Floyd Warshall All Pairs Shortest Path in DSA Javascript - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Floyd Warshall on a simple graph

What is the output distance matrix after running Floyd Warshall on this graph?

DSA Javascript

const INF = 99999;
const graph = [
  [0, 3, INF, 7],
  [8, 0, 2, INF],
  [5, INF, 0, 1],
  [2, INF, INF, 0]
];

function floydWarshall(graph) {
  const dist = graph.map(row => row.slice());
  const V = graph.length;
  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];
        }
      }
    }
  }
  return dist;
}

console.log(floydWarshall(graph));

A [[0, 3, 5, 6],[8, 0, 2, 3],[5, 8, 0, 1],[2, 5, 7, 0]]

B [[0, 3, 5, 6],[7, 0, 2, 3],[3, 6, 0, 1],[2, 5, 7, 0]]

C [[0, 3, 5, 7],[8, 0, 2, 3],[5, 8, 0, 1],[2, 5, 7, 0]]

D [[0, 3, 7, 7],[8, 0, 2, 3],[5, 8, 0, 1],[2, 5, 7, 0]]

💡 Hint

Check if going through intermediate nodes reduces the path cost.

Option B shows the shortest distances between all pairs after considering all intermediate nodes. Other options have incorrect distances due to missing shorter paths.

🧠 Conceptual

intermediate

Understanding Floyd Warshall Algorithm Complexity

What is the time complexity of the Floyd Warshall algorithm for a graph with V vertices?

A O(V^3)

B O(V^2)

C O(V^4)

D O(V log V)

💡 Hint

Consider the three nested loops over vertices.

Floyd Warshall uses three nested loops each running V times, resulting in O(V^3) time complexity.

🔧 Debug

advanced

Identify the error in this Floyd Warshall implementation

What error will this code produce when run?

DSA Javascript

const INF = 99999;
const graph = [
  [0, 1, INF],
  [INF, 0, 2],
  [3, INF, 0]
];

function floydWarshall(graph) {
  const dist = graph.map(row => row.slice());
  const V = graph.length;
  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];
        }
      }
    }
  }
  return dist;
}

console.log(floydWarshall(graph));

A ReferenceError: dist is not defined

B TypeError: Cannot read property of undefined

C RangeError: Maximum call stack size exceeded

D No error, outputs shortest path matrix

💡 Hint

Check if the code uses valid indices and initialization.

The code correctly initializes and updates the distance matrix without errors.

❓ Predict Output

advanced

Detecting negative cycles with Floyd Warshall

After running Floyd Warshall, how can you detect if the graph contains a negative weight cycle?

DSA Javascript

const INF = 99999;
const graph = [
  [0, 1, INF],
  [INF, 0, -1],
  [-1, INF, 0]
];

function floydWarshall(graph) {
  const dist = graph.map(row => row.slice());
  const V = graph.length;
  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];
        }
      }
    }
  }
  return dist;
}

const dist = floydWarshall(graph);
const hasNegativeCycle = dist.some((row, i) => row[i] < 0);
console.log(hasNegativeCycle);

A undefined

B false

C true

D TypeError

💡 Hint

Check the diagonal elements of the distance matrix after running the algorithm.

If any diagonal element dist[i][i] is negative, it means a negative cycle exists reachable from vertex i.

🧠 Conceptual

expert

Why Floyd Warshall is preferred over repeated Dijkstra for dense graphs?

Which reason best explains why Floyd Warshall is often preferred over running Dijkstra from each vertex in dense graphs?

A Dijkstra cannot handle negative weights, Floyd Warshall can

B Floyd Warshall uses less memory than Dijkstra

C Floyd Warshall is easier to implement than Dijkstra

D Floyd Warshall has better time complexity O(V^3) compared to running Dijkstra V times with O(V^2 log V) in dense graphs

💡 Hint

Consider edge weights and algorithm capabilities.

Floyd Warshall can handle graphs with negative edge weights (but no negative cycles), while Dijkstra cannot handle negative weights correctly.

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