scholarly journals The design of Maple's sum-of-products and POLY data structures for representing mathematical objects

2014 ◽  
Author(s):  
Michael M Monagan
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Computer Algebra Systems ◽  
Improve Performance ◽  
Mathematical Expressions ◽  
Sum Of Products ◽  
Pros And Cons ◽  
Work Done ◽  
Purpose Computer ◽  
Special Purpose Computer

The principal data structure in Maple used to represent polynomials and general mathematical expressions involving functions like sqrt(x), sin x, exp(2x), y'(x) etc., is known to the Maple developers as the sum-of-products data structure. Gaston Gonnet, as the primary author of the Maple kernel, designed and implemented this data structure in the early 80s. As part of the process of simplifying a mathematical formula, he represented every Maple object and every sub-object uniquely in memory. This makes testing for equality, which is used in many operations, very fast. In this article, on occasion of Gaston's retirement, we present details of his design, its pros and cons, and changes we have made to it over the years. One of the cons is the sum-of-products data structure is not nearly as efficient for multiplying multivariate polynomials as other special purpose computer algebra systems. We describe the new data structure called POLY which we added to Maple 17 (released 2013) to improve performance for polynomials in Maple, and recent work done for Maple 18 (released 2014).

scholarly journals The design of Maple's sum-of-products and POLY data structures for representing mathematical objects

2014 ◽  
Author(s):  
Michael M Monagan
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Computer Algebra Systems ◽  
Improve Performance ◽  
Mathematical Expressions ◽  
Sum Of Products ◽  
Pros And Cons ◽  
Work Done ◽  
Purpose Computer ◽  
Special Purpose Computer

The principal data structure in Maple used to represent polynomials and general mathematical expressions involving functions like sqrt(x), sin x, exp(2x), y'(x) etc., is known to the Maple developers as the sum-of-products data structure. Gaston Gonnet, as the primary author of the Maple kernel, designed and implemented this data structure in the early 80s. As part of the process of simplifying a mathematical formula, he represented every Maple object and every sub-object uniquely in memory. This makes testing for equality, which is used in many operations, very fast. In this article, on occasion of Gaston's retirement, we present details of his design, its pros and cons, and changes we have made to it over the years. One of the cons is the sum-of-products data structure is not nearly as efficient for multiplying multivariate polynomials as other special purpose computer algebra systems. We describe the new data structure called POLY which we added to Maple 17 (released 2013) to improve performance for polynomials in Maple, and recent work done for Maple 18 (released 2014).

scholarly journals A Comparative Study about Data Structures Used for Efficient Management of Voxelised Full-Waveform Airborne LiDAR Data during 3D Polygonal Model Creation

2021 ◽  
Vol 13 (4) ◽  
pp. 559
Author(s):  
Milto Miltiadou ◽  
Neill D. F. Campbell ◽  
Darren Cosker ◽  
Michael G. Grant
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Airborne Lidar ◽  
Memory Allocation ◽  
Lidar Data ◽  
Full Waveform ◽  
Waveform Lidar ◽  
Airborne Lidar Data ◽  
Full Waveform Lidar ◽  
Polygonal Model

In this paper, we investigate the performance of six data structures for managing voxelised full-waveform airborne LiDAR data during 3D polygonal model creation. While full-waveform LiDAR data has been available for over a decade, extraction of peak points is the most widely used approach of interpreting them. The increased information stored within the waveform data makes interpretation and handling difficult. It is, therefore, important to research which data structures are more appropriate for storing and interpreting the data. In this paper, we investigate the performance of six data structures while voxelising and interpreting full-waveform LiDAR data for 3D polygonal model creation. The data structures are tested in terms of time efficiency and memory consumption during run-time and are the following: (1) 1D-Array that guarantees coherent memory allocation, (2) Voxel Hashing, which uses a hash table for storing the intensity values (3) Octree (4) Integral Volumes that allows finding the sum of any cuboid area in constant time, (5) Octree Max/Min, which is an upgraded octree and (6) Integral Octree, which is proposed here and it is an attempt to combine the benefits of octrees and Integral Volumes. In this paper, it is shown that Integral Volumes is the more time efficient data structure but it requires the most memory allocation. Furthermore, 1D-Array and Integral Volumes require the allocation of coherent space in memory including the empty voxels, while Voxel Hashing and the octree related data structures do not require to allocate memory for empty voxels. These data structures, therefore, and as shown in the test conducted, allocate less memory. To sum up, there is a need to investigate how the LiDAR data are stored in memory. Each tested data structure has different benefits and downsides; therefore, each application should be examined individually.

scholarly journals Shape Neutral Analysis of Graph-based Data-structures

2018 ◽  
Vol 18 (3-4) ◽  
pp. 470-483 ◽  
Author(s):  
GREGORY J. DUCK ◽  
JOXAN JAFFAR ◽  
ROLAND H. C. YAP
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Program Analysis ◽  
Constraint Handling ◽  
C Programs ◽  
Wide Range ◽  
Target Program ◽  
Target Data ◽  
Structure Graph ◽  
Structure Properties

AbstractMalformed data-structures can lead to runtime errors such as arbitrary memory access or corruption. Despite this, reasoning over data-structure properties for low-level heap manipulating programs remains challenging. In this paper we present a constraint-based program analysis that checks data-structure integrity, w.r.t. given target data-structure properties, as the heap is manipulated by the program. Our approach is to automatically generate a solver for properties using the type definitions from the target program. The generated solver is implemented using a Constraint Handling Rules (CHR) extension of built-in heap, integer and equality solvers. A key property of our program analysis is that the target data-structure properties are shape neutral, i.e., the analysis does not check for properties relating to a given data-structure graph shape, such as doubly-linked-lists versus trees. Nevertheless, the analysis can detect errors in a wide range of data-structure manipulating programs, including those that use lists, trees, DAGs, graphs, etc. We present an implementation that uses the Satisfiability Modulo Constraint Handling Rules (SMCHR) system. Experimental results show that our approach works well for real-world C programs.

Integrating Image Computation in Undergraduate Level Data-Structure Education

1998 ◽  
Vol 12 (08) ◽  
pp. 1071-1080 ◽  
Author(s):  
Sudeep Sarkar ◽  
Dmitry Goldgof
Keyword(s):  
Image Processing ◽  
Image Analysis ◽  
Data Structure ◽  
Data Structures ◽  
Science Curriculum ◽  
Early Stage ◽  
Real Life ◽  
South Florida ◽  
The University ◽  
Structure Education

There is a growing need for expertise both in image analysis and in software engineering. To date, these two areas have been taught separately in an undergraduate computer and information science curriculum. However, we have found that introduction to image analysis can be easily integrated in data-structure courses without detracting from the original goal of teaching data structures. Some of the image processing tasks offer a natural way to introduce basic data structures such as arrays, queues, stacks, trees and hash tables. Not only does this integrated strategy expose the students to image related manipulations at an early stage of the curriculum but it also imparts cohesiveness to the data-structure assignments and brings them closer to real life. In this paper we present a set of programming assignments that integrates undergraduate data-structure education with image processing tasks. These assignments can be incorporated in existing data-structure courses with low time and software overheads. We have used these assignment sets thrice: once in a 10-week duration data-structure course at the University of California, Santa Barbara and the other two times in 15-week duration courses at the University of South Florida, Tampa.

scholarly journals DenseZDD: A Compact and Fast Index for Families of Sets

Algorithms ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 128 ◽  
Author(s):  
Shuhei Denzumi ◽  
Jun Kawahara ◽  
Koji Tsuda ◽  
Hiroki Arimura ◽  
Shin-ichi Minato ◽  
...  
Keyword(s):  
Data Structure ◽  
Data Structures ◽  
Information Integration ◽  
Decision Diagrams ◽  
Web Information Retrieval ◽  
Binary Decision ◽  
Web Information ◽  
Set Operations ◽  
Succinct Data Structure ◽  
The Family

In this article, we propose a succinct data structure of zero-suppressed binary decision diagrams (ZDDs). A ZDD represents sets of combinations efficiently and we can perform various set operations on the ZDD without explicitly extracting combinations. Thanks to these features, ZDDs have been applied to web information retrieval, information integration, and data mining. However, to support rich manipulation of sets of combinations and update ZDDs in the future, ZDDs need too much space, which means that there is still room to be compressed. The paper introduces a new succinct data structure, called DenseZDD, for further compressing a ZDD when we do not need to conduct set operations on the ZDD but want to examine whether a given set is included in the family represented by the ZDD, and count the number of elements in the family. We also propose a hybrid method, which combines DenseZDDs with ordinary ZDDs. By numerical experiments, we show that the sizes of our data structures are three times smaller than those of ordinary ZDDs, and membership operations and random sampling on DenseZDDs are about ten times and three times faster than those on ordinary ZDDs for some datasets, respectively.

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