Weighted graphs in Data Structures Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to represent a weighted edge between two nodes.

Data Structures Theory

edge = ([1], 5)

Drag options to blanks, or click blank then click option'

A"node"

B"A-B"

C"weight"

D"A"

Attempts:

3 left

💡 Hint

Common Mistakes

Using the weight value instead of the node name.

Putting the entire edge as a string instead of a tuple.

The first element in the tuple represents the node name, so "A" is correct.

2fill in blank

medium

Complete the code to add a weighted edge from node 'A' to node 'B' with weight 7 in an adjacency list.

Data Structures Theory

graph["A"] = [("B", [1])]

Drag options to blanks, or click blank then click option'

A5

B7

C10

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong weight value.

Confusing node names with weights.

The weight of the edge from 'A' to 'B' is 7, so 7 is the correct value.

3fill in blank

hard

Fix the error in the weighted graph representation by completing the missing weight value.

Data Structures Theory

edges = {"A-B": [1]

Drag options to blanks, or click blank then click option'

A3

B"weight"

C"3"

DNone

Attempts:

3 left

💡 Hint

Common Mistakes

Using a string instead of a number for weight.

Leaving the weight as None.

The weight should be a number, so 3 is correct. Using a string or None is incorrect.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps nodes to their weighted edges with weights greater than 4.

Data Structures Theory

{node: edges for node, edges in graph.items() if edges[0][[1]] [2] 4}

Drag options to blanks, or click blank then click option'

A1

B>

C<

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong index for weight.

Using less than instead of greater than.

The weight is at index 1 in the edge tuple, and we want weights greater than 4, so use 1 and >.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps each node to the weight of its first edge if the weight is less than 10.

Data Structures Theory

{node: edges[[1]][[2]] for node, edges in graph.items() if edges[0][[3]] < 10}

Drag options to blanks, or click blank then click option'

A0

B1

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing up indices for nodes and weights.

Using the wrong index in the condition.

The first blank selects the first edge (index 0), the second blank selects the weight (index 1), and the third blank checks the weight (index 1) for the condition.

Practice

1. What does the weight on an edge in a weighted graph usually represent?

easy

A. The cost or distance between two connected points

B. The color of the edge

C. The number of vertices in the graph

D. The direction of the edge

Solution

1. Step 1: Understand the role of weights in graphs

   Weights on edges represent values like cost, distance, or time between two connected points (vertices).

2. Step 2: Differentiate weights from other graph properties

   Weights are not about color, number of vertices, or direction but about measurable values on edges.

3. Final Answer:

   The cost or distance between two connected points -> Option A

4. Quick Check:

   Weight = cost/distance [OK]

Hint: Weights show cost or distance between points [OK]

Common Mistakes:

• Confusing weight with edge color
• Thinking weight counts vertices
• Mixing weight with edge direction

2. Which of the following is the correct way to represent a weighted edge between vertices A and B with weight 5?

easy

A. (A, B, 5)

B. {A: B = 5}

C. [A, B, weight=5]

D. A - B : 5

Solution

1. Step 1: Recognize common weighted edge notation

   Weighted edges are often represented as tuples like (vertex1, vertex2, weight).

2. Step 2: Check each option's format

   (A, B, 5) uses tuple format (A, B, 5), which is standard. Others are incorrect syntax or informal.

3. Final Answer:

   (A, B, 5) -> Option A

4. Quick Check:

   Weighted edge = (vertex1, vertex2, weight) [OK]

Hint: Use tuple (A, B, weight) for weighted edges [OK]

Common Mistakes:

• Using incorrect symbols like braces or colons
• Confusing syntax with dictionaries
• Writing weight as a keyword inside list

3. Consider the weighted graph edges: (A, B, 3), (B, C, 4), (A, C, 10). What is the shortest path weight from A to C?

medium

A. 4

B. 10

C. 3

D. 7

Solution

1. Step 1: Identify possible paths from A to C

   Paths: Direct (A to C) with weight 10, or via B: A to B (3) + B to C (4).

2. Step 2: Calculate total weights for each path

   Direct path weight = 10; via B = 3 + 4 = 7.

3. Final Answer:

   7 -> Option D

4. Quick Check:

   Shortest path weight = 7 [OK]

Hint: Sum weights on all paths, pick smallest [OK]

Common Mistakes:

• Choosing direct edge without checking alternatives
• Adding weights incorrectly
• Ignoring intermediate vertices

4. Given the weighted graph edges: (X, Y, 2), (Y, Z, 5), (X, Z, 4), a student claims the shortest path from X to Z is 7 by going through Y. What is wrong with this claim?

medium

A. They confused vertices Y and Z

B. They ignored the direct edge from X to Z with weight 4

C. They added weights incorrectly; 2 + 5 is not 7

D. They assumed edges are unweighted

Solution

1. Step 1: Analyze the paths from X to Z

   Paths: Direct edge (X, Z) with weight 4, and path via Y with weights 2 + 5 = 7.

2. Step 2: Identify the shortest path

   The direct edge weight 4 is less than 7, so shortest path is direct, not via Y.

3. Final Answer:

   They ignored the direct edge from X to Z with weight 4 -> Option B

4. Quick Check:

   Shortest path uses smallest weight edge [OK]

Hint: Check all edges before choosing path [OK]

Common Mistakes:

• Ignoring direct edges
• Incorrectly adding weights
• Mixing up vertex names

5. You have a weighted graph representing cities connected by roads with distances. To find the cheapest route from city A to city D considering toll costs on roads, which approach is best?

hard

A. Select the path with the most edges to maximize tolls

B. Count the number of roads between cities ignoring weights

C. Use a shortest path algorithm like Dijkstra's considering weights as toll costs

D. Use a depth-first search without considering weights

Solution

1. Step 1: Understand the problem context

   We want the cheapest route considering toll costs, which are weights on edges.

2. Step 2: Choose an appropriate algorithm

   Dijkstra's algorithm finds shortest paths in weighted graphs by minimizing total weight (cost).

3. Final Answer:

   Use a shortest path algorithm like Dijkstra's considering weights as toll costs -> Option C

4. Quick Check:

   Weighted shortest path = Dijkstra's algorithm [OK]

Hint: Use Dijkstra's for weighted shortest path problems [OK]

Common Mistakes:

• Ignoring weights and counting edges only
• Using DFS which ignores weights
• Choosing longest path mistakenly