The file djikstraData.txt contains an adjacency list representation of an undirected weighted graph with 200 vertices labeled 1 to 200. Each row consists of the node tuples that are adjacent to that particular vertex along with the length of that edge. For example, the 6th row has 6 as the first entry indicating that this row corresponds to the vertex labeled 6. The next entry of this row "141,8200" indicates that there is an edge between vertex 6 and vertex 141 that has length 8200. The rest of the pairs of this row indicate the other vertices adjacent to vertex 6 and the lengths of the corresponding edges.
Your task is to run Dijkstra's shortest-path algorithm on this graph, using 1 (the first vertex) as the source vertex, and to compute the shortest-path distances between 1 and every other vertex of the graph. If there is no path between a vertex v and vertex 1, we'll define the shortest-path distance between 1 and v to be 1000000.
You should report the shortest-path distances to the following ten vertices, in order: 7,37,59,82,99,115,133,165,188,197. You should encode the distances as a comma-separated string of integers. So if you find that all ten of these vertices except 115 are at distance 1000 away from vertex 1 and 115 is 2000 distance away, then your answer should be 1000,1000,1000,1000,1000,2000,1000,1000,1000,1000. Remember the order of reporting DOES MATTER, and the string should be in the same order in which the above ten vertices are given. The string should not contain any spaces. Please type your answer in the space provided.
IMPLEMENTATION NOTES: This graph is small enough that the straightforward O(mn) time implementation of Dijkstra's algorithm should work fine. OPTIONAL: For those of you seeking an additional challenge, try implementing the heap-based version. Note this requires a heap that supports deletions, and you'll probably need to maintain some kind of mapping between vertices and their positions in the heap.
**TODO**:
In [3]:
class djikstra(object):
MAX_WEIGHT = 1000000
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 = {} # 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 = 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
In [400]:
# heap structure. a binary tree where the key values of children is larger than the key value of the parent
class myHeapArray(object):
def __init__(self):
self.heap = []
self.elem_idx = {}
self.idx_to_elem = {}
"""
Utilities to handle information associated with the tree nodes
"""
def move_element_info(self, src_i, dst_i):
"""
Move elements info from src_i index to dst_i index and delete src_i information
Before:
elem_idx[src] = src_i
idx_to_elem[src_i] = src
elem_idx[dst] = dst_i
idx_to_elem[dst_i] = dst
After:
delete elem_idx[src]
delete idx_to_elem[src_i]
elem_idx[src] = dst_i
idx_to_elem[dst_i] = src
"""
src = self.idx_to_elem[src_i]
del self.idx_to_elem[src_i]
del self.elem_idx[src]
self.idx_to_elem[dst_i] = src
self.elem_idx[src] = dst_i
def add_element_info(self, elem, info):
self.heap.append(elem)
n = len(self.heap) - 1
self.elem_idx[info] = n
self.idx_to_elem[n] = info
def remove_element_info(self, a_i):
a = self.idx_to_elem[a_i]
del self.idx_to_elem[a_i]
del self.elem_idx[a]
def swap_elements_info(self, a_i, b_i):
"""
We want to switch elements at index a and index b
Before:
elem_idx[a] = a_i
idx_to_elem[a_i] = a
elem_idx[b] = b_i
elem_idx[b_i] = b
After:
elem_idx[a] = b_i
idx_to_elem[a_i] = b
elem_idx[b] = a_i
elem_idx[b_i] = a
"""
a = self.idx_to_elem[a_i]
b = self.idx_to_elem[b_i]
self.elem_idx[a] = b_i
self.idx_to_elem[a_i] = b
self.elem_idx[b] = a_i
self.idx_to_elem[b_i] = a
"""
Main Routines for Heap Structure
"""
def bubbleDown(self, p=None):
n = len(self.heap)-1
if not p:
p = 0
while (2*p <= n):
c1 = 2*p
if 2*p + 1 <= n:
c2 = 2*p + 1
# Second child exists
if self.heap[c1] < self.heap[c2]:
c_val, c = (self.heap[c1], c1)
else:
c_val, c = (self.heap[c2], c2)
else:
c_val, c = (self.heap[c1], c1)
# Swap parent with the child with the smallest key value
if self.heap[p] > c_val:
tmp = self.heap[p]
self.heap[p] = c_val
self.heap[c] = tmp
# swap the ids
self.swap_elements_info(p, c)
p = c
else:
break
def extractMin(self):
if not self.heap:
return None, None
val = self.heap[0]
elem = self.idx_to_elem[0]
# copy last value to root, then remove and discard.
last = len(self.heap) - 1
self.heap[0] = self.heap[last]
self.heap.pop()
self.move_element_info(last, 0)
self.bubbleDown()
return val,elem
def bubbleUp(self, c=None):
"""
for parent at node i, children is at node 2*i and 2*i + 1
for child i, parent is at i/2 if i is even or floor(i/2) if i is odd
Bubble Up starting the newly added key at the end of the heap
"""
if not c:
c = len(self.heap) - 1
while c!= 0:
p = int(c / 2)
if self.heap[p] > self.heap[c]:
tmp = self.heap[p]
self.heap[p] = self.heap[c]
self.heap[c] = tmp
self.swap_elements_info(p, c)
c = p
else:
break
def remove(self, info):
n = len(self.heap) - 1
dst_i = self.elem_idx[info]
print ("remove elem {} dst_i {}".format(info, dst_i))
if dst_i == n:
self.remove_element_info(n)
self.heap.pop()
else:
self.heap[dst_i] = self.heap[n]
self.remove_element_info(dst_i)
self.move_element_info(n, dst_i)
self.heap.pop()
self.bubbleUp(dst_i)
def insert(self, elem, info):
self.add_element_info(elem, info)
self.bubbleUp()
def insertList(self, elemList, elemListInfo):
for elem, info in zip(elemList, elemListInfo):
self.add_element_info(elem, info)
self.bubbleUp()
def get_ordered_list(self):
ordered = []
elements = []
while True:
n , elem = self.extractMin()
if n:
ordered.append(n)
elements.append(elem)
else:
break
return ordered, elements
In [416]:
arr = [1, 10, 9, 14, 7 ,19, 24, 16]
#arr =[1,9,11,33,27,21,19,17, 22]
info = [str(i) for i in range(len(arr))]
h = myHeapArray()
h.insertList(arr, info)
print ("arr",arr)
print ("heap", h.heap)
print ("elem_idx", h.elem_idx)
print ("idx_to_elem", h.idx_to_elem)
h.remove("4")
print ("heap", h.heap)
print ("elem_idx", h.elem_idx)
print ("idx_to_elem", h.idx_to_elem)
#sortedHS, infoList = h.get_ordered_list()
#print ("sorted list with indices from original array", sortedHS, infoList)
In [12]:
class djikstraHEAP_old(object):
MAX_WEIGHT = 1000000
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 = {} # computed shortest path from source s to key
def compute_next_min_edge(self):
minD = djikstra.MAX_WEIGHT
minE = ""
minS = ""
minD, minE = self.heap.extractMin()
def reinit(self, s):
self.X = [s]
self.unprocessed_vertices = vertices.copy()
self.unprocessed_vertices.remove(s)
self.A[s] = 0
self.heap = myHeapArray() # heap over vertices in V-X
for v in self.unprocessed_vertices:
src = v[0]
dst = v[1]
w = v[2]
if src == s:
self.heap.insert(w, elem=dst)
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
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_old(graph, vertices, edges)
print(d.run("1", "4"))
In [13]:
import heapq
import itertools
import numpy as np
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 [5]:
#Example
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 = djikstra(graph, vertices, edges)
print(d.run("1", "4"))
In [16]:
FILE = "dijkstraData.txt"
fp = open(FILE, 'r')
graph = {}
vertices = set()
edges = []
for line in fp.readlines():
v = line.strip().split("\t")
vertices.add(v[0])
graph.setdefault(v[0], [])
for u in v[1:]:
t = u.split(",")
graph[v[0]].append([t[0], int(t[1])])
edges.append([v[0], t[0], int(t[1])])
print ("Vertex 1 adj:", graph["1"])
print ("First 5 Edges:", edges[:5])
In [17]:
d = djikstra(graph, vertices, edges)
d.run("1", "7")
print (d.A["7"])
res = []
for k in ["7","37","59","82","99","115","133","165","188","197"]:
# print (d.A[k])
res.append(str(d.A[k]))
print (",".join(res))
In [18]:
d = djikstraHEAP(graph, vertices, edges)
d.run("1")
print (d.A["7"])
res = []
for k in ["7","37","59","82","99","115","133","165","188","197"]:
# print (d.A[k])
res.append(str(d.A[k]))
print (",".join(res))
In [220]:
# compare to SelectionSort
arr = [1, 10, 5, 7, 20, 3, 100, 99, 45, 13]
def InsertSort(arr):
arr = arr.copy()
n = len(arr)
i=1
while i < n:
val = arr[i]
for j in range(0, i):
if arr[j] >= val:
# insert at location j, and shift all by one
arr[j+1:i+1] = arr[j:i]
arr[j]=val
break
i += 1
return arr
s = datetime.now()
sortedArray = InsertSort(arr)
print (sortedArray)
In [332]:
import numpy as np
arr = [1, 10, 5, 7, 20, 3, 100, 99, 45, 13]
N = 10
arr = np.random.randint(0, 10000, size=N)
s = datetime.now()
h = myHeapArray()
h.insertList(arr)
sortedHS = h.get_ordered_list()
e = datetime.now()
print ("heapSort(ms)", (e-s).microseconds / 1000, "ms")
s = datetime.now()
sortedIS = InsertSort(arr)
e = datetime.now()
print ("InsertSort(ms)", (e-s).microseconds / 1000, "ms")
arrsort = arr.copy()
arrsort.sort()
assert sum(sortedHS == arrsort) == N, "Don't match! HS {} arr {}, orig {}".format(sortedHS, arrsort, arr)
assert sum(sortedIS == arrsort) == N, "Don't match! HS {} arr {}, orig {}".format(sortedHS, arrsort, arr)
In [253]:
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