diff --git a/data_structures/Graph/dijkstra_algorithm.py b/data_structures/Graph/dijkstra_algorithm.py new file mode 100644 index 000000000..c43ff37f5 --- /dev/null +++ b/data_structures/Graph/dijkstra_algorithm.py @@ -0,0 +1,211 @@ +# Title: Dijkstra's Algorithm for finding single source shortest path from scratch +# Author: Shubham Malik +# References: https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm + +import math +import sys +# For storing the vertex set to retreive node with the lowest distance + + +class PriorityQueue: + # Based on Min Heap + def __init__(self): + self.cur_size = 0 + self.array = [] + self.pos = {} # To store the pos of node in array + + def isEmpty(self): + return self.cur_size == 0 + + def min_heapify(self, idx): + lc = self.left(idx) + rc = self.right(idx) + if lc < self.cur_size and self.array(lc)[0] < self.array(idx)[0]: + smallest = lc + else: + smallest = idx + if rc < self.cur_size and self.array(rc)[0] < self.array(smallest)[0]: + smallest = rc + if smallest != idx: + self.swap(idx, smallest) + self.min_heapify(smallest) + + def insert(self, tup): + # Inserts a node into the Priority Queue + self.pos[tup[1]] = self.cur_size + self.cur_size += 1 + self.array.append((sys.maxsize, tup[1])) + self.decrease_key((sys.maxsize, tup[1]), tup[0]) + + def extract_min(self): + # Removes and returns the min element at top of priority queue + min_node = self.array[0][1] + self.array[0] = self.array[self.cur_size - 1] + self.cur_size -= 1 + self.min_heapify(1) + del self.pos[min_node] + return min_node + + def left(self, i): + # returns the index of left child + return 2 * i + 1 + + def right(self, i): + # returns the index of right child + return 2 * i + 2 + + def par(self, i): + # returns the index of parent + return math.floor(i / 2) + + def swap(self, i, j): + # swaps array elements at indices i and j + # update the pos{} + self.pos[self.array[i][1]] = j + self.pos[self.array[j][1]] = i + temp = self.array[i] + self.array[i] = self.array[j] + self.array[j] = temp + + def decrease_key(self, tup, new_d): + idx = self.pos[tup[1]] + # assuming the new_d is atmost old_d + self.array[idx] = (new_d, tup[1]) + while idx > 0 and self.array[self.par(idx)][0] > self.array[idx][0]: + self.swap(idx, self.par(idx)) + idx = self.par(idx) + + +class Graph: + def __init__(self, num): + self.adjList = {} # To store graph: u -> (v,w) + self.num_nodes = num # Number of nodes in graph + # To store the distance from source vertex + self.dist = [0] * self.num_nodes + self.par = [-1] * self.num_nodes # To store the path + + def add_edge(self, u, v, w): + # Edge going from node u to v and v to u with weight w + # u (w)-> v, v (w) -> u + # Check if u already in graph + if u in self.adjList.keys(): + self.adjList[u].append((v, w)) + else: + self.adjList[u] = [(v, w)] + + # Assuming undirected graph + if v in self.adjList.keys(): + self.adjList[v].append((u, w)) + else: + self.adjList[v] = [(u, w)] + + def show_graph(self): + # u -> v(w) + for u in self.adjList: + print(u, '->', ' -> '.join(str("{}({})".format(v, w)) + for v, w in self.adjList[u])) + + def dijkstra(self, src): + # Flush old junk values in par[] + self.par = [-1] * self.num_nodes + # src is the source node + self.dist[src] = 0 + Q = PriorityQueue() + Q.insert((0, src)) # (dist from src, node) + for u in self.adjList.keys(): + if u != src: + self.dist[u] = sys.maxsize # Infinity + self.par[u] = -1 + + while not Q.isEmpty(): + u = Q.extract_min() # Returns node with the min dist from source + # Update the distance of all the neighbours of u and + # if their prev dist was INFINITY then push them in Q + for v, w in self.adjList[u]: + new_dist = self.dist[u] + w + if self.dist[v] > new_dist: + if self.dist[v] == sys.maxsize: + Q.insert((new_dist, v)) + else: + Q.decrease_key((self.dist[v], v), new_dist) + self.dist[v] = new_dist + self.par[v] = u + + # Show the shortest distances from src + self.show_distances(src) + + def show_distances(self, src): + print("Distance from node: {}".format(src)) + for u in range(self.num_nodes): + print('Node {} has distance: {}'.format(u, self.dist[u])) + + def show_path(self, src, dest): + # To show the shortest path from src to dest + # WARNING: Use it *after* calling dijkstra + path = [] + cost = 0 + temp = dest + # Backtracking from dest to src + while self.par[temp] != -1: + path.append(temp) + if temp != src: + for v, w in self.adjList[temp]: + if v == self.par[temp]: + cost += w + break + temp = self.par[temp] + path.append(src) + path.reverse() + + print('----Path to reach {} from {}----'.format(dest, src)) + for u in path: + print('{}'.format(u), end=' ') + if u != dest: + print('-> ', end='') + + print('\nTotal cost of path: ', cost) + + +if __name__ == '__main__': + graph = Graph(9) + graph.add_edge(0, 1, 4) + graph.add_edge(0, 7, 8) + graph.add_edge(1, 2, 8) + graph.add_edge(1, 7, 11) + graph.add_edge(2, 3, 7) + graph.add_edge(2, 8, 2) + graph.add_edge(2, 5, 4) + graph.add_edge(3, 4, 9) + graph.add_edge(3, 5, 14) + graph.add_edge(4, 5, 10) + graph.add_edge(5, 6, 2) + graph.add_edge(6, 7, 1) + graph.add_edge(6, 8, 6) + graph.add_edge(7, 8, 7) + graph.show_graph() + graph.dijkstra(0) + graph.show_path(0, 4) + +# OUTPUT +# 0 -> 1(4) -> 7(8) +# 1 -> 0(4) -> 2(8) -> 7(11) +# 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7) +# 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4) +# 3 -> 2(7) -> 4(9) -> 5(14) +# 8 -> 2(2) -> 6(6) -> 7(7) +# 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2) +# 4 -> 3(9) -> 5(10) +# 6 -> 5(2) -> 7(1) -> 8(6) +# Distance from node: 0 +# Node 0 has distance: 0 +# Node 1 has distance: 4 +# Node 2 has distance: 12 +# Node 3 has distance: 19 +# Node 4 has distance: 21 +# Node 5 has distance: 11 +# Node 6 has distance: 9 +# Node 7 has distance: 8 +# Node 8 has distance: 14 +# ----Path to reach 4 from 0---- +# 0 -> 7 -> 6 -> 5 -> 4 +# Total cost of path: 21