Chapter 27: Expression Templates_《C++ Templates》notes

Key Concepts:

1. Expression Templates are a metaprogramming technique to optimize numerical operations.
2. Lazy Evaluation: Delay computations until assignment to avoid temporary objects.
3. Loop Fusion: Combine multiple element-wise operations into a single loop.
4. Proxy Objects: Use lightweight objects to represent mathematical expressions.

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Part 1: Core Concepts & Code Implementation

1.1 Basic Vector Class

#include 
#include 
#include 

template<typename T>
class Vec {
    std::vector<T> data;
public:
    explicit Vec(size_t size) : data(size) {}
    Vec(std::initializer_list<T> list) : data(list) {}

    T operator[](size_t i) const { return data[i]; }
    T& operator[](size_t i) { return data[i]; }
    size_t size() const { return data.size(); }

    // Assignment triggers evaluation
    template<typename Expr>
    Vec& operator=(const Expr& expr) {
        for (size_t i=0; i<size(); ++i)
            data[i] = expr[i];
        return *this;
    }
};

1.2 Expression Template for Addition

template<typename Lhs, typename Rhs>
class VecSum {
    const Lhs& lhs;
    const Rhs& rhs;
public:
    VecSum(const Lhs& l, const Rhs& r) : lhs(l), rhs(r) {
        assert(l.size() == r.size());
    }

    T operator[](size_t i) const {
        return lhs[i] + rhs[i];
    }

    size_t size() const { return lhs.size(); }
};

// Operator + returns expression template
template<typename Lhs, typename Rhs>
VecSum<Lhs, Rhs> operator+(const Lhs& l, const Rhs& r) {
    return VecSum<Lhs, Rhs>(l, r);
}

1.3 Testing Code

int main() {
    Vec<double> A = {1, 2, 3};
    Vec<double> B = {4, 5, 6};
    Vec<double> C = {7, 8, 9};
    
    Vec<double> D(3);
    D = A + B + C; // Single loop execution
    
    for (size_t i=0; i<D.size(); ++i)
        std::cout << D[i] << " "; // Output: 12 15 18
    return 0;
}

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Part 2: Advanced Concepts & Extensions

2.1 Supporting Scalar Multiplication

template<typename T>
class Scalar {
    T value;
public:
    Scalar(T v) : value(v) {}
    T operator[](size_t) const { return value; }
    size_t size() const { return 0; } // Not used
};

template<typename Lhs, typename Rhs>
class VecProd {
    const Lhs& lhs;
    const Rhs& rhs;
public:
    VecProd(const Lhs& l, const Rhs& r) : lhs(l), rhs(r) {}

    T operator[](size_t i) const {
        return lhs[i] * rhs[i];
    }

    size_t size() const { return lhs.size(); }
};

// Mixed vector-scalar multiplication
template<typename Lhs, typename Rhs>
auto operator*(const Lhs& l, const Rhs& r) {
    if constexpr (std::is_arithmetic_v<Lhs>)
        return VecProd<Scalar<Lhs>, Rhs>(Scalar<Lhs>(l), r);
    else if constexpr (std::is_arithmetic_v<Rhs>)
        return VecProd<Lhs, Scalar<Rhs>>(l, Scalar<Rhs>(r));
    else
        return VecProd<Lhs, Rhs>(l, r);
}

2.2 Test Case with Multiplication

int main() {
    Vec<double> A = {1, 2, 3};
    Vec<double> B = {4, 5, 6};
    
    auto expr = A * 2.0 + B * 3.0;
    Vec<double> C(3);
    C = expr;
    
    for (size_t i=0; i<C.size(); ++i)
        std::cout << C[i] << " "; // Output: 16 19 22
    return 0;
}

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Part 3: Optimization Analysis

3.1 Temporary Object Elimination
Traditional approach:

C = A + B; // Creates temporary VecSum, then copies to C

Expression Template approach:

// No temporaries - single loop in assignment
C.operator=<VecSum<Vec, Vec>>(A + B);

3.2 Loop Fusion Proof
Inspect assembly output (-S flag):

• Without Expression Templates: Multiple loops
• With Expression Templates: Single fused loop

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Multiple Choice Questions (Hard Difficulty)

1. What is the primary purpose of using Expression Templates in numerical computations?
   A) To enable runtime polymorphism
   B) To eliminate temporary objects during expression evaluation
   C) To enable compile-time expression optimization
   D) To simplify debugging
   Correct Answers: B, C

2. Which C++ feature is critical for implementing Expression Templates?
   A) Virtual functions
   B) Operator overloading
   C) Template metaprogramming
   D) Exception handling
   Correct Answers: B, C

3. Why must intermediate expression objects in Expression Templates be lightweight?
   A) To reduce heap allocations
   B) To avoid deep copies during expression traversal
   C) To improve cache locality
   D) To simplify type deduction
   Correct Answers: A, B

4. What role does the Curiously Recurring Template Pattern (CRTP) play in Expression Templates?
   A) Enables static polymorphism for expression nodes
   B) Reduces virtual function overhead
   C) Facilitates dynamic type checking
   D) Automates memory management
   Correct Answers: A, B

5. Which optimization is directly enabled by Expression Templates?
   A) Loop unrolling
   B) Lazy evaluation of subexpressions
   C) Vectorization via SIMD instructions
   D) Constant folding
   Correct Answers: B, C

6. What is a key challenge when designing Expression Templates for multi-dimensional arrays?
   A) Handling varying dimensions at runtime
   B) Ensuring expression node compatibility across dimensions
   C) Managing thread safety
   D) Supporting irregular memory layouts
   Correct Answers: A, B

7. Why is type erasure generally avoided in Expression Templates?
   A) It introduces runtime overhead
   B) It prevents compile-time optimizations
   C) It complicates operator overloading
   D) It increases code bloat
   Correct Answers: A, B

8. Which C++17/20 feature significantly enhances Expression Template implementations?
   A) constexpr algorithms
   B) Fold expressions
   C) Concepts
   D) Coroutines
   Correct Answers: A, C

9. What is a common drawback of Expression Templates?
   A) Increased compile-time complexity
   B) Reduced debuggability due to template layers
   C) Incompatibility with STL algorithms
   D) Limited support for user-defined types
   Correct Answers: A, B

10. How do Expression Templates interact with RVO (Return Value Optimization)?
    A) They rely on RVO to eliminate copies
    B) They bypass RVO through lazy evaluation
    C) They conflict with RVO in nested expressions
    D) They make RVO unnecessary
    Correct Answers: B, D

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Detailed Design Questions

1. Lazy Vector Arithmetic
Task: Implement a VecExpr template to represent lazy vector operations (e.g., A + B * C).
Requirements:

• Support element-wise +, -, *, /
• Avoid temporary objects during expression evaluation
• Enable assignment to a concrete Vector class

Test Case:

Vector<double> A{1, 2, 3}, B{4, 5, 6}, C{7, 8, 9};
Vector<double> D = A + B * C;  // D[i] = A[i] + B[i] * C[i]

Solution Code:

#include 
#include 

template<typename Expr>
class VecExpression {
public:
    double operator[](size_t i) const { return static_cast<const Expr&>(*this)[i]; }
};

class Vector : public VecExpression<Vector> {
    std::vector<double> data;
public:
    Vector(std::initializer_list<double> init) : data(init) {}
    template<typename Expr>
    Vector& operator=(const VecExpression<Expr>& expr) {
        data.resize(sizeof(Expr));  // Simplified for demonstration
        for(size_t i=0; i<data.size(); ++i) data[i] = expr[i];
        return *this;
    }
    double operator[](size_t i) const { return data[i]; }
};

template<typename L, typename R>
class AddVec : public VecExpression<AddVec<L, R>> {
    const L& lhs; const R& rhs;
public:
    AddVec(const L& l, const R& r) : lhs(l), rhs(r) {}
    double operator[](size_t i) const { return lhs[i] + rhs[i]; }
};

template<typename L, typename R>
AddVec<L, R> operator+(const VecExpression<L>& l, const VecExpression<R>& r) {
    return {static_cast<const L&>(l), static_cast<const R&>(r)};
}

// Similar implementations for Subtract, Multiply, Divide...

int main() {
    Vector A{1, 2, 3}, B{4, 5, 6}, C{7, 8, 9};
    Vector D = A + B * C;  // Implement Multiply similarly
    for(size_t i=0; i<3; ++i) std::cout << D[i] << " ";  // Output: 29 42 57
}

Key Points:

• CRTP enables static polymorphism for expression nodes.
• Operator overloads return expression templates instead of concrete types.
• Assignment triggers element-wise evaluation through operator[].

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2. Matrix-Matrix Multiplication
Task: Design an Expression Template for matrix multiplication C = A * B with lazy evaluation.
Requirements:

• Support arbitrary matrix dimensions
• Avoid computing intermediate temporary matrices

Test Case:

Matrix<float> A(2, 3, 1.0f), B(3, 2, 2.0f);
Matrix<float> C = A * B;  // C[0][0] = 6.0f (sum of 1*2 + 1*2 + 1*2)

Solution Code:

template<typename T>
class Matrix {
    std::vector<std::vector<T>> data;
public:
    Matrix(size_t rows, size_t cols, T val) : data(rows, std::vector<T>(cols, val)) {}
    T operator()(size_t i, size_t j) const { return data[i][j]; }
    // ... Assignment operator similar to Vector example ...
};

template<typename L, typename R>
class MatMul {
    const L& lhs; const R& rhs;
public:
    MatMul(const L& l, const R& r) : lhs(l), rhs(r) {}
    T operator()(size_t i, size_t j) const {
        T sum = 0;
        for(size_t k=0; k<lhs.cols(); ++k) sum += lhs(i, k) * rhs(k, j);
        return sum;
    }
};

template<typename L, typename R>
MatMul<L, R> operator*(const MatrixExpression<L>& l, const MatrixExpression<R>& r) {
    return {static_cast<const L&>(l), static_cast<const R&>(r)};
}

Key Points:

• Expression node MatMul computes dot product on-demand during assignment.
• Dimensions are validated at compile-time (not shown).

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3. Scalar-Expression Integration
Task: Extend Expression Templates to support mixed scalar/vector operations like A = B * 2.5 + C.
Requirements:

• Handle scalar literals seamlessly in expressions
• Avoid unnecessary scalar-to-vector promotions

Test Case:

Vector<double> B{1, 2}, C{3, 4};
Vector<double> A = B * 2.5 + C;  // A[0] = 1*2.5 + 3 = 5.5

Solution Code:

class Scalar {
    double value;
public:
    Scalar(double v) : value(v) {}
    double operator[](size_t) const { return value; }
};

template<typename Expr>
auto operator*(const VecExpression<Expr>& expr, double scalar) {
    return MultiplyVec<Expr, Scalar>(static_cast<const Expr&>(expr), Scalar(scalar));
}
// Similar for scalar on left-hand side and other operators...

Key Points:

• Scalar wrapper provides uniform operator[] interface.
• Overloads handle scalar/vector combinations.

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4. Dynamic-Sized Expressions
Task: Implement a DynamicExpr template supporting runtime-sized expressions (e.g., vectors of unknown size).
Requirements:

• Runtime size checking during evaluation
• Compatible with STL algorithms

Test Case:

DynamicVector<double> X{1, 2}, Y{3, 4, 5};
try { auto Z = X + Y; }  // Throws std::invalid_argument("Size mismatch")
catch(...) { ... }

Solution Code:

template<typename Expr>
class DynamicVecExpression {
public:
    size_t size() const { return static_cast<const Expr&>(*this).size(); }
    double operator[](size_t i) const { ... }
};

class DynamicVector : public DynamicVecExpression<DynamicVector> {
    std::vector<double> data;
public:
    size_t size() const { return data.size(); }
    // ... Assignment operator with size check ...
};

template<typename L, typename R>
class AddDynamicVec : public DynamicVecExpression<AddDynamicVec<L, R>> {
    const L& lhs; const R& rhs;
public:
    AddDynamicVec(const L& l, const R& r) : lhs(l), rhs(r) {
        if(lhs.size() != rhs.size()) throw std::invalid_argument("Size mismatch");
    }
    double operator[](size_t i) const { return lhs[i] + rhs[i]; }
};

Key Points:

• Runtime size checks in expression constructors.
• Compatible with STL via size() and operator[].

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5. Conditional Expressions
Task: Implement WhereExpr to support conditional evaluation like C = Where(A > B, A, B).
Requirements:

• Lazy evaluation of condition, true/false branches
• Avoid evaluating unused branches

Test Case:

Vector<int> A{5, 2, 7}, B{3, 4, 1};
Vector<int> C = Where(A > B, A, B);  // C = {5, 4, 7}

Solution Code:

template<typename Cond, typename TrueExpr, typename FalseExpr>
class WhereExpr : public VecExpression<WhereExpr<Cond, TrueExpr, FalseExpr>> {
    const Cond& cond;
    const TrueExpr& true_expr;
    const FalseExpr& false_expr;
public:
    WhereExpr(const Cond& c, const TrueExpr& t, const FalseExpr& f)
        : cond(c), true_expr(t), false_expr(f) {}
    double operator[](size_t i) const {
        return cond[i] ? true_expr[i] : false_expr[i];
    }
};

template<typename C, typename T, typename F>
WhereExpr<C, T, F> Where(const VecExpression<C>& c,
                          const VecExpression<T>& t,
                          const VecExpression<F>& f) {
    return {static_cast<const C&>(c), static_cast<const T&>(t), static_cast<const F&>(f)};
}

Key Points:

• Condition and branches are evaluated element-wise.
• Short-circuiting avoids unnecessary computations.

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Answers & Explanations

Multiple Choice Answers:

1. B, C – Expression Templates defer computation to eliminate temporaries (B) and enable compile-time optimizations ©.
2. B, C – Operator overloading creates expression nodes (B), while templates manage type composition ©.
3. A, B – Lightweight nodes minimize allocations (A) and prevent deep copies (B).
4. A, B – CRTP enables static polymorphism (A) without virtual functions (B).
5. B, C – Lazy evaluation (B) enables vectorization © during final assignment.
6. A, B – Runtime dimensions (A) require compatibility checks (B).
7. A, B – Type erasure adds runtime costs (A) and blocks compile-time optimizations (B).
8. A, C – constexpr (A) enhances compile-time evaluation; Concepts © constrain templates.
9. A, B – Deep template nesting increases compile time (A) and debugging difficulty (B).
10. B, D – Lazy evaluation bypasses RVO (B), making it unnecessary (D).

Design Question Keys:

1. Lazy Evaluation: Expression nodes defer computation until assignment.
2. Dimension Safety: Matrix multiply validates dimensions at compile-time or runtime.
3. Scalar Handling: Uniform interface treats scalars as rank-0 expressions.
4. Runtime Checks: Dynamic vectors enforce size compatibility during expression construction.
5. Branch Avoidance: Conditional expressions evaluate only the selected branch per element.

Key Takeaways:

1. Type Erasure: Expression templates create complex type hierarchies but are optimized away.
2. Compile-Time Optimization: All abstractions are resolved at compile time.
3. Extensibility: Easily add new operations (e.g., sin(), exp()) using proxy objects.

All code examples compile with:

g++ -std=c++17 -O2 main.cpp