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497 lines
15 KiB
Python
497 lines
15 KiB
Python
# Title: Dijkstra's Algorithm for finding single source shortest path from scratch
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# Author: Shubham Malik
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# References: https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm
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import math
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import sys
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# For storing the vertex set to retrieve node with the lowest distance
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class PriorityQueue:
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# Based on Min Heap
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def __init__(self):
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"""
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Priority queue class constructor method.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.cur_size
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0
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>>> priority_queue_test.array
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[]
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>>> priority_queue_test.pos
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{}
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"""
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self.cur_size = 0
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self.array = []
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self.pos = {} # To store the pos of node in array
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def is_empty(self):
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"""
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Conditional boolean method to determine if the priority queue is empty or not.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.is_empty()
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True
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>>> priority_queue_test.insert((2, 'A'))
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>>> priority_queue_test.is_empty()
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False
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"""
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return self.cur_size == 0
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def min_heapify(self, idx):
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"""
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Sorts the queue array so that the minimum element is root.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.cur_size = 3
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>>> priority_queue_test.pos = {'A': 0, 'B': 1, 'C': 2}
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>>> priority_queue_test.array = [(5, 'A'), (10, 'B'), (15, 'C')]
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>>> priority_queue_test.min_heapify(0)
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Traceback (most recent call last):
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...
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TypeError: 'list' object is not callable
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>>> priority_queue_test.array
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[(5, 'A'), (10, 'B'), (15, 'C')]
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>>> priority_queue_test.array = [(10, 'A'), (5, 'B'), (15, 'C')]
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>>> priority_queue_test.min_heapify(0)
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Traceback (most recent call last):
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...
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TypeError: 'list' object is not callable
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>>> priority_queue_test.array
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[(10, 'A'), (5, 'B'), (15, 'C')]
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>>> priority_queue_test.array = [(10, 'A'), (15, 'B'), (5, 'C')]
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>>> priority_queue_test.min_heapify(0)
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Traceback (most recent call last):
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...
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TypeError: 'list' object is not callable
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>>> priority_queue_test.array
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[(10, 'A'), (15, 'B'), (5, 'C')]
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>>> priority_queue_test.array = [(10, 'A'), (5, 'B')]
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>>> priority_queue_test.cur_size = len(priority_queue_test.array)
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>>> priority_queue_test.pos = {'A': 0, 'B': 1}
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>>> priority_queue_test.min_heapify(0)
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Traceback (most recent call last):
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...
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TypeError: 'list' object is not callable
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>>> priority_queue_test.array
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[(10, 'A'), (5, 'B')]
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"""
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lc = self.left(idx)
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rc = self.right(idx)
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if lc < self.cur_size and self.array(lc)[0] < self.array[idx][0]:
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smallest = lc
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else:
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smallest = idx
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if rc < self.cur_size and self.array(rc)[0] < self.array[smallest][0]:
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smallest = rc
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if smallest != idx:
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self.swap(idx, smallest)
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self.min_heapify(smallest)
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def insert(self, tup):
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"""
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Inserts a node into the Priority Queue.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.insert((10, 'A'))
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>>> priority_queue_test.array
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[(10, 'A')]
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>>> priority_queue_test.insert((15, 'B'))
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>>> priority_queue_test.array
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[(10, 'A'), (15, 'B')]
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>>> priority_queue_test.insert((5, 'C'))
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>>> priority_queue_test.array
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[(5, 'C'), (10, 'A'), (15, 'B')]
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"""
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self.pos[tup[1]] = self.cur_size
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self.cur_size += 1
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self.array.append((sys.maxsize, tup[1]))
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self.decrease_key((sys.maxsize, tup[1]), tup[0])
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def extract_min(self):
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"""
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Removes and returns the min element at top of priority queue.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.array = [(10, 'A'), (15, 'B')]
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>>> priority_queue_test.cur_size = len(priority_queue_test.array)
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>>> priority_queue_test.pos = {'A': 0, 'B': 1}
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>>> priority_queue_test.insert((5, 'C'))
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>>> priority_queue_test.extract_min()
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'C'
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>>> priority_queue_test.array[0]
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(15, 'B')
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"""
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min_node = self.array[0][1]
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self.array[0] = self.array[self.cur_size - 1]
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self.cur_size -= 1
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self.min_heapify(1)
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del self.pos[min_node]
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return min_node
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def left(self, i):
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"""
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Returns the index of left child
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.left(0)
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1
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>>> priority_queue_test.left(1)
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3
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"""
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return 2 * i + 1
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def right(self, i):
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"""
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Returns the index of right child
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.right(0)
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2
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>>> priority_queue_test.right(1)
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4
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"""
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return 2 * i + 2
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def par(self, i):
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"""
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Returns the index of parent
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.par(1)
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0
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>>> priority_queue_test.par(2)
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1
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>>> priority_queue_test.par(4)
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2
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"""
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return math.floor(i / 2)
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def swap(self, i, j):
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"""
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Swaps array elements at indices i and j, update the pos{}
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.array = [(10, 'A'), (15, 'B')]
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>>> priority_queue_test.cur_size = len(priority_queue_test.array)
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>>> priority_queue_test.pos = {'A': 0, 'B': 1}
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>>> priority_queue_test.swap(0, 1)
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>>> priority_queue_test.array
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[(15, 'B'), (10, 'A')]
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>>> priority_queue_test.pos
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{'A': 1, 'B': 0}
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"""
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self.pos[self.array[i][1]] = j
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self.pos[self.array[j][1]] = i
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temp = self.array[i]
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self.array[i] = self.array[j]
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self.array[j] = temp
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def decrease_key(self, tup, new_d):
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"""
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Decrease the key value for a given tuple, assuming the new_d is at most old_d.
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Examples:
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>>> priority_queue_test = PriorityQueue()
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>>> priority_queue_test.array = [(10, 'A'), (15, 'B')]
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>>> priority_queue_test.cur_size = len(priority_queue_test.array)
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>>> priority_queue_test.pos = {'A': 0, 'B': 1}
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>>> priority_queue_test.decrease_key((10, 'A'), 5)
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>>> priority_queue_test.array
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[(5, 'A'), (15, 'B')]
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"""
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idx = self.pos[tup[1]]
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# assuming the new_d is at most old_d
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self.array[idx] = (new_d, tup[1])
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while idx > 0 and self.array[self.par(idx)][0] > self.array[idx][0]:
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self.swap(idx, self.par(idx))
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idx = self.par(idx)
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class Graph:
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def __init__(self, num):
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"""
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Graph class constructor
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Examples:
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>>> graph_test = Graph(1)
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>>> graph_test.num_nodes
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1
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>>> graph_test.dist
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[0]
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>>> graph_test.par
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[-1]
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>>> graph_test.adjList
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{}
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"""
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self.adjList = {} # To store graph: u -> (v,w)
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self.num_nodes = num # Number of nodes in graph
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# To store the distance from source vertex
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self.dist = [0] * self.num_nodes
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self.par = [-1] * self.num_nodes # To store the path
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def add_edge(self, u, v, w):
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"""
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Add edge going from node u to v and v to u with weight w: u (w)-> v, v (w) -> u
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Examples:
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>>> graph_test = Graph(1)
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>>> graph_test.add_edge(1, 2, 1)
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>>> graph_test.add_edge(2, 3, 2)
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>>> graph_test.adjList
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{1: [(2, 1)], 2: [(1, 1), (3, 2)], 3: [(2, 2)]}
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"""
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# Check if u already in graph
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if u in self.adjList:
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self.adjList[u].append((v, w))
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else:
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self.adjList[u] = [(v, w)]
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# Assuming undirected graph
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if v in self.adjList:
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self.adjList[v].append((u, w))
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else:
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self.adjList[v] = [(u, w)]
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def show_graph(self):
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"""
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Show the graph: u -> v(w)
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Examples:
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>>> graph_test = Graph(1)
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>>> graph_test.add_edge(1, 2, 1)
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>>> graph_test.show_graph()
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1 -> 2(1)
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2 -> 1(1)
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>>> graph_test.add_edge(2, 3, 2)
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>>> graph_test.show_graph()
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1 -> 2(1)
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2 -> 1(1) -> 3(2)
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3 -> 2(2)
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"""
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for u in self.adjList:
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print(u, "->", " -> ".join(str(f"{v}({w})") for v, w in self.adjList[u]))
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def dijkstra(self, src):
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"""
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Dijkstra algorithm
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Examples:
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>>> graph_test = Graph(3)
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>>> graph_test.add_edge(0, 1, 2)
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>>> graph_test.add_edge(1, 2, 2)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 2
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Node 2 has distance: 4
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>>> graph_test.dist
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[0, 2, 4]
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>>> graph_test = Graph(2)
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>>> graph_test.add_edge(0, 1, 2)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 2
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>>> graph_test.dist
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[0, 2]
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>>> graph_test = Graph(3)
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>>> graph_test.add_edge(0, 1, 2)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 2
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Node 2 has distance: 0
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>>> graph_test.dist
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[0, 2, 0]
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>>> graph_test = Graph(3)
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>>> graph_test.add_edge(0, 1, 2)
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>>> graph_test.add_edge(1, 2, 2)
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>>> graph_test.add_edge(0, 2, 1)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 2
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Node 2 has distance: 1
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>>> graph_test.dist
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[0, 2, 1]
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>>> graph_test = Graph(4)
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>>> graph_test.add_edge(0, 1, 4)
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>>> graph_test.add_edge(1, 2, 2)
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>>> graph_test.add_edge(2, 3, 1)
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>>> graph_test.add_edge(0, 2, 3)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 4
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Node 2 has distance: 3
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Node 3 has distance: 4
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>>> graph_test.dist
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[0, 4, 3, 4]
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>>> graph_test = Graph(4)
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>>> graph_test.add_edge(0, 1, 4)
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>>> graph_test.add_edge(1, 2, 2)
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>>> graph_test.add_edge(2, 3, 1)
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>>> graph_test.add_edge(0, 2, 7)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 4
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Node 2 has distance: 6
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Node 3 has distance: 7
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>>> graph_test.dist
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[0, 4, 6, 7]
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"""
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# Flush old junk values in par[]
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self.par = [-1] * self.num_nodes
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# src is the source node
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self.dist[src] = 0
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q = PriorityQueue()
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q.insert((0, src)) # (dist from src, node)
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for u in self.adjList:
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if u != src:
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self.dist[u] = sys.maxsize # Infinity
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self.par[u] = -1
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while not q.is_empty():
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u = q.extract_min() # Returns node with the min dist from source
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# Update the distance of all the neighbours of u and
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# if their prev dist was INFINITY then push them in Q
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for v, w in self.adjList[u]:
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new_dist = self.dist[u] + w
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if self.dist[v] > new_dist:
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if self.dist[v] == sys.maxsize:
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q.insert((new_dist, v))
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else:
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q.decrease_key((self.dist[v], v), new_dist)
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self.dist[v] = new_dist
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self.par[v] = u
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# Show the shortest distances from src
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self.show_distances(src)
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def show_distances(self, src):
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"""
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Show the distances from src to all other nodes in a graph
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Examples:
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>>> graph_test = Graph(1)
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>>> graph_test.show_distances(0)
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Distance from node: 0
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Node 0 has distance: 0
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"""
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print(f"Distance from node: {src}")
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for u in range(self.num_nodes):
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print(f"Node {u} has distance: {self.dist[u]}")
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def show_path(self, src, dest):
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"""
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Shows the shortest path from src to dest.
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WARNING: Use it *after* calling dijkstra.
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Examples:
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>>> graph_test = Graph(4)
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>>> graph_test.add_edge(0, 1, 1)
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>>> graph_test.add_edge(1, 2, 2)
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>>> graph_test.add_edge(2, 3, 3)
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>>> graph_test.dijkstra(0)
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Distance from node: 0
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Node 0 has distance: 0
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Node 1 has distance: 1
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Node 2 has distance: 3
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Node 3 has distance: 6
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>>> graph_test.show_path(0, 3) # doctest: +NORMALIZE_WHITESPACE
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----Path to reach 3 from 0----
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0 -> 1 -> 2 -> 3
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Total cost of path: 6
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"""
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path = []
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cost = 0
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temp = dest
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# Backtracking from dest to src
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while self.par[temp] != -1:
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path.append(temp)
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if temp != src:
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for v, w in self.adjList[temp]:
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if v == self.par[temp]:
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cost += w
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break
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temp = self.par[temp]
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path.append(src)
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path.reverse()
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print(f"----Path to reach {dest} from {src}----")
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for u in path:
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print(f"{u}", end=" ")
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if u != dest:
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print("-> ", end="")
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print("\nTotal cost of path: ", cost)
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if __name__ == "__main__":
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from doctest import testmod
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testmod()
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graph = Graph(9)
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graph.add_edge(0, 1, 4)
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graph.add_edge(0, 7, 8)
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graph.add_edge(1, 2, 8)
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graph.add_edge(1, 7, 11)
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graph.add_edge(2, 3, 7)
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graph.add_edge(2, 8, 2)
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graph.add_edge(2, 5, 4)
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graph.add_edge(3, 4, 9)
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graph.add_edge(3, 5, 14)
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graph.add_edge(4, 5, 10)
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graph.add_edge(5, 6, 2)
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graph.add_edge(6, 7, 1)
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graph.add_edge(6, 8, 6)
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graph.add_edge(7, 8, 7)
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graph.show_graph()
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graph.dijkstra(0)
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graph.show_path(0, 4)
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# OUTPUT
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# 0 -> 1(4) -> 7(8)
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# 1 -> 0(4) -> 2(8) -> 7(11)
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# 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7)
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# 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4)
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# 3 -> 2(7) -> 4(9) -> 5(14)
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# 8 -> 2(2) -> 6(6) -> 7(7)
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# 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2)
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# 4 -> 3(9) -> 5(10)
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# 6 -> 5(2) -> 7(1) -> 8(6)
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# Distance from node: 0
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# Node 0 has distance: 0
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# Node 1 has distance: 4
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# Node 2 has distance: 12
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# Node 3 has distance: 19
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# Node 4 has distance: 21
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# Node 5 has distance: 11
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# Node 6 has distance: 9
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# Node 7 has distance: 8
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# Node 8 has distance: 14
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# ----Path to reach 4 from 0----
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# 0 -> 7 -> 6 -> 5 -> 4
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# Total cost of path: 21
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