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1 Paths in Graphs Oscar Miguel Alonso M

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2 Outline The problem to be solved Breadth first search Dijkstra's Algorithm Bellman-Ford Algorithm Shortest path in DAGs Efficient implementation of Dijkstra (Heap) Bonus: graph diameter, all to all shortest path (Floyd algorithm)

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3 Paths in graphs Find the shortest path between two nodes Are all the edges of the same cost? Are there edges with negative cost? Is the graph directed? Are there cycles of negative cost? Given a graph, which are the most distant nodes?

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4 Breadth first search Idea: You begin with one node S Then, visit all the nodes at distance 1 from S Then, visit the all nodes at distance 2 from S You continue until you meet your target node or you have reached all the nodes

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5 s Breadth first search

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6 s 1 1 1

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7 s

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8 BFS(G,s) Input: Graph G=(V,E), node s Output: array dist[] // dist[u] is the distance from s to u Vars: Queue Q for-each u in V dist[u]=∞ dist [s]=0 Q.add(s) while(!Q.empty()) u = Q.get() for-each edge ( u, v ) in E if dist[v]==∞ Q.add(v) dist[v]=dist[u]+1 O(E)

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9 s Breadth first search Application of the algorithm: see whiteboard

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10 Dijkstra's algorithm Edge Relaxation if(dist[u] > dist[k]+W[k][u]) dist[u] = dist[k]+W[k][u] Used in weighted graphs Do not use it if there are negative weighs!

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11 Dijkstra's algorithm Idea: Keep a list of nodes which you already know the shortest path to Such list begin with only one node (s), and you add one node in each iteration of the algorithm The node added is connected to one node in the list, and is the one with minimum distance to the origin from all candidates

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Dijkstra's algorithm

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Dijkstra's algorithm

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Dijkstra's algorithm

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Dijkstra's algorithm

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16 Dijkstra's algorithm For the implementation Keep an array indicating the minimum distance so far to each node (initialize with infinite) Update the distance each time you add a node (relax the edges from that node) Keep a list of the nodes not reached, instead of the ones reached Now, instead of searching the node with minimum distance, just pick the one with lower distance

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17 ∞ ∞ ∞ ∞ ∞ Dijkstra's algorithm

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∞ Dijkstra's algorithm

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Dijkstra's algorithm

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Dijkstra's algorithm

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Dijkstra's algorithm

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Dijkstra's algorithm

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23 Dijkstra's algorithm DIJKSTRA(G,s) for-each i in V d[ i ] ← ∞ Q.add(i) d[s] ← 0 while not empty(queue) do u = Q.extract_min() // take node with minimum dist for-each v where (u, v) in E if d[v] > d[u] + w uv then d[v] = d[u] + w uv Note that d[v] represents the known shortest distance of v from s during the process and may not be the real shortest distance until v is marked black.

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24 Dijkstra's algorithm Complexity: Depends on Queue implementation E lg VVHeap E lg VEHeap (Lazy) VEEArray (Lazy) VEVSorted Array E + V lg VVFibonacci Heap V2V2 VArray TimeMemoryWorst Case Big-Oh

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25 Bellman-Ford algorithm Useful when there are negative-weighted edges Idea: relax all the edges n times A path can be found with max n-i edge relaxations If after that, the graph allow other edge relaxation, the graph has a negative weight cycle 5 -3

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26 Bellman-Ford algorithm Do n-1 times for-each (u, v) in E if d[v] > d[u] + w uv then d[v] ← d[u] + w uv Complexity = O(VE)

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27 Shortest path in DAGs In DAGs the problem is easier Idea: sort the nodes (topological sort), and traverse the nodes, relaxing all their edges ShortestPath-DAG(G,s) for-each i in V d[ i ] ← ∞ d[s] ← 0 TS = TopologicalSort(G) for-each u in TS for-each v where (u, v) is and edge if d[v] > d[u] + w uv then d[v] = d[u] + w uv O(E)

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28 Shortest path in DAGs

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29 Heap data structure A heap is a comlete binary tree in which every parent is greater (smaller) than its child(ren). extract-min and decrease-key operations are performed in O(log(n)) constructed in O(n) stored efficiently in an array: no need for pointers parent of element i is in pos (i-1)/2 left son is in pos 2*i+1 left son is in pos 2*i

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30 Heap data structure

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31 Heap data structure Example: extract-min

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32 Heap data structure

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33 Heap data structure

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34 Heap data structure

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35 Warshall-Floyd Algorithm All to all shortest path d[i][j] is the minimum distance from vertex i to vertex j the diameter of the graph is the maximum d[i][j] after the execution of the algorithm

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36 Warshall-Floyd Algorithm Floyd(G) for-each (u, v) in E d[u][v] ← w uv for-each k in V for-each i in V for-each j in V if d[i][j] > d[i][k] + d[k][j] d[i][j] ← d[i][k] + d[k][j] Time Complexity: O(V 3 )

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37 THANK YOU !!

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