undirected graphs that performs O ( mn / B log M / B n / B ) block transfers. This matches the cache-aware bound and is a substantial improvement over the previous cache-oblivious bound for the problem.