Unit 4: Recurrence Relations, Generating Functions & Divide-and-Conquer

1. General Form of Linear Homogeneous Recurrences

A linear homogeneous recurrence relation of degree $k$ with constant coefficients has the canonical algebraic form:

where $c_1, c_2, \dots, c_k$ are real or complex constants with $c_k \ne 0$, accompanied by $k$ initial conditions $a_0, a_1, \dots, a_{k-1}$.

2. Characteristic Polynomial and Root Multiplicities

We seek non-trivial geometric sequence solutions of the form $a_n = r^n$ with $r \ne 0$. Substituting $a_n = r^n$ into the recurrence relation:

Dividing both sides by $r^{n-k}$ yields the fundamental characteristic equation:

Theorem 4.1 (Distinct Roots Solution): If the characteristic polynomial has $k$ distinct real or complex roots $r_1, r_2, \dots, r_k$, then the general solution is:

where constants $\alpha_1, \dots, \alpha_k$ are uniquely determined by the initial conditions via the non-singular Vandermonde matrix:

Theorem 4.2 (Multiple Roots / Degeneracy): If a characteristic root $r_j$ occurs with multiplicity $m_j > 1$, then its contribution to the general solution is:

Summing across all distinct roots yields the full $k$-parameter vector space of solutions.

1. Structure of Linear Non-Homogeneous Recurrences

A linear non-homogeneous recurrence relation has the form:

where $F(n)$ is a non-zero driving function.

Theorem 4.3 (Superposition Principle): The general solution of the non-homogeneous recurrence is the sum:

where $a_n^{(h)}$ is the general solution of the associated homogeneous equation, and $a_n^{(p)}$ is any particular solution of the non-homogeneous equation.

2. Method of Undetermined Coefficients

For standard forcing functions, trial particular solutions $a_n^{(p)}$ are selected based on the form of $F(n)$:

| Forcing Term $F(n)$ | Root Condition | Form of Particular Solution $a_n^{(p)}$ | | :--- | :--- | :--- | | Polynomial $P_d(n)$ of degree $d$ | $r = 1$ is not a root of char. eq. | $Q_d(n) = A_0 + A_1 n + \dots + A_d n^d$ | | Polynomial $P_d(n)$ of degree $d$ | $r = 1$ is a root of multiplicity $s$ | $n^s Q_d(n) = n^s (A_0 + A_1 n + \dots + A_d n^d)$ | | Exponential $c \cdot \lambda^n$ | $\lambda$ is not a characteristic root | $A \cdot \lambda^n$ | | Exponential $c \cdot \lambda^n$ | $\lambda$ is a root of multiplicity $s$ | $A \cdot n^s \lambda^n$ | | Mixed $P_d(n) \cdot \lambda^n$ | $\lambda$ is a root of multiplicity $s$ | $n^s (A_0 + A_1 n + \dots + A_d n^d) \lambda^n$ |

The unknown coefficients $A_i$ are determined by substituting $a_n^{(p)}$ directly into the recurrence equation and equating corresponding powers of $n$ and $\lambda^n$.

1. Divide-and-Conquer Algorithm Complexity

Many foundational computer science algorithms (such as MergeSort, Karatsuba Integer Multiplication, and Strassen's Matrix Multiplication) divide a problem of size $n$ into $a$ subproblems of size $n/b$, solve them recursively, and combine their results in $f(n)$ work:

where $a \ge 1$, $b > 1$, and $f(n) \ge 0$ asymptotically.

2. The Master Theorem

The critical threshold exponent is:

which represents the asymptotic growth rate of work done at the leaf level of the recursion tree.

Theorem 4.4 (Master Theorem for Divide-and-Conquer): Let $T(n) = a T(n/b) + f(n)$.

1. Case 1 (Leaf Dominant / Subproblems Dominate):

If $f(n) = O(n^{\log_b a - \epsilon})$ for some constant $\epsilon > 0$, then:

1. Case 2 (Balanced / Even Work Across Tree Levels):

If $f(n) = \Theta(n^{\log_b a} \log^k n)$ for some integer $k \ge 0$, then:

(Standard case $k=0$ yields $T(n) = \Theta(n^{\log_b a} \log n)$).

1. Case 3 (Root Dominant / Division & Combine Work Dominates):

If $f(n) = \Omega(n^{\log_b a + \epsilon})$ for some constant $\epsilon > 0$, and if $f(n)$ satisfies the regularity condition:

then:

1. Formal Power Series and Ordinary Generating Functions

The ordinary generating function (OGF) of a sequence $(a_0, a_1, a_2, \dots)$ is the formal power series:

In formal algebraic power series, $x$ serves as a mathematical placeholder; questions of analytic radius of convergence are secondary to algebraic identities.

Fundamental Algebraic Operations:

• Sum: $\sum (a_n + b_n) x^n = A(x) + B(x)$.
• Cauchy Convolution (Product):

• Shifting:

• Differentiation:

2. The Catalan Sequence via Generating Functions

The Catalan numbers count the number of valid Dyck paths, binary search trees with $n$ nodes, and ways to parenthesize a string of $n+1$ factors. They satisfy Segner's non-linear convolution recurrence:

Multiplying by $x^n$ and summing over $n \ge 1$:

Thus, $C(x)$ satisfies the quadratic equation:

Solving via the quadratic formula:

(choosing the negative square root to satisfy $C(0) = \lim_{x \to 0} C(x) = 1$). Expanding via the generalized binomial theorem yields the celebrated formula:

1. Exponential Generating Functions (EGF)

When counting ordered structures (such as labeled graphs, surjections, or permutations with restrictions), the exponential generating function (EGF) is standard:

Product Rule for EGF:

The binomial coefficient $\binom{n}{k}$ automatically counts the ways to partition $n$ labeled elements between the two substructures!

2. Interactive Recurrence Tree Explorer

The interactive simulation below renders the recursion tree for any divide-and-conquer parameters $a, b, f(n)$:

• Visual Depth Progression: Displays individual tree levels, node counts ($a^k$), subproblem dimensions ($n/b^k$), and local non-recursive overhead $a^k f(n/b^k)$.
• Dynamic Regime Classifier: Highlights whether the recurrence belongs to Case 1 (Leaf Dominant), Case 2 (Balanced), or Case 3 (Root Dominant), displaying the exact Master Theorem complexity bound.