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Improve Kruskals materials
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‎Seminars/sem_14/README.md‎

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@@ -75,18 +75,59 @@ prim(5, 5, graph) # 13
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![Prim's algorithm creating a MST of a graph, step by step example.](media/prims_algorithm_example.png)
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## Алгоритъм на Крускал (Kruskal's algorithm)
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![Alt text](media/mst_vs_dijkstra.png)
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- Намира минимално покриващо дърво на граф.
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- Сортира ребрата по минимална тежест, като на всяка стъпка добавя реброто с най-малка тежест, което няма да създаде цикъл в графа.
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- Използва структурата *Disjoint set* за оптимална проверка за цикличност.
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- Сложност по време *O(E\*logE)* заради сортирането на всички ребра.
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- При *dense* граф, когато *Е = V<sup>2</sup>*, *O(ElogE) = O(ElogV<sup>2</sup>) = O(2ElogV) = O(ElogV)*
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-
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Подробно описание как работи Disjoint-set (Union-find) структурата в [playground-а](playground_14.ipynb).
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```python
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def find(x, parents):
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if parents[x] == x:
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return x
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furthest_parent = find(parents[x], parents)
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parents[x] = furthest_parent
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return furthest_parent
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def union(x, y, parents, rank):
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x_root = find(x, parents)
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y_root = find(y, parents)
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if rank[x_root] < rank[y_root]:
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parents[x_root] = y_root
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elif rank[x_root] > rank[y_root]:
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parents[y_root] = x_root
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else:
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parents[x_root] = y_root
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rank[y_root] += 1
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def kruskal(V, edges):
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edges.sort(key=lambda x: x[2])
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parents = [i for i in range(V + 1)]
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rank = [0] * (V + 1)
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mst_weight = 0
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for x, y, w in edges:
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if find(x, parents) != find(y, parents):
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mst_weight += w
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union(x, y, parents, rank)
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return mst_weight
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kruskal(5 , graph_list_of_edges) # 13
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```
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![Kruskal's algorithm creating a MST of a graph, step by step example.](media/kruskals_algorithm_example.png)
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## Задачи за упражнение
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- [Prim's (MST) : Special Subtree](https://www.hackerrank.com/challenges/primsmstsub/problem)
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- [Kruskal (MST): Really Special Subtree](https://www.hackerrank.com/challenges/kruskalmstrsub/problem)
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Решения на задачите: [тук](/Tasks/tasks_14)

‎Seminars/sem_14/playground_14.ipynb‎

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},
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{
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"cell_type": "code",
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"execution_count": 479,
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"execution_count": 20,
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"metadata": {},
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"outputs": [],
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"source": [
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},
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{
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"cell_type": "code",
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"execution_count": 480,
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"metadata": {},
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"outputs": [
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},
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{
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"cell_type": "code",
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"outputs": [
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},
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"cell_type": "code",
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"outputs": [
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},
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"outputs": [
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},
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},
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"13"
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]
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},
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}
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" rank[y_root] += 1\n",
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"\n",
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"def kruskal(V, edges):\n",
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" edges.sort(key=lambda x: x[2]) # Sort edges by weight\n",
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" edges.sort(key=lambda x: x[2]) # Sorts edges by weight\n",
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" parents = [i for i in range(V + 1)]\n",
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" mst_weight = 0\n",
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},
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]
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},
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"metadata": {},
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"output_type": "execute_result"
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}

‎Tasks/tasks_14/kruskals_really_special_subtree/solution_kruskal_best.py‎

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def find(x, parents):
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if parents[x] == x:
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return x
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furthest_parent = find(parents[x], parents)
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parents[x] = furthest_parent
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return furthest_parent
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def union(x, y, parents, rank):
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x_root = find(x, parents)
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y_root = find(y, parents)
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if rank[x_root] < rank[y_root]:
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parents[x_root] = y_root
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elif rank[x_root] > rank[y_root]:
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parents[y_root] = x_root
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else:
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parents[x_root] = y_root
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rank[y_root] += 1
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def kruskal(V, edges):
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edges.sort(key=lambda x: x[2])
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parents = [i for i in range(V + 1)]
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rank = [0] * (V + 1)
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mst_weight = 0
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for x, y, w in edges:
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if find(x, parents) != find(y, parents):
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mst_weight += w
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union(x, y, parents, rank)
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return mst_weight
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V, E = map(int, input().split())
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edges = [tuple(map(int, input().split())) for i in range(E)]
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result = kruskal(V, edges)
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print(result)

‎Tasks/tasks_14/prims_special_subtree/solution_kruskal_better.py‎ renamed to ‎Tasks/tasks_14/kruskals_really_special_subtree/solution_kruskal_better.py‎

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@@ -32,6 +32,5 @@ def kruskal(V, edges):
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N, M = map(int, input().split())
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edges = [tuple(map(int, input().split())) for _ in range(M)]
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start = int(input())
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mst_weight = kruskal(N, edges)
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print(mst_weight)

‎Tasks/tasks_14/prims_special_subtree/solution_kruskal_medium.py‎ renamed to ‎Tasks/tasks_14/kruskals_really_special_subtree/solution_kruskal_medium.py‎

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N, M = map(int, input().split())
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edges = [tuple(map(int, input().split())) for _ in range(M)]
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start = int(input())
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mst_weight = kruskal(N, edges)
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print(mst_weight)

‎Tasks/tasks_14/prims_special_subtree/solution_kruskal_worst.py‎ renamed to ‎Tasks/tasks_14/kruskals_really_special_subtree/solution_kruskal_worst.py‎

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N, M = map(int, input().split())
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edges = [tuple(map(int, input().split())) for _ in range(M)]
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start = int(input())
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mst_weight = kruskal(N, edges)
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print(mst_weight)

‎Tasks/tasks_14/kruskals_really_special_subtree/solution_prim.py‎

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from heapq import heappush, heappop
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def prim(start, V, graph):
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visited = set()
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pq = [(0, start)]
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mst_weight = 0
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while len(visited) != V:
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current_weight, current = heappop(pq)
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if current in visited:
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continue
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visited.add(current)
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mst_weight += current_weight
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for neighb, weight in graph[current]:
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if neighb in visited:
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continue
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heappush(pq, (weight, neighb))
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return mst_weight
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N, M = map(int, input().split())
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graph = {v: set() for v in range(1, N + 1)}
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for _ in range(M):
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x, y, w = map(int, input().split())
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graph[x].add((y, w))
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graph[y].add((x, w))
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mst_weight = prim(1, N, graph)
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print(mst_weight)

‎Tasks/tasks_14/prims_special_subtree/solution_kruskal_best.py‎ renamed to ‎Tasks/tasks_14/prims_special_subtree/solution_kruskal.py‎

File renamed without changes.

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