# Setup code
import networkx as nx
def check(actual, expected):
if expected != actual:
print(f"The expected value is {expected}, but we have {actual}")
else:
print(f"Congratulations, the test case passed!")
def draw(graph):
g = nx.DiGraph(graph)
nx.draw(g, with_labels=True, node_size=1000, node_color='#cfc8f4', arrowsize=20)
Depth-First Search (DFS) is an algorithm used for traversing or searching graphs. Starting from a designated starting node, it explores as far as possible along each branch (a neighbor of a node) before backtracking and explore another branch.
Key Concepts:
n
nodes labelled from 0, 1, ..., n - 1), then we can use a list
to keep track of which is visited.Time Complexity: The time complexity for DFS on a graph is $O(V+E)$, where $V$ is the number of vertices and $E$ is the number of edges.
Consider the following undirected graph:
0 -- 1
| |
2 -- 3
You can represent this graph in code using an adjacency list/dictionary:
graph = { 0: [1, 2], 1: [0, 3], 2: [0, 3], 3: [1, 2] }
Step-by-Step Execution of DFS (Starting from Node 0):
The DFS traversal order might be: 0 -> 1 -> 3 -> 2
.
Consider the following undirected graph:
# A 6-node graph
G = {0: [1, 2, 3, 4],
1: [0, 4],
2: [0, 3, 4, 5],
3: [0, 2, 4, 5],
4: [0, 1, 2, 3, 5],
5: [2, 3, 4]}
draw(G)
Let's start from 0 and just take 6 steps randomly!
import random
def random_walk(G, current_node, path, steps_remaining):
if steps_remaining == 0:
return path
# Add the current node to the path
path.append(current_node)
# Pick a random neighbor of current node as the next node
next_node = random.choice(G[current_node])
# Recurse from the next node
return random_walk(G, next_node, path, steps_remaining - 1)
# Let's start from node 0
num_steps = 10
path = random_walk(G, current_node=0, path=[], steps_remaining=num_steps)
path_as_str = ' -> '.join(map(str, path))
# Prin the path
print('Random walk path:', path_as_str)
Random walk path: 0 -> 2 -> 4 -> 1 -> 0 -> 2 -> 0 -> 2 -> 5 -> 4
Try running the above multiple times.
Notice that:
Now, what if we don't want to visit the same node twice? We then want to keep track of which nodes have been visited as part of the traversal. Consider the following modification:
def random_walk_once(G, current_node, path, steps_remaining, visited):
if steps_remaining == 0:
return path
path.append(current_node)
visited[current_node] = True
# NOTE! We stop early if all of current node's neighbors have been visited
if all([visited[n] for n in G[current_node]]):
return path
# NOTE! we check whether the next node is visited, and only move on if unvisited
next_node = random.choice(G[current_node])
while visited[next_node]:
next_node = random.choice(G[current_node])
return random_walk_once(G, next_node, path, steps_remaining - 1, visited)
# NOTE! Now we keep track of visited nodes. Since nodes are just indices from 0 to n-1,
# we can just use a list (instead of things like sets or dictionaries).
visited = [False] * len(G)
# Let's start from node 0
num_steps = 10
path = random_walk_once(G, current_node=0, path=[], steps_remaining=num_steps, visited=visited)
path_as_str = ' -> '.join(map(str, path))
# Prin the path
print('Random walk path:', path_as_str)
Random walk path: 0 -> 2 -> 5 -> 4 -> 3
Try running the above multiple times. Notice that:
num_steps
, and the last node must have all of its neighbors already visited!
Write a function that given a graph G
and starting node start_node
performs DFS starting at start_node
and prints every node it visits.
For example, on the following graph, starting from 0 can produce 0, 1, 2, 4, 5, 3
.
G = {0: [1, 2, 3], 1: [0], 2: [0, 4, 5], 3: [0], 4: [2, 5], 5: [2, 4]}
draw(G)
def my_dfs(G, start_node):
# TODO: your code here
pass
# Hint:
# - Recall the warm up above. DFS can be implemented in a similar way.
# - You should have some way of keep track of which nodes have been visited.
# - You should also have some way of keeping track of the order in which nodes
# are visited.
# - You may need a **helper function** to do the actual recursion.
# - You only need to print the path. You can either have DFS return the list
# of traversed nodes and print them, or just print the nodes as you traverse them.
# - You can randomly pick a neighbor to visit next, or you can visit them in
# increasing order. Either way is fine.
G = {0: [1, 2, 3], 1: [0], 2: [0, 4, 5], 3: [0], 4: [2, 5], 5: [2, 4]}
s = 0
my_dfs(G, s)
def my_dfs(G, start_node):
n = len(G)
visited = [False] * n
traversed = []
# The DFS modifies the traversed list in place, so we don't need to return it.
dfs_helper(G, start_node, visited, traversed)
return traversed
def dfs_helper(G, node, visited, traversed):
traversed.append(node)
visited[node] = True
for neighbor in G[node]:
if not visited[neighbor]:
dfs_helper(G, neighbor, visited)
Task: Write a function that counts the number of connected components of an undirected graph using DFS.
A connected component is a subset of the nodes in which every node can reach every other node. A graph may have more than one connected component! For example, the following graph has two connected components:
G = {
0: [1, 3],
1: [0, 2],
2: [3, 1],
3: [0, 2],
4: [5],
5: [4],
}
draw(G)
def count_connected_components(G: dict) -> int:
# TODO: your code here
return -1
# Test cases
# You may use `draw(G)` to help you visualize the graph
G = {
0: [1, 3],
1: [0, 2],
2: [3, 1],
3: [0, 2],
4: [5],
5: [4],
}
check(count_connected_components(G), expected=2)
G = {0: [4, 5], 1: [2, 4], 2: [1, 4, 5], 3: [4], 4: [0, 1, 2, 3, 5], 5: [0, 2, 4]}
check(count_connected_components(G), expected=1)
G = {0: [1, 2], 1: [0, 2], 2: [0, 1], 3: [4], 4: [3]}
check(count_connected_components(G), expected=2)
G = {0: [1, 2], 1: [0, 2], 2: [0, 1], 3: [4], 4: [3], 5: [6], 6: [5]}
check(count_connected_components(G), expected=3)
G = {0: [2, 3, 5, 6, 7, 9, 16, 17, 18, 19],
1: [5, 6, 9, 16],
2: [0, 3, 17],
3: [0, 2, 11, 12, 14, 16, 19],
4: [8, 15],
5: [0, 1, 6, 8, 9, 11, 13, 15],
6: [0, 1, 5, 7, 8, 10, 12, 13, 17],
7: [0, 6, 9, 14, 19],
8: [4, 5, 6, 9, 16, 17, 18],
9: [0, 1, 5, 7, 8, 12, 13, 17, 18, 19],
10: [6, 12, 13, 17, 18],
11: [3, 5, 18],
12: [3, 6, 9, 10],
13: [5, 6, 9, 10, 18],
14: [3, 7, 15],
15: [4, 5, 14, 16],
16: [0, 1, 3, 8, 15, 18],
17: [0, 2, 6, 8, 9, 10, 18],
18: [0, 8, 9, 10, 11, 13, 16, 17, 19],
19: [0, 3, 7, 9, 18]}
check(count_connected_components(G), expected=1)
The expected value is 2, but we have -1 The expected value is 1, but we have -1 The expected value is 2, but we have -1 The expected value is 3, but we have -1 The expected value is 1, but we have -1
A cycle in a graph is a path where you travel from a node back to itself through a series of edges.
Consider the following graphs with and without cycles:
# A graph without cycles
G = {0: [1], 1: [0, 2, 3], 2: [1], 3: [1]}
draw(G)
# A graph with cycle
G = {0: [1], 1: [0, 2, 3], 2: [1, 3], 3: [1, 2]}
draw(G)
Task: Write a function that detects if there's a cycle in a given graph using DFS.
Hint: You can use the visited
set to keep track of the nodes you've visited so far. If you visit a node that's already in the visited
set and it is NOT the node that you came from, then you've found a cycle.
def has_cycle(G: dict) -> bool:
# TODO: your implementation here
return None
# Test cases
G = {0: [1], 1: [0, 2, 3], 2: [1, 3], 3: [1, 2]}
check(has_cycle(G), expected=True)
G = {0: [1], 1: [0, 2, 3], 2: [1], 3: [1]}
check(has_cycle(G), expected=False)
G = {0: [1], 1: [0, 2, 3], 2: [1, 3], 3: [1, 2], 4: [5], 5: [4]}
check(has_cycle(G), expected=True)
G = {
0: [4, 7, 9],
1: [3, 7, 8],
2: [],
3: [5],
4: [1],
5: [2, 4, 7, 8],
6: [0, 1, 4, 7, 8],
7: [4, 9],
8: [0, 4],
9: [4, 7, 8]
}
check(has_cycle(G), expected=True)
The expected value is True, but we have None The expected value is False, but we have None The expected value is True, but we have None The expected value is True, but we have None
For the following example graph, there are many possible paths from node 0 to node 9:
G = {
0: [1, 2, 6, 8, 3],
1: [0, 3, 4, 5, 8, 6],
2: [0, 7, 8],
3: [1, 0, 6, 8],
4: [1, 9, 5, 6, 8],
5: [1, 4, 9, 6],
6: [0, 3, 1, 4, 5, 7],
7: [2, 6, 8, 9],
8: [0, 1, 2, 3, 7, 4, 9],
9: [4, 5, 7, 8]
}
draw(G)
For example, valid paths include:
[0, 1, 3, 6, 4, 9]
[0, 6, 3, 1, 5, 9]
[0, 8, 2, 7, 6, 3, 1, 4, 5, 9]
[0, 8, 9]
Task: Given a graph G
, a starting node, and an ending node, use DFS to find a path between them. Any path is fine (does not have to be the shortest path)! If there is no path, return an empty list.
def find_path(G: dict, start: int, end: int) -> list:
# TODO: your code here
return [-1]
# Test cases
G = {
0: [1, 3],
1: [0, 2],
2: [3, 1],
3: [0, 2],
4: [5],
5: [4],
}
start, end = 0, 5
check(find_path(G, start, end), expected=[]) # no paths!
G = {
0: [1],
1: [0, 2],
2: [1, 3],
3: [2, 4],
4: [3],
}
start, end = 0, 4
check(find_path(G, start, end), expected=[0, 1, 2, 3, 4]) # only 1 path!
G = {0: [1], 1: [0, 2, 3], 2: [1, 3], 3: [1, 2]}
start, end = 0, 2
possible_paths = list(nx.all_simple_paths(nx.Graph(G), start, end))
check(find_path(G, start, end) in possible_paths, expected=True) # any path is fine
G = {
0: [1, 2, 6, 8, 3],
1: [0, 3, 4, 5, 8, 6],
2: [0, 7, 8],
3: [1, 0, 6, 8],
4: [1, 9, 5, 6, 8],
5: [1, 4, 9, 6],
6: [0, 3, 1, 4, 5, 7],
7: [2, 6, 8, 9],
8: [0, 1, 2, 3, 7, 4, 9],
9: [4, 5, 7, 8]
}
start, end = 0, 9
possible_paths = list(nx.all_simple_paths(nx.Graph(G), start, end))
check(find_path(G, start, end) in possible_paths, expected=True) # any path is fine
The expected value is [], but we have [-1] The expected value is [0, 1, 2, 3, 4], but we have [-1] The expected value is True, but we have False The expected value is True, but we have False
You are given a connected graph of cities having a football stadium and undirected roads (can travel both ways) between some of them.
Whenever a game is happening in a city (a node) it becomes impossible to drive through it and all adjacent roads close (cannot move through edges).
We call a city important if after closing this city (node) and all neighboring roads (edges), it becomes impossible to get from one city to another city and hold a football game. For example, for the following graph we should return 2
(node 0
and 2
are important):
G = {0: [1, 2, 3], 1: [0], 2: [0, 4, 5], 3: [0], 4: [2, 5], 5: [2, 4]}
draw(G)
Task: Find the number of important cities given a graph G
.
*Hint: A simple but perhaps slow approach is the following. Try removing a node t
from the graph along with its edges, and traversing the graph with DFS. If you can reach all the nodes in the graph (except t
which is removed), then t
is not an important node. Otherwise, it is an important node.*
# For your convenience, we have implemented the following function to remove a
# node from an undirected graph.
def remove_node(G: dict, node) -> dict:
# Note that the index of the remaining nodes are NOT changed!
# So if you remove node `1`, node `2` will still have index `2`.
nx_graph = nx.Graph(G)
nx_graph.remove_node(node)
return nx.to_dict_of_lists(nx_graph)
def count_important(G: dict) -> int:
# TODO: your code here
return -1
G = {0: [1, 2, 3], 1: [0], 2: [0, 4, 5], 3: [0], 4: [2, 5], 5: [2, 4]}
check(count_important(G), expected=2)
G = {0: [1, 2], 1: [0, 2], 2: [0, 1]}
check(count_important(G), expected=0)
G = {0: [1], 1: [0, 2], 2: [1, 3], 3: [2, 4], 4: [3]}
check(count_important(G), expected=3)
G = {
0: [1, 2, 7, 6],
1: [0, 3, 4, 2, 7],
2: [0, 6, 4, 1, 3],
3: [1, 5, 8, 7, 6, 2],
4: [1, 8, 9, 7, 2, 5],
5: [3, 4],
6: [2, 0, 3],
7: [0, 9, 4, 3, 1],
8: [3, 4],
9: [7, 4]
}
check(count_important(G), expected=0)
G = {0: [1, 2, 4, 5, 3], 1: [0, 2], 2: [0, 3, 4, 1], 3: [2, 5, 0], 4: [0, 2, 5], 5: [3, 0, 4]}
check(count_important(G), expected=0)
The expected value is 2, but we have -1 The expected value is 0, but we have -1 The expected value is 3, but we have -1 The expected value is 0, but we have -1 The expected value is 0, but we have -1
Can you solve the count_important
problem above using $O(N+M)$ time complexity where $N$ is number of cities (nodes) and $M$ is number of roads (edges)?
Hint: draw how the dfs travels the graph
def count_important_fast(G):
# TODO
# Complete the function as in the previous question
return -1
G={0:[1,2,3], 1:[0], 2:[0,4,5], 3:[0], 4:[2,5], 5:[2,4]}
check(count_important_fast(G), expected=2)
G=[[1,2],[0,2],[0,1]]
check(count_important_fast(G), expected=0)
G=[[1],[0,2],[1,3],[2,4],[3]]
check(count_important_fast(G), expected=3)
G=[[1, 2, 7, 6], [0, 3, 4, 2, 7], [0, 6, 4, 1, 3], [1, 5, 8, 7, 6, 2], [1, 8, 9, 7, 2, 5], [3, 4], [2, 0, 3], [0, 9, 4, 3, 1], [3, 4], [7, 4]]
check(count_important_fast(G), expected=0)
G=[[1, 2, 4, 5, 3], [0, 2], [0, 3, 4, 1], [2, 5, 0], [0, 2, 5], [3, 0, 4]]
check(count_important_fast(G), expected=0)
G=[[1, 8, 25, 64, 73, 89, 423, 1534, 2133, 2353, 2433, 2875, 729, 3021], [0, 2, 3, 7, 11, 24, 27, 58, 112, 236, 435, 3911, 3045, 829], [1, 4, 6, 9, 12, 14, 28, 181, 671, 3665, 4564, 637], [1, 5, 10, 21, 39, 106, 532, 3002, 323, 4809, 4496], [2, 13, 15, 38, 100, 118, 400, 3421, 910, 2952], [3, 17, 32, 201, 3188, 4731], [2, 16, 29, 74, 291, 874, 4037, 4213], [1, 103, 458, 2452, 1622], [0, 94, 125, 228, 238, 3742], [2, 50, 564, 1494, 1693, 4121, 692], [3, 18, 23, 30, 43, 413, 2543, 3015, 1931, 1973], [1, 20, 193, 2323, 2131], [2, 77, 194, 863, 2523, 2582, 3385, 2150], [4, 61, 92, 245, 284, 726, 1352, 2537, 4240], [2, 41, 51, 158, 296, 1523, 1585], [4, 34, 144, 174, 268, 280, 853, 946, 3999], [6, 79, 110, 411, 1109, 1780, 1105, 3124], [5, 19, 117, 272], [10, 49, 152, 1556, 4032, 1701, 4192], [17, 40, 54, 81, 176, 179, 204, 349, 3499, 3833, 2238, 4359, 1053, 2018], [11, 22, 33, 42, 60, 1589, 4081, 3768, 2003], [3, 113, 127, 1761, 2688, 4479, 4977, 2531], [20, 242, 518, 974, 4727, 3549, 1964, 2178, 2701], [10, 26, 111, 260, 609, 2706, 4772], [1, 53, 62, 330, 2255, 2359, 712, 4943], [0, 108, 239, 312, 1845], [23, 31, 44, 86, 177, 293, 995, 1062, 2779, 451, 873], [1, 66, 364, 883], [2, 36, 45, 155, 547, 4561, 4699, 1121, 2438], [6, 35, 85, 154, 164, 691, 702, 928, 2330, 976, 4155, 1161, 2565], [10, 37, 52, 71, 84, 96, 207, 1922, 4565], [26, 145, 263, 351, 820, 1443, 4089, 4194, 4563], [5, 59, 78, 1470, 3585, 1081, 1113, 4290], [20, 56, 233, 867, 1042], [15, 93, 215, 249, 1095, 3140], [29, 87, 134, 327, 544, 1512, 1570, 484], [28, 55, 314, 682, 4603], [30, 47, 4503, 3674, 3347], [4, 163, 222, 417, 2373, 293], [3, 185, 762, 1159, 3515], [19, 190, 271, 2627, 4123, 4295], [14, 383, 2654, 3288, 4061, 3234, 1787, 3961], [20, 109, 130, 153, 357], [10, 4086, 882, 3526], [26, 46, 48, 65, 133, 589, 1268, 1898, 3494, 696], [28, 57, 160, 437, 490, 605, 630, 1738, 4886], [44, 114, 235, 1192, 1529, 1565, 2934, 3198, 4294, 2596], [37, 1321, 1467, 2321, 3208, 694, 4785], [44, 63, 717, 1818, 4583, 4120, 1672], [18, 126, 195, 231, 876, 1305, 4375, 4454, 863, 464, 2036], [9, 76, 2308, 2885, 4683, 2722, 4109], [14, 4393, 939], [30, 115, 213, 531, 2820, 1072, 448], [24, 72, 102, 227, 545, 923, 978, 2124, 4821, 4960], [19, 83, 116, 147, 389, 559, 1277, 2188], [36, 465, 515, 824, 970, 2407, 4659, 1563, 3992, 3174, 1648, 1815, 4268], [33, 97, 159, 1133, 1798, 4838, 4363, 3001, 436, 3579, 3582], [45, 146, 241, 320, 634, 1721, 2429, 3043, 4204, 717, 3762], [1, 191, 290, 426, 1256, 1525, 4054, 4166], [32, 69, 119, 166, 419, 610, 613, 2269, 2438, 2869, 3501, 1546, 2302, 1625, 1782], [20], [13, 186, 358, 567, 4281, 2160, 1614, 1562], [24, 371, 548, 936, 1855, 3001, 2250, 2875, 3478], [48, 68, 2008], [0, 178, 576, 3316, 4592, 1555], [44, 392, 1018, 1557, 1765, 3231, 4514, 3933], [27, 67, 4101, 1068], [66, 70, 98, 208, 246, 1685, 2127, 1526, 1776], [63, 99, 775, 1358, 1484, 1680, 2013, 2450, 4894, 3227], [59, 80, 172, 196, 2174], [67, 75, 88, 578, 736, 899, 1584, 3241], [30, 90, 275, 4239], [53, 151, 898, 962, 2319, 2982, 2669, 3695, 760], [0, 123, 386, 436, 1892, 2034, 4011, 4815, 3224, 3598, 2684, 2295], [6, 353, 462, 1455, 4327, 108], [70, 91, 439, 958, 2250, 4794], [50, 124, 307, 463, 2345, 1451], [12, 148, 229, 2497], [32, 107, 862, 579, 3003], [16, 180, 225, 343, 467, 780, 785, 1374, 1843, 1939], [69, 82, 101, 232, 575, 823, 1102, 1587, 2776, 2118], [19, 95, 598, 1364, 1062, 257], [80, 167, 1617, 3976, 3989, 2587], [54, 135, 3907, 4098], [30, 137, 701, 2620, 2856, 4151, 1235, 3597], [29, 149, 1771, 2639, 3119, 3384, 2837, 2185], [26, 104, 1168, 1635, 2612, 4333, 2977, 1676], [35, 580, 113, 1199, 2194], [70, 161, 1729, 1921, 2461, 3167, 3361, 369, 179, 2819, 1836], [0, 120, 306, 308, 336, 344, 377, 454, 603, 1232, 1408, 1810, 2811, 4388, 2101, 4394, 2350, 4208], [71, 175, 421, 512, 734, 2143, 2366, 4558], [75, 128, 162, 432, 815, 1106, 1493, 3624, 3654], [13, 3664, 3362], [34, 270, 295, 387, 857, 1398, 3618, 4361, 1573, 3585], [8, 292, 2076, 4990, 1485, 3284], [81, 169, 517, 2669, 3395, 4363], [30, 373, 1251], [56, 131, 226, 338, 728, 1210, 4772, 4852], [67, 122, 129, 187, 381, 896, 2555, 4686, 1244, 4690, 1480], [68, 234, 4845, 368], [4, 216, 807, 2930, 3014, 1062, 2448], [80, 171, 572, 873, 1000, 3866, 2042, 3501, 3776, 2158], [53, 105, 346, 655, 725, 4123, 3038, 4074], [7, 325, 466, 615, 232], [86, 665, 2151, 4277, 4574], [102, 258, 315, 2170, 1521, 545, 1023], [3, 253, 560, 1519, 2122, 2136, 2416], [78, 140, 1049, 4335, 4087, 3234], [25, 214, 692, 74, 1132, 1891], [42, 378, 1029, 1438, 1537, 2628, 3674, 3940, 2886, 1340], [16, 132], [23, 901, 1385, 4690, 2352, 2897, 2978], [1, 205, 488, 4851], [21, 1688, 87, 2221, 2468], [46, 1423, 2113, 3698, 1230], [52, 369, 3276, 2935, 1364, 1951, 1416], [54, 141, 626, 843, 971, 1396, 1984, 3641, 4912, 4134, 2966], [17, 121, 142, 202, 259, 628, 1826], [4, 301, 3130, 3463, 765, 2868], [59, 1279, 3470, 1547, 4232, 4131, 4819], [89, 183, 472, 637, 1123, 953, 2509], [117, 157, 313, 326, 407, 520, 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2946, 4700], [1710], [1853, 4936, 1856, 2915, 4482], [2509, 2407], [4598, 874, 3082, 263, 4468, 1997, 1239], [3771, 3589], [4249, 3654], [1893, 3055, 4907, 1570, 4362], [2803, 2919, 847], [1731, 2349, 3844, 3451, 2953], [850, 408], [22, 2584, 4812], [1480, 2578, 2506], [1471, 4245], [933, 3608, 2957], [2735, 462, 5, 1742, 3448], [4167, 325, 1225, 4804], [3718, 3415, 2258], [3220, 3521], [4324, 4536, 1048, 4823], [3992, 1387], [1744, 2699, 849], [2520, 4808, 3660, 2472], [1465, 1381], [636, 3797, 2107], [1284, 3568, 2108, 1528], [4304, 4811, 4284, 3173], [1047], [1843], [699, 206, 2940, 624, 1834], [2344, 2381], [4242, 254, 1568, 1105, 1078], [1630, 3362, 3308, 1437, 1680], [1798, 2948], [753, 3408, 3968], [3271, 1300, 4790], [4237, 4971, 4215], [1143, 2489], [3051, 768, 205, 3778], [836, 2037, 3548, 2749], [2869, 3128, 4458], [942, 276], [4258, 2968, 1159, 552, 2391], [2918, 575], [2801, 4653], [3560, 4283, 2992, 4356, 482], [3514, 3552, 543], [2292, 2457], [3011], [3682, 4933, 460, 4858, 4128, 1808, 4246, 3120], [3149], [2220, 613, 1412], [2376, 1849], [2677], [467, 1994], [635, 4834, 3810, 4097], [97, 23, 303, 2473], [1885, 671, 3685], [3878, 4950, 1769, 4064], [3447, 1201, 1190], [2560, 554, 714, 4156, 1473, 4866, 2222], [1459, 1917, 2136, 1153], [4589, 1604, 2879], [4413, 540, 4670], [2771, 3874, 3090], [691, 2293, 4449, 782, 3234, 3572, 4329, 2420, 2722], [1137], [4584], [2221], [1103, 205, 457, 47, 3785, 2083], [1767, 1096], [3528, 2978], [3981, 4077], [2011, 1237, 3356, 753], [3831, 2575, 4751], [751, 4419, 2275, 3895], [1656, 324, 1885, 2196], [1311, 1508, 2190, 159], [3233, 3614, 75, 3147], [4178, 322], [3678, 3023, 3166], [1984, 3200, 3337], [3035], [748, 4556], [2977, 4418, 3746, 3640, 2517, 2790], [4181, 2205, 1768], [3127, 438], [4644, 3571, 1866], [160, 1932, 123, 2957, 453, 4732], [1416, 4055, 2054, 1081], [1552, 1733], [2095, 2153], [4738, 4834, 3774, 1638], [4270, 3, 917], [4338, 4132, 1945, 2235], [1989, 4742], [1289, 219, 4727], [4198, 1858], [2945, 1641, 3764, 1038], [73, 217, 3899], [401, 1845], [817, 4901, 3902, 3069, 1702], [3243, 1920], [894, 376, 119, 4350], [1995, 3250], [53, 1469, 2611, 1331], [219, 1676, 474, 4158], [2790, 3931, 4735, 3684, 3287], [3985, 3798, 2003, 720], [3235, 700, 1610], [1427, 2310, 3377, 2109, 3458], [1261, 2516, 2351, 3594], [2779, 868, 3798], [3639, 2182, 472, 3901], [2654, 3370, 1852], [1442, 3440, 697, 3248], [2325, 3360, 3545], [1417, 1610, 4297], [4808, 3631, 3465, 4771, 1641, 4044], [3259], [1909, 3935], [665, 624, 2823, 1248], [2345, 56, 3234, 4683], [4538, 2953, 4648], [3211, 2924], [1243, 3753], [670, 2275, 2907, 1442], [2519, 3387, 474, 602, 3038, 457], [623, 3076, 656, 1856, 1685], [99, 2627, 4197, 3517], [2053, 3909], [4040], [1034, 3674], [1981, 2299], [3277, 1501, 4182, 4628, 4300], [4088, 1600, 3117, 3706, 112, 3488], [97, 4426, 1797], [3743, 1709], [717, 1315], [3815, 3120, 3939], [2557, 2523, 4225], [1989, 1720, 3874, 4494, 303], [3641, 3876, 406, 3032, 4765, 2847], [4356, 1160], [323, 985, 4706, 3780, 4184], [4036, 2517, 4652], [4385, 519, 2639, 1740, 836], [2328, 4878], [1936, 4937, 983, 896, 3142, 3108], [3360, 4516, 1167, 430, 4667], [2177, 4164, 4776, 2469, 2220], [1247, 1478], [978, 3697], [4448, 4375], [2342, 3074, 3666, 4334, 4466], [4316, 1481, 881], [3076, 4193, 3211], [2290], [514, 4126, 201, 4374], [3196, 281], [1762], [517, 3909, 3490, 2539], [1620, 4863, 4099], [2197, 3486, 2340], [2583, 3212, 3316], [3021, 1930, 1579], [3940, 2839], [1915], [3979, 1061], [1614, 4648], [1308, 1236, 4288, 45, 915], [171, 683, 3860], [451, 4356, 2309, 3175], [4265], [2751, 4167], [3063, 4051, 296, 743], [4450, 2684, 4526, 2349, 3821, 2038, 2160, 4357], [1719, 1311, 1630, 3806, 3745, 3798, 1707], [4459, 3495, 68], [3725], [3540, 1785, 1957], [2893, 3685, 2725, 2826], [3537, 1762, 148], [2816, 4951, 3266], [3750, 4575, 2464, 419], [3861, 2513, 4817], [2905, 1525, 2227, 4360, 4923, 3629], [602, 4612, 2124], [1314, 1533, 4565], [1608, 376, 318], [253, 861, 958, 1988], [3683, 240, 1959, 4723], [3615, 1901, 1625, 1180, 3729], [2295, 286, 3311, 234], [2857], [2570, 1771], [116, 1656, 4977, 4530], [2345], [4689], [3638, 3239, 4932], [1694, 1985, 1460, 199, 3742], [2217], [2870, 2019], [3781, 775, 3696, 2021], [2966, 4536], [445, 3915, 2195], [2486, 2846], [3045, 3907, 4902], [911, 2862, 1595], [4224, 2076, 4600, 3696], [1139, 2578], [2956], [199, 620, 1818, 2131], [3937], [4416, 425, 2549], [1102], [1270, 1658, 4915], [4765, 3524, 1887], [4322, 3175, 3852, 2019, 368], [4564], [4718, 330], [4864, 976, 2952, 3080, 2579, 1336], [1651, 2246, 4633, 3425], [2693, 678, 3907, 1929], [3191], [1606, 409], [2254, 4233, 4077, 2605, 3312], [1795, 1798, 4460, 24], [2831], [1949, 413, 462], [2986, 2098, 1059], [3185, 4213, 1827, 1881, 3546], [2151, 1882, 2860, 2150], [2238, 4241, 1699], [1344, 4774, 1562, 652, 1679, 3218], [4899], [2263, 660], [2766, 3309, 3050, 4372], [3499, 4237], [2639, 2218, 1367, 2625, 2985], [1501, 955], [2688, 2602], [3425], [1306, 2328, 4077, 3248, 1803, 2265], [53, 1327], [4024, 4582, 444, 2602, 3378], [2466, 4112], [4670, 4137], [3302, 660, 1636], [1710, 231, 380], [1426, 1697], [460, 4068], [3738, 4651, 1538, 1069, 2955, 589], [2089], [729, 4259, 1450], [2732, 2390, 4752, 2268, 3012], [1471], [2153, 1418, 1492], [1356, 2281, 3540], [3534, 4149, 2294, 2172, 4494], [1448, 2819, 2969], [21, 448, 2984, 4912], [3994, 614], [3177, 3391, 404, 4482], [3164, 3013], [1550], [992], [2269], [4356, 4516, 3479], [4149, 1956, 821, 2443], [3635, 2985, 2524, 4065], [1334, 3922, 388, 4177, 1261], [3703, 3945, 1298, 1573], [4278, 2249, 4042, 1804, 3764], [94], [2746, 1970, 1978], [1967], [1266], [767], [2379, 888, 2487], [515, 3385, 1376], [3337, 2910, 678], [2285], [3518]]
check(count_important_fast(G), expected=23)
The expected value is 2, but we have -1 The expected value is 0, but we have -1 The expected value is 3, but we have -1 The expected value is 0, but we have -1 The expected value is 0, but we have -1 The expected value is 23, but we have -1