In this assignment you will implement one or more algorithms for the all-pairs shortest-path problem. There are data files describing three graphs:
g1.txt g2.txt g3.txt
The first line indicates the number of vertices and edges, respectively. Each subsequent line describes an edge (the first two numbers are its tail and head, respectively) and its length (the third number). NOTE: some of the edge lengths are negative. NOTE: These graphs may or may not have negative-cost cycles.
Your task is to compute the "shortest shortest path". Precisely, you must first identify which, if any, of the three graphs have no negative cycles. For each such graph, you should compute all-pairs shortest paths and remember the smallest one (i.e., compute , where denotes the shortest-path distance from to ).
If each of the three graphs has a negative-cost cycle, then enter "NULL" in the box below. If exactly one graph has no negative-cost cycles, then enter the length of its shortest shortest path in the box below. If two or more of the graphs have no negative-cost cycles, then enter the smallest of the lengths of their shortest shortest paths in the box below.
OPTIONAL: You can use whatever algorithm you like to solve this question. If you have extra time, try comparing the performance of different all-pairs shortest-path algorithms!
OPTIONAL: For fun, try computing the shortest shortest path of the graph on a larger graph in file large.txt.
see test cases here: https://github.com/beaunus/stanford-algs/tree/master/testCases/course4/assignment1AllPairsShortestPath
In [ ]:
import numpy as np
import collections
graph1 = "g1.txt"
graph2 = "g2.txt"
graph3 = "g3.txt"
fp1 = open(graph1, 'r')
data = fp1.readlines()
n1, m1 = data[0].strip().split(" ")
n1, m1 = int(n1), int(m1)
# edges1 = collections.defaultdict(list)
edges1 = [] # node1, node2, weight
nodes1 = set()
print (n1, m1)
for line in data[1:]:
a,b,w = line.split(" ")
edges1.append([a,b,int(w)])
# edges1[b].append([a,int(w)])
nodes1.add(a)
nodes1.add(b)
nodes1 = list(nodes1)
nodes1_to_idx = {nodes1[i]:i for i in range(len(nodes1))}
In [ ]:
print (edges1[:5])
print (nodes1[999], nodes1_to_idx['725'])
In [ ]:
A = np.full([n1,n1,n1], np.nan)
for i in range(n1):
A[i,i,0] = 0
for e in edges1:
i = nodes1_to_idx[e[0]]
j = nodes1_to_idx[e[1]]
A[i,j,0] = e[2]
In [ ]:
for k in range(1, n1):
for i in range(1,n1):
for j in range(1,n1):
A[i,j,k] = min(A[i,j,k-1], A[i,k,k-1] + A[k,j,k-1])
In [ ]:
In [56]:
import numpy as np
import collections
from tqdm import tqdm
import copy
def bellman(s, n, edges, nodes, nodes_to_idx):
# n = len(nodes)
A = np.zeros([n+1,n+1])
for v in range(n):
A[0,v] = np.inf
A[0, nodes_to_idx[s]] = 0
for i in tqdm(range(1,n+1)):
for v in nodes:
mindist = np.inf
# go through all incoming edges to v
for w in edges[v]:
dist = A[i-1, nodes_to_idx[w[0]]] + w[1]
if dist < mindist:
mindist = dist
A[i, nodes_to_idx[v]] = min(A[i-1, nodes_to_idx[v]] , mindist)
# print (A[i,nodes_to_idx[v]])
return A
file = "input_random_10_8.txt"
file = "input_random_14_16.txt"
file = "input_random_20_32.txt"
file = "input_random_36_512.txt"
file = "g3.txt"
fp1 = open(file, 'r')
data = fp1.readlines()
n, m = data[0].strip().split(" ")
n, m = int(n), int(m)
edges = collections.defaultdict(list) # stores incoming edges for each node
nodes = set()
print (n, m)
for line in data[1:]:
a,b,w = line.strip().split(" ")
edges[b].append([a,int(w)])
nodes.add(a)
nodes.add(b)
# create graph G'
edges_prime = copy.deepcopy(edges)
nodes_prime = nodes
# add extra node S connected to all
for v in nodes_prime:
edges_prime[v].append(["S",0])
nodes_prime.update("S")
nodes_prime = list(nodes_prime)
nodes_to_idx_prime = {nodes_prime[i]:i for i in range(len(nodes_prime))}
n_prime = n+1
# Run Bellman-Ford on G' tarting from new node S
A = bellman("S", n_prime, edges_prime, nodes_prime, nodes_to_idx_prime)
p = A[n_prime-1,:]
# Check if there is a negative loop
if (False in (A[n_prime-1,:] == A[n_prime-2,:])):
print("Negative cycle present!")
else:
print ("no negative cycles!")
# Run the reweighting steps on the original graph
nodes = list(nodes)
nodes_to_idx = {nodes[i]:i for i in range(len(nodes))}
for v in nodes:
p_v = p[nodes_to_idx[v]]
for _u in edges[v]:
p_u = p[nodes_to_idx[_u[0]]]
_u[1] = _u[1] + p_u - p_v
# print (v, _u, p_u - p_v)
# remove S from nodes
nodes.remove("S")
# Create the graph data structure for djikstra input
graph = collections.defaultdict(list)
for v, val in edges.items(): # edge[v] -> v
for u in val:
graph[u[0]].append([v, u[1]])
print (graph)
# Run Djikstra n times
1000 47978
100%|██████████| 1001/1001 [00:38<00:00, 28.28it/s]
no negative cycles!
defaultdict(<class 'list'>, {'707': [['91', 37.0], ['271', 38.0], ['400', 38.0], ['80', 15.0], ['539', 22.0], ['638', 25.0], ['769', 36.0], ['817', 53.0], ['392', 40.0], ['802', 12.0], ['600', 46.0], ['373', 45.0], ['635', 38.0], ['633', 17.0], ['226', 11.0], ['334', 49.0], ['692', 16.0], ['290', 2.0], ['145', 8.0], ['714', 20.0], ['348', 33.0], ['355', 19.0], ['517', 31.0], ['332', 39.0], ['124', 17.0], ['884', 45.0], ['102', 23.0], ['152', 27.0], ['27', 29.0], ['698', 0.0], ['826', 8.0], ['554', 25.0], ['492', 38.0], ['828', 38.0], ['677', 17.0], ['239', 6.0], ['120', 28.0], ['424', 20.0], ['472', 28.0], ['903', 26.0], ['642', 28.0], ['208', 5.0], ['659', 23.0], ['606', 26.0], ['44', 29.0], ['435', 28.0], ['335', 31.0]], '301': [['679', 15.0], ['944', 24.0], ['198', 36.0], ['551', 7.0], ['621', 10.0], ['150', 48.0], ['41', 18.0], ['106', 32.0], ['901', 29.0], ['246', 22.0], ['146', 26.0], ['699', 53.0], ['708', 39.0], ['670', 14.0], ['260', 34.0], ['978', 4.0], ['794', 33.0], ['968', 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20.0], ['315', 1.0], ['596', 23.0], ['656', 13.0], ['805', 16.0], ['853', 12.0], ['581', 19.0], ['408', 24.0], ['43', 33.0], ['995', 55.0], ['147', 13.0], ['173', 27.0], ['357', 23.0], ['794', 19.0], ['741', 53.0], ['282', 5.0], ['475', 48.0], ['219', 34.0], ['175', 37.0], ['380', 45.0], ['567', 53.0], ['71', 0.0], ['563', 39.0], ['112', 4.0], ['837', 15.0], ['234', 17.0], ['504', 23.0], ['665', 38.0], ['471', 47.0], ['129', 43.0], ['393', 22.0], ['838', 27.0], ['677', 11.0], ['888', 43.0], ['889', 16.0], ['70', 21.0], ['994', 44.0], ['867', 31.0], ['164', 21.0], ['489', 44.0], ['485', 12.0], ['69', 3.0]], '934': [['280', 0.0], ['163', 11.0], ['806', 11.0], ['805', 17.0], ['553', 23.0], ['814', 32.0], ['583', 29.0], ['360', 18.0], ['500', 19.0], ['981', 45.0], ['683', 7.0], ['851', 5.0], ['973', 0.0], ['928', 7.0], ['154', 22.0], ['955', 0.0], ['625', 38.0], ['676', 19.0], ['276', 25.0], ['441', 15.0], ['543', 48.0], ['933', 10.0], ['355', 20.0], ['616', 42.0], ['346', 5.0], ['496', 14.0], ['215', 38.0], ['42', 5.0], ['471', 22.0], ['800', 23.0], ['671', 10.0], ['652', 11.0], ['279', 44.0], ['715', 16.0], ['462', 40.0], ['213', 31.0], ['385', 44.0], ['18', 6.0], ['694', 1.0], ['693', 21.0], ['568', 43.0], ['659', 11.0], ['743', 39.0], ['45', 28.0], ['21', 15.0], ['759', 5.0], ['298', 11.0], ['5', 6.0], ['807', 19.0], ['571', 32.0], ['417', 40.0], ['439', 22.0], ['103', 23.0]], '433': [['970', 8.0], ['50', 4.0], ['49', 32.0], ['295', 10.0], ['742', 54.0], ['247', 22.0], ['886', 5.0], ['58', 45.0], ['706', 17.0], ['525', 47.0], ['408', 26.0], ['633', 51.0], ['670', 6.0], ['990', 52.0], ['794', 8.0], ['692', 40.0], ['618', 1.0], ['932', 35.0], ['118', 23.0], ['291', 32.0], ['915', 18.0], ['367', 32.0], ['937', 49.0], ['302', 23.0], ['272', 24.0], ['337', 6.0], ['410', 48.0], ['376', 48.0], ['248', 24.0], ['608', 19.0], ['54', 32.0], ['92', 20.0], ['580', 7.0], ['374', 43.0], ['333', 41.0], ['214', 18.0], ['953', 19.0], ['471', 35.0], ['432', 46.0], ['96', 33.0], ['803', 40.0], ['459', 14.0], ['209', 16.0], ['903', 18.0], ['855', 25.0], ['139', 8.0], ['164', 14.0], ['501', 34.0], ['659', 48.0], ['473', 11.0], ['980', 33.0], ['510', 9.0], ['878', 31.0], ['576', 0.0], ['533', 2.0], ['270', 1.0]], '485': [['548', 24.0], ['243', 31.0], ['690', 49.0], ['55', 37.0], ['414', 2.0], ['729', 30.0], ['586', 15.0], ['222', 21.0], ['890', 19.0], ['16', 1.0], ['100', 11.0], ['852', 45.0], ['541', 43.0], ['879', 10.0], ['373', 9.0], ['304', 4.0], ['505', 26.0], ['166', 2.0], ['602', 27.0], ['464', 24.0], ['301', 14.0], ['949', 21.0], ['268', 13.0], ['65', 29.0], ['277', 21.0], ['695', 40.0], ['124', 11.0], ['11', 9.0], ['608', 33.0], ['476', 31.0], ['148', 26.0], ['99', 29.0], ['420', 1.0], ['318', 11.0], ['293', 26.0], ['516', 23.0], ['619', 31.0], ['672', 43.0], ['671', 27.0], ['561', 50.0], ['855', 23.0], ['385', 14.0], ['68', 33.0], ['867', 51.0], ['709', 14.0], ['730', 26.0], ['510', 41.0], ['717', 9.0], ['658', 26.0], ['571', 5.0], ['292', 13.0]]})
In [ ]:
class djikstra(object):
MAX_WEIGHT = np.inf
def __init__(self, graph, vertices, edges):
self.graph = graph
self.vertices = vertices
self.edges = edges
self.X = [] # Vertices processed so far
self.unprocessed_vertices = vertices.copy() # V-X
self.A = {v:np.inf for v in vertices} # computed shortest path from source s to key
def compute_next_min_edge(self):
minD = djikstra.MAX_WEIGHT
minE = ""
minS = ""
#print ("X", self.X, "V-X", self.unprocessed_vertices )
for edge in self.edges:
src = edge[0]
dst = edge[1]
weight = edge[2]
if src in self.X and dst in self.unprocessed_vertices:
d = self.A[src] + weight
#print ("Consider edge", src, dst, d)
if d < minD:
#print ("add Edge", src, dst, weight)
minD = d
minE = dst
minS = src
#print ("final choice", minS, minE, self.A[minS], minD)
if minE:
self.A[minE] = minD
return minE
else:
return None
def reinit(self, s):
self.X = [s]
self.unprocessed_vertices = self.vertices.copy()
self.unprocessed_vertices.remove(s)
self.A[s] = 0
def run(self, s, d):
self.reinit(s)
n = len(self.vertices)
v = s
while (n > 0):
w = self.compute_next_min_edge()
if w is None:
# No more edges between X and V-X to process. Set all other edges to MAX
#print ("unprocessed", self.unprocessed_vertices)
for i in self.unprocessed_vertices:
self.A[i] = djikstra.MAX_WEIGHT
break
#print ("pick", w)
#print ("processed", self.X, self.A)
self.unprocessed_vertices.remove(w)
self.X.append(w)
n -= 1
# if w == d:
# break
return self.A
#Example
def get_edges(graph):
edges = []
for s, adj in graph.items():
for v in adj:
edges.append([s, v[0], int(v[1])])
return edges
In [2]:
import heapq
import itertools
class djikstraHEAP(object):
MAX_WEIGHT = np.inf
def __init__(self, graph, vertices, edges):
self.graph = graph
self.vertices = vertices
self.edges = edges
self.X = [] # Vertices processed so far
self.unprocessed_vertices = vertices.copy() # V-X
self.A = dict() # computed shortest path from source s to key
#### Init Heap Functions
def init_heap(self):
self.pq = [] # list of entries arranged in a heap
self.entry_finder = {} # mapping of tasks to entries
self.REMOVED = '<removed-task>' # placeholder for a removed task
self.counter = itertools.count() # unique sequence count
def add_task(self, task, priority=0):
'Add a new task or update the priority of an existing task'
if task in self.entry_finder:
self.remove_task(task)
count = next(self.counter)
entry = [priority, count, task]
self.entry_finder[task] = entry
heapq.heappush(self.pq, entry)
def remove_task(self, task):
'Mark an existing task as REMOVED. Raise KeyError if not found.'
entry = self.entry_finder.pop(task)
entry[-1] = self.REMOVED
def pop_task(self):
'Remove and return the lowest priority task. Raise KeyError if empty.'
while self.pq:
priority, count, task = heapq.heappop(self.pq)
if task is not self.REMOVED:
del self.entry_finder[task]
return task, priority
return None, None
# raise KeyError('pop from an empty priority queue')
#####
def compute_next_min_edge(self):
minD = djikstraHEAP.MAX_WEIGHT
minE = ""
minS = ""
# minD, minE = heapq.heappop(self.heap)
minE, minD = self.pop_task() # returns closest edge to X task, priority
# print ("Next Min Edge:", minE, minD)
if not minE:
return None
for edges in self.graph[minE]:
u = edges[0]
w = edges[1]
if u in self.X:
# don't handle destination that are alreayd in X
continue
elif u not in self.A:
self.add_task(u, w)
self.A[u] = self.A[minE] + w
elif (self.A[u] > self.A[minE] + w):
self.A[u] = self.A[minE] + w
# heapq.decrease-key(self.heap, )
self.remove_task(u)
self.add_task(u, self.A[u])
return minE
def reinit(self, s):
self.X = [s]
self.unprocessed_vertices = self.vertices.copy()
self.unprocessed_vertices.remove(s)
self.A[s] = 0
# self.heap = [] # minheap for the edges out of X
self.init_heap()
for v in self.edges:
src = v[0]
dst = v[1]
w = v[2]
if src == s:
# heapq.heappush(self.heap, (w, dst))
self.add_task(dst, w)
self.A[dst] = w
# print ("self.pq", self.pq)
def run(self, s):
self.reinit(s)
n = len(self.vertices)
v = s
while (n > 0):
w = self.compute_next_min_edge()
if w is None:
# No more edges between X and V-X to process. Set all other edges to MAX
#print ("unprocessed", self.unprocessed_vertices)
for i in self.unprocessed_vertices:
self.A[i] = djikstraHEAP.MAX_WEIGHT
break
#print ("pick", w)
#print ("processed", self.X, self.A)
self.unprocessed_vertices.remove(w)
self.X.append(w)
n -= 1
return self.A
def get_edges(graph):
edges = []
for s, adj in graph.items():
for v in adj:
edges.append([s, v[0], v[1]])
return edges
# graph ={
# "1": [["2",1], ["3", 4]],
# "2": [["3", 2], ["4",6]],
# "3": [["4",3]],
# "4": []
# }
# vertices = ["1", "3", "2", "4"]
# edges = get_edges(graph)
# d = djikstraHEAP(graph, vertices, edges)
# print(d.run("1"))
In [57]:
distances = []
for u in tqdm(nodes):
# print ("Process node", u)
_graph = copy.deepcopy(graph)
_nodes = copy.deepcopy(nodes)
edges = get_edges(_graph)
d = djikstraHEAP(_graph, _nodes, edges)
F = d.run(u)
# print (F)
p_u = p[nodes_to_idx[u[0]]]
for v, val in F.items():
if u == v:
continue
p_v = p[nodes_to_idx[v]]
ce = val - p_u + p_v
# print (u,"-",v,"=",ce, "p_u", p_u, "p_v", p_v)
distances.append(ce)
print ("Min:", min(distances))
100%|██████████| 1000/1000 [14:09<00:00, 1.36it/s]
Min: -19.0
In [ ]:
# _graph = copy.deepcopy(graph)
# _nodes = copy.deepcopy(nodes)
# edges = get_edges(_graph)
# d = djikstraHEAP(_graph, _nodes, edges)
# F = d.run('756')
In [ ]:
print (distances)
In [ ]:
Content source: jazracherif/algorithms
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