Question

Suppose you have a computer that requires 1 minute to solve problem instances of size $n=1,000.$ Suppose you buy a new computer that runs 1,000 times faster than the old one. What instance sizes can be run in 1 minute, assuming the following time complexities $T(n)$ for our algorithm? a. $T(n)=n$ b. $T(n)=n^3$ c. $T(n)={10}^n$

Short answer

The new computer can run these instance sizes in 1 minute: for $T(n)=n$, $n=1,000,000$; for $T(n)=n^3$, $n=10,000$; and finally for $T(n)={10}^n$, $n=3000$.

Step-by-step solution

Solving for $T(n)=n$

Given that the old computer takes 1 minute to solve a problem of size $n=1000$ and the new computer is 1000 times faster, it means in 1 minute, the new computer can solve a problem of size $n=1000\times 1000=1,000,000$.

Solving for $T(n)=n^3$

Here, the rate at which the computer solves the problem is cubed. Since the new computer is 1000 times faster than the old one, we can say that $1000=n^3$. Solving for $n$, we then take the cube root of both sides to get $n=10$. We need to multiply this by the original problem size on the old computer to get $n=10\times 1000=10,000$.

Solving for $T(n)={10}^n$

For this exponential function, we can say the new computer can solve a problem of size $n={\log }_{10}1000=3$ in a minute. Again, multiplying this by the original problem size on the old computer, gives $n=3\times 1000=3000$.

Key concepts

Algorithm Analysis

When we talk about algorithm analysis, we refer to the process of understanding how an algorithm performs, particularly in terms of time and space. This step is crucial for determining the efficiency of an algorithm and its suitability for various applications. Several factors must be considered during algorithm analysis:

• Time Complexity: Measures the time an algorithm takes to complete as the input size grows.
• Space Complexity: Assesses the amount of memory required by an algorithm as it processes.
• Correctness and Robustness: Ensures the algorithm produces the correct output and can handle unexpected situations without failure.

In our exercise, we look at three different time complexities: linear, cubic, and exponential. Understanding how these grow with larger inputs helps us select the best algorithm for a task.

Big O Notation

Big O Notation is a mathematical concept used to describe the upper bound of an algorithm's growth. It tells us how the runtime or space requirements scale with input size, helping to predict performance on large data sets.For instance:

• Linear ($O(n)$): Indicates the runtime grows directly with input size. Efficient when processing small or moderate-sized datasets.
• Cubic ($O(n^3)$): Shows that the runtime grows with the cube of the input size. Often becomes inefficient with larger inputs.
• Exponential ($O({10}^n)$): Indicates extremely rapid growth, making it impractical beyond small inputs.

Employing Big O Notation helps in making decisions such as hardware upgrades and algorithm refinement as it points out potential bottlenecks.

Computational Efficiency

Computational efficiency refers to how effectively an algorithm performs its intended function within certain resource constraints, usually time and space. Efficient algorithms solve problems quickly and use minimal resources.In the exercise example:

• The new computer runs 1000 times faster. If our algorithm is linear $(T(n)=n)$, it will handle a million inputs in the time the old one handled a thousand.
• For cubic growth $(T(n)=n^3)$, only a tenfold increase in efficient size is realized. Meaning improvements diminish quickly for larger inputs.
• With exponential algorithms $(T(n)={10}^n)$, even a faster computer hardly makes a difference, as shown by handling only up to 3000 inputs.

Thus, understanding computational efficiency is key to knowing when a faster computer will help, or when algorithmic changes are necessary. It ensures that the newly available computational resources are optimally utilized while highlighting limitations for long-term scalability.