Deepak Ajwani

Characterizing the performance of flash memory storage devices and its impact on algorithm design

Initially used in digital audio players, digital cameras, mobile phones, and USB memory sticks, flash memory may become the dominant form of end-user storage in mobile computing, either completely replacing the magnetic hard disks or being an additional secondary storage. We study the design of algorithms and data structures that can exploit the flash memory devices better. For this, we characterize the performance of NAND flash based storage devices, including many solid state disks. We show that these devices have better random read performance than hard disks, but much worse random write performance. We also analyze the effect of misalignments, aging and past I/O patterns etc. on the performance obtained on these devices. We show that despite the similarities between flash memory and RAM (fast random reads) and between flash disk and hard disk (both are block based devices), the algorithms designed in the RAM model or the external memory model do not realize the full potential of the flash memory devices. We later give some broad guidelines for designing algorithms which can exploit the comparative advantages of both a flash memory device and a hard disk, when used together.

A computational study of external-memory BFS algorithms

Breadth First Search (BFS) traversal is an archetype for many important graph problems. However, computing a BFS level decomposition for massive graphs was considered nonviable so far, because of the large number of I/Os it incurs. This paper presents the first experimental evaluation of recent external-memory BFS algorithms for general graphs. With our STXXL based implementations exploiting pipelining and disk-parallelism, we were able to compute the BFS level decomposition of a web-crawl based graph of around 130 million nodes and 1.4 billion edges in less than 4 hours using single disk and 2.3 hours using 4 disks. We demonstrate that some rather simple external-memory algorithms perform significantly better (minutes as compared to hours) than internal-memory BFS, even if more than half of the input resides internally.

Improved external memory BFS implementations

Breadth first search (BFS) traversal on massive graphs in external memory was considered non-viable until recently, because of the large number of I/Os it incurs. Ajwani et al. [3] showed that the randomized variant of the o(n) I/O algorithm of Mehlhorn and Meyer [24] (MM_BFS) can compute the BFS level decomposition for large graphs (around a billion edges) in a few hours for small diameter graphs and a few days for large diameter graphs. We improve upon their implementation of this algorithm by reducing the overhead associated with each BFS level, thereby improving the results for large diameter graphs which are more difficult for BFS traversal in external memory. Also, we present the implementation of the deterministic variant of MM_BFS and show that in most cases, it outperforms the randomized variant. The running time for BFS traversal is further improved with a heuristic that preserves the worst case guarantees of MM_BFS. Together, they reduce the time for BFS on large diameter graphs from days shown in [3] to hours. In particular, on line graphs with random layout on disks, our implementation of the deterministic variant of MM_BFS with the proposed heuristic is more than 75 times faster than the previous best result for the randomized variant of MM_BFS in [3].

Conflict-free coloring for rectangle ranges using O ( n .382 ) colors

Given a set of points <i>P</i> ⊆ R², a <i>conflict-free coloring</i> of <i>P</i> w.r.t. rectangle ranges is an assignment of colors to points of <i>P</i>, such that each non-empty axis-parallel rectangle <i>T</i> in the plane contains a point whose color is distinct from all other points in <i>P</i> ∩ <i>T</i>. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of <i>n</i> points in R² can be conflict-free colored with <i>Õ</i>(<i>n</i>^β+ε) colors in expected polynomial time, for any arbitrarily small ε > 0 and β = 3?√5<over>2 < 0.382. This improves upon the previously known bound of <i>O</i>(√<i>n</i>log log <i>n</i>/ log <i>n</i>).

On Computational Models for Flash Memory Devices

Flash memory-based solid-state disks are fast becoming the dominant form of end-user storage devices, partly even replacing the traditional hard-disks. Existing two-level memory hierarchy models fail to realize the full potential of flash-based storage devices. We propose two new computation models, the general flash model and the unit-cost model, for memory hierarchies involving these devices. Our models are simple enough for meaningful algorithm design and analysis. In particular, we show that a broad range of existing external-memory algorithms and data structures based on the merging paradigm can be adapted efficiently into the unit-cost model. Our experiments show that the theoretical analysis of algorithms on our models corresponds to the empirical behavior of algorithms when using solid-state disks as external memory.

Geometric algorithms for private-cache chip multiprocessors

We study techniques for obtaining efficient algorithms for geometric problems on private-cache chip multiprocessors. We show how to obtain optimal algorithms for interval stabbing counting, 1-D range counting, weighted 2-D dominance counting, and for computing 3-D maxima, 2-D lower envelopes, and 2-D convex hulls. These results are obtained by analyzing adaptations of either the PEM merge sort algorithm or PRAM algorithms. For the second group of problems—orthogonal line segment intersection reporting, batched range reporting, and related problems—more effort is required. What distinguishes these problems from the ones in the previous group is the variable output size, which requires I/O-efficient load balancing strategies based on the contribution of the individual input elements to the output size. To obtain nearly optimal algorithms for these problems, we introduce a parallel distribution sweeping technique inspired by its sequential counterpart.

An O(n 2.75 ) algorithm for online topological ordering

We present a simple algorithm which maintains the topological order of a directed acyclic graph with n nodes under an online edge insertion sequence in

Average-case analysis of online topological ordering

{\cal O}(n^{2.75})

Design and Engineering of External Memory Traversal Algorithms for General Graphs

time, independent of the number of edges m inserted. For dense DAGs, this is an improvement over the previous best result of

An O ( n 2.75 ) algorithm for incremental topological ordering

{\cal O}(\min\{m^{\frac{3}{2}} \log{n}, m^{\frac{3}{2}} + n^2 \log{n}\})