ALGO-1 区间k大数查询

1. Introduction

This article discusses the ALGO-1 problem of finding the k-th largest number in a given interval. This problem arises in various scenarios where we need to quickly query the k-th largest number within a range, such as in statistical analysis, data processing, and algorithmic optimizations. ALGO-1 provides an efficient solution for this problem using a data structure called interval trees.

2. ALGO-1: Interval K-th Largest Number Query

An interval tree is a specialized data structure that allows efficient querying of the k-th largest number in a given interval. It is constructed by recursively splitting the input array into smaller intervals and storing the maximum number within each interval. This allows for quick identification of the k-th largest number within any given interval.

2.1 Construction of Interval Tree

To construct the interval tree, we first divide the input array into equal-sized intervals. Then, for each interval, we calculate the maximum number and store it in the interval tree. This process is repeated recursively for each interval until we reach the base case of having single-element intervals in the tree.

The construction of the interval tree can be described by the following steps:

function constructIntervalTree(array, start, end):
    if start == end:
        tree[node] = array[start]
        return
    mid = (start + end) / 2
    constructIntervalTree(array, start, mid)  // Left child
    constructIntervalTree(array, mid+1, end)  // Right child
    tree[node] = max(tree[left_child], tree[right_child])

In the above code, "array" represents the input array, "start" and "end" represent the start and end indices of the current interval, and "node" represents the current node in the interval tree. The "left_child" and "right_child" variables represent the indices of the left and right child nodes, respectively.

2.2 Querying the K-th Largest Number

Once the interval tree is constructed, we can easily query the k-th largest number within a given interval. This can be done by traversing the interval tree and comparing the desired k-th largest number with the maximum number stored in each interval.

The process of querying the k-th largest number can be described by the following steps:

function queryKthLargest(node, start, end, left, right, k):
    if left > end or right < start:
        return 0   // Invalid interval, return 0
    if left <= start and right >= end:
        return tree[node]   // Fully covered interval, return maximum number
    mid = (start + end) / 2
    left_max = queryKthLargest(left_child, start, mid, left, right, k)
    right_max = queryKthLargest(right_child, mid+1, end, left, right, k)
    return max(left_max, right_max)

In the above code, "left" and "right" represent the start and end indices of the desired interval, and "k" represents the desired k-th largest number. The function recursively checks if the current interval is fully covered by the desired interval. If so, it returns the maximum number stored in that interval. Otherwise, it splits the current interval into left and right child intervals and continues the query process.

3. Conclusion

In conclusion, the ALGO-1 problem of finding the k-th largest number in a given interval can be efficiently solved using interval trees. The construction of the interval tree allows for quick identification of the k-th largest number, while the querying process efficiently traverses the tree to find the desired number. This algorithm has numerous applications in various fields and provides a scalable solution for interval-based k-th largest number queries.