Lecture 7

SoSe24/lec03_sort_alg/algdat_heap.py

@@ -0,0 +1,52 @@
+from SoSe24.algodat.foundation import AlgoDatArray
+
+
+class AlgoDatHeap():
+
+    def __init__(self, values: AlgoDatArray):
+        self.heap = values
+        self.size = len(values)
+
+    def left_child(self, i):
+        return 2*i + 1
+
+    def right_child(self, i):
+        return 2*i + 2
+
+    def parent(self, i):
+        return (i-1)//2
+
+    def max_heapify(self, i):
+        l = self.left_child(i)
+        r = self.right_child(i)
+        largest = i
+        if l <= self.size and self.heap[l-1] > self.heap[i-1]:
+            largest = l
+        if r <= self.size and self.heap[r-1] > self.heap[largest-1]:
+            largest = r
+        if largest != i:
+            self.heap[i-1], self.heap[largest-1] = self.heap[largest-1], self.heap[i-1]
+            self.max_heapify(largest)
+
+    def min_heapify(self, i):
+        l = self.left_child(i)
+        r = self.right_child(i)
+        smallest = i
+        if l <= self.size and self.heap[l-1] < self.heap[i-1]:
+            smallest = l
+        if r <= self.size and self.heap[r-1] < self.heap[smallest-1]:
+            smallest = r
+        if smallest != i:
+            self.heap[i-1], self.heap[smallest-1] = self.heap[smallest-1], self.heap[i-1]
+            self.min_heapify(smallest)
+
+    def build_max_heap(self):
+        for i in range(self.size//2, 0, -1):
+            self.max_heapify(i)
+
+    def build_min_heap(self):
+        for i in range(self.size//2, 0, -1):
+            self.min_heapify(i)
+
+    def as_list(self):
+        return [self.heap[i].value for i in range(self.size)]

SoSe24/lec06_graph/disjoint.py

@@ -0,0 +1,18 @@
+
+
+class DisjointValue():
+
+    def __init__(self, value):
+        self.value = value
+        self.parent = None
+
+    def canonical(self):
+        if self.parent:
+            return self.parent.canonical()
+        return self
+
+    def same_set(self, other):
+        return self.canonical() == other.canonical()
+
+    def union(self, other):
+        self.canonical().parent = other.canonical()

SoSe24/lec06_graph/graph.py

@@ -39,6 +39,8 @@ class Graph:
     def get_adjacent_vertices_with_weight(self, name: str) -> List[tuple[Vertex, float]]:
         raise NotImplementedError("Please implement this method in subclass")
 
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        raise NotImplementedError("Please implement this method in subclass")
 
     def bfs(self, start_name: str):
         """
@@ -124,6 +126,12 @@ class AdjacencyListGraph(Graph):
 
     def get_adjacent_vertices_with_weight(self, name: str) -> List[tuple[Vertex, float]]:
         return self.adjacency_map[name]
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        result = []
+        for name in self.adjacency_map:
+            for (dest, weight) in self.adjacency_map[name]:
+                result.append((name, dest.value, weight))
+        return result
 
 
 
@@ -173,7 +181,13 @@ class AdjacencyMatrixGraph(Graph):
                 result.append((self.get_vertex(name), self.adjacency_matrix[index][i]))
         return result
 
-
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        result = []
+        for i in range(len(self.vertex_list)):
+            for j in range(len(self.vertex_list)):
+                if self.adjacency_matrix[i][j] is not None:
+                    result.append((self.vertex_list[i].value, self.vertex_list[j].value, self.adjacency_matrix[i][j]))
+        return result
 
 
 

SoSe24/lec06_graph/mst.py

@@ -1,13 +1,16 @@
 import math
 import re
-from graph import Graph, AdjacencyListGraph, AdjacencyMatrixGraph, NodeColor, Vertex
+
+from SoSe24.lec06_graph.disjoint import DisjointValue
+from graph import Graph, AdjacencyListGraph, AdjacencyMatrixGraph, Vertex
 import heapq
 
 
 def mst_prim(self, start_name: str = None):
+    """ Compute the minimum spanning tree of the graph using Prim's algorithm. """
 
-    distance_map = {}
+    distance_map = {}   # maps vertices to their current distance from the spanning tree
-    parent_map = {}
+    parent_map = {}     # maps vertices to their predecessor in the spanning tree
 
     Vertex.__lt__ = lambda self, other: distance_map[self] < distance_map[other]
 
@@ -20,26 +23,60 @@ def mst_prim(self, start_name: str = None):
     for vertex in self.all_vertices():
         distance_map[vertex] = 0 if vertex.value == start_name else math.inf
         parent_map[vertex] = None
-        heapq.heappush(queue, vertex)
+        queue.append(vertex)
+
+    heapq.heapify(queue)    # Convert the list into a heap
 
     # Process the queue
-    cost = 0
+    cost = 0 # The cost of the minimum spanning tree
     while len(queue) > 0:
         vertex = heapq.heappop(queue)
-        cost += distance_map[vertex]
+        cost += distance_map[vertex] # Add the cost of the edge to the minimum spanning tree
         for (dest, w) in self.get_adjacent_vertices_with_weight(vertex.value):
             if dest in queue and distance_map[dest] > w:
+                # Update the distance and parent maps
                 queue.remove(dest)
                 distance_map[dest] = w
                 parent_map[dest] = vertex
-                heapq.heappush(queue, dest)
+                queue.append(dest)      # Add the vertex back to the queue
+                heapq.heapify(queue)    # Re-heapify the queue
 
     # Return the distance and predecessor maps
     return parent_map, cost
 
+
+def mst_kruskal(self, start_name: str = None):
+    """ Compute the minimum spanning tree of the graph using Kruskal's algorithm. """
+
+    cost = 0
+    result = []
+    edges = self.all_edges()
+
+    # Create a disjoint set for each vertex
+    vertex_map = {v.value: DisjointValue(v) for v in self.all_vertices()}
+
+    # Sort the edges by weight
+    edges.sort(key=lambda edge: edge[2])
+
+    # Process the edges
+    for edge in edges:
+        start_name, end_name, weight = edge
+        # Check if the edge creates a cycle
+        if not vertex_map[start_name].same_set(vertex_map[end_name]):
+            result.append(edge)
+            vertex_map[start_name].union(vertex_map[end_name])
+            cost += weight
+
+    return result, cost
+
+
+
 AdjacencyListGraph.mst_prim = mst_prim
 AdjacencyMatrixGraph.mst_prim = mst_prim
 
+AdjacencyListGraph.mst_kruskal = mst_kruskal
+AdjacencyMatrixGraph.mst_kruskal = mst_kruskal
+
 
 def read_elektro_into_graph(graph: Graph, filename: str):
     pattern = re.compile(r'"([^"]+)";"([^"]+)";(\d+)')
@@ -59,8 +96,16 @@ if __name__ == "__main__":
 
     graph = AdjacencyMatrixGraph()
     read_elektro_into_graph(graph, "../../elektro.txt")
+
     parents, cost = graph.mst_prim()
-    print(f"Kosten {cost}")
+    print(f"Kosten nach Prim: {cost}")
     for node, parent in parents.items():
         if parent is not None:
-            print(f"{node} - {parent}")
+            print(f"{node} - {parent}")
+
+    edges, cost = graph.mst_kruskal()
+    print(f"Kosten nach Kruskal: {cost}")
+    for start_name, end_name, _ in edges:
+        print(f"{start_name} - {end_name}")
+
+