shortest path algorithms
DESCRIPTION
Shortest Path Algorithms. Shortest Path Finding. Unweighted Shortest Path All-pairs Shortest-path Single-source Shortest-path Single-source-destination Shortest-path. Single-source Shortest Path. Bellman-Ford algorithm Dijkstra’s algorithm. Dijkstra’s Algorithm. Dijkstra(DG, source) - PowerPoint PPT PresentationTRANSCRIPT
04/21/23//pages.cpsc.ucalgary.ca/~verwaal/335
Shortest Path Algorithms
Shortest Path Finding
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Unweighted Shortest PathAll-pairs Shortest-pathSingle-source Shortest-pathSingle-source-destination Shortest-path
Single-source Shortest Path
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Bellman-Ford algorithmDijkstra’s algorithm
Dijkstra’s Algorithm
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Dijkstra(DG, source) initialize(DG, source) verticesFound queue.addAll(DG.vertices) while (queue not empty) v queue.deleteMin() add v to verticesFound relaxEdges(v)
Example
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Initialize a
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queue: a(0), b(), c(), d(), e()verticesFound: {}
Example – 1st Iteration
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1st Iter a
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Example – 2nd Iteration
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2nd Iter a
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queue: d(5), b(10), c(), e()verticesFound: {a}
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queue: b(10), c(), e()verticesFound: {a, d}
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Example – 3rd Iteration
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3rd Iter a
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queue: e(7), b(8), c(14)verticesFound: {a, d}
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queue: b(8), c(14)verticesFound: {a, d, e}
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Example – 4th Iteration
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4th Iter a
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queue: b(8), c(13)verticesFound: {a, d, e}
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queue: c(13)verticesFound: {a, d, e, b}
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Example – 5th Iteration
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5th Iter a
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queue: c(9)verticesFound: {a, d, e, b}
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queue: verticesFound: {a, d, e, b, c}
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Analysis of Dijkstra’s Algorithm
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Loop executed |V| timescall removeMin in queue |V| times
decreaseKey in queue called |E| timesPriority Queue implementation:
Array – O(|V|2)Binary Heap – O(|E|log|V|) Fibonacci Heap – O(|V|log|V| + |E|)
All-pairs Shortest-path
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Run Dijkstra’s algorithm on all verticesRecursive Matrix MultiplicationFloyd-Warshall Algorithm
Recursive Solution
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Considers length of shortest pathAt most |V| - 1Path must contain sub path which is also
shortestThe shortest path from u to v either
goes directly from u to v (path length 1), orgoes through some other vertex w
Divide and Conquer Algorithm
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APSP-Recursive(G, u, v) if (u = v) return 0 minPathLength for each vertex w in G a APSP-Recursive(G, u, w) if (a+weight(w,v)< minPathLength)
minPath a + weight(w,v) return minPathLength
Analysis of Divide and Conquer
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Run time O(2n)Duplicating workInstead compute from the bottom-upDynamic Programming AlgorithmIn table store shortest path considering
each vertex as intermediate
Dynamic Programming Algorithm
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APSP-DP(G ) // G is adjacency matrix
D G for k 1 to |V| do for i 1 to |V| do for j 1 to |V| do if D[i,k] + D(k,j) < D[i,j] D[i,j] = D[i,k] + D(k,j)