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Shortest paths and spanning trees in graphs Lyzhin Ivan, 2015

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Shortest path problem The problem of finding a path between two vertices such that the sum of the weights of edges in path is minimized. Known algorithms: Dijkstra Floyd–Warshall Bellman–Ford and so on...

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Dijkstra algorithm There are two sets of vertices – visited and unvisited. For visited vertices we know minimal distance from start. For unvisited vertices we know some distance which can be not minimal. Initially, all vertices are unvisited and distance to each vertex is INF. Only distance to start node is equal 0. On each step choose unvisited vertex with minimal distance. Now it’s visited vertex. And try to relax distance of neighbors. Complexity: trivial implementation O(|V|^2+|E|) implementation with set O(|E|log|V|+|V|log|V|)

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Trivial implementation

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Implementation with set

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Implementation with priority queue

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Floyd–Warshall algorithm Initially, dist[u][u]=0 and for each edge (u, v): dist[u][v]=weight(u, v) On iteration k we let use vertex k as intermediate vertex and for each pair of vertices we try to relax distance. dist[u][v] = min(dist[u][v], dist[u][k]+dist[k][v]) Complexity: O(|V|^3)

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Implementation

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Bellman–Ford algorithm |V|-1 iterations, on each we try relax distance with all edges. If we can relax distance on |V| iteration then negative cycle exists in graph Why |V|-1 iterations? Because the longest way without cycles from one node to another one contains no more |V|-1 edges. Complexity O(|V||E|)

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Implementation

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Minimal spanning tree A spanning tree T of an undirected graph G is a subgraph that includes all of the vertices of G that is a tree. A minimal spanning tree is a spanning tree and sum of weights is minimized.

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Prim’s algorithm Initialize a tree with a single vertex, chosen arbitrarily from the graph. Grow the tree by one edge: of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, transfer it to the tree and try to relax distance for neighbors. Repeat step 2 (until all vertices are in the tree). Complexity: trivial implementation O(|V|^2+|E|) implementation with set O(|E|log|V|+|E|)

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Implementation

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Kruskal’s algorithm Create a forest F (a set of trees), where each vertex in the graph is a separate tree Create a set S containing all the edges in the graph While S is nonempty and F is not yet spanning: remove an edge with minimum weight from S if the removed edge connects two different trees then add it to the forest F, combining two trees into a single tree Complexity: trivial O(|V|^2+|E|log|E|) with DSU O(|E|log|E|)

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Trivial implementation

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Implementation with DSU

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Disjoint-set-union (DSU) Two main operations: Find(U) – return root of set, which contains U, complexity O(1) Union(U, V) – join sets, which contain U and V, complexity O(1) After creating DSU: After some operations: