Dijkstra's Algorithm Single Source Shortest Path in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of Dijkstra's Algorithm?

To find the shortest path from a single starting point (source) to all other points (vertices) in a graph with non-negative edge weights.

intermediate

In Dijkstra's Algorithm, what data structure is commonly used to select the next vertex with the smallest tentative distance?

A priority queue (often implemented as a min-heap) is used to efficiently pick the vertex with the smallest current distance estimate.

intermediate

Why can't Dijkstra's Algorithm handle graphs with negative edge weights?

Because it assumes that once a vertex's shortest distance is found, it won't change. Negative edges can create shorter paths later, breaking this assumption.

advanced

What is the time complexity of Dijkstra's Algorithm using a min-heap and adjacency list?

O((V + E) log V), where V is the number of vertices and E is the number of edges.

beginner

Explain the relaxation step in Dijkstra's Algorithm.

Relaxation means checking if the current known distance to a neighbor can be improved by going through the current vertex. If yes, update the neighbor's distance.

What type of graph edge weights does Dijkstra's Algorithm require?

A Any weights including negative cycles

B Negative weights allowed

C Only zero weights

D Non-negative weights

Dijkstra's Algorithm only works correctly with non-negative edge weights.

Which data structure helps efficiently pick the next vertex with the smallest distance in Dijkstra's Algorithm?

A Priority queue (min-heap)

B Stack

C Queue

D Linked list

A priority queue (min-heap) is used to select the vertex with the smallest tentative distance efficiently.

What happens during the relaxation step in Dijkstra's Algorithm?

A Edges are reversed

B Distances to neighbors are updated if a shorter path is found

C Vertices are removed from the graph

D All distances are reset to infinity

Relaxation updates the distance to neighbors if a shorter path through the current vertex is found.

What is the initial distance to the source vertex in Dijkstra's Algorithm?

A Infinity

B Negative infinity

C Zero

D One

The source vertex distance is set to zero because the distance from the source to itself is zero.

What is the worst-case time complexity of Dijkstra's Algorithm using a simple array instead of a priority queue?

A O(V^2)

B O(E log V)

C O(V + E)

D O(log V)

Using a simple array to find the minimum distance vertex leads to O(V^2) time complexity.

Describe the main steps of Dijkstra's Algorithm for finding the shortest path from a source vertex.

Think about how distances are updated and how the next vertex is chosen.

Explain why Dijkstra's Algorithm does not work with graphs containing negative edge weights.

Consider what happens if a shorter path appears after a vertex is finalized.

Practice

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

easy

A. To sort the vertices of a graph topologically

B. To find the minimum spanning tree of a graph

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 goal of Dijkstra's Algorithm

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

2. Step 2: Compare with other graph algorithms

   Minimum spanning tree algorithms find a tree connecting all vertices with minimum total edge weight, cycle detection finds loops, and topological sort orders vertices in directed acyclic graphs. These are different goals.

3. Final Answer:

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

4. Quick Check:

   Dijkstra's Algorithm = shortest paths from source [OK]

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

Common Mistakes:

• Confusing with minimum spanning tree algorithms
• Thinking it detects cycles
• Assuming it sorts vertices

2. Which of the following is the correct way to initialize the distance array in Dijkstra's Algorithm in C?

easy

A. int dist[V]; for(int i=0; i

B. int dist[V] = {INT_MAX};

C. int dist[V] = {0};

D. int dist[V] = NULL;

Solution

1. Step 1: Understand distance initialization

   Distances should start as infinity (or a very large number) except the source vertex, to represent unknown shortest paths.

2. Step 2: Check each option

   int dist[V] = {0}; initializes all distances to 0, which is incorrect. int dist[V]; for(int i=0; i

3. Final Answer:

   int dist[V]; for(int i=0; i -> Option A

4. Quick Check:

   Initialize distances to INT_MAX with loop [OK]

Hint: Use loop to set all distances to INT_MAX [OK]

Common Mistakes:

• Using {INT_MAX} initializes only first element
• Setting all distances to 0
• Assigning NULL to int array

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

int graph[4][4] = {
  {0, 1, 4, 0},
  {1, 0, 2, 6},
  {4, 2, 0, 3},
  {0, 6, 3, 0}
};

medium

A. [0, 1, 4, 7]

B. [0, 1, 3, 5]

C. [0, 1, 3, 6]

D. [0, 2, 4, 6]

Solution

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

   Distances: dist[0]=0, dist[1]=∞, dist[2]=∞, dist[3]=∞. Update neighbors of 0: dist[1]=1, dist[2]=4.

2. Step 2: Pick vertex with smallest tentative distance and update neighbors

   Pick vertex 1 (dist=1). Update dist[2] = min(4, 1+2=3) = 3, dist[3] = min(∞, 1+6=7) = 7. Next pick vertex 2 (dist=3). Update dist[3] = min(7, 3+3=6) = 6. Finally pick vertex 3 (dist=6).

3. Final Answer:

   [0, 1, 3, 6] -> Option C

4. Quick Check:

   Distances after relaxation = [0,1,3,6] [OK]

Hint: Relax edges stepwise, pick smallest distance vertex next [OK]

Common Mistakes:

• Not updating distances after relaxation
• Choosing wrong next vertex
• Ignoring shorter paths through intermediate vertices

4. Identify the error in the following snippet of Dijkstra's Algorithm in C:

for (int count = 0; count < V-1; count++) {
  int u = minDistance(dist, sptSet);
  sptSet[u] = 1;
  for (int v = 0; v < V; v++) {
    if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX && dist[u] + graph[u][v] < dist[v]) {
      dist[v] = dist[u] + graph[u][v];
    }
  }
}

medium

A. The loop should run from 1 to V instead of 0 to V-1

B. The condition 'graph[u][v]' should be 'graph[u][v] != 0' to check edge existence

C. The sptSet array should be initialized with 1 instead of 0

D. The dist[u] check should be 'dist[u] == INT_MAX' instead of '!='

Solution

1. Step 1: Analyze edge existence condition

   The condition 'graph[u][v]' relies on implicit truthiness of non-zero values. In C, this works but is less clear. Using 'graph[u][v] != 0' explicitly checks for edge presence and improves readability.

2. Step 2: Check other conditions

   Loop range from 0 to V-1 is correct for V vertices. sptSet should start with 0 (false). The dist[u] check must be '!= INT_MAX' to ensure u is reachable before relaxing edges.

3. Final Answer:

   The condition 'graph[u][v]' should be 'graph[u][v] != 0' to check edge existence -> Option B

4. Quick Check:

   Explicit edge check improves clarity [OK]

Hint: Use explicit '!= 0' to check edges in adjacency matrix [OK]

Common Mistakes:

• Assuming loop range is wrong
• Initializing sptSet incorrectly
• Reversing dist[u] check logic

5. Given a weighted graph with vertices 0 to 5 and edges:

0-1: 7, 0-2: 9, 0-5: 14, 1-2: 10, 1-3: 15, 2-3: 11, 2-5: 2, 3-4: 6, 4-5: 9

Using Dijkstra's Algorithm from source 0, what is the shortest distance to vertex 4?

hard

A. 20

B. 26

C. 21

D. 19

Solution

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

   dist[0]=0, others = ∞. Update neighbors: dist[1]=7, dist[2]=9, dist[5]=14.

2. Step 2: Pick vertex with smallest dist and update neighbors

   Pick 1 (7): update dist[3] = min(∞, 7+15=22) = 22. Pick 2 (9): update dist[3] = min(22, 9+11=20) = 20, dist[5] = min(14, 9+2=11) = 11. Pick 5 (11): update dist[4] = min(∞, 11+9=20) = 20. Pick 3 (20): update dist[4] = min(20, 20+6=26) = 20. Pick 4 (20).

3. Final Answer:

   20 -> Option A

4. Quick Check:

   Shortest path to 4 = 20 [OK]

Hint: Relax edges carefully, update shortest distances stepwise [OK]

Common Mistakes:

• Not updating dist[5] to smaller value
• Ignoring shorter path through vertex 2
• Adding edges incorrectly