Dijkstra's Algorithm Single Source Shortest Path in DSA Python - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Dijkstra's Algorithm on a Simple Graph

What is the output distances dictionary after running Dijkstra's algorithm from source node 0 on the given graph?

DSA Python

graph = {
    0: {1: 4, 2: 1},
    1: {3: 1},
    2: {1: 2, 3: 5},
    3: {}
}

import heapq

def dijkstra(graph, start):
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    heap = [(0, start)]

    while heap:
        current_distance, current_node = heapq.heappop(heap)
        if current_distance > distances[current_node]:
            continue
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(heap, (distance, neighbor))
    return distances

result = dijkstra(graph, 0)
print(result)

A {"0": 0, "1": 4, "2": 1, "3": 5}

B {"0": 0, "1": 2, "2": 1, "3": 3}

C {"0": 0, "1": 3, "2": 1, "3": 4}

D {"0": 0, "1": 3, "2": 2, "3": 4}

💡 Hint

Trace the shortest paths from node 0 to others step by step.

Starting from node 0, the shortest path to node 1 is via node 2 with total weight 3 (1 + 2). Node 3 is reached via node 1 with total weight 4.

🧠 Conceptual

intermediate

Understanding Dijkstra's Algorithm Behavior

Which statement correctly describes why Dijkstra's algorithm does not work with negative edge weights?

A Because it uses a stack which cannot handle negative weights.

B Because it assumes once a node's shortest distance is found, it cannot be improved later.

C Because it requires all nodes to be visited exactly twice.

D Because it only works on graphs with cycles.

💡 Hint

Think about how the algorithm finalizes distances.

Dijkstra's algorithm finalizes the shortest distance to a node when it is first popped from the priority queue. Negative edges can later reduce this distance, breaking this assumption.

🔧 Debug

advanced

Identify the Bug in Dijkstra's Implementation

What error or issue will occur when running this Dijkstra's algorithm code on a graph with nodes 0 to 3?

DSA Python

graph = {
    0: {1: 7, 2: 9, 5: 14},
    1: {0: 7, 2: 10, 3: 15},
    2: {0: 9, 1: 10, 3: 11, 5: 2},
    3: {1: 15, 2: 11, 4: 6},
    4: {3: 6, 5: 9},
    5: {0: 14, 2: 2, 4: 9}
}

def dijkstra(graph, start):
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    visited = set()
    while len(visited) < len(graph):
        min_node = None
        for node in graph:
            if node not in visited:
                if min_node is None or distances[node] < distances[min_node]:
                    min_node = node
        for neighbor, weight in graph[min_node].items():
            if distances[min_node] + weight < distances[neighbor]:
                distances[neighbor] = distances[min_node] + weight
        visited.add(min_node)
    return distances

print(dijkstra(graph, 0))

A KeyError because min_node can be None if all distances are infinity initially.

B Infinite loop because visited set never grows.

C TypeError due to wrong data type in distances dictionary.

D Correct output with shortest distances from node 0.

💡 Hint

Check what happens if all unvisited nodes have infinite distance.

If all unvisited nodes have distance infinity, min_node remains None. Accessing graph[min_node] causes KeyError.

🚀 Application

advanced

Number of Nodes with Distance Less Than or Equal to 5

After running Dijkstra's algorithm from node 0 on the graph below, how many nodes have a shortest distance less than or equal to 5?

DSA Python

graph = {
    0: {1: 2, 2: 4},
    1: {2: 1, 3: 7},
    2: {3: 3},
    3: {4: 1},
    4: {}
}

import heapq

def dijkstra(graph, start):
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    heap = [(0, start)]

    while heap:
        current_distance, current_node = heapq.heappop(heap)
        if current_distance > distances[current_node]:
            continue
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(heap, (distance, neighbor))
    return distances

result = dijkstra(graph, 0)
count = sum(1 for dist in result.values() if dist <= 5)
print(count)

A 2

B 4

C 5

D 3

💡 Hint

Calculate shortest distances and count nodes with distance ≤ 5.

Distances: 0->0=0, 0->1=2, 0->2=3 (via 1), 0->3=6 (via 2), 0->4=7 (via 3). Nodes with distance ≤5 are 0,1,2. Node 3 is 6, so excluded.

❓ Predict Output

expert

Output of Dijkstra's Algorithm on a Directed Weighted Graph

What is the output distances dictionary after running Dijkstra's algorithm from source node 'A' on the graph below?

DSA Python

graph = {
    'A': {'B': 1, 'C': 4},
    'B': {'C': 2, 'D': 5},
    'C': {'D': 1},
    'D': {}
}

import heapq

def dijkstra(graph, start):
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    heap = [(0, start)]

    while heap:
        current_distance, current_node = heapq.heappop(heap)
        if current_distance > distances[current_node]:
            continue
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(heap, (distance, neighbor))
    return distances

result = dijkstra(graph, 'A')
print(result)

A {"A": 0, "B": 1, "C": 3, "D": 4}

B {"A": 0, "B": 1, "C": 4, "D": 5}

C {"A": 0, "B": 1, "C": 2, "D": 3}

D {"A": 0, "B": 2, "C": 3, "D": 4}

💡 Hint

Calculate shortest paths stepwise from A to D.

Shortest path A->B=1, A->B->C=3, A->B->C->D=4 is shorter than A->C->D=5 or A->B->D=6.

Practice

1. What is the main purpose of Dijkstra's Algorithm in graph theory?

easy

A. To sort vertices in topological order

B. To find the maximum flow in a network

C. To detect cycles in a graph

D. To find the shortest path from a single source to all other vertices

Solution

1. Step 1: Understand the algorithm's goal

   Dijkstra's Algorithm is designed to find the shortest distance from one starting point (source) to every other point in a graph.

2. Step 2: Compare with other graph algorithms

   Other algorithms like max flow or cycle detection serve different purposes, so they do not match Dijkstra's goal.

3. Final Answer:

   To find the shortest path from a single source to all other vertices -> Option D

4. Quick Check:

   Shortest path = find minimum distances [OK]

Hint: Dijkstra finds shortest paths from one start point [OK]

Common Mistakes:

• Confusing with max flow algorithms
• Thinking it detects cycles
• Assuming it sorts vertices

2. Which Python data structure is commonly used to efficiently select the next vertex with the smallest tentative distance in Dijkstra's Algorithm?

easy

A. List

B. Heap (Priority Queue)

C. Queue

D. Set

Solution

1. Step 1: Identify the need for efficient minimum extraction

   Dijkstra's Algorithm repeatedly picks the vertex with the smallest tentative distance, so an efficient way to get the minimum is needed.

2. Step 2: Choose the right data structure

   A heap or priority queue allows quick extraction of the smallest element, unlike lists or queues which are slower for this task.

3. Final Answer:

   Heap (Priority Queue) -> Option B

4. Quick Check:

   Priority queue = efficient min extraction [OK]

Hint: Use a heap to get smallest distance vertex fast [OK]

Common Mistakes:

• Using a simple list which is slow
• Using a queue which is FIFO, not min-based
• Using a set which has no order

3. What will be the output distances array after running Dijkstra's Algorithm from source vertex 0 on this graph represented by adjacency list with weights?

graph = {
  0: [(1, 4), (2, 1)],
  1: [(3, 1)],
  2: [(1, 2), (3, 5)],
  3: []
}

medium

A. [0, 4, 1, 6]

B. [0, 2, 1, 3]

C. [0, 3, 1, 4]

D. [0, 1, 2, 5]

Solution

1. Step 1: Initialize distances and start from vertex 0

   Distances: 0 to 0 = 0, others infinity initially. Update neighbors: vertex 1 = 4, vertex 2 = 1.

2. Step 2: Pick vertex with smallest distance (2), update neighbors

   From vertex 2, update vertex 1: min(4, 1+2=3) = 3, vertex 3 = 1+5=6.

3. Step 3: Pick vertex with smallest distance (1), update neighbors

   From vertex 1, update vertex 3: min(6, 3+1=4) = 4.

4. Step 4: Final distances

   Distances array is [0, 3, 1, 4].

5. Final Answer:

   [0, 3, 1, 4] -> Option C

6. Quick Check:

   Distances updated stepwise = [0, 3, 1, 4] [OK]

Hint: Update neighbors with min distance each step [OK]

Common Mistakes:

• Not updating vertex 1 distance from vertex 2
• Ignoring better path to vertex 3
• Mixing up edge weights

4. Consider this Dijkstra's Algorithm snippet in Python:

def dijkstra(graph, source):
    dist = {v: float('inf') for v in graph}
    dist[source] = 0
    visited = set()
    while len(visited) < len(graph):
        u = min((v for v in graph if v not in visited), key=lambda x: dist[x])
        visited.add(u)
        for v, w in graph[u]:
            if dist[u] + w < dist[v]:
                dist[v] = dist[u] + w
    return dist

What is the main issue with it?

medium

A. The algorithm does not use a priority queue for efficiency

B. The visited set is not updated properly

C. The algorithm does not handle unreachable vertices

D. The algorithm does not update distances correctly

Solution

1. Step 1: Analyze the selection of next vertex

   The code uses min() over all unvisited vertices to find the smallest distance vertex, which is correct but inefficient.

2. Step 2: Identify efficiency problem

   Without a priority queue, each min() call takes O(V) time, making the algorithm slower but still correct logically.

3. Step 3: Check for correctness vs efficiency

   Though inefficient, the code updates distances and visited set correctly, so the main issue is lack of priority queue for efficiency.

4. Final Answer:

   The algorithm does not use a priority queue for efficiency -> Option A

5. Quick Check:

   Missing priority queue = inefficient but correct [OK]

Hint: Dijkstra needs priority queue for speed, else slow [OK]

Common Mistakes:

• Thinking visited set is not updated
• Assuming distance updates are wrong
• Confusing unreachable vertex handling

5. Given a weighted graph with negative edge weights but no negative cycles, why can't Dijkstra's Algorithm be used to find shortest paths from a source vertex?

hard

A. Because Dijkstra's Algorithm assumes all edge weights are non-negative

B. Because Dijkstra's Algorithm cannot handle graphs with cycles

C. Because Dijkstra's Algorithm requires the graph to be directed acyclic

D. Because Dijkstra's Algorithm only works on unweighted graphs

Solution

1. Step 1: Understand Dijkstra's assumption on edge weights

   Dijkstra's Algorithm assumes all edge weights are zero or positive to guarantee correctness.

2. Step 2: Explain why negative weights break the algorithm

   Negative weights can cause the algorithm to miss shorter paths found later, as it greedily picks the smallest tentative distance.

3. Step 3: Clarify other options

   The algorithm can handle cycles if no negative cycles exist, and it works on weighted graphs, not just unweighted.

4. Final Answer:

   Because Dijkstra's Algorithm assumes all edge weights are non-negative -> Option A

5. Quick Check:

   Negative weights break Dijkstra's assumptions [OK]

Hint: Dijkstra fails if any edge weight is negative [OK]

Common Mistakes:

• Thinking it can't handle cycles at all
• Believing it only works on unweighted graphs
• Confusing negative cycles with negative edges