New and improved search algorithms and precise analysis of their average-case complexity

Highlights

• We propose improved ternary search (ITS) algorithm.

• We also propose a new Binary–Quaternary Search (BQS) algorithm.

• We discuss weak and correct implementations of the binary search (BS) algorithm.

• We calculate average number of comparisons for weak and correct implementations of the BS algorithm precisely.

• We calculate average number of comparisons for the ITS and BQS algorithms precisely.

Abstract

In this paper, we consider the searching problem over ordered sequences. It is well known that Binary Search (BS) algorithm solves this problem with very efficient complexity, namely with the complexity $\theta \left({log}_{2}n\right)$. The developments of the BS algorithm, such as Ternary Search (TS) algorithm do not improve the efficiency. The rapid increase in the amount of data has made the search problem more important than in the past. And this made it important to reduce average number of comparisons in cases where the asymptotic improvement is not achieved. In this paper, we identify and analyze an implementation issue of BS. Depending on the location of the conditional operators, we classify two different implementations for BS which are widely used in the literature. We call these two implementations weak and correct implementations. We calculate precise number of comparisons in average case for both implementations. Moreover, we transform the TS algorithm into an improved ternary search (ITS) algorithm. We also propose a new Binary–Quaternary Search (BQS) algorithm by using a novel dividing strategy. We prove that an average number of comparisons for both presented algorithms ITS and BQS is less than for the case of correct implementation of the BS algorithm. We also provide the experimental results.

Introduction

Searching and sorting problems are classical problems of computer science. Due to excessive increase in the amount of data in recent years, these problems keep attracting the attention of researchers. In our previous work [1], we have made a short summary of the related works about sorting algorithms published recently [2], [3], [4], [5], [6], [7], [8], [9]. The study [10] conducted after our publication proposes two novel sorting algorithms, called as Brownian Motus insertion sort and Clustered Binary Insertion Sort. Both algorithms are based on the concept of classical Insertion Sort. Marszałek [11] describes how to use the parallelization of the sorting processes for the modified method of sorting by merging for large data sets.

Besides of these studies Woźniak et al. [12] modify Merge Sort algorithm for large scale data sets. Marszałek [13] proposes a new recursive version of fast sort algorithm for large data sets. Woźniak et al. [14] examine quick sort algorithm in two versions for large data sets. Dymora et al. [15] calculate the rate of existence of long-term correlations in processing dynamics of the quicksort algorithm basing on Hurst coefficient. Napoli et al. [16] propose the idea of applying the simplified firefly algorithm to search for key-areas in 2D images. Woźniak and Marszałek [17] use classic firefly algorithm to search for special areas in test images. Das and Khilar [18] propose a Randomized Searching Algorithm and compare its performance with the Binary Search and Linear Search Algorithms. They show that the performance of the algorithm lies between Binary Search and Linear Search. Ambainis et al. [19] study the classic binary search problem, with a delay between query and answer. They give upper and lower bounds of the matching depending on the number of queries for the constant delays. Finocchi and Italiano [20] investigate the design and analysis of the sorting and searching algorithms resilient to memory faults. Chadha et al. [21] propose a modification to the binary search algorithm in which it checks the presence of the input element at each iteration. Rahim et al. [22] provide the experimental comparison the linear, binary and interpolation search algorithms by testing to search data with different length with pseudo process approach. Kumar [23] proposes a new quadratic search algorithm based on binary search algorithm and he experimentally shows that this algorithm better than binary search algorithm.

Carmo et al. [24] consider the problem of searching for a given element in a partially ordered set. Bonasera et al. [25] propose an adaptive search algorithm over ordered sets. Proposed by Mohammed et al. [26] hybrid search algorithm on ordered data sets is similar to the adaptive search algorithm. Bender et al. [27] develop a library sort algorithm, which is developed based on insertion sort and binary search (BS) algorithm.

It is well known that BS algorithm is one of the widely used algorithms in computer applications due to obtaining a good performance for different data types and key distributions. It works on the principle of the divide-and-conquer approach [28]. This algorithm is used in solving several problems. For instance, Gao et al. [29] propose a scheduling algorithm for ridesharing using binary search strategy. Hatamlou [30] presents a binary search algorithm for data clustering. BS is a simple and understandable algorithm, although it may contain some tricks in implementation. Donald Knuth emphasized: “Although the basic idea of binary search is comparatively straightforward, the details can be surprisingly tricky” [31]. Most of the implementation issues in the binary search were described in the literature. Pattis [32] notes five implementation errors. The study [33] involves a program to compute the semi-sum of two integers. In turn, this approach solves the problem of overflow that happens in binary search for very large arrays. Bentley discusses some errors in the implementation of the binary search in the section titled the challenge of binary search [34].

In this paper, we discuss two different implementations of the BS algorithm, which we call as weak and correct implementations. We calculate an average number of comparisons for both implementations precisely. We discuss the TS algorithm which is known as slower than BS, and then we present an improved ternary search (ITS) algorithm which is faster than the correct implementation of the BS algorithm. We prove this fact by calculating an average number of comparisons for ITS algorithm precisely. Moreover, we offer a new searching algorithm called as Binary–Quaternary Search (BQS) algorithm. We calculate an average number of comparisons for the BQS algorithm and we show that this algorithm is better than the correct implementation of BS algorithm. Theoretically, BQS slightly shows more average comparisons number compared with presented ITS algorithm.

The rest of the paper is organized as follows: In Section 2 we discuss the weak and correct implementation of the BS algorithm. In this section we also calculate average number of the comparisons for weak and correct implementation of the BS algorithm. In Section 3 we discuss the TS algorithm. In Section 4 we propose ITS algorithm and we calculate average number of comparisons for this algorithm. In Section 5 we develop a new searching algorithm BQS and we find precisely average number of comparisons for BQS algorithm. In Section 6 we compare the implementations of the ITS and BQS algorithms. In Section 7 we demonstrate experimental results and comparison of these searching algorithms. Finally, we summarize our results in Section 8.