---
title: "Dijkstra shortest path (2072 Asoj)"
subject: "Discrete Structure"
chapter: 4
key: "semester_4/discrete_structure:dijkstra-2072-asoj"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/dijkstra-2072-asoj
site: BCT Engineering
---

# Dijkstra shortest path (2072 Asoj)

Discrete Structure · Chapter 4: Graph Theory

- Asked in: 2072 Asoj, 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-2072-asoj
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-4

## Question

Use Dijkstra's algorithm to find the length of shortest path in the following weighted graph. Also highlight the shortest path/paths in the graph. [Figure: weighted graph on vertices a, b, c, d, e, z with edges a-b 2, a-c 3, b-c 5, b-e 7, c-e 10, c-d 8, e-d 6, e-z 3, d-z 4]

## Short answer

Dijkstra's algorithm gives a shortest path length of 12 along a–b–e–z.

## 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 2, a–c 3, b–c 5, b–e 7, c–e 10, c–d 8, e–d 6, e–z 3, d–z 4.

#### 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** | 2(a) | 3(a) | ∞ | ∞ | ∞ |
| 2 | b | ✓ | **2**(a) | 3(a) | ∞ | 9(b) | ∞ |
| 3 | c | ✓ | ✓ | **3**(a) | 11(c) | 9(b) | ∞ |
| 4 | e | ✓ | ✓ | ✓ | 11(c) | **9**(b) | 12(e) |
| 5 | d | ✓ | ✓ | ✓ | **11**(c) | ✓ | 12(e) |
| 6 | z | ✓ | ✓ | ✓ | ✓ | ✓ | **12**(e) |

#### Result
Shortest distance from a to z = **12**.

Shortest path (highlight these edges): **a → b → e → z** with edge weights 2 + 7 + 3 = 12.

**Answer:** shortest path length = 12 along a → b → e → z (2 + 7 + 3).

## 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 path length from a to z:
   A. 12, along a-b-e-z
   B. 15 along a-c-d-z
   C. 13 along a-c-e-z
   D. 16
   Answer: A. 2 + 7 + 3 = 12; a-c-d-z costs 15 and a-c-e-z costs 16. (Key point: Result)

4. When b (distance 2) is fixed, what label does e get?
   A. 10
   B. 9 = 2 + 7
   C. 7
   D. 12
   Answer: B. e is reached through b with cost 2 + 7 = 9, better than the later c-e route (13). (Key point: Labels)

## Related in Discrete Structure

- [Dijkstra shortest path (2074 Bhadra)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2074-bhadra) — Discrete Structure, chapter 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×
- [Dijkstra shortest path (2075 Bhadra)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2075-bhadra) — Discrete Structure, chapter 4
- [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 (2069 Bhadra)](https://www.bctengineer.com/question.md?key=semester_4%2Fdiscrete_structure%3Adijkstra-2069-bhadra) — Discrete Structure, chapter 4, asked 4×

## Related in other subjects

- [Dijkstra's shortest paths from source F](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Adijkstra-graph-source-f) — Data Structure and Algorithm, chapter 10
- [Dijkstra's shortest path algorithm](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Adijkstra-algorithm-example) — Data Structure and Algorithm, chapter 10, asked 2×