Graphs Lesson – Popcorn Hack and Homework Hack Answers

Popcorn Hack Answer

To represent cities and paths as graphs:

  • Nodes (Vertices): Each city is a node. For example, cities A, B, C, D, and E can be represented as five separate nodes.
  • Edges: Roads between cities form the edges. If there is a direct road from city A to city B, there is an edge between nodes A and B.
  • Weights: These can represent either the physical distance or the cost of travel between cities. For example, the edge from A to B might have a weight of 5, indicating 5 miles or a $5 cost.

Example 5-City Graph

  • A connected to B and C
  • B connected to D
  • C connected to D and E
  • D connected to E

Possible Routes from A to E

  • A → C → E
  • A → B → D → E
  • A → C → D → E

This graph can be used to explore shortest paths, cheapest travel, or to apply algorithms like Dijkstra’s or A*.

Homework Hack Answer

Q1: In which of the configurations is it possible to have redundant routing between devices Q and V?

Answer: C) Both configuration I and configuration II
Explanation: Redundant routing means there are multiple independent paths between the devices. Both configurations provide alternative paths from Q to V, allowing communication even if one path is broken.

Q2: In configuration I, what is the minimum number of connections that must be broken or removed before device T can no longer communicate with device U?

Answer: B) Two
Explanation: There are at least two independent paths connecting T and U. To completely disconnect T from U, both paths would need to be severed. This is characteristic of fault-tolerant or redundant network designs.

Heuristics Lesson – Popcorn Hack and Homework Hack Answers

Popcorn Hack Answer: Reverse the Greedy Strategy

We are given a greedy algorithm that starts with the largest coin first, typically coins = [25, 10, 5, 1]. The idea is to change this to coins = [1, 5, 10, 25] and observe how the result changes.

Experiment

Using a target amount of, say, 63 cents:

  • Greedy (largest to smallest):
    • 2 x 25 = 50
    • 1 x 10 = 10
    • 3 x 1 = 3
    • Total coins used: 6
  • Reversed order (smallest to largest):
    • 63 x 1 = 63
    • Total coins used: 63

Reflection

Reversing the coin order makes the algorithm much less efficient. It uses many more coins because it picks the smallest option every time. This demonstrates that a heuristic like the greedy algorithm can be fast and “good enough” in some cases (like U.S. currency), but not optimal when the order is not ideal.

Heuristics Lesson – Popcorn Hack and Homework Hack Answers

Popcorn Hack Answer: Reverse the Greedy Strategy

Task:

We are examining the effect of reversing the order of coins in the Greedy Coin Change Algorithm. The standard approach is coins = [25, 10, 5, 1], which picks the largest coin first. In this activity, we reverse it to coins = [1, 5, 10, 25] and see how this affects efficiency.

Example:

Target Amount: 63 cents

Greedy Strategy ([25, 10, 5, 1]):

  • 2 x 25¢ = 50¢
  • 1 x 10¢ = 10¢
  • 3 x 1¢ = 3¢
  • Total coins used: 6

Reversed Strategy ([1, 5, 10, 25]):

  • 63 x 1¢ = 63¢
  • Total coins used: 63

Reflection:

  • Efficiency: The reversed strategy is much less efficient in terms of the number of coins.
  • Conclusion: The order in which coins are chosen affects how quickly and efficiently the algorithm arrives at a solution.
  • Takeaway: While the greedy strategy is not always optimal, it provides a good-enough solution very efficiently in many practical situations, especially when the denominations allow it.

Homework Hack Answer: Reflection Summary

Prompt:

Reflect on the effect of changing the coin order, efficiency of different approaches, where greedy algorithms work well or fail, and what you learned from modifying the algorithm.

Response:

Reversing the order of coins in the greedy algorithm greatly increased the number of coins used. The standard greedy strategy used only 6 coins to make 63 cents, while the reversed order used 63 coins. This shows that greedy algorithms work well when the problem structure fits the greedy assumption — such as U.S. coin denominations — but they can fail badly in other cases.

Greedy algorithms are useful for fast, approximate solutions when computing the perfect solution is too expensive. Changing the order of coin values helped me understand that heuristics trade off optimality for speed and practicality. It highlighted how important problem structure is in determining algorithm performance.