Optimal Time-Space Trade-Offs for Sorting

Given an unordered array of N elements drawn from a totally ordered set and an integer k in the range from 1 to N , in the classic selection problem the task is to find the k-th smallest element in the array. We study the complexity of this problem in the space-restricted random-access model: The input array is stored on read-only memory, and the algorithm has access to a limited amount of workspace. We prove that the linear-time prune-and-search algorithmpresented in most textbooks on algorithms-can be modified to use Θ(N ) bits instead of Θ(N ) words of extra space. Prior to our work, the best known algorithm by Frederickson could perform the task with Θ(N ) bits of extra space in O(N lg * N ) time. Our result separates the space-restricted random-access model and the multi-pass streaming model, since we can surpass the Ω(N lg * N ) lower bound known for the latter model. We also generalize our algorithm for the case when the size of the workspace is Θ(S) bits, where lg 3 N ≤ S ≤ N . The running time of our generalized algorithm is O(N lg * (N/S)+N (lg N )/ lg S), slightly improving over the O(N lg * (N (lg N )/S) + N (lg N )/ lg S) bound of Frederickson's algorithm. To obtain the improvements mentioned above, we developed a new data structure, called the wavelet stack, that we use for repeated pruning. We expect the wavelet stack to be a useful tool in other applications as well.

Section: Discussionmentioning

confidence: 99%

Selection from Read-Only Memory with Limited Workspace

Elmasry

Juhl

Katajainen

et al. 2013

“…We encourage the reader to compare our solution to that of Pagter and Rauhe [24]. In the basic setting, Pagter and Rauhe proved that the running time of their sorting algorithm is O(N 2 /S + N lg 2 S) using O(S) bits of workspace.…”

Section: Reference Spacementioning

confidence: 99%

“…The main result of this paper is a simplification of the memory-adjustable priority queue by Pagter and Rauhe [24] that is a precursor of all later constructions. First, we devise a memory-adjustable priority queue that we call an adjustable navigation pile.…”

Section: Reference Spacementioning

confidence: 99%

“…Analogous with the sequential-access machine, we have a read-only array for input, a write-only array for output, and a limited workspace that allows random access. Over the years, starting by a seminal paper of Munro and Paterson [20], the space-time trade-offs have been studied in this model for many problems including: sorting [4,12,24], selection [11,12], and various geometric tasks [2,3,7]. The practical motivation for some of the previous work has been the appearance of special devices, where the size of working space is limited (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Sorting is one of the few problems for which the optimal space-time product has been settled: Beame showed [4] (see also [25,Theorem 10.13.8]) that Ω(N 2 ) is a lower bound, and Pagter and Rauhe showed [24] that an O(N 2 /S +N lg S) running time is achievable for any S when lg N ≤ S ≤ N/ lg N .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Priority Queues and Sorting for Read-Only Data

Asano

Elmasry

Katajainen

2013

Abstract. We revisit the random-access-machine model in which the input is given on a read-only random-access media, the output is to be produced to a write-only sequential-access media, and in addition there is a limited random-access workspace. The length of the input is N elements, the length of the output is limited by the computation itself, and the capacity of the workspace is O(S + w) bits, where S is a parameter specified by the user and w is the number of bits per machine word. We present a state-of-the-art priority queue-called an adjustable navigation pile-for this model. Under some reasonable assumptions, our priority queue supports minimum and insert in O(1) worst-case time and extract in O(N/S +lg S) worst-case time, where lg N ≤ S ≤ N/ lg N . We also show how to use this data structure to simplify the existing optimal O(N 2 /S + N lg S)-time sorting algorithm for this model.

Optimal In-place Algorithms for Basic Graph Problems

Chakraborty

Sadakane

Satti

2020

We present linear time in-place algorithms for several fundamental graph problems including the well-known graph search methods (like depth-first search, breadth-first search, maximum cardinality search), connectivity problems (like biconnectivity, 2-edge connectivity), decomposition problem (like chain decomposition) among various others, improving the running time (by polynomial multiplicative factor) of the recent results of Chakraborty et al. [ESA, 2018] who designed O(n 3 lg n) time in-place algorithms for some of the above mentioned problems. The running times of all our algorithms are essentially optimal as they run in linear time. One of the main ideas behind obtaining these algorithms is the detection and careful exploitation of sortedness present in the input representation for any graph without loss of generality. This observation alone is powerful enough to design some basic linear time in-place algorithms, but more non-trivial graph problems require extra techniques which, we believe, may find other applications while designing in-place algorithms for different graph problems in future.