Data Structures and algorithms (IS ZC361) Weighted Graphs – Shortest path algorithms – MST S.P.Vimal BITS-Pilani http://discovery.bits-pilani.ac.in/discipline/csis/vimal/ Source :This presentation is composed from the presentation materials provided by the authors (GOODRICH and TAMASSIA) of text book -1 specified in the handout Data Structures and Algorithms 1 Topics today 1. 2. DAGs, topological sorting Weighted Graphs 3. Dijkstra’s algorithm The Bellman-Ford algorithm Shortest paths in dags All-pairs shortest paths Minimum Spanning Trees Data Structures and Algorithms 2 DAGS, Topological Sorting (Text book Reference 6.4.4) Data Structures and Algorithms 3 DAGs and Topological Ordering D • A directed acyclic graph (DAG) is a digraph that has no directed cycles • A topological ordering of a digraph is a numbering v1 , …, vn of the vertices such that for every edge (vi , vj), we have i < j • Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints Theorem A digraph admits a topological ordering if and only if it is a DAG E B C DAG G A v2 v1 D B C A Data Structures and Algorithms v4 E v5 v3 Topological ordering of G 4 Topological Sorting • Number vertices, so that (u,v) in E implies u < v wake up 1 A typical student day 3 2 study computer sci. eat 4 7 play nap 8 write c.s. program 9 make cookies for professors 5 more c.s. 6 work out 10 11 sleep dream about graphs Data Structures and Algorithms 5 Algorithm for Topological Sorting • Note: This algorithm is different than the one in Goodrich-Tamassia Method TopologicalSort(G) HG // Temporary copy of G n G.numVertices() while H is not empty do Let v be a vertex with no outgoing edges Label v n nn-1 Remove v from H • Running time: O(n + m). How…? Data Structures and Algorithms 6 Topological Sorting Algorithm using DFS • Simulate the algorithm by using depthfirst search Algorithm topologicalDFS(G) Input dag G Output topological ordering of G n G.numVertices() for all u G.vertices() setLabel(u, UNEXPLORED) for all e G.edges() setLabel(e, UNEXPLORED) for all v G.vertices() if getLabel(v) = UNEXPLORED topologicalDFS(G, v) • O(n+m) time. Algorithm topologicalDFS(G, v) Input graph G and a start vertex v of G Output labeling of the vertices of G in the connected component of v setLabel(v, VISITED) for all e G.incidentEdges(v) if getLabel(e) = UNEXPLORED w opposite(v,e) if getLabel(w) = UNEXPLORED setLabel(e, DISCOVERY) topologicalDFS(G, w) else {e is a forward or cross edge} Label v with topological number n nn-1 Data Structures and Algorithms 7 Topological Sorting Example Data Structures and Algorithms 8 Topological Sorting Example 9 Data Structures and Algorithms 9 Topological Sorting Example 8 9 Data Structures and Algorithms 10 Topological Sorting Example 7 8 9 Data Structures and Algorithms 11 Topological Sorting Example 6 7 8 9 Data Structures and Algorithms 12 Topological Sorting Example 6 5 7 8 9 Data Structures and Algorithms 13 Topological Sorting Example 4 6 5 7 8 9 Data Structures and Algorithms 14 Topological Sorting Example 3 4 6 5 7 8 9 Data Structures and Algorithms 15 Topological Sorting Example 2 3 4 6 5 7 8 9 Data Structures and Algorithms 16 Topological Sorting Example 2 1 3 4 6 5 7 8 9 Data Structures and Algorithms 17 Weighted graphs (Minimum Spanning Tree) (Text book Reference 7.3) Data Structures and Algorithms 18 Minimum Spanning Tree Spanning subgraph – Subgraph of a graph G containing all the vertices of G Spanning tree – Spanning subgraph that is itself a (free) tree Minimum spanning tree (MST) – Spanning tree of a weighted graph with minimum total edge weight • Applications – Communications networks – Transportation networks ORD 10 1 DEN PIT 9 6 STL 4 8 DFW Data Structures and Algorithms 7 3 DCA 5 2 ATL 19 Partition Property U f Partition Property: V 7 4 – Consider a partition of the vertices of G 9 into subsets U and V 5 2 8 – Let e be an edge of minimum weight across the partition 3 8 e – There is a minimum spanning tree of G 7 containing edge e Proof: Replacing f with e yields – Let T be an MST of G another MST – If T does not contain e, consider the cycle U V C formed by e with T and let f be an edge 7 f of C across the partition 4 – weight(f) weight(e) 9 – Thus, weight(f) = weight(e) 5 2 – We obtain another MST by replacing f 8 with e 8 Data Structures and Algorithms e 7 3 20 Prim-Jarnik’s Algorithm • We pick an arbitrary vertex s and we grow the MST as a cloud of vertices, starting from s • We store with each vertex v a label d(v) = the smallest weight of an edge connecting v to a vertex in the cloud At each step: We add to the cloud the vertex u outside the cloud with the smallest distance label We update the labels of the vertices adjacent to u Data Structures and Algorithms 21 Prim-Jarnik’s Algorithm (cont.) • A priority queue stores the vertices outside the cloud – Key: distance – Element: vertex • Locator-based methods – insert(k,e) returns a locator – replaceKey(l,k) changes the key of an item • We store three labels with each vertex: – Distance – Parent edge in MST – Locator in priority queue Algorithm PrimJarnikMST(G) Q new heap-based priority queue s a vertex of G for all v G.vertices() if v = s setDistance(v, 0) else setDistance(v, ) setParent(v, ) l Q.insert(getDistance(v), v) setLocator(v,l) while Q.isEmpty() u Q.removeMin() for all e G.incidentEdges(u) z G.opposite(u,e) r weight(e) if r < getDistance(z) setDistance(z,r) setParent(z,e) Q.replaceKey(getLocator(z),r) Data Structures and Algorithms 22 Example 2 7 B 0 2 B 5 C 0 2 0 A 4 9 5 C 5 F 8 8 7 E 7 7 2 4 F 8 7 B 7 D 7 3 9 8 A D 7 5 F E 7 2 2 4 8 8 A 9 8 C 5 2 D E 3 7 7 B 2 0 Data Structures and Algorithms 3 7 7 4 9 5 C 5 F 8 8 A D 7 E 3 7 23 4 Example (contd.) 2 2 B 0 4 9 5 C 5 F 8 8 A D 7 7 7 E 4 3 3 2 2 B A Data Structures and Algorithms 4 9 5 C 5 F 8 8 0 D 7 7 7 E 4 3 3 24 Analysis • Graph operations – Method incidentEdges is called once for each vertex • Label operations – We set/get the distance, parent and locator labels of vertex z O(deg(z)) times – Setting/getting a label takes O(1) time • Priority queue operations – Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time – The key of a vertex w in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time • Prim-Jarnik’s algorithm runs in O((n + m) log n) time provided the graph is represented by the adjacency list structure – Recall that Sv deg(v) = 2m • The running time is O(m log n) since the graph is connected Data Structures and Algorithms 25 Kruskal’s Algorithm A priority queue stores the edges outside the cloud Key: weight Element: edge At the end of the algorithm We are left with one cloud that encompasses the MST A tree T which is our MST Algorithm KruskalMST(G) for each vertex V in G do define a Cloud(v) of {v} let Q be a priority queue. Insert all edges into Q using their weights as the key T while T has fewer than n-1 edges do edge e = T.removeMin() Let u, v be the endpoints of e if Cloud(v) Cloud(u) then Add edge e to T Merge Cloud(v) and Cloud(u) return T Data Structures and Algorithms 26 Data Structure for Kruskal Algortihm • The algorithm maintains a forest of trees • An edge is accepted it if connects distinct trees • We need a data structure that maintains a partition, i.e., a collection of disjoint sets, with the operations: -find(u): return the set storing u -union(u,v): replace the sets storing u and v with their union Data Structures and Algorithms 27 Representation of a Partition • Each set is stored in a sequence • Each element has a reference back to the set – operation find(u) takes O(1) time, and returns the set of which u is a member. – in operation union(u,v), we move the elements of the smaller set to the sequence of the larger set and update their references – the time for operation union(u,v) is min(nu,nv), where nu and nv are the sizes of the sets storing u and v • Whenever an element is processed, it goes into a set of size at least double, hence each element is processed at most log n times Data Structures and Algorithms 28 Partition-Based Implementation • A partition-based version of Kruskal’s Algorithm performs cloud merges as unions and tests as finds. Algorithm Kruskal(G): Input: A weighted graph G. Output: An MST T for G. Let P be a partition of the vertices of G, where each vertex forms a separate set. Let Q be a priority queue storing the edges of G, sorted by their weights Let T be an initially-empty tree while Q is not empty do (u,v) Q.removeMinElement() if P.find(u) != P.find(v) then Running time: Add (u,v) to T O((n+m)log n) P.union(u,v) return T Data Structures and Algorithms 29 Kruskal Example 2704 BOS 867 849 ORD LAX 1391 1464 1235 144 JFK 1258 184 802 SFO 337 187 740 621 1846 PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 30 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 31 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 32 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 33 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 34 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 35 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 36 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 37 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 38 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 39 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 40 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 41 Example 2704 BOS 867 849 ORD 740 621 1846 337 LAX 1391 1464 1235 187 144 JFK 1258 184 802 SFO PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 42 Example 2704 BOS 867 849 ORD LAX 1391 1464 1235 144 JFK 1258 184 802 SFO 337 187 740 621 1846 PVD BWI 1090 DFW 946 1121 MIA 2342 Data Structures and Algorithms 43 Weighted graphs (Shortest Path Algorithms) (Text book Reference 6.44) Data Structures and Algorithms 44 Weighted Graphs • • • In a weighted graph, each edge has an associated numerical value, called the weight of the edge Edge weights may represent, distances, costs, etc. Example: – In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports PVD ORD SFO LGA HNL LAX DFW Data Structures and Algorithms MIA 45 Shortest Path Problem • Given a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v. – Length of a path is the sum of the weights of its edges. • Example: – Shortest path between Providence and Honolulu • Applications – Internet packet routing – Flight reservations – Driving directions PVD ORD SFO LGA HNL LAX DFW Data Structures and Algorithms MIA 46 Shortest Path Properties Property 1: A subpath of a shortest path is itself a shortest path Property 2: There is a tree of shortest paths from a start vertex to all the other vertices Example: Tree of shortest paths from Providence PVD ORD SFO LGA HNL LAX DFW Data Structures and Algorithms MIA 47 Dijkstra’s Algorithm • • • The distance of a vertex v from a vertex s is the length of a shortest path between s and v Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex s Assumptions: – the graph is connected – the edges are undirected – the edge weights are nonnegative • • • We grow a “cloud” of vertices, beginning with s and eventually covering all the vertices We store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent vertices At each step – We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u) – We update the labels of the vertices adjacent to u Data Structures and Algorithms 48 Edge Relaxation • Consider an edge e = (u,z) such that – u is the vertex most recently added to the cloud – z is not in the cloud • d(u) = 50 s u e d(z) = 75 z The relaxation of edge e updates distance d(z) as follows: d(z) min{d(z),d(u) + weight(e)} d(u) = 50 s Data Structures and Algorithms u d(z) = 60 e z 49 Example A 8 0 4 A 8 2 B 8 7 2 C 2 1 D 9 E F A 8 4 5 B 8 7 5 E 2 C 3 B 2 7 5 E A 8 3 D 8 5 0 4 2 C 3 1 9 0 4 2 F 2 8 4 2 3 0 2 1 9 D 11 F 5 3 B 7 2 Data Structures and Algorithms 7 5 E C 3 2 1 9 D 8 F 3 5 50 Example (cont.) A 8 0 4 2 B 2 7 7 C 3 5 E 2 1 9 D 8 F 3 5 A 8 0 4 2 B 2 7 7 3 5 Data Structures and Algorithms C E 2 1 9 D 8 F 3 5 51 Dijkstra’s Algorithm • A priority queue stores the vertices outside the cloud – Key: distance – Element: vertex • Locator-based methods – insert(k,e) returns a locator – replaceKey(l,k) changes the key of an item • We store two labels with each vertex: – Distance (d(v) label) – locator in priority queue Algorithm DijkstraDistances(G, s) Q new heap-based priority queue for all v G.vertices() if v = s setDistance(v, 0) else setDistance(v, ) l Q.insert(getDistance(v), v) setLocator(v,l) while Q.isEmpty() u Q.removeMin() for all e G.incidentEdges(u) { relax edge e } z G.opposite(u,e) r getDistance(u) + weight(e) if r < getDistance(z) setDistance(z,r) Q.replaceKey(getLocator(z),r) Data Structures and Algorithms 52 Analysis • • • • • Graph operations – Method incidentEdges is called once for each vertex Label operations – We set/get the distance and locator labels of vertex z O(deg(z)) times – Setting/getting a label takes O(1) time Priority queue operations – Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes O(log n) time – The key of a vertex in the priority queue is modified at most deg(w) times, where each key change takes O(log n) time Dijkstra’s algorithm runs in O((n + m) log n) time provided the graph is represented by the adjacency list structure – Recall that Sv deg(v) = 2m The running time can also be expressed as O(m log n) since the graph is connected Data Structures and Algorithms 53 Extension • • Using the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other vertices We store with each vertex a third label: – parent edge in the shortest path tree • In the edge relaxation step, we update the parent label Algorithm DijkstraShortestPathsTree(G, s) … for all v G.vertices() … setParent(v, ) … for all e G.incidentEdges(u) { relax edge e } z G.opposite(u,e) r getDistance(u) + weight(e) if r < getDistance(z) setDistance(z,r) setParent(z,e) Q.replaceKey(getLocator(z),r) Data Structures and Algorithms 54 Bellman-Ford Algorithm • • • • • Works even with negative-weight Algorithm BellmanFord(G, s) edges for all v G.vertices() Must assume directed edges (for if v = s otherwise we would have setDistance(v, 0) negative-weight cycles) else Iteration i finds all shortest paths setDistance(v, ) that use i edges. for i 1 to n-1 do Running time: O(nm). for each e G.edges() { relax edge e } Can be extended to detect a u G.origin(e) negative-weight cycle if it exists z G.opposite(u,e) – How? r getDistance(u) + weight(e) if r < getDistance(z) setDistance(z,r) Data Structures and Algorithms 55 Bellman-Ford Example Nodes are labeled with their d(v) values 0 8 4 0 8 -2 7 -2 1 3 -2 8 7 9 3 5 0 8 -2 4 7 1 -2 6 1 5 4 9 0 8 4 -2 1 -2 3 -2 -2 5 8 4 9 4 9 -1 5 7 3 5 -2 Data Structures and Algorithms 1 1 -2 9 4 9 -1 5 56 DAG-based Algorithm • • • • • Works even with negativeweight edges Uses topological order Doesn’t use any fancy data structures Is much faster than Dijkstra’s algorithm Running time: O(n+m). Algorithm DagDistances(G, s) for all v G.vertices() if v = s setDistance(v, 0) else setDistance(v, ) Perform a topological sort of the vertices for u 1 to n do {in topological order} for each e G.outEdges(u) { relax edge e } z G.opposite(u,e) r getDistance(u) + weight(e) if r < getDistance(z) setDistance(z,r) Data Structures and Algorithms 57 1 DAG Example Nodes are labeled with their d(v) values 1 1 0 8 4 0 8 -2 3 2 7 -2 4 1 3 -5 3 8 2 7 9 5 6 -5 5 0 8 -5 2 1 3 6 4 1 4 9 6 4 5 5 0 8 -2 7 -2 1 8 5 3 1 3 4 4 -2 4 1 -2 4 -1 3 5 2 7 9 7 5 -5 5 Data Structures and Algorithms 0 1 3 6 4 1 -2 9 4 7 (two steps) -1 5 5 58 This presentation & the practice problems will be available in the course page. http://discovery.bits-pilani.ac.in/discipline/csis/vimal/course%2007-08%20First/DSA_Offcampus/DSA.htm Questions ? Data Structures and Algorithms 59