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---
title: "secret note jfgp3d7rrx0"
date: 2025-10-28T16:09:00+05:30
draft: false
---
There will be an explanation for non-trivial questions.
# Activity 1
## 1
Dijkstra's algorithm guarantees finding the shortest path from a single source to all other vertices under which of the following conditions?
**Answer:** All edge weights must be non-negative.
## 2
Consider an undirected graph with 5 vertices $(V_0, V_1, V_2, V_3, V_4)$. At a certain point
in Dijkstra's algorithm (starting from $V_0$), the current tentative distances are:
$dist = \{V_0:0, V_1:5, V_2:3, V_3:8, V_4:10\}$. And the processed set is: $\{V_0:True,
V_1:False, V_2:False, V_3:False, V_4:False\}$. Assuming the next step is to select an unvisited
vertex to mark as processed, which vertex will be chosen?
**Answer:** $V_2$
Explanation: Dijkstra will choose the next unprocessed vertex with the least weight.
## 3
Which of the following components is/are essential for a standard implementation of Dijkstra's algorithm?
**Answers:**
- A way to store the tentative shortest distance to each vertex.
- A set to keep track of vertices whose shortest paths have been finalized.
- A mechanism to select the unvisited vertex with the smallest current distance.
## 4
Consider an undirected graph with vertices S, A, B, C, D and edges with weights: S-A (4), S-B (2), A-C (5), B-A (1), B-D (8), C-D (3). If Dijkstra's algorithm starts from vertex 'S', what is the value of dist[D] immediately after vertex 'A' has been marked as processed?
**Answer:** 10
## 5
When Dijkstra's algorithm is examining an edge (u, v) from a newly processed vertex u , and it finds that the path source -> ... -> u -> v offers a shorter route to v than its currently recorded shortest distance ( dist[v] ), what is the direct consequence for dist[v] and v 's status?
**Answer:**
dist[v] is updated, and v's status remains un-processed, awaiting its turn in a future selection step
## 6
Given a weighted graph where weights of all edges are unique, there is always a unique shortest path from a source to destination in such a graph.
**Answer:** False
Explanation: There can be multiple sequence of edges from vertex $A$ to $B$ representing the shortest path. The only necessary property is that the sum of the weights on these edges is the least.
# Activity 2
## 1
What is a fundamental capability of the Bellman-Ford algorithm that distinguishes it from Dijkstra's algorithm for finding shortest paths?
**Answer:** It is able to correctly find shortest paths in graphs containing negative edge weights.
## 2
For a graph with 7 vertices and 10 edges, if it contains no negative cycles, what is the minimum number of passes over all edges that the Bellman-Ford algorithm needs to perform to guarantee that all shortest path distances are finalized?
**Answer:** 6
## 3
Consider the following directed graph. If the Bellman-Ford algorithm starts from vertex 'S', what are the shortest distances to vertices 'A', 'B', and 'C' (in that order: dist[A] , dist[B] , dist[C] ) after exactly 2 passes over all edges?
**Answer:** 6
Explanation: It takes $n-1$ iterations for a graph with $n$ edges to stabilize via Bellman Ford if there are no negative edges.
If the graph stabilizes further from the $n$ iteration onwards, it contains negative cycles.
## 4
When Bellman-Ford performs a pass over all edges, does the order in which edges are relaxed within that single pass affect the final shortest path distances after all V−1 passes (assuming no negative cycles are present)?
**Answer:** No, the order of edge relaxation within a pass does not affect the final distances after all V−1 passes.
## 5
Which of the following statements accurately describe properties or uses of the Bellman-Ford algorithm?
**Answers:**
- If a negative cycle is detected, the algorithm reports its presence and may not produce valid shortest paths for affected vertices.
- The algorithm is suitable for distributed shortest path computation.
## 6
Given a graph where all edges have positive weights, the shortest path produced by Dijkstra's and Bellman Ford algorithm may be different but path weight would be same.
**Answer:** True
# Activity 3
## 1
What type of shortest path problem is the Floyd-Warshall algorithm designed to solve?
**Answer:** All-pairs shortest paths in graphs with negative edge weights or positive edge weights (but no negative cycles).
## 2
The core update rule in Floyd-Warshall is $$dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])$$
What does the variable k fundamentally represent in this context?
**Answer:** An intermediate vertex that might lie on a shortest path from i to j
## 3
What is the time complexity of the Floyd-Warshall algorithm for a graph with N vertices?
**Answer:** $O(N^3)$
## 4
How does the Floyd-Warshall algorithm detect the presence of a negative weight cycle in the graph?
**Answer:** By observing if any dist[i][i] (distance from a vertex to itself) becomes negative after all iterations
## 5
Consider a graph with 3 vertices {1, 2, 3}. The initial distance matrix dist is given as: dist[1][1] = 0, dist[1][2] = 2, dist[1][3] = 7 dist[2][1] = inf, dist[2][2] = 0, dist[2][3] = 3 dist[3][1] = inf, dist[3][2] = inf, dist[3][3] = 0
What is the value of dist[1][3] after the iteration where k = 2 is considered as the intermediate vertex?
**Answer:** 5
## 6
Consider a graph with 3 vertices {1, 2, 3}. The initial distance matrix dist is set up. dist[1][1]=0, dist[1] [2]=5, dist[1][3]=inf dist[2][1]=inf, dist[2][2]=0, dist[2][3]=2 dist[3][1]=inf, dist[3] [2]=inf, dist[3][3]=0
After all iterations of the Floyd-Warshall algorithm, what will be the value of $dist[3][1]$?
**Answer:** inf
Explanation: The distance of 3 to 1 is never updated.
# Activity 4
## 1
Prim's algorithm builds the Minimum Spanning Tree (MST) by iteratively adding edges. At each step, which type of edge does it always select?
**Answer:** The edge with the smallest weight that connects a vertex already in the MST to a vertex not yet in the MST
## 2
Consider the following undirected graph. If Prim's algorithm starts from vertex 'A', what is the total weight of the Minimum Spanning Tree (MST) it finds?
Graph: A-B (4), A-C (2), B-C (5), B-D (10), C-D (3), C-E (7), D-E (1)
**Answer:** 10
## 3
A connected, undirected graph has 6 vertices and 7 edges. If Prim's algorithm is used to find its Minimum Spanning Tree, how many edges will be included in the final MST?
**Answer:** 5
## 4
When is the Minimum Spanning Tree (MST) of a connected, weighted graph guaranteed to be unique?
**Answer:** If all edge weights in the graph are distinct.
## 5
Suppose Prim's algorithm has constructed an MST for a graph. If the weight of an edge not currently in the MST is decreased, will the existing MST always change?
**Answer:** Not necessarily; the MST will change only if the decreased edge becomes the new minimum edge across some cut that was previously crossed by a different (heavier) MST edge.
## 6
If Prim's algorithm has generated a Minimum Spanning Tree (MST) for a connected graph, the unique path between any two vertices within that MST is always the shortest path between those two vertices in the original graph
**Answer:** False
# Activity 5
## 1
Which of the following statements about how Kruskal's algorithm constructs the MST are true?
**Answer:** (Yes there is only one I think is correct)
- It adds an edge only if it connects two vertices that belong to different existing components.
## 2
A graph has 5 vertices {V1, V2, V3, V4, V5}. Kruskal's algorithm is applied. Initially, each vertex is in its own set: {V1}, {V2}, {V3}, {V4}, {V5}. Consider the following sequence of edges processed by Kruskal's algorithm:
(V1, V2) - weight 2
(V3, V4) - weight 3
(V1, V3) - weight 4
(V2, V4) - weight 5
(V5, V2) - weight 6
How many distinct connected components are there after these 5 edges have been processed and decisions made?
**Answer:** 1
## 3
If Kruskal's algorithm considers an edge e and decides not to include it in the MST, what is the precise reason for this exclusion?
**Answer:** Adding edge e would connect two vertices that are already in the same connected component within the set of edges already chosen for the MST.
## 4
A graph has 7 vertices and consists of 3 distinct connected components. Kruskal's algorithm is executed on this graph. Assuming all components are non-empty and connected internally, how many edges will Kruskal's algorithm include in the resulting Minimum Spanning Forest (MSF)?
**Answer:** 4
## 5
If Kruskal's algorithm rejects an edge e (meaning e is not included in the MST), it implies that e is the heaviest edge in any cycle that e forms with edges already selected for the MST.
**Answer:** True
## 6
In a connected undirected graph with more than three vertices and all distinct edge weights, the two edges with the smallest weights will always be part of its Minimum Spanning Tree (MST).
**Answer:** True
# GrPAs
## 1
We are implementing Kruskal's algorithm because it's just a tad bit easier than Prim's algorithm.
```python
def FiberLink(wl):
sum, edges, component = 0, [], {}
for u, u_edges in wl.items():
component[u] = u
for v, d in u_edges:
edges.append((d, u, v))
edges.sort()
for d, u, v in edges:
if component[v] == component[u]:
continue
sum += d
c = component[u]
for ckey, cval in component.items():
if cval == c:
component[ckey] = component[v]
return sum
```
## 2
The shorted walk from `src` to `dst` while bouncing off the `bounce` node. We are using Bellman-Ford because simple is good.
We can find the shortest path from `bounce` to each of `src` and `dst` and then add up those two paths (both the path sequence and the weight).
```python
def min_cost_walk(wl, src, dest, bounce):
dist, parent = {}, {}
for u in wl.keys():
dist[u] = 1e6 # close to infinity
dist[bounce] = 0
for _ in range(len(wl)):
for u, edges in wl.items():
for v, weight in edges:
if dist[u] + weight < dist[v]:
dist[v] = dist[u] + weight
parent[v] = u
path = []
revpath = []
distance = dist[src] + dist[dest]
while src != bounce:
path.append(src)
src = parent[src]
path.append(bounce)
while dest != bounce:
revpath.append(dest)
dest = parent[dest]
return distance, path + revpath[::-1]
```
## 3
Here again, we are using Bellman-Ford. Remember from the activity question answers that if there are any relaxations in the $n_{th}$ iteration,
there exists a negative cycle in the graph.
```python
def IsNegativeWeightCyclePresent(wl):
n = len(wl)
dist = {}
parent = {}
for u in wl:
dist[u] = 1e6
dist[u] = 0 # the node to start with
for i in range(n):
for u, edges in wl.items():
for v, weight in edges:
if dist[u] + weight < dist[v]:
if i == n - 1:
return True
dist[v] = dist[u] + weight
parent[v] = u
return False
```
# GA