Minimum spanning tree (Kruskal's) in Data Structures Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to select the next edge with the smallest weight in Kruskal's algorithm.

Data Structures Theory

edges.sort(key=lambda edge: edge.[1])

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

Aweight

Bcost

Clength

Dvalue

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'length' or 'cost' instead of 'weight' which is the standard term.

Trying to sort by 'value' which is not a property of edges.

In Kruskal's algorithm, edges are sorted by their weight to pick the smallest edge first.

2fill in blank

medium

Complete the code to check if adding an edge creates a cycle using the union-find data structure.

Data Structures Theory

if find(parent, u) != [1](parent, v):

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

Amerge

Bunion

Cconnect

Dfind

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'union' instead of 'find' to check parents.

Using unrelated functions like 'connect' or 'merge'.

To check if two vertices belong to different sets, we use the find function on both.

3fill in blank

hard

Fix the error in the union operation to correctly merge two sets in Kruskal's algorithm.

Data Structures Theory

def union(parent, rank, u, v):
    root_u = find(parent, u)
    root_v = find(parent, v)
    if root_u != root_v:
        if rank[root_u] < rank[root_v]:
            parent[root_u] = [1]
        else:
            parent[root_v] = root_u
            if rank[root_u] == rank[root_v]:
                rank[root_u] += 1

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

Au

Broot_u

Croot_v

Dv

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning parent[root_u] = root_u which causes a cycle.

Using the original vertices u or v instead of their roots.

When the rank of root_u is less, we attach root_u's parent to root_v to keep the tree shallow.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each vertex to its parent after union-find operations.

Data Structures Theory

parent_map = {vertex: [1](parent, vertex) for vertex in [2]

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

Afind

Bvertices

Cnodes

Dget_parent

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'get_parent' which is not a standard function.

Using 'nodes' instead of the variable that holds all vertices.

The dictionary maps each vertex in the set vertices to its root parent found by the find function.

5fill in blank

hard

Fill all three blanks to complete the Kruskal's algorithm loop that adds edges to the MST.

Data Structures Theory

for edge in [1]:
    u, v, weight = edge
    if find(parent, u) != find(parent, v):
        [2](parent, rank, u, v)
        mst_edges.append([3])

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

Aedges

Bunion

Cedge

Dgraph

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'graph' instead of 'edges' to iterate.

Appending 'u' or 'v' instead of the whole edge.

The loop goes through all edges, uses union to merge sets, and adds the current edge to the MST list.

Practice

1. What is the main goal of Kruskal's algorithm in graph theory?

easy

A. To detect if a graph contains cycles

B. To find the shortest path between two vertices

C. To find the minimum spanning tree connecting all vertices without cycles

D. To sort vertices by their degree

Solution

1. Step 1: Understand the purpose of Kruskal's algorithm

   Kruskal's algorithm is designed to connect all points in a graph with the smallest total edge weight without forming any loops.

2. Step 2: Compare with other graph problems

   Unlike shortest path or cycle detection alone, Kruskal's focuses on building a minimum spanning tree, which is a tree connecting all vertices with minimal cost.

3. Final Answer:

   To find the minimum spanning tree connecting all vertices without cycles -> Option C

4. Quick Check:

   Kruskal's goal = minimum spanning tree [OK]

Hint: Kruskal's connects all points cheaply without loops [OK]

Common Mistakes:

• Confusing minimum spanning tree with shortest path
• Thinking it detects cycles only
• Assuming it sorts vertices instead of edges

2. Which of the following is the correct first step in Kruskal's algorithm?

easy

A. Pick any vertex and start a depth-first search

B. Sort all edges in increasing order of their weights

C. Remove all vertices with degree 1

D. Sort vertices by their labels alphabetically

Solution

1. Step 1: Identify the initial step of Kruskal's algorithm

   Kruskal's algorithm begins by sorting all edges by their weight from smallest to largest to ensure the cheapest edges are considered first.

2. Step 2: Eliminate incorrect options

   Depth-first search and vertex removal are not part of Kruskal's initial steps, and sorting vertices alphabetically is irrelevant.

3. Final Answer:

   Sort all edges in increasing order of their weights -> Option B

4. Quick Check:

   First step = sort edges by weight [OK]

Hint: Always start by sorting edges by weight ascending [OK]

Common Mistakes:

• Sorting vertices instead of edges
• Starting with a search instead of sorting
• Removing vertices prematurely

3. Given the edges with weights: (A-B: 1), (B-C: 4), (A-C: 3), (C-D: 2), what edges will Kruskal's algorithm select for the minimum spanning tree?

medium

A. (A-C), (B-C), (C-D)

B. (A-B), (C-D), (B-C)

C. (A-B), (C-D), (A-C), (B-C)

D. (A-B), (C-D), (A-C)

Solution

1. Step 1: Sort edges by weight

   Edges sorted: (A-B:1), (C-D:2), (A-C:3), (B-C:4).

2. Step 2: Add edges without creating cycles

   Add (A-B), then (C-D). Adding (A-C) connects A and C without cycle. Adding (B-C) would create a cycle, so skip it.

3. Final Answer:

   (A-B), (C-D), (A-C) -> Option D

4. Quick Check:

   Edges chosen = smallest weights without cycles [OK]

Hint: Pick edges from smallest weight, skip if cycle forms [OK]

Common Mistakes:

• Including all edges regardless of cycles
• Choosing heavier edges before lighter ones
• Not checking for cycles before adding edges

4. In Kruskal's algorithm, what is the common error if the algorithm adds an edge that creates a cycle?

medium

A. The disjoint set structure is not used or updated correctly

B. Edges are not sorted by weight

C. Vertices are not labeled properly

D. The graph is directed instead of undirected

Solution

1. Step 1: Understand cycle prevention in Kruskal's

   Kruskal's algorithm uses a disjoint set (union-find) data structure to track connected components and avoid cycles.

2. Step 2: Identify the cause of cycle creation

   If an edge creates a cycle, it means the disjoint set was not used or updated properly to detect that the two vertices are already connected.

3. Final Answer:

   The disjoint set structure is not used or updated correctly -> Option A

4. Quick Check:

   Cycle error = faulty disjoint set use [OK]

Hint: Use disjoint sets to detect cycles before adding edges [OK]

Common Mistakes:

• Ignoring disjoint set updates
• Assuming sorting fixes cycles
• Confusing graph direction with cycle detection

5. You have a graph with edges: (X-Y: 2), (Y-Z: 2), (X-Z: 3), (Z-W: 1), (W-X: 4). Using Kruskal's algorithm, which set of edges forms the minimum spanning tree?

hard

A. (Z-W), (X-Y), (Y-Z)

B. (X-Y), (Y-Z), (X-Z)

C. (Z-W), (X-Z), (W-X)

D. (Z-W), (X-Y), (Y-Z), (W-X)

Solution

1. Step 1: Sort edges by weight

   Edges sorted: (Z-W:1), (X-Y:2), (Y-Z:2), (X-Z:3), (W-X:4).

2. Step 2: Add edges without cycles until all vertices connected

   Add (Z-W), then (X-Y), then (Y-Z). Now all vertices X, Y, Z, W are connected. Adding (W-X) or (X-Z) would create cycles, so skip them.

3. Final Answer:

   (Z-W), (X-Y), (Y-Z) -> Option A

4. Quick Check:

   Minimum spanning tree = edges with smallest weights connecting all nodes [OK]

Hint: Add smallest edges until all vertices connected, avoid cycles [OK]

Common Mistakes:

• Adding extra edges causing cycles
• Skipping necessary edges to connect all vertices
• Choosing heavier edges before lighter ones