Priority Queues and Sorting for Read-Only Data

Asano, Tomohiko; Elmasry, Amr; Katajainen, Jyrki

doi:10.1007/978-3-642-38236-9_4

“…The time-space trade-off for sorting is known to be Θ(N 2 /S +N lg S) [2,19], where S is the size of the workspace in bits, lg N ≤ S ≤ N/ lg N . The optimal bound for sorting can even be realized using a natural priority-queue-based algorithm [1].…”

Section: Discussionmentioning

confidence: 99%

Selection from read-only memory with limited workspace

Elmasry

¹

,

Juhl

²

,

Katajainen

³

et al. 2014

Theoretical Computer Science

Self Cite

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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.

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“…The time-space trade-off for sorting is known to be Θ(N 2 /S +N lg S) [2,19], where S is the size of the workspace in bits, lg N ≤ S ≤ N/ lg N . The optimal bound for sorting can even be realized using a natural priority-queue-based algorithm [1].…”

Section: Discussionmentioning

confidence: 99%

Selection from Read-Only Memory with Limited Workspace

Elmasry

¹

,

Juhl

²

,

Katajainen

³

et al. 2013

Lecture Notes in Computer Science

Self Cite

5

0

View full text Add to dashboard Cite

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.

show abstract

“…Since the seminal paper by Munro and Patterson [102] on sorting and selection for read-only input memory with a limited read-write working space (and writeonly output memory for the case of sorting), a sequence of papers have presented priority queues for sorting in this model. Frederickson [72] achieved a time-space product of O(n 2 lg n) for sorting, and [107] and [7] achieved an O(n 2 ) time-space product for a wide-range of working space sizes, which was proven to be optimal by Beame [9].…”

Section: Priority Queues For Sorting With Limited Spacementioning

confidence: 99%