Lecture 9

Dynamic tables

Question

How large should a hash table or an array be?

Goal: Make the table as small as possible, but large enough so that it won't overflow.

Dynamic tables

• Example: vector in STL

Analysis

Let \(c_i\) be the cost of the \(i\)th insertion.

\[ c_i = \begin{cases} &i \quad \text{if $i - 1 = 2^m$} \\ &1 \quad \text{otherwise} \end{cases} \]

\[ Cost = \sum_{i = 1}^n c_i \leq n + \sum_{j=0}^{\lfloor \lg n\rfloor} 2^j < 3n = \Theta(n) \]

The average cost of each dynamic-table operation is \(\Theta(n)/ n = \Theta(1)\)

Amortized analysis

Definition

An amortized analysis is any strategy for analyzing a sequence of operations to show that the average cost per operation is small, even though a single operation with in the sequence might be expensive.

• the aggregate method
• the accounting method
• the potential method

Aggregate analysis

Incrementing a binary counter

Definition

incrementing a k-bit binary counter that counts upward from 0.

The accounting method

Definition

The credit in the bank must not go negative, we require

\[ \sum_{i=1}^n \hat{c_i} \geq \sum_{i=1}^n c_i \]

for all \(n\)

Accounting analysis of dynamic tables

• Charge an amortized cost of \(\hat{c_i} = 3\) for the \(i\)th insertion.
  • \(1\) pays for the immediate insertion
  • \(2\) is stored for later table doubling

Incrementing a binary counter

• Charge an amortized cost of \(\hat{c_i}\) to set a bit to \(1\)
  • the other for flip back
• At any point in time, every \(1\) in the counter has a dollar of credit on it, and thus we needn't charge anything to reset a bit to 0.

The potential method

Idea

View credit stored as the potential energy of the dynamic set.

• Start with an initial data structure \(D_0\)
• Operation \(i\) transforms \(D_{i - 1}\) to \(D_i\)
• The cost of operation \(i\) is \(c_i\)
• Define a potential function \(\phi\), \(\phi(D_0) = 0\), \(\phi(D_i) \geq 0 \forall i\)
• The amortized cost $\hat{c_i} $ with respect to \(\phi\) is defined to be \(\hat{c_i} = c_i + \phi(D_i) - \phi(D_{i -1})\)

If \(\Delta \phi_i > 0\), Operation \(i\) stores work in the data structure for later use.

The amortized costs bound the true costs

\[\sum_{i=1}^n \hat{c_i} \geq \sum_{i=1}^{n} c_i\]

Dynamic table insertion

\[\phi(D_i) = 2 num_i - size_i\]

It's easy to check that it is a potential function.

We calculate \(\hat{c_i} = c_i + \phi_{D_i} - \phi_{D_{i-1}} = 3\)

Table contraction

When to contract

\(\alpha(T) = num(T)/size(T) < 1/4\) is a good choice

Potential function

\[ \phi(T) = \begin{cases} 2 num[T] - size[T] \quad &\text{if $\alpha(T) \geq 1/2$} \\ size[T]/2 -num[T] \quad &\text{if $\alpha(T) < 1/2$} \end{cases} \]

Incrementing a binary counter

We define the potential of the counter after the \(i\)th Increament operation to be \(b_i\), the number of 1's in the counter after the \(i\)th operation

Fibonacci Heaps