AlgorithmsI Exercises: Analysis of Algorithms

Question 1

Suppose that you time a program as a function of N and produce
the following table.

N seconds
-------------------
1024 0.000
2048 0.001
4096 0.007
8192 0.029
16384 0.121
32768 0.519
65536 2.156
131072 9.182
262144 38.784
524288 164.585
1048576 698.592
2097152 2958.141

Estimate the order of growth of the running time as a function of N.
Assume that the running time obeys a power law T(N) ~ a N^b. For your
answer, enter the constant b. Your answer will be marked as correct
if it is within 1% of the target answer - we recommend using
two digits after the decimal separator, e.g., 2.34.

Answer: 2.08

Question Explanation

The theoretical order-of-growth is N ^ (25/12) = 2.08
The empirical order-of-growth is N ^ (log_2 ratio)

log_2
N seconds ratio ratio
---------------------------------------
1024 0.000 - -
2048 0.001 - -
4096 0.007 7.00 2.81
8192 0.029 4.14 2.05
16384 0.121 4.17 2.06
32768 0.519 4.29 2.10
65536 2.156 4.15 2.05
131072 9.182 4.26 2.09
262144 38.784 4.22 2.08
524288 164.585 4.24 2.09
1048576 698.592 4.24 2.09
2097152 2958.141 4.23 2.08

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Question 2

What is the order of growth of the worst case running time of the following code fragment
as a function of N?

int sum = 0;
for (int i = 0; i < N; i++)
    for (int j = i+1; j < N; j++)
　　　　for (int k = j+1; k < N; k++)
　　　　　　for (int h = k+1; h < N; h++)
　　　　　　　　　sum++;
Answer: N^4

int sum = 0;

for (int i = N*N; i > 1; i = i/2)
     sum++;

Answer: logN 

The i loops iterates ~ lg (N^2) ~ 2 lg N times.

int sum = 0;
for (int i = 1; i <= N*N; i = i*2)
    for (int j = 0; j < i; j++)
        sum++;

Answer: N^2

The body of the innermost loop executes 1 + 2 + 4 + 8 + ... + N^2 ~ 2 N^2 times.

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Question 3

Given the following definition of a MysteryBox object:

public class MysteryBox {
private final long x0, x1, x2, x3;
private final double y0, y1, y2, y3;
private final boolean z0;
private final int[] a = new int[320];

...
}

Using the 64-bit memory cost model from lecture, how many bytes does
each object of type MysteryBox use? Include all memory allocated when the
client calls new MysteryBox().

Answer: 1400

Question Explanation:

The correct answer is: 1400

public class MysteryBox {                           //   16 (object overhead)
    private final long x0, x1, x2, x3;              //   32 (4 long)
    private final double y0, y1, y2, y3;            //   32 (4 double)
    private final boolean z0;                       //    1 (1 boolean)
    private final int[] a = new int[320];           //    8 (reference to array)
                                                    // 1304 (int array of size 320)
    ...                                                   7 (padding to round up to a multiple of 8)
}                                                      ----
                                                       1400