Selection Algorithm - Using Data Structures To Select in Sublinear Time

Using Data Structures To Select in Sublinear Time

Given an unorganized list of data, linear time (Ω(n)) is required to find the minimum element, because we have to examine every element (otherwise, we might miss it). If we organize the list, for example by keeping it sorted at all times, then selecting the kth largest element is trivial, but then insertion requires linear time, as do other operations such as combining two lists.

The strategy to find an order statistic in sublinear time is to store the data in an organized fashion using suitable data structures that facilitate the selection. Two such data structures are tree-based structures and frequency tables.

When only the minimum (or maximum) is needed, a good approach is to use a heap, which is able to find the minimum (or maximum) element in constant time, while all other operations, including insertion, are O(log n) or better. More generally, a self-balancing binary search tree can easily be augmented to make it possible to both insert an element and find the kth largest element in O(log n) time. We simply store in each node a count of how many descendants it has, and use this to determine which path to follow. The information can be updated efficiently since adding a node only affects the counts of its O(log n) ancestors, and tree rotations only affect the counts of the nodes involved in the rotation.

Another simple strategy is based on some of the same concepts as the hash table. When we know the range of values beforehand, we can divide that range into h subintervals and assign these to h buckets. When we insert an element, we add it to the bucket corresponding to the interval it falls in. To find the minimum or maximum element, we scan from the beginning or end for the first nonempty bucket and find the minimum or maximum element in that bucket. In general, to find the kth element, we maintain a count of the number of elements in each bucket, then scan the buckets from left to right adding up counts until we find the bucket containing the desired element, then use the expected linear-time algorithm to find the correct element in that bucket.

If we choose h of size roughly sqrt(n), and the input is close to uniformly distributed, this scheme can perform selections in expected O(sqrt(n)) time. Unfortunately, this strategy is also sensitive to clustering of elements in a narrow interval, which may result in buckets with large numbers of elements (clustering can be eliminated through a good hash function, but finding the element with the kth largest hash value isn't very useful). Additionally, like hash tables this structure requires table resizings to maintain efficiency as elements are added and n becomes much larger than h2. A useful case of this is finding an order statistic or extremum in a finite range of data. Using above table with bucket interval 1 and maintaining counts in each bucket is much superior to other methods. Such hash tables are like frequency tables used to classify the data in descriptive statistics.