Floyd Warshall All Pairs Shortest Path in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Floyd Warshall algorithm?

To find the shortest paths between all pairs of vertices in a weighted graph, including graphs with negative edge weights but no negative cycles.

intermediate

What is the time complexity of the Floyd Warshall algorithm?

The time complexity is O(V³), where V is the number of vertices in the graph.

intermediate

In Floyd Warshall, what does the matrix entry dist[i][j] represent during the algorithm?

It represents the shortest known distance from vertex i to vertex j considering intermediate vertices up to a certain index k.

advanced

Why can Floyd Warshall handle negative edge weights but not negative cycles?

Because it updates shortest paths by checking all intermediate vertices, but if a negative cycle exists, distances can decrease indefinitely, making shortest paths undefined.

beginner

What is the initial setup of the distance matrix in Floyd Warshall?

Set dist[i][j] to the edge weight if there is an edge from i to j, 0 if i equals j, and Infinity if there is no direct edge.

What does Floyd Warshall algorithm compute?

A Shortest paths between all pairs of vertices

B Shortest path from a single source to all vertices

C Minimum spanning tree

D Topological sort

Floyd Warshall finds shortest paths between every pair of vertices.

What is the time complexity of Floyd Warshall algorithm?

A O(E log V)

B O(V²)

C O(V + E)

D O(V³)

Floyd Warshall runs in cubic time relative to the number of vertices.

Which type of graphs can Floyd Warshall handle that Dijkstra's algorithm cannot?

A Graphs with only positive weights

B Graphs with negative edge weights

C Graphs with cycles

D Unweighted graphs

Floyd Warshall can handle negative edge weights as long as there are no negative cycles.

What does dist[i][i] equal at the start of Floyd Warshall?

A Edge weight from i to i

B Infinity

C 0

D -1

Distance from a vertex to itself is zero initially.

If a negative cycle exists in the graph, what happens to Floyd Warshall's output?

A Distances can become infinitely small, making results invalid

B It correctly finds shortest paths

C It detects the cycle and stops

D It ignores negative cycles

Negative cycles cause distances to decrease indefinitely, so shortest paths are undefined.

Explain how Floyd Warshall algorithm updates the distance matrix to find shortest paths.

Think about checking all possible paths through each vertex step by step.

Describe the limitations of Floyd Warshall algorithm and when it should not be used.

Consider graph size and presence of negative cycles.

Practice

1. What is the main purpose of the Floyd Warshall algorithm in graph theory?

easy

A. To detect cycles in a graph

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

C. To find the shortest paths between all pairs of vertices in a weighted graph

D. To sort vertices in topological order

Solution

1. Step 1: Understand Floyd Warshall's goal

   The Floyd Warshall algorithm calculates shortest paths between every pair of vertices in a weighted graph.

2. Step 2: Differentiate from other algorithms

   Unlike Dijkstra's algorithm which finds shortest paths from one source, Floyd Warshall finds paths for all pairs.

3. Final Answer:

   To find the shortest paths between all pairs of vertices in a weighted graph -> Option C

4. Quick Check:

   All pairs shortest path = Floyd Warshall [OK]

Hint: Floyd Warshall finds shortest paths for all vertex pairs [OK]

Common Mistakes:

• Confusing with Dijkstra's single source shortest path
• Thinking it detects cycles
• Assuming it sorts vertices

2. Which of the following is the correct way to initialize the distance matrix dist for Floyd Warshall in JavaScript given a graph adjacency matrix graph?

easy

A. const dist = graph.map(row => row.slice());

B. const dist = graph.slice();

C. const dist = graph.map(row => row);

D. const dist = [...graph];

Solution

1. Step 1: Understand deep copy necessity

   We need a deep copy of the 2D array so changes to dist do not affect graph.

2. Step 2: Check each option for deep copy

   const dist = graph.map(row => row.slice()); uses map and slice to copy each row array, creating a deep copy. Others create shallow copies.

3. Final Answer:

   const dist = graph.map(row => row.slice()); -> Option A

4. Quick Check:

   Deep copy with map and slice = const dist = graph.map(row => row.slice()); [OK]

Hint: Use map with slice for deep copy of 2D arrays [OK]

Common Mistakes:

• Using shallow copy causing original graph to change
• Using spread operator which copies only top-level array
• Using map without slice copying references

3. What will be the output distance matrix after running Floyd Warshall on the following graph adjacency matrix where Infinity means no direct edge?

const graph = [
  [0, 3, Infinity],
  [2, 0, 5],
  [Infinity, 1, 0]
];

medium

A. [[0, 3, 7], [2, 0, 5], [3, 1, 0]]

B. [[0, 3, 5], [2, 0, 5], [Infinity, 1, 0]]

C. [[0, 3, Infinity], [2, 0, 5], [3, 1, 0]]

D. [[0, 3, 8], [2, 0, 5], [3, 1, 0]]

Solution

1. Step 1: Initialize dist matrix same as graph

   dist = [[0,3,∞],[2,0,5],[∞,1,0]]

2. Step 2: Update dist using Floyd Warshall iterations

   Check paths through each vertex k=0,1,2:
   k=0: dist[2][0] = min(∞, dist[2][0]+dist[0][0])=∞ no change
   k=1: dist[0][2] = min(∞, dist[0][1]+dist[1][2])=3+5=8
   dist[2][0] = min(∞, dist[2][1]+dist[1][0])=1+2=3
   k=2: no shorter paths found
   Final dist = [[0,3,8],[2,0,5],[3,1,0]]

3. Final Answer:

   [[0, 3, 8], [2, 0, 5], [3, 1, 0]] -> Option D

4. Quick Check:

   Updated shortest paths matrix = [[0, 3, 8], [2, 0, 5], [3, 1, 0]] [OK]

Hint: Check all intermediate vertices to update shortest paths [OK]

Common Mistakes:

• Not updating paths through intermediate vertices
• Confusing Infinity with zero
• Missing updates for indirect paths

4. Identify the error in this Floyd Warshall implementation snippet:

for (let k = 0; k < V; k++) {
  for (let i = 0; i < V; i++) {
    for (let j = 0; j < V; j++) {
      if (dist[i][k] + dist[k][j] < dist[i][j]) {
        dist[i][j] = dist[i][k] + dist[k][j];
      }
    }
  }
}

medium

A. Missing check if dist[i][k] or dist[k][j] is Infinity before addition

B. The loops should be reversed: i, j, then k

C. No error, the code is correct

D. The condition should use <= instead of <

Solution

1. Step 1: Analyze addition of distances

   Adding dist[i][k] + dist[k][j] without checking if either is Infinity can cause incorrect results or overflow.

2. Step 2: Identify missing Infinity checks

   Before adding, must ensure dist[i][k] !== Infinity and dist[k][j] !== Infinity to avoid invalid sums.

3. Final Answer:

   Missing check if dist[i][k] or dist[k][j] is Infinity before addition -> Option A

4. Quick Check:

   Check Infinity before sum to avoid errors [OK]

Hint: Always check for Infinity before adding distances [OK]

Common Mistakes:

• Ignoring Infinity causing wrong updates
• Changing loop order unnecessarily
• Using <= instead of < causing redundant updates

5. Given a weighted directed graph with possible negative edges but no negative cycles, how does Floyd Warshall detect the presence of a negative cycle after running the algorithm?

hard

A. By checking if any dist[i][j] is Infinity

B. By checking if any diagonal element dist[i][i] becomes negative

C. By verifying if the sum of all distances is zero

D. By checking if any dist[i][j] is negative for i != j

Solution

1. Step 1: Understand negative cycle effect on dist matrix

   Negative cycles cause the shortest path from a vertex to itself to become negative.

2. Step 2: Check diagonal elements after Floyd Warshall

   If any dist[i][i] < 0, it means a negative cycle exists involving vertex i.

3. Final Answer:

   By checking if any diagonal element dist[i][i] becomes negative -> Option B

4. Quick Check:

   Negative cycle detected if dist[i][i] < 0 [OK]

Hint: Check diagonal dist[i][i] for negativity to find negative cycles [OK]

Common Mistakes:

• Checking off-diagonal elements for negative cycles
• Confusing Infinity with negative cycles
• Assuming sum of distances indicates cycles