【矩阵乘法】C++实现外部矩阵乘法

问题描述

​ 使用文件和内存模拟系统缓存，并利用矩阵乘法验证实际和理论情况。

算法思想

设计一个Matrix类，其中Matrix是存在磁盘中的一个二进制文件，类通过保存的矩阵属性来读取磁盘。前八个字节为两个int32，保存矩阵的行列数。

Matrix中有一个buffer成员为读取到的数据缓存，通过pos属性来确定其在矩阵中的位置。其映射逻辑为$row=\lfloor pos÷colSize\rfloor ,col=pos\mathrm{mod}colSize$。而缓存的管理模型则是模仿CPU的内存缓存模型，采用写回法。当读取矩阵中row行col列时，判断逻辑如下。

在缓存内

不再缓存内

没有脏数据

有脏数据

接受row,col变量

判断所需变量是否在缓存内

读取数据

判断缓存内是否有脏数据

从磁盘内读取到缓存

写入脏数据

​ 完成磁盘交互后，其余操作即为正常的矩阵操作

功能模块设计

矩阵类的成员变量，：

    Buffer buffer;
    const int rowSize;
    const int colSize;
    long cacheMissCount; // cache miss 次数
    long cacheCount;     // cache 访问次数
    int pos;             // pos = -1 means invalid buffer
    const int offset = 2 * INT_BYTE;
    fstream file;

​ 矩阵值获取和修改的函数

    int get(int row, int col)
    {
        int index = row * colSize + col;
        assert(index < rowSize * colSize && index >= 0);
        cacheCount++;
        if (index >= pos && index < pos + buffer.size && pos != -1)
            return buffer.get(index - pos);
        else
        {
            cacheMissCount++;
            readBuffer(index);
            return buffer.get(0);
        }
    }
    void set(int row, int col, int val)
    {
        assert(row <= rowSize && col <= colSize);
        if (pos <= indexOf(row, col) && indexOf(row, col) < pos + buffer.size)
            buffer.set(indexOf(row, col) - pos, val);
        else
        {
            cacheMissCount++;
            readBuffer(indexOf(row, col));
            buffer.set(0, val);
        }
        buffer.dirty = true;
    }

​ 磁盘操作函数

    void writeBuffer()
    {
        file.seekp(pos * INT_BYTE + offset, ios::beg);
        for (int i = 0; i < buffer.size; i++)
        {
            file.write((char *)&buffer.get(i), INT_BYTE);
        }
        buffer.dirty = false;
    }

    void readBuffer(int startIndex)
    {
        if (buffer.dirty)
        {
            writeBuffer();
        }

        file.seekg(startIndex * INT_BYTE + offset, ios::beg);
        buffer.size = 0;
        for (int i = 0; i < buffer.capacity && startIndex + i < rowSize * colSize; i++)
        {
            int value;
            file.read((char *)&value, INT_BYTE);
            buffer.set(i, value);
            buffer.size++;
        }
        pos = startIndex;
    }

​ 矩阵乘法函数

    Matrix *multiple_ijk(Matrix &right)
    {
        Matrix *t = new Matrix(rowSize, right.colSize, buffer.capacity);
        int i, j, k;
        for (i = 0; i < rowSize; i++)
        {
            for (j = 0; j < right.colSize; j++)
            {
                for (k = 0; k < right.rowSize; k++)
                {
                    t->set(i, j, t->get(i, j) + get(i, k) * right.get(k, j));
                }
            }
        }
        return t;
    }

    Matrix *multiple_ikj(Matrix &right)
    {
        Matrix *t = new Matrix(rowSize, right.colSize, buffer.capacity);
        int i, j, k;
        for (i = 0; i < rowSize; i++)
        {
            for (k = 0; k < right.rowSize; k++)
            {
                for (j = 0; j < right.colSize; j++)
                {
                    t->set(i, j, t->get(i, j) + get(i, k) * right.get(k, j));
                }
            }
        }
        return t;
    }

测试结果与分析

测试用例

class MatrixTest
{
public:
    void test1()
    {
        Matrix m(10, 10, 3, "m.txt");
        m.randomize();
        cout << m << endl;
        cout << "cache miss:" << m.getCacheMissCount() << endl;
        Matrix m1(10, 10, 3);
        m1.randomize();
        cout << m1 << endl;
        cout << "cache miss:" << m1.getCacheMissCount() << endl;
    }

    void test2()
    {
        Matrix m(10, 10, 3);
        cout << m << endl;
        m.set(0, 0, 9999);
        cout << m << endl;
        m.set(5, 0, 9999);
        cout << m << endl;
        m.set(3, 7, 9999);
        cout << m << endl;
    }
    void test3()
    {
        Matrix m1(2, 3, 5, "m1.txt");
        m1.randomize(100);
        cout << m1 << endl;
        Matrix m2(3, 4, 7, "m2.txt");
        m2.randomize(100);
        cout << m2 << endl;
        Matrix *m3 = m1.multiple_ijk(m2);
        cout << *m3 << endl;
        Matrix *m4 = m1.multiple_ikj(m2);
        cout << *m4 << endl;
        assert(m3->toString() == m4->toString());
    }
    void test4(int n, int cacheSize)
    {
        cout << "cache size is " << cacheSize << endl;
        Matrix m1(n, n, cacheSize, "m1.txt");
        cout << "initial m1(size is " << n << ")finished" << endl;
        Matrix m2(n, n, cacheSize, "m2.txt");
        cout << "initial m2(size is " << n << ")finished" << endl;
        Matrix *m3 = m1.multiple_ijk(m2);
        cout << "m1 ijk m2 finished" << endl;
        cout << "cache coutnt of m1: " << m1.getCacheCount() << ",	cache miss count of m1:" << m1.getCacheMissCount() << endl;
        cout << "cache coutnt of m2: " << m2.getCacheCount() << ",	cache miss count of m2:" << m2.getCacheMissCount() << endl;
        cout << "cache coutnt of m3: " << m3->getCacheCount() << ",	cache miss count of m3:" << m3->getCacheMissCount() << endl;
        cout << endl;
    }
    void test5(int n, int cacheSize)
    {
        cout << "cache size is " << cacheSize << endl;
        Matrix m1(n, n, cacheSize, "m1.txt");
        cout << "initial m1(size is " << n << ")finished" << endl;
        Matrix m2(n, n, cacheSize, "m2.txt");
        cout << "initial m2 finished" << endl;
        Matrix *m3 = m1.multiple_ikj(m2);
        cout << "m1 ijk m2(size is " << n << ")finished" << endl;
        cout << "cache coutnt of m1: " << m1.getCacheCount() << ",	cache miss count of m1:" << m1.getCacheMissCount() << endl;
        cout << "cache coutnt of m2: " << m2.getCacheCount() << ",	cache miss count of m2:" << m2.getCacheMissCount() << endl;
        cout << "cache coutnt of m3: " << m3->getCacheCount() << ",	cache miss count of m3:" << m3->getCacheMissCount() << endl;
        cout << endl;
    }
    void runTest()
    {
        for (int i = 5; i < 35; i += 5)
        {
            for (int j = 1; j < 5; j++)
            {
                test4(i, i / j);
                test5(i, i / j);
            }
        }
    }
};

实验测试

实验分析

假设矩阵为$C=A\times B$，且cache大小小于矩阵的一行或一列,且A、B、C都是大小为n的方阵

对于ijk的情况：

​ 每次读取时发生一次磁盘访问，而若cache当中有脏数据，则有另一次的写入访问。对于矩阵C，因为其作为操作矩阵*（总是进行+=操作）*，所以只要是C矩阵发生了cache miss其实是进行了两次磁盘访问，而对于AB矩阵，发生cache miss 时只发生一次磁盘访问。

​ cache miss 发生次数如下
$$\begin{gathered} C_{cachemiss}=\frac{n^2}{cachesize} \\ {A}_{cachemiss}=\frac{n^3}{cachesize} \\ {B}_{cachemiss}=n^3 \end{gathered}$$
​ 所消耗的总时间
$$\begin{gathered} t_{total}=2T_{diskaccesstime}{C}_{cachemiss}+T_{diskaccesstime}{A}_{cachemiss}+T_{diskaccesstime}{B}_{cachemiss} \\ =\frac{n^3{T}_{diskaccesstime}}{cachesize(1+\frac{1}{n}+cachesize)} \end{gathered}$$
​ 而当顺序化为ijk的时候，B矩阵的cache hit 率有所提高
$$\begin{gathered} C_{cachemiss}=\frac{n^3}{cachesize} \\ {A}_{cachemiss}=\frac{n^2}{cachesize} \\ {B}_{cachemiss}=\frac{n^3}{cachesize} \end{gathered}$$
所消耗的总时间变为了
$$\begin{gathered} t_{total}=2T_{diskaccesstime}{C}_{cachemiss}+T_{diskaccesstime}{A}_{cachemiss}+T_{diskaccesstime}{B}_{cachemiss} \\ =\frac{n^3{T}_{diskaccesstime}}{cachesize(2+\frac{1}{n})} \end{gathered}$$