Karger's Algorithm (#2237)

* Add implementation of Karger's Algorithm

* Remove print statement from karger's algorithm function

* Fix style issues in graphs/karger.py

* Change for loops to set comprehensions where appropriate in graphs/karger.py

graphs/karger.py

@@ -0,0 +1,83 @@
+"""
+An implementation of Karger's Algorithm for partitioning a graph.
+"""
+
+import random
+from typing import Dict, List, Set, Tuple
+
+
+# Adjacency list representation of this graph:
+# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
+TEST_GRAPH = {
+    '1': ['2', '3', '4', '5'],
+    '2': ['1', '3', '4', '5'],
+    '3': ['1', '2', '4', '5', '10'],
+    '4': ['1', '2', '3', '5', '6'],
+    '5': ['1', '2', '3', '4', '7'],
+    '6': ['7', '8', '9', '10', '4'],
+    '7': ['6', '8', '9', '10', '5'],
+    '8': ['6', '7', '9', '10'],
+    '9': ['6', '7', '8', '10'],
+    '10': ['6', '7', '8', '9', '3']
+}
+
+
+def partition_graph(graph: Dict[str, List[str]]) -> Set[Tuple[str, str]]:
+    """
+    Partitions a graph using Karger's Algorithm. Implemented from
+    pseudocode found here:
+    https://en.wikipedia.org/wiki/Karger%27s_algorithm.
+    This function involves random choices, meaning it will not give
+    consistent outputs.
+
+    Args:
+        graph: A dictionary containing adacency lists for the graph.
+            Nodes must be strings.
+
+    Returns:
+        The cutset of the cut found by Karger's Algorithm.
+
+    >>> graph = {'0':['1'], '1':['0']}
+    >>> partition_graph(graph)
+    {('0', '1')}
+    """
+    # Dict that maps contracted nodes to a list of all the nodes it "contains."
+    contracted_nodes = {node: {node} for node in graph}
+
+    graph_copy = {node: graph[node][:] for node in graph}
+
+    while len(graph_copy) > 2:
+
+        # Choose a random edge.
+        u = random.choice(list(graph_copy.keys()))
+        v = random.choice(graph_copy[u])
+
+        # Contract edge (u, v) to new node uv
+        uv = u + v
+        uv_neighbors = list(set(graph_copy[u] + graph_copy[v]))
+        uv_neighbors.remove(u)
+        uv_neighbors.remove(v)
+        graph_copy[uv] = uv_neighbors
+        for neighbor in uv_neighbors:
+            graph_copy[neighbor].append(uv)
+
+        contracted_nodes[uv] = {contracted_node for contracted_node in
+                                contracted_nodes[u].union(contracted_nodes[v])}
+
+        # Remove nodes u and v.
+        del graph_copy[u]
+        del graph_copy[v]
+        for neighbor in uv_neighbors:
+            if u in graph_copy[neighbor]:
+                graph_copy[neighbor].remove(u)
+            if v in graph_copy[neighbor]:
+                graph_copy[neighbor].remove(v)
+
+    # Find cutset.
+    groups = [contracted_nodes[node] for node in graph_copy]
+    return {(node, neighbor) for node in groups[0]
+            for neighbor in graph[node] if neighbor in groups[1]}
+
+
+if __name__ == "__main__":
+    print(partition_graph(TEST_GRAPH))