An Analysis of Las Vegas Algorithms for Linear Search

In this paper, we analyze two Las Vegas algorithms for the problem of checking if a search item is present in an unordered array. This problem, henceforth called the Linear Search (LS) problem, is well-studied and has applications in a host of domains, including computer graphics and computational biology. While the problem can be solved deterministically by scanning the entire array, this becomes impractical for large datasets that cannot fit into the main memory. To address this issue, algorithms that operate on a portion of the input array are necessary. Likewise, minimizing array accesses in computational biology is crucial, due to the high cost of accessing an element. This paper presents two Las Vegas algorithms for the LS problem that are optimal in their respective models. We show that a partition-based Las Vegas algorithm for the LS problem (with an additional condition that will be discussed) with one bit must have expected comparison complexity at least [Formula: see text]. We also prove that any Las Vegas algorithm for the LS problem with an arbitrary number of random bits has expected comparison complexity at least [Formula: see text]. It is important to note the latter of our two algorithms is analyzed in an extremely restrictive computational model. Despite the limitations on the computational model, our approach is optimal over all Las Vegas algorithms, when the number of permitted random bits is unbounded. Finally, we analyze the expected comparison complexity of our approach when the partition size is an arbitrary value [Formula: see text], for [Formula: see text]. Our algorithms offer an efficient approach for the LS problem in various practical applications.