NAG CL Interface
m01ndc (realvec_​vec_​search)

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1 Purpose

m01ndc searches a strictly ordered vector of double numbers and returns the indices of the largest numbers in the vector which are smaller than or equal to the sought-after items.

2 Specification

The function may be called by the names: m01ndc or nag_sort_realvec_vec_search.

3 Description

m01ndc searches an array, X, of n strictly-increasing double numbers, Xi<Xi+1, i=1(n-1), for the elements of an unordered array of m double numbers, Z. If a value equal to a sought-after item Zi is not found in X, the index of the immediate lower value is returned. If Zi is less than X1, −1 is returned.
The function implements the direct search algorithm presented as Algorithm 8 in Cannizzo (2018). It precomputes a scalar, H, and a reference vector of indices, K, that it uses to speed up searches of X. The length of K is a function of the number of entries in X and their values, and the function provides a mechanism to compute what this length should be.
If the amount of memory required for K is infeasible, m01ndc implements offset-based binary search (Algorithm 5 in Cannizzo (2018)) as an alternative that does not require K to be used.
See Section 9 for more information on the relative time complexities of the two search algorithms.

4 References

Cannizzo F (2018) A fast and vectorizable alternative to binary search in O(1) with wide applicability to arrays of floating point numbers Journal of Parallel and Distributed Computing 113 37–54

5 Arguments

1: validate Nag_Boolean Input
On entry: if validate is set to Nag_TRUE argument checking will be performed. If validate is set to Nag_FALSE m01ndc will be called without argument checking (which includes checking that array rv is sorted in strictly ascending order). See Section 9 for further details.
2: mode Integer Input
On entry: a code for selecting the operation to be performed by the function. The first call to m01ndc must be made with mode=0. This must be followed by a second setup call with mode=1 or a combined setup and search call with mode=4. Thereafter repeated searches of rv can be made through repeated calls with mode=2.
mode=0
Compute h and lk. Following a call with mode=0, k must be allocated to hold lk elements and then the function must be called again with mode=1 or 4.
mode=1
Set up k (the reference vector) using the values of h and lk returned from a prior call to m01ndc with mode=0.
mode=2
Direct search using h and k set up in prior calls to m01ndc with mode=0, 1 or 4.
mode=3
Search using offset-based binary search, which does not require h or k to be used.
mode=4
Set up as with mode=1 and follow by a single direct search as with mode=2.
Constraint: mode=0, 1, 2, 3 or 4.
3: rv[n] const double Input
On entry: X, the double values to be searched.
Constraint: elements of rv must be sorted in strictly ascending order.
4: n Integer Input
On entry: n, the number of double values to be searched.
Constraint: n2.
5: m1 Integer Input
On entry: the index of the first element of rv to be searched.
Constraint: m10.
6: m2 Integer Input
On entry: the index of the last element of rv to be searched.
Constraint: m1m2n-1.
7: item[m] const double Input
On entry: Z, the sought-after items.
8: m Integer Input
On entry: m, the number of items sought.
Constraint: m1.
9: idx[m] Integer Output
On exit: if mode0, idx[i-1] contains the index of the largest entry in rv which is smaller than or equal to item[i-1].
10: h double * Communication
On entry: H, the reference scalar required by the direct search algorithm. If mode=0, m01ndc calculates the value of h required by the direct search function for the given values in rv. Otherwise, the value of h as declared in the function from which m01ndc is called. If mode=0, h is not referenced.
On exit: the value of h required by the direct search function for the given values in rv.
Constraint: if mode=1, 2 or 4, h must be unchanged from the previous call to m01ndc.
11: k[lk] Integer Communication Array
On entry: if mode=2, k, the reference vector from the previous call to m01ndc. If mode=0 or 3, k is not referenced and may be NULL.
On exit: if mode=1, 2 or 4, the reference vector.
Constraint: if mode=2, k must be unchanged from the previous call to m01ndc.
12: lk Integer * Input/Output
On entry: if mode=0, m01ndc calculates the dimension of k required by the direct search function for the given values in rv. Otherwise, the dimension of the array k as declared in the function from which m01ndc is called. If mode=3, lk is not referenced.
On exit: if mode=0, the dimension of the array k required by the direct search function for the given values in rv. Otherwise, the dimension of the array k as declared in the function from which m01ndc is called.
Constraint: if mode=1, 2 or 4, lk must be unchanged from the previous call to m01ndc.
13: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
Either m01ndc was not called with mode=0, 1 or 4, or the values of h, k or lk may have become corrupted.
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, m1=value.
Constraint: m10.
On entry, mode=value.
Constraint: mode=0, 1, 2, 3 or 4.
On entry, n=value.
Constraint: n2.
NE_INT_2
On entry, m1=value, m2=value, n=value.
Constraint: m1m2n-1.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING
On entry, rv must be sorted in strictly ascending order: rv​ element ​value>​ element ​value.
NE_OVERFLOW
On entry, the values in rv are such that the required value of lk would overflow the current platform's largest positive integer value. This error should only appear when m01ndc is called with mode=0 (i.e., when the function is asked to calculate the required value of lk for the given rv).

7 Accuracy

Not applicable.

8 Parallelism and Performance

m01ndc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The argument validate should be used with caution. Set it to Nag_FALSE only if you are confident that the other arguments are correct, in particular that array rv is in fact arranged in strictly ascending order. If you wish to search the same array rv many times, you are recommended to set validate to Nag_TRUE on the first call of m01ndc and to Nag_FALSE on subsequent calls, in order to minimize the amount of time spent checking rv, which may be significant if rv is large.
The time taken by m01ndc to construct the reference vector (i.e., when the function is called with mode=1 or 4) is O(n). Note that the values stored in k depend on all entries of rv, and not just those in the interval [m1,m2]. Thereafter, searching for m items using direct search (i.e., when the function is called with mode=2 or 4) is O(m). In contrast offset-based binary search (i.e., when the function is called with mode=3), does not require construction of the reference vector and has time complexity O(mlog(n)). Although this is the same as classical binary search, offset-based binary search may be faster in practice because the implementation does not feature conditional branches. Direct search is preferable when the overhead of constructing the reference vector can be offset by conducting multiple searches on an unchanged rv.

10 Example

This example reads a list of double numbers and a list of sought-after items and performs the search for these items. The example demonstrates how to use m01ndc for both direct search (i.e., mode=4) and offset-based binary search (mode=3).

10.1 Program Text

Program Text (m01ndce.c)

10.2 Program Data

Program Data (m01ndce.d)

10.3 Program Results

Program Results (m01ndce.r)