15.1-1

Show that equation $\text{(15.4)}$ follows from equation $\text{(15.3)}$ and the initial condition $T(0) = 1$.

  • For $n = 0$, this holds since $2^0 = 1$.

  • For $n > 0$, substituting into the recurrence, we have

    $$ \begin{aligned} T(n) & = 1 + \sum_{j = 0}^{n - 1} 2^j \\ & = 1 + (2^n - 1) \\ & = 2^n. \end{aligned} $$

15.1-2

Show, by means of a counterexample, that the following "greedy" strategy does not always determine an optimal way to cut rods. Define the density of a rod of length $i$ to be $p_i / i$, that is, its value per inch. The greedy strategy for a rod of length $n$ cuts off a first piece of length $i$, where $1 \le i \le n$, having maximum density. It then continues by applying the greedy strategy to the remaining piece of length $n - i$.

The counterexample:

$$ \begin{array}{c|cccc} \text{length $i$} & 1 & 2 & 3 & 4 \\ \hline \text{price $p_i$} & 1 & 20 & 33 & 36 \\ p_i / i & 1 & 10 & 11 & 9 \end{array} $$

15.1-3

Consider a modification of the rod-cutting problem in which, in addition to a price $p_i$ for each rod, each cut incurs a fixed cost of $c$. The revenue associated with a solution is now the sum of the prices of the pieces minus the costs of making the cuts. Give a dynamic-programming algorithm to solve this modified problem.

We can modify $\text{BOTTOM-UP-CUT-ROD}$ algorithm from section 15.1 as follows:

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MODIFIED-CUT-ROD(p, n, c)
    let r[0..n] be a new array
    r[0] = 0
    for j = 1 to n
        q = p[j]
        for i = 1 to j - 1
            q = max(q, p[i] + r[j - i] - c)
        r[j] = q
    return r[n]

We need to account for cost $c$ on every iteration of the loop in lines 5-6 but the last one, when $i = j$ (no cuts).

We make the loop run to $j - 1$ instead of $j$, make sure $c$ is subtracted from thecandidate revenue in line 6, then pick the greater of current best revenue $q$ and $p[j]$ (no cuts) in line 7.

15.1-4

Modify $\text{MEMOIZED-CUT-ROD}$ to return not only the value but the actual solution, too.

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MEMOIZED-CUT-ROD(p, n)
    let r[0..n] and s[0..n] be new arrays
    for i = 0 to n
        r[i] = -∞
    (val, s) = MEMOIZED-CUT-ROD-AUX(p, n, r, s)
    print "The optimal value is" val "and the cuts are at"
    j = n
    while j > 0
        print s[j]
        j = j - s[j]

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MEMOIZED-CUT-ROD-AUX(p, n, r, s)
    if r[n] ≥ 0
        return r[n]
    if n == 0
        q = 0
    else q = -∞
        for i = 1 to n
            (val, s) = MEMOIZED-CUT-ROD-AUX(p, n - i, r, s)
            if q < p[i] + val
                q = p[i] + val
                s[n] = i
    r[n] = q
    return (q, s)

15.1-5

The Fibonacci numbers are defined by recurrence $\text{(3.22)}$. Give an $O(n)$-time dynamic-programming algorithm to compute the nth Fibonacci number. Draw the subproblem graph. How many vertices and edges are in the graph?

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FIBONACCI(n)
    let fib[0..n] be a new array
    fib[0] = 1
    fib[1] = 1
    for i = 2 to n
        fib[i] = fib[i - 1] + fib[i - 2]
    return fib[n]

There are $n + 1$ vertices in the subproblem graph, i.e., $v_0, v_1, \dots, v_n$.

  • For $v_0, v_1$, each has $0$ leaving edge.
  • For $v_2, v_3, \dots, v_n$, each has $2$ leaving edges.

Thus, there are $2n - 2$ edges in the subproblem graph.