Space-Efficient DFS and Applications: Simpler, Leaner, Faster

The problem of space-efficient depth-first search (DFS) is reconsidered. A particularly simple and fast algorithm is presented that, on a directed or undirected input graph $G=(V,E)$ with $n$ vertices and $m$ edges, carries out a DFS in $O(n+m)$ time with $n+\sum_{v\in V_{\ge 3}}\lceil{\log_2(d_v-1)}\rceil +O(\log n)\le n+m+O(\log n)$ bits of working memory, where $d_v$ is the (total) degree of $v$, for each $v\in V$, and $V_{\ge 3}=\{v\in V\mid d_v\ge 3\}$. A slightly more complicated variant of the algorithm works in the same time with at most $n+({4/5})m+O(\log n)$ bits. It is also shown that a DFS can be carried out in a graph with $n$ vertices and $m$ edges in $O(n+m\log^*\! n)$ time with $O(n)$ bits or in $O(n+m)$ time with either $O(n\log\log(4+{m/n}))$ bits or, for arbitrary integer $k\ge 1$, $O(n\log^{(k)}\! n)$ bits. These results among them subsume or improve most earlier results on space-efficient DFS. Some of the new time and space bounds are shown to extend to applications of DFS such as the computation of cut vertices, bridges, biconnected components and 2-edge-connected components in undirected graphs

Answering Spatial Multiple-Set Intersection Queries Using 2-3 Cuckoo Hash-Filters

We show how to answer spatial multiple-set intersection queries in O(n(log w)/w + kt) expected time, where n is the total size of the t sets involved in the query, w is the number of bits in a memory word, k is the output size, and c is any fixed constant. This improves the asymptotic performance over previous solutions and is based on an interesting data structure, known as 2-3 cuckoo hash-filters. Our results apply in the word-RAM model (or practical RAM model), which allows for constant-time bit-parallel operations, such as bitwise AND, OR, NOT, and MSB (most-significant 1-bit), as exist in modern CPUs and GPUs. Our solutions apply to any multiple-set intersection queries in spatial data sets that can be reduced to one-dimensional range queries, such as spatial join queries for one-dimensional points or sets of points stored along space-filling curves, which are used in GIS applications.Comment: Full version of paper from 2017 ACM SIGSPATIAL International Conference on Advances in Geographic Information System

Fast deterministic processor allocation

Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given $n$ nonnegative integers $x_1,\ldots,x_n$, to allocate $n$ nonoverlapping subarrays of sizes $x_1,\ldots,x_n$ from within a base array of $O(\sum_{j=1}^n x_j)$ cells. We show that interval allocation problems of size $n$ can be solved in $O((\log\log n)^3)$ time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of $\Theta({{\log n}/{\log\log n}})$. We also describe an application to the problem of computing the connected components of an undirected graph

On a compaction theorem of ragde

Ragde demonstrated that in constant time a PRAM with $n$ processors can move at most $k$ items, stored in distinct cells of an array of size $n$, to distinct cells in an array of size at most $k^4$. We show that the exponent of 4 in the preceding sentence can be replaced by any constant greater than~2