From 5a65dcbab95e94b92881b4bff121cea85c4d14f1 Mon Sep 17 00:00:00 2001 From: trekhleb <3000285+trekhleb@users.noreply.github.com> Date: Fri, 7 Nov 2025 20:59:10 +0100 Subject: [PATCH 1/3] Add Dijkstra algorithm illustrations and explanations --- src/algorithms/graph/dijkstra/README.md | 80 +++++++++++++++--- .../graph/dijkstra/images/dijkstra-01.png | Bin 0 -> 42807 bytes .../graph/dijkstra/images/dijkstra-02.png | Bin 0 -> 84611 bytes .../graph/dijkstra/images/dijkstra-03.png | Bin 0 -> 87505 bytes .../graph/dijkstra/images/dijkstra-04.png | Bin 0 -> 92117 bytes .../graph/dijkstra/images/dijkstra-05.png | Bin 0 -> 90401 bytes .../graph/dijkstra/images/dijkstra-06.png | Bin 0 -> 97616 bytes .../graph/dijkstra/images/dijkstra-07.png | Bin 0 -> 97971 bytes .../graph/dijkstra/images/dijkstra-08.png | Bin 0 -> 91641 bytes .../graph/dijkstra/images/dijkstra-09.png | Bin 0 -> 101622 bytes .../graph/dijkstra/images/dijkstra-10.png | Bin 0 -> 101375 bytes .../graph/dijkstra/images/dijkstra-11.png | Bin 0 -> 91179 bytes .../graph/dijkstra/images/dijkstra-12.png | Bin 0 -> 95242 bytes .../graph/dijkstra/images/dijkstra-13.png | Bin 0 -> 93539 bytes .../graph/dijkstra/images/dijkstra-14.png | Bin 0 -> 101418 bytes .../graph/dijkstra/images/dijkstra-15.png | Bin 0 -> 92631 bytes .../graph/dijkstra/images/dijkstra-16.png | Bin 0 -> 88368 bytes 17 files changed, 70 insertions(+), 10 deletions(-) create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-01.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-02.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-03.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-04.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-05.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-06.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-07.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-08.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-09.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-10.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-11.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-12.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-13.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-14.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-15.png create mode 100644 src/algorithms/graph/dijkstra/images/dijkstra-16.png diff --git a/src/algorithms/graph/dijkstra/README.md b/src/algorithms/graph/dijkstra/README.md index d94e2a3cdf..2f6ad06fb9 100644 --- a/src/algorithms/graph/dijkstra/README.md +++ b/src/algorithms/graph/dijkstra/README.md @@ -3,24 +3,84 @@ _Read this in other languages:_ [_한국어_](README.ko-KR.md) -Dijkstra's algorithm is an algorithm for finding the shortest -paths between nodes in a graph, which may represent, for example, -road networks. - -The algorithm exists in many variants; Dijkstra's original variant -found the shortest path between two nodes, but a more common -variant fixes a single node as the "source" node and finds -shortest paths from the source to all other nodes in the graph, +Dijkstra's algorithm is an algorithm for finding the shortest +paths between nodes in a graph, which may represent, for example, +road networks. + +The algorithm exists in many variants; Dijkstra's original variant +found the shortest path between two nodes, but a more common +variant fixes a single node as the "source" node and finds +shortest paths from the source to all other nodes in the graph, producing a shortest-path tree. ![Dijkstra](https://upload.wikimedia.org/wikipedia/commons/5/57/Dijkstra_Animation.gif) Dijkstra's algorithm to find the shortest path between `a` and `b`. -It picks the unvisited vertex with the lowest distance, -calculates the distance through it to each unvisited neighbor, +It picks the unvisited vertex with the lowest distance, +calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors. +## Practical applications of Dijkstra's algorithm + +- GPS / navigation systems +- Public transit and airline route optimization +- Internet routing (OSPF, IS-IS protocols) +- Network traffic & latency optimization +- Game pathfinding (shortest path on maps) +- Logistics & delivery route optimization +- Supply chain & transportation network design + +## Step-by-step Dijkstra's algorithm example + +Let's say we have a graph of nodes, where each edge has a distance parameter between nodes. Let's say the distance from node `A` and node `B` is `7 meters`, and so on. + +The algorithm uses [Priority Queue](../../../data-structures/priority-queue/) to always pull the next unvisited vertex that has the smallest distance from the origin node. + +The start node by definition has a distance of `0m` from itself. So we start from it, the only node in the priority queue. + +![Dijkstra step 1](./images/dijkstra-01.png) + +Each neighbor of the pulled (from the queue) node is being traversed to calculate the distance to it from the origin. For example the distance from `A` to `B` is `0 + 7 = 7m`, and so on. + +Every time we visit a not yet seen neighbor, we add it to the priority queue, where the priority is a distance to the neighbor node from the origin node. + +If the node is already in the queue, it means we already calculated the distance to it before but from another node/path. If the current distance to it (from the current neighbor node) is shorter than the one that was calculated before, we update the distance (in the priority queue) to the shortest one, since we're searching for the shortest paths. + +![Dijkstra step 2](./images/dijkstra-02.png) + +![Dijkstra step 3](./images/dijkstra-03.png) + +![Dijkstra step 4](./images/dijkstra-04.png) + +Once all the neighbors of the current nodes were checked, the current node is being put to the `visited` set. We don't want to visit such nodes ones again during the upcoming traverses. + +![Dijkstra step 5](./images/dijkstra-05.png) + +![Dijkstra step 6](./images/dijkstra-06.png) + +![Dijkstra step 7](./images/dijkstra-07.png) + +![Dijkstra step 8](./images/dijkstra-08.png) + +![Dijkstra step 9](./images/dijkstra-09.png) + +![Dijkstra step 10](./images/dijkstra-10.png) + +![Dijkstra step 11](./images/dijkstra-11.png) + +![Dijkstra step 12](./images/dijkstra-12.png) + +![Dijkstra step 13](./images/dijkstra-13.png) + +![Dijkstra step 14](./images/dijkstra-14.png) + +![Dijkstra step 15](./images/dijkstra-15.png) + +![Dijkstra step 16](./images/dijkstra-16.png) + +Now we now the shortest distances to ech node from the start node `A`. In practice, during the distance calculations we also record the the `previousVertices` for each node to be able to show the exact sequence of nodes that forms the shortest path. + ## References - [Wikipedia](https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm) diff --git a/src/algorithms/graph/dijkstra/images/dijkstra-01.png b/src/algorithms/graph/dijkstra/images/dijkstra-01.png new file mode 100644 index 0000000000000000000000000000000000000000..3f6fba1dc704ee9ee32f9f084967a7d515214633 GIT binary patch literal 42807 zcmeFYWn5I<_dZMvFhfWRC;~%w3rKfMcXvyJbazOX(n?8pcS<8j3IfvIO7lPCec#`D z-aapX^O_xKj8syP!a#kF3I_*=AtNoW0tbf}1P2G74@L&gn3-mn z!oi`-TZ@S)$%u(Tm7E=6X=XUM$go%Xv;-R>cNB0Q(Zcj2C;_ygZP#T!pK&UKwWN>{ zLz*NbIn$@9>-xk(?kT@@xT zAcz0FboANmq4Ic)Bb7$@RXL>qJF1BtoRS{niwjK?i&XNLBAaoG;0--3j;;`okNn)R z+51X)N-1R`w3&vKi{a&T*&Rrj77jKn+f zh~24nykSKyc(9aEK#(ekv$l-%5W7h?HP-OK=~Ih9MC*%8z_Jy7K{pYJMtUa0H2PEW z$?X@s>R~-bXlf)S89cpj8NV3*3%-p`K1IPyzPIFZqtP#VvaYpH^GbA#<3M-aSH{d* 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