From 14f425286e720804c7b1e8c606aa50c5cdee279c Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Sun, 2 Jun 2024 15:58:06 +0530 Subject: [PATCH 01/14] Create Dijkstra's_algorithm.md --- contrib/ds-algorithms/Dijkstra's_algorithm.md | 88 +++++++++++++++++++ 1 file changed, 88 insertions(+) create mode 100644 contrib/ds-algorithms/Dijkstra's_algorithm.md diff --git a/contrib/ds-algorithms/Dijkstra's_algorithm.md b/contrib/ds-algorithms/Dijkstra's_algorithm.md new file mode 100644 index 0000000..18845f6 --- /dev/null +++ b/contrib/ds-algorithms/Dijkstra's_algorithm.md @@ -0,0 +1,88 @@ + +# Dijkstra's Algorithm +Dijkstra's algorithm is a graph algorithm that gives the shortest distance of each node from the given node in a weighted, undirected graph. It operates by continually choosing the closest unvisited node and determining the distance to all its unvisited neighboring nodes. This algorithm is similar to BFS in graphs, with the difference being it gives priority to nodes with shorter distances by using a priority queue(min-heap) instead of a FIFO queue. The data structures required would be a distance list (to store the minimum distance of each node), a priority queue or a set, and we assume the adjacency list will be provided. + +## Working +- We will store the minimum distance of each node in the distance list, which has a length equal to the number of nodes in the graph. Thus, the minimum distance of the 2nd node will be stored in the 2nd index of the distance list. We initialize the list with the maximum number possible, say infinity. + +- We now start the traversal from the starting node given and mark its distance as 0. We push this node to the priority queue along with its minimum distance, which is 0, so the structure pushed will be (0, node), a tuple. + +- Now, with the help of the adjacency list, we will add the neighboring nodes to the priority queue with the distance equal to (edge weight + current node distance), and this should be less than the distance list value. We will also update the distance list in the process. + +- When all the nodes are added, we will select the node with the shortest distance and repeat the process. + +## Dry Run +We will now do a manual simulation using an example graph given. First, (0, a) is pushed to the priority queue (pq). +![gssoc photo 1](https://github.com/animator/learn-python/assets/154299201/713cf7dd-fa8a-4f81-829f-00ac33eb12f4) +- **Step1:** The lowest element is popped from the pq, which is (0, a), and all its neighboring nodes are added to the pq while simultaneously checking the distance list. Thus (3, b), (7, c), (1, d) are added to the pq. +![gssoc photo2](https://github.com/animator/learn-python/assets/154299201/9456b701-7877-44dc-854b-da416c5e2e9b) +- **Step2:** Again, the lowest element is popped from the pq, which is (1, d). It has two neighboring nodes, a and e, from which + (0 + 1, a) will not be added to the pq as dist[a] = 0 is less than 1. +![gssoc photo3](https://github.com/animator/learn-python/assets/154299201/83990a40-f3fc-4173-9e21-10eb9e2003cb) + +- **Step3:** Now, the lowest element is popped from the pq, which is (3, b). It has two neighboring nodes, a and c, from which + (0 + 1, a) will not be added to the pq. But the new distance to reach c is 5 (3 + 2), which is less than dist[c] = 7. So (5, c) is added to the pq. +![gssoc photo4](https://github.com/animator/learn-python/assets/154299201/b47eb063-8ce7-4e68-85a5-da98f5ac9a3d) + +- **Step4:** The next smallest element is (5, c). It has neighbors a and e. The new distance to reach a will be 5 + 7 = 12, which is more than dist[a], so it will not be considered. Similarly, the new distance for e is 5 + 3 = 8, which again will not be considered. So, no new tuple has been added to the pq. +![gssoc photo5](https://github.com/animator/learn-python/assets/154299201/c0ef3c2a-1938-4c33-9e07-0929e8e5e631) + +- **Step5:** Similarly, both the elements of the pq will be popped one by one without any new addition. +![gssoc photo6](https://github.com/animator/learn-python/assets/154299201/828739c6-413b-41f6-a09f-b702802b20e1) +![gssoc photo7](https://github.com/animator/learn-python/assets/154299201/488d4c7b-d7bf-4887-a9c5-04accc6c6c89) + +- The distance list we get at the end will be our answer. +- `Output` `dist=[1, 3, 7, 1, 6]` + +## Python Code +```python +import heapq + +def dijkstra(graph, start): + # Create a priority queue + pq = [] + heapq.heappush(pq, (0, start)) + + # Create a dictionary to store distances to each node + dist = {node: float('inf') for node in graph} + dist[start] = 0 + + while pq: + # Get the node with the smallest distance + current_distance, current_node = heapq.heappop(pq) + + # If the current distance is greater than the recorded distance, skip it + if current_distance > dist[current_node]: + continue + + # Update the distances to the neighboring nodes + for neighbor, weight in graph[current_node].items(): + distance = current_distance + weight + # Only consider this new path if it's better + if distance < dist[neighbor]: + dist[neighbor] = distance + heapq.heappush(pq, (distance, neighbor)) + + return dist + +# Example usage: +graph = { + 'A': {'B': 1, 'C': 4}, + 'B': {'A': 1, 'C': 2, 'D': 5}, + 'C': {'A': 4, 'B': 2, 'D': 1}, + 'D': {'B': 5, 'C': 1} +} + +start_node = 'A' +dist = dijkstra(graph, start_node) +print(dist) +``` + +## Complexity Analysis + +- **Time Complexity**: \(O((V + E) log V)\) +- **Space Complexity**: \(O(V + E)\) + + + + From cc85ae3802f2f015a353a81b48f33f1ff5cb6a10 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Sun, 2 Jun 2024 22:54:56 +0530 Subject: [PATCH 02/14] Update index.md --- contrib/ds-algorithms/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/contrib/ds-algorithms/index.md b/contrib/ds-algorithms/index.md index 1d7293b..2263620 100644 --- a/contrib/ds-algorithms/index.md +++ b/contrib/ds-algorithms/index.md @@ -13,3 +13,4 @@ - [Stacks in Python](stacks.md) - [Sliding Window Technique](sliding-window.md) - [Trie](trie.md) +- [Dijkstra's Algorithm](Dijkstra's_algorithm.md) From 2647dbc3678d8380da938bc2ff02cb2243fc37b1 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Sat, 8 Jun 2024 11:48:48 +0530 Subject: [PATCH 03/14] Update index.md --- contrib/ds-algorithms/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/contrib/ds-algorithms/index.md b/contrib/ds-algorithms/index.md index 2263620..b18b79a 100644 --- a/contrib/ds-algorithms/index.md +++ b/contrib/ds-algorithms/index.md @@ -13,4 +13,6 @@ - [Stacks in Python](stacks.md) - [Sliding Window Technique](sliding-window.md) - [Trie](trie.md) 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9961a0c9f80160a2b28db22b802b45c77ff22d1f Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:30:27 +0530 Subject: [PATCH 06/14] Rename Dijkstra's_algorithm photo2.png to Dijkstra's_algorithm_photo2.png --- ...m photo2.png => Dijkstra's_algorithm_photo2.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo2.png => Dijkstra's_algorithm_photo2.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo2.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo2.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo2.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo2.png From 3e7e46969390235b3c12fbef8be0522f52dc4155 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:31:20 +0530 Subject: [PATCH 07/14] Rename Dijkstra's_algorithm photo3.png to Dijkstra's_algorithm_photo3.png --- ...m photo3.png => Dijkstra's_algorithm_photo3.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo3.png => Dijkstra's_algorithm_photo3.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo3.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo3.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo3.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo3.png From 77f20e9908d4be3e616e9ccefd4f0e5c9c672648 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:31:43 +0530 Subject: [PATCH 08/14] Rename Dijkstra's_algorithm photo4.png to Dijkstra's_algorithm_photo4.png --- ...m photo4.png => Dijkstra's_algorithm_photo4.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo4.png => Dijkstra's_algorithm_photo4.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo4.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo4.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo4.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo4.png From c32dc12b1a7ea1dd1bcf6e1576f308b01d4e3d56 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:32:03 +0530 Subject: [PATCH 09/14] Rename Dijkstra's_algorithm photo5.png to Dijkstra's_algorithm_photo5.png --- ...m photo5.png => Dijkstra's_algorithm_photo5.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo5.png => Dijkstra's_algorithm_photo5.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo5.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo5.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo5.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo5.png From 01207f441c592d8a00411aa41b89f533458da6ef Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:32:22 +0530 Subject: [PATCH 10/14] Rename Dijkstra's_algorithm photo6.png to Dijkstra's_algorithm_photo6.png --- ...m photo6.png => Dijkstra's_algorithm_photo6.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo6.png => Dijkstra's_algorithm_photo6.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo6.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo6.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo6.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo6.png From d98376b7da3417e9b1b8bb60b00e0e0092c86267 Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:32:40 +0530 Subject: [PATCH 11/14] Rename Dijkstra's_algorithm photo7.png to Dijkstra's_algorithm_photo7.png --- ...m photo7.png => Dijkstra's_algorithm_photo7.png} | Bin 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/images/{Dijkstra's_algorithm photo7.png => Dijkstra's_algorithm_photo7.png} (100%) diff --git a/contrib/ds-algorithms/images/Dijkstra's_algorithm photo7.png b/contrib/ds-algorithms/images/Dijkstra's_algorithm_photo7.png similarity index 100% rename from contrib/ds-algorithms/images/Dijkstra's_algorithm photo7.png rename to contrib/ds-algorithms/images/Dijkstra's_algorithm_photo7.png From 85ecb2cabcda3c16c72480e6268f1e946d36c76d Mon Sep 17 00:00:00 2001 From: yashasvi232 <154299201+yashasvi232@users.noreply.github.com> Date: Thu, 13 Jun 2024 22:33:49 +0530 Subject: [PATCH 12/14] Update Dijkstra's_algorithm.md --- contrib/ds-algorithms/Dijkstra's_algorithm.md | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) diff --git a/contrib/ds-algorithms/Dijkstra's_algorithm.md b/contrib/ds-algorithms/Dijkstra's_algorithm.md index 18845f6..cea6da4 100644 --- a/contrib/ds-algorithms/Dijkstra's_algorithm.md +++ b/contrib/ds-algorithms/Dijkstra's_algorithm.md @@ -13,23 +13,25 @@ Dijkstra's algorithm is a graph algorithm that gives the shortest distance of ea ## Dry Run We will now do a manual simulation using an example graph given. First, (0, a) is pushed to the priority queue (pq). -![gssoc photo 1](https://github.com/animator/learn-python/assets/154299201/713cf7dd-fa8a-4f81-829f-00ac33eb12f4) +![Photo 1](images/Dijkstra's_algorithm_photo1.png) + - **Step1:** The lowest element is popped from the pq, which is (0, a), and all its neighboring nodes are added to the pq while simultaneously checking the distance list. Thus (3, b), (7, c), (1, d) are added to the pq. -![gssoc photo2](https://github.com/animator/learn-python/assets/154299201/9456b701-7877-44dc-854b-da416c5e2e9b) +![Photo 2](images/Dijkstra's_algorithm_photo2.png) + - **Step2:** Again, the lowest element is popped from the pq, which is (1, d). It has two neighboring nodes, a and e, from which (0 + 1, a) will not be added to the pq as dist[a] = 0 is less than 1. -![gssoc photo3](https://github.com/animator/learn-python/assets/154299201/83990a40-f3fc-4173-9e21-10eb9e2003cb) +![Photo 3](images/Dijkstra's_algorithm_photo3.png) - **Step3:** Now, the lowest element is popped from the pq, which is (3, b). It has two neighboring nodes, a and c, from which (0 + 1, a) will not be added to the pq. But the new distance to reach c is 5 (3 + 2), which is less than dist[c] = 7. So (5, c) is added to the pq. -![gssoc photo4](https://github.com/animator/learn-python/assets/154299201/b47eb063-8ce7-4e68-85a5-da98f5ac9a3d) +![Photo 4](images/Dijkstra's_algorithm_photo4.png) - **Step4:** The next smallest element is (5, c). It has neighbors a and e. The new distance to reach a will be 5 + 7 = 12, which is more than dist[a], so it will not be considered. Similarly, the new distance for e is 5 + 3 = 8, which again will not be considered. So, no new tuple has been added to the pq. -![gssoc photo5](https://github.com/animator/learn-python/assets/154299201/c0ef3c2a-1938-4c33-9e07-0929e8e5e631) +![Photo 5](images/Dijkstra's_algorithm_photo5.png) - **Step5:** Similarly, both the elements of the pq will be popped one by one without any new addition. -![gssoc photo6](https://github.com/animator/learn-python/assets/154299201/828739c6-413b-41f6-a09f-b702802b20e1) -![gssoc photo7](https://github.com/animator/learn-python/assets/154299201/488d4c7b-d7bf-4887-a9c5-04accc6c6c89) +![Photo 6](images/Dijkstra's_algorithm_photo6.png) +![Photo 7](images/Dijkstra's_algorithm_photo7.png) - The distance list we get at the end will be our answer. - `Output` `dist=[1, 3, 7, 1, 6]` From 848af819957763fa9a3231cd62265e2fb6081057 Mon Sep 17 00:00:00 2001 From: Ashita Prasad Date: Sat, 15 Jun 2024 23:34:33 +0530 Subject: [PATCH 13/14] Rename Dijkstra's_algorithm.md to dijkstra.md --- contrib/ds-algorithms/{Dijkstra's_algorithm.md => dijkstra.md} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename contrib/ds-algorithms/{Dijkstra's_algorithm.md => dijkstra.md} (100%) diff --git a/contrib/ds-algorithms/Dijkstra's_algorithm.md b/contrib/ds-algorithms/dijkstra.md similarity index 100% rename from contrib/ds-algorithms/Dijkstra's_algorithm.md rename to contrib/ds-algorithms/dijkstra.md From fbc528b00bb794b358064b0585cbc1da23bd0d93 Mon Sep 17 00:00:00 2001 From: Ashita Prasad Date: Sat, 15 Jun 2024 23:34:50 +0530 Subject: [PATCH 14/14] Update index.md --- contrib/ds-algorithms/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/contrib/ds-algorithms/index.md b/contrib/ds-algorithms/index.md index e1d12ce..047b3dd 100644 --- a/contrib/ds-algorithms/index.md +++ b/contrib/ds-algorithms/index.md @@ -18,4 +18,4 @@ - [Hashing through Chaining](hashing-chaining.md) - [AVL Trees](avl-trees.md) - [Splay Trees](splay-trees.md) -- [Dijkstra's Algorithm](Dijkstra's_algorithm.md) \ No newline at end of file +- [Dijkstra's Algorithm](dijkstra.md)