Visual explanation of Dijkstra's algorithm using Python step by step

1. Dijkstra's Algorithm
#Función para construir adyacencia
def constructAdj(edges, V):
adj = [[] for _ in range(V)]
for edge in edges:
u, v, wt = edge
adj[u].append([v, wt])
adj[v].append([u, wt])
return adj
def dijkstra(V, edges, src):
adj = constructAdj(edges, V)
pq = [] # Cola de prioridad (min-heap)
dist = [sys.maxsize] * V
heapq.heappush(pq, [0, src])
dist[src] = 0
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
1 2
3
4 5
7
0
6
Source
1
5
2 3 1
9
2 9
4
9
3

2. Dijkstra's Algorithm
#Función para construir adyacencia
def constructAdj(edges, V):
adj = [[] for _ in range(V)]
for edge in edges:
u, v, wt = edge
adj[u].append([v, wt])
adj[v].append([u, wt])
return adj
def dijkstra(V, edges, src):
adj = constructAdj(edges, V)
pq = [] # Cola de prioridad (min-heap)
dist = [sys.maxsize] * V
heapq.heappush(pq, [0, src])
dist[src] = 0
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
Edsger W. Dijkstra (1930-2002)

3. 1 2
3
4 5
7
0
6
Source
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
adj = [[ ],
[ ],
[ ],
[ ],
[ ],
[ ],
[ ],
[ ],]
edges = [[0, 1, 9], [0, 2, 4], [0, 3, 3], [1, 4, 2], [1, 6, 1],
[2, 5, 1], [2, 7, 9], [3, 4, 2], [3, 5, 9], [4, 6, 5], [5, 7, 3]]
u v wt
#Función para construir adyacencia
def constructAdj(edges, V):
adj = [[] for _ in range(V)]
for edge in edges:
u, v, wt = edge
adj[u].append([v, wt])
adj[v].append([u, wt])
return adj
Input
v = 8
adj = [[1,9],
[0,9],
[ ],
[ ],
[ ],
[ ],
[ ],
[ ],]
adj[0]
adj[1]
adj = [[1,9], [2,4],
[0,9],
[0,4],
[ ],
[ ],
[ ],
[ ],
[ ],]
adj = [[1,9], [2,4], [3,3]
[0,9],
[0,4],
[0,3],
[ ],
[ ],
[ ],
[ ],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2]
[0,4],
[0,3],
[1,2],
[ ],
[ ],
[ ],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4],
[0,3],
[1,2],
[ ],
[1,1],
[ ],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1]
[0,3],
[1,2],
[2,1],
[1,1],
[ ],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1], [7,9]
[0,3],
[1,2],
[2,1],
[1,1],
[2,9],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1], [7,9]
[0,3], [4, 2]
[1,2], [3, 2]
[2,1],
[1,1],
[2,9],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1], [7,9]
[0,3], [4, 2], [5, 9]
[1,2], [3, 2]
[2,1], [3, 9]
[1,1],
[2,9],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1], [7,9]
[0,3], [4, 2], [5, 9]
[1,2], [3, 2], [6, 5]
[2,1], [3, 9]
[1,1], [4, 5]
[2,9],]
adj = [[1,9], [2,4], [3,3]
[0,9], [4,2], [6,1]
[0,4], [5,1], [7,9]
[0,3], [4, 2], [5, 9]
[1,2], [3, 2], [6, 5]
[2,1], [3, 9], [7, 3]
[1,1], [4, 5]
[2,9], [5, 3]]

4. 1 2
3
4 5
7
0
6
∞ Source ∞
∞
∞ ∞
∞
∞
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 ∞ ∞ ∞ ∞ ∞ ∞ ∞
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
def dijkstra(V, edges, src):
adj = constructAdj(edges, V)
pq = []# Cola de prioridad (min-heap)
dist = [sys.maxsize] * V
heapq.heappush(pq, [0, src])
dist[src] = 0
pq = [0, 0]
dist nodo

5. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
∞
∞
∞
adj[0] = [1,9], [2,4], [3,3]
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
∞ ∞ ∞ ∞ ∞ ∞ ∞
pq = [0, 0]

6. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
pq = [0, 0], [9, 1]
∞
∞
adj[0] = [1,9], [2,4], [3,3]
0 9
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
∞ ∞ ∞ ∞ ∞ ∞
9
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist

7. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
pq = [0, 0], [9, 1]
∞
adj[0] = [1,9], [2,4], [3,3]
0 9 3
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
∞ ∞ ∞ ∞ ∞
9
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
∞

8. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
pq = [0, 0], [4, 2], [9, 1]
adj[0] = [1,9], [2,4], [3,3]
0 9 3
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
∞ ∞ ∞ ∞ ∞
9
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
4
∞

9. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
adj[0] = [1,9], [2,4], [3,3]
0 9 3
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
∞ ∞ ∞ ∞ ∞
9
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist
4
∞
pq = [0, 0], [4, 2], [9, 1]

10. 1 2
3
4 5
7
0
6
Source
∞ ∞
∞
∞
0
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
pq = [0, 0], [3, 3], [4, 2], [9, 1]
adj[0] = [1,9], [2,4], [3,3]
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
9
3
4
0 9 4 3 ∞ ∞ ∞ ∞
while pq:
u = heapq.heappop(pq)[1]
for x in adj[u]:
v, weight = x[0], x[1]
if dist[v] > dist[u] + weight:
# Actualizar la distancia de v
dist[v] = dist[u] + weight
heapq.heappush(pq, [dist[v], v])
return dist

11. 1 2
3
4 5
7
0
6
9 Source 4
3
∞ ∞
∞
∞
0
0 9 4 3
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{0,0}
{4,2}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{9,1}
{3,3}

12. 1 2
3
4 5
7
0
6
9 Source 4
3
∞ ∞
∞
∞
0
0 9 4 3
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{3,3}
{4,2}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{9,1}

13. 1 2
3
4 5
7
0
6
9 Source 4
3
5 11
∞
∞
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{9,1}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{11,5}
{5,4}
0 9 4 3 5 11
{4,2}

14. 1 2
3
4 5
7
0
6
9 Source 4
3
5 11
∞
∞
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{9,1}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{11,5}
{5,4}
0 9 4 3 5 11
{4,2}

15. 1 2
3
4 5
7
0
6
9 Source 4
3
5 5
13
∞
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{9,1}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{13,7}
{5,5}
0 9 4 3 5 5 13
{5,4}

16. 1 2
3
4 5
7
0
6
9 Source 4
3
5 5
13
∞
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{9,1}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{13,7}
{5,5}
0 9 4 3 5 5 13
{5,4}

17. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
13
10
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{10,6}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{13,7}
{7,1}
0 7 4 3 5 5 10 13
{5,5}

18. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
13
10
0
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
push
pop
1
5
2 3
{10,6}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{13,7}
{7,1}
0 7 4 3 5 5 10 13
{5,5}

19. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
10
0
push
pop
1
5
2 3
{8,7}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{10,6}
{7,1}
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 10 8

20. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
10
0
push
pop
1
5
2 3
{8,7}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{10,6}
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 10 8
{7,1}

21. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
push
pop
1
5
2 3
{8,6}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{8,7}
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8

22. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
push
pop
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
{8,7}
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8
{8,6}

23. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
push
pop
1
5
2 3
{8,6}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{8,7}
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8

24. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
push
pop
1
5
2 3
{8,6}
Dijkstra's Algorithm
1
9
2 9
4
9
3
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8
{8,7}

25. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
push
pop
1
5
2 3
Dijkstra's Algorithm
1
9
2 9
4
9
3
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8

26. 1 2
3
4 5
7
0
6
7 Source 4
3
5 5
8
8
0
1
2 3
{8,7}
Dijkstra's Algorithm
1
2
4
3
{5,5
dist[ ] =
vértice = 0 1 2 3 4 5 6 7
0 7 4 3 5 5 8 8
5
9
9
9