CBMS-2019-09

题目来源：[[做题/文字版题库/CBMS/2019#Question 9|2019#Question 9]] 日期：2024-07-26 题目主题：CS-算法-归并排序

解题思路

这道题目主要考察归并排序算法及其复杂度分析，尤其是并行计算的应用。题目分为五个部分，从简单的两元素排序开始，到合并两个已排序数组，再到复杂度分析和并行计算应用。

Solution

1. Sorting an array of two elements

For $n=1$, the array has only two elements, $a_1$ and $a_2$. We can sort this array with a simple comparison and swap if needed.

Algorithm:

1. Compare $a_1$ and $a_2$.
2. If $a_1>a_2$, swap them.

Pseudocode:

if a_1 > a_2 then
    swap(a_1, a_2)

2. Merging two sorted arrays

Given two sorted arrays $\mathbf{x}=[x_1,x_2,\dots ,x_p]$ and $\mathbf{y}=[y_1,y_2,\dots ,y_q]$, we merge them into a single sorted array $\mathbf{z}=[z_1,z_2,\dots ,z_{p+q}]$.

Algorithm:

1. Initialize pointers $i$, $j$, and $k$ to $1$.
2. While both arrays have elements to be compared:
   • Compare $\mathbf{x}[i]$ and $\mathbf{y}[j]$.
   • Append the smaller element to $\mathbf{z}[k]$ and increment the corresponding pointer.
   • Increment $k$.
3. If one array is exhausted, append the remaining elements of the other array to $\mathbf{z}$.

Pseudocode:

i, j, k = 1, 1, 1
while i <= p and j <= q do
    if x_i < y_j then
        z_k = x_i
        i = i + 1
    else
        z_k = y_j
        j = j + 1
    k = k + 1
 
while i <= p do
    z_k = x_i
    i = i + 1
    k = k + 1
 
while j <= q do
    z_k = y_j
    j = j + 1
    k = k + 1

The time complexity of this algorithm is $O(p+q)$.

3. Recurrence for $T(2^n)$

We sort the first and second halves of the array separately and then merge them. The recurrence relation is:

$$T(2^n)=2T(2^{n-1})+O(2^n)$$

The $O(2^n)$ term comes from the merging step.

To solve this recurrence, we can use the Master Theorem for divide-and-conquer recurrences of the form $T(n)=aT(n/b)+f(n)$. Here, $a=2$, $b=2$, and $f(n)=O(n)$.

According to the Master Theorem:

• If $f(n)=O(n^{{\log }_{b}a})$, then $T(n)=O(n^{{\log }_{b}a}\log n)$.
• ${\log }_{b}a={\log }_{2}2=1$.

Thus, $f(n)=O(n)$ matches $O(n^{{\log }_{b}a})$. Therefore,

$$T(2^n)=O(2^n\log 2^n)=O(n{2}^n)$$

4. Finding $t$ in $O(\log (p+q))$ time

We need to find the position $t$ such that the first $\lceil \frac{p+q}{2}\rceil$ elements of the merged array come from the first $t$ elements of $\mathbf{x}$ and the first $\lceil \frac{p+q}{2}\rceil -t$ elements of $\mathbf{y}$.

Algorithm:

1. Perform a binary search on $\mathbf{x}$ to find $t$.
2. Initialize low and high pointers: $low=0$ and $high=p$.
3. While $low\le high$:
   • Set $t=\frac{low+high}{2}$.
   • Set $s=\lceil \frac{p+q}{2}\rceil -t$.
   • Check the conditions to adjust the pointers:
     • If $\mathbf{x}[t-1]\le \mathbf{y}[s]$ and $\mathbf{y}[s-1]\le \mathbf{x}[t]$, then $t$ is found.
     • If $\mathbf{x}[t-1]>\mathbf{y}[s]$, adjust $high$.
     • Otherwise, adjust $low$.

Pseudocode:

low, high = 1, p+1
while low <= high do
    t = (low + high) / 2
    s = ceil((p + q) / 2) - t
    if x_t <= y_s+1 and y_s <= x_t+1 then
        return t
    else if x_t > y_s+1 then
        high = t - 1
    else
        low = t + 1

The time complexity of this algorithm is $O(\log (p+q))$ due to the binary search.

5. Parallel merge sort complexity

Assuming $O(2^n)$ CPU cores and ignoring synchronization costs, we can sort each half of the array in parallel.

For each level of recursion, the work is divided equally among available cores. Therefore, the time complexity at each level is determined by the depth of the recursion tree.

The depth of the recursion tree is $n$, and each merge operation at each level takes $O(2^n)$ divided by the number of available cores.

Overall Time Complexity:

$$O\left(\frac{2^n\log 2^n}{2^n}\right)=O(\log 2^n)=O(n)$$

Thus, the parallel merge sort algorithm with $O(2^n)$ CPU cores has a time complexity of $O(n)$.

知识点

归并排序二分查找并行计算排序算法

难点思路

在第四部分，找到合适的 $t$ 使得合并的前半部分数组满足特定条件是一个难点。利用二分查找可以有效减少时间复杂度。

解题技巧和信息

• 归并排序是一种分治算法，其时间复杂度为 $O(n\log n)$。
• 合并两个已排序数组的时间复杂度为 $O(p+q)$。
• 利用二分查找可以在 $O(\log n)$ 时间内找到特定位置。
• 并行计算可以显著加速大规模数据的排序。

重点词汇

• merge 合并
• binary search 二分查找
• parallel computation 并行计算

参考资料

1. Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Chapter 2: Getting Started.
2. The Art of Computer Programming, Donald E. Knuth, Volume 3: Sorting and Searching, Section 5.2.4: Merge Sort.