---
title: "Dijkstra shortest path (2069 Bhadra)"
subject: "Discrete Structure"
chapter: 4
key: "semester_4/discrete_structure:dijkstra-2069-bhadra"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/dijkstra-2069-bhadra
site: BCT Engineering
---

# Dijkstra shortest path (2069 Bhadra)

Discrete Structure · Chapter 4: Graph Theory

- Asked in: 2069 Bhadra, 8 marks
- Video: [Dijkstras Shortest Path Algorithm Explained | With Example | Graph Theory](https://www.youtube.com/watch?v=bZkzH5x0SKU) — FelixTechTips
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/dijkstra-2069-bhadra
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-4

## Question

Use Dijkstra's algorithm to find the length of the shortest path between the vertices a and z in the weighted graph displayed below. [Figure: weighted graph on vertices a, b, c, d, e, z with edges a-b 4, a-c 5, b-d 3, b-e 4, d-e 1, d-z 4, c-e 3, e-z 2]

## Short answer

Dijkstra's algorithm iteratively fixes the vertex with smallest tentative distance, updating neighbors. The shortest distance from a to z is 10, via paths like a-b-e-z (4+4+2) or a-c-e-z (5+3+2).

## Answer

Dijkstra's algorithm (non-negative weights) labels every vertex with the best distance known so far and fixes one vertex at a time:
1. Give the source distance 0 and every other vertex ∞. All vertices are temporary.
2. Make the temporary vertex with the smallest label permanent.
3. For each temporary neighbour v of the new permanent vertex u: if d(u) + w(u,v) < d(v), set d(v) = d(u) + w(u,v) and record u as predecessor.
4. Repeat steps 2 and 3 until the target (or all vertices) is permanent.

#### Graph as read
Source a, target z.

Edges: a–b 4, a–c 5, b–d 3, b–e 4, d–e 1, d–z 4, c–e 3, e–z 2.

#### Trace
Each cell shows label = distance(predecessor). **Bold** marks the vertex made permanent in that step; ✓ means already permanent.

| Step | Fixed | a | b | c | d | e | z |
|---|---|---|---|---|---|---|---|
| 1 | a | **0** | 4(a) | 5(a) | ∞ | ∞ | ∞ |
| 2 | b | ✓ | **4**(a) | 5(a) | 7(b) | 8(b) | ∞ |
| 3 | c | ✓ | ✓ | **5**(a) | 7(b) | 8(b) | ∞ |
| 4 | d | ✓ | ✓ | ✓ | **7**(b) | 8(b) | 11(d) |
| 5 | e | ✓ | ✓ | ✓ | ✓ | **8**(b) | 10(e) |
| 6 | z | ✓ | ✓ | ✓ | ✓ | ✓ | **10**(e) |

#### Result
Shortest distance from a to z = **10**.

There are 3 shortest paths of the same length 10 (highlight all of them):
- a → b → d → e → z
- a → b → e → z
- a → c → e → z

**Answer:** shortest path length = 10. (Edges a–b–e–z: 4 + 4 + 2; a–c–e–z: 5 + 3 + 2; a–b–d–e–z: 4 + 3 + 1 + 2.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At each step Dijkstra's algorithm makes permanent:
   A. The vertex with the most neighbours
   B. The vertex with fewest edges to the target
   C. The temporary vertex with the smallest label
   D. A vertex chosen at random
   Answer: C. The smallest temporary label is final when all weights are non-negative. (Key point: Dijkstra step)

2. A neighbour v of the newly fixed vertex u gets a new label when:
   A. w(u,v) < d(u)
   B. d(v) is already permanent
   C. d(u) > d(v)
   D. d(u) + w(u,v) < d(v)
   Answer: D. The label is replaced only if the route through u is shorter. (Key point: Relaxation)

3. Shortest distance from a to z:
   A. 10
   B. 8
   C. 11
   D. 12
   Answer: A. Several paths all cost 10. (Key point: Result)

4. How many shortest paths of length 10 exist?
   A. 1
   B. 3 (a-b-d-e-z, a-b-e-z, a-c-e-z)
   C. 2
   D. 4
   Answer: B. 4+3+1+2, 4+4+2 and 5+3+2 all equal 10. (Key point: Ties)

## Close variants set in other exams

- [Dijkstra shortest path (2071 Bhadra)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2071-bhadra) — Discrete Structure, chapter 4, asked 4×
- [Dijkstra shortest path (2070 Magh)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2070-magh) — Discrete Structure, chapter 4, asked 4×
- [Dijkstra shortest path (2068 Bhadra)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2068-bhadra) — Discrete Structure, chapter 4, asked 4×

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## Related in other subjects

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