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SFO CHAPTER 13 GRAPH ALGORITHMS LAX ORD DFW ACKNOWLEDGEMENT: THESE SLIDES ARE ADAPTED FROM SLIDES PROVIDED WITH DATA STRUCTURES AND ALGORITHMS IN C++, GOODRICH, TAMASSIA AND MOUNT (WILEY 2004) AND SLIDES FROM NANCY M. AMATO 2704 BOS 867 849 PVD ORD 740 621 1846 MINIMUM SPANNING TREES LAX 1391 1464 1235 144 JFK 1258 184 802 SFO 337 187 BWI 1090 DFW 946 1121 MIA 2342 REMINDER 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 SFO PVD ORD LGA HNL LAX DFW MIA MINIMUM SPANNING TREE ORD • Spanning subgraph • Subgraph of a graph πΊ containing all the vertices of πΊ 1 DEN • Spanning tree • Spanning subgraph that is itself a (free) tree • Minimum spanning tree (MST) 10 PIT 9 STL 4 8 • Spanning tree of a weighted graph with 6 7 3 DCA 5 2 minimum total edge weight • Applications • Communications networks • Transportation networks DFW ATL EXERCISE MST • Show an MST of the following graph. SFO PVD ORD LGA HNL LAX DFW MIA CYCLE PROPERTY 8 f • Cycle Property: 2 6 • Let π be a minimum spanning tree of a C 3 e 7 7 • Let π be an edge of πΊ that is not in π and πΆ let be the cycle formed by π with π Replacing f with e yields a better spanning tree • For every edge π of πΆ, π€πππβπ‘ π ≤ π€πππβπ‘ π f • Proof: spanning tree of smaller weight by replacing π with π 9 8 weighted graph πΊ • By contradiction • If π€πππβπ‘ π > π€πππβπ‘(π) we can get a 4 2 6 8 4 C 9 3 8 7 e 7 PARTITION PROPERTY • • Partition Property: • Consider a partition of the vertices of πΊ into subsets π and π • • Let π be an edge of minimum weight across the partition U f 9 8 8 Replacing f with e yields another MST U Let π be an MST of πΊ • By the cycle property, π€πππβπ‘ π ≤ π€πππβπ‘(π) • • Thus, π€πππβπ‘ π = π€πππβπ‘ π We obtain another MST by replacing π with π 3 e 7 There is a minimum spanning tree of G containing edge π If π does not contain π, consider the cycle πΆ formed by π with π and let π be an edge of πΆ across the partition 4 5 2 Proof: • • V 7 f 2 V 7 4 9 5 8 8 e 7 3 PRIM-JARNIK’S ALGORITHM • We pick an arbitrary vertex π and we grow the MST as a cloud of vertices, starting from π • We store with each vertex π£ a label π π£ = the smallest weight of an edge connecting π£ to a vertex in the cloud • At each step: • • We add to the cloud the vertex π’ outside the cloud with the smallest distance label We update the labels of the vertices adjacent to π’ PRIM-JARNIK’S ALGORITHM • An adaptable priority queue stores the vertices outside the cloud • Key: distance, π·[π£] • Element: vertex π£ • π. ππππππππΎππ¦ π, π changes the key of an item • We store three labels with each vertex π£: • Distance π·[π£] • Parent edge in MST π[π£] • Locator in priority queue Algorithm PrimJarnikMST(πΊ) Input: A weighted connected graph πΊ Output: A minimum spanning tree π of πΊ 1. Pick any vertex π£ of πΊ 2. π· π£ ← 0; π π£ ← ∅ 3. for each vertex π’ ≠ π£ do 4. π· π’ ← ∞; π π’ ← ∅ 5. π ← ∅ 6. Priority queue π of vertices with π·[π’] as the key 7. while ¬π. empty( ) do 8. π’ ← π. removeMin( ) 9. Add vertex π’ and edge π[π’] to π 10. for each π ∈ π’. incidentEdges do 11. π§ ← π. opposite π’ 12. if π. weight( ) < π·[π§] 13. π· π§ ← π. weight( ); π π§ ← π 14. π. replaceKey π§, π·[π§] 15. return π 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 ο₯ B 7 D 7 3 9 8 A D 7 5 F E 7 2 2 4 8 8 A ο₯ 9 8 C 5 2 D E ο₯ 2 3 7 7 B 0 3 7 7 4 9 5 C 5 F 8 8 A D 7 ο₯ E 3 7 4 EXAMPLE 2 2 B 0 4 9 5 C 5 F 8 8 A D 7 7 7 E 4 3 3 2 2 B 0 4 9 5 C 5 F 8 8 A D 7 7 7 E 3 3 4 EXERCISE PRIM’S MST ALGORITHM • Show how Prim’s MST algorithm works on the following graph, assuming you start with SFO • Show how the MST evolves in each iteration (a separate figure for each iteration). SFO PVD ORD LGA HNL LAX DFW MIA ANALYSIS • Graph operations • Method incidentEdges is called once for each vertex • Label operations • We set/get the distance, parent and locator labels of vertex π§ π deg π§ times • Setting/getting a label takes π 1 time • Priority queue operations • Each vertex is inserted once into and removed once from the priority queue, where each insertion or • removal takes π log π time The key of a vertex π€ in the priority queue is modified at most deg π€ times, where each key change takes π log π time • Prim-Jarnik’s algorithm runs in π π + π log π time provided the graph is represented by the adjacency list structure • Recall that Σπ£ deg π£ = 2π • The running time is π π log π since the graph is connected KRUSKAL’S ALGORITHMS • 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(πΊ) 1. for each vertex π£ ∈ πΊ. vertices do 2. Define a cluster πΆ π£ ← π£ 3. Initialize a priority queue π of edges using the weights as keys 4. π ← ∅ 5. while π has fewer than π − 1 edges do 6. π’, π£ ← π. removeMin( ) 7. if πΆ π£ ≠ πΆ(π’) then 8. Add π’, π£ to π 9. Merge πΆ π£ and πΆ π’ 10. return π DATA STRUCTURE FOR KRUSKAL’S ALGORITHM • 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 π’ : return the set storing u • union π’, π£ : replace the sets storing u and v with their union REPRESENTATION OF A PARTITION • Each set is stored in a sequence • Each element has a reference back to the set • Operation ππππ π’ takes π 1 time, and returns the set of which π’ is a member. • In operation π’ππππ π’, π£ , we move the elements of the smaller set to the sequence of the larger set and • update their references The time for operation π’ππππ(π’, π£) is π min ππ’ , ππ£ , where ππ’ and ππ£ are the sizes of the sets storing π’ and π£ • Whenever an element is processed, it goes into a set of size at least double, hence each element is processed at most log π times ANALYSIS • A partition-based version of Kruskal’s Algorithm performs cluster merges as unions and tests as finds. • Running time • There will be at most π removals from the priority queue - π π log π • Each vertex can be merged at most log π times, as the clouds tend to “double” in size π π log π • Total: π π + π log π 2704 EXAMPLE 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 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 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 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 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 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 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 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 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 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 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 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 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 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 EXERCISE KRUSKAL’S MST ALGORITHM • Show how Kruskal’s MST algorithm works on the following graph. • Show how the MST evolves in each iteration (a separate figure for each iteration). SFO PVD ORD LGA HNL LAX DFW MIA A 8 SHORTEST PATHS 0 4 2 B 2 8 7 5 E C 3 2 1 9 D 8 F 5 3 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 SFO PVD ORD LGA HNL LAX DFW MIA SHORTEST PATH PROBLEM • Given a weighted graph and two vertices π’ and π£, we want to find a path of minimum total weight between π’ and π£. • • Example: • • Length of a path is the sum of the weights of its edges. Shortest path between Providence and Honolulu Applications • • • Internet packet routing Flight reservations Driving directions HNL SFO PVD ORD LGA LAX DFW MIA SHORTEST PATH PROBLEM • If there is no path from π£ to π’, we denote the distance between them by π π£, π’ = ∞ • What if there is a negative-weight cycle in the graph? SFO PVD ORD LGA HNL LAX DFW MIA 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 SFO LGA from Providence HNL PVD ORD LAX DFW MIA DIJKSTRA’S ALGORITHM • The distance of a vertex π£ from a vertex π is the length of a shortest path between π and π£ • Dijkstra’s algorithm computes the distances of all the vertices from a given start vertex π (single-source shortest paths) • Assumptions: • The graph is connected • The edges are undirected • The edge weights are nonnegative • Extremely similar to Prim-Jarnik’s MST Algorithm • We grow a “cloud” of vertices, beginning with π and eventually covering all the vertices • We store with each vertex π£ a label π· π£ representing the distance of π£ from π in the subgraph consisting of the cloud and its adjacent vertices • The label π· π£ is initialized to positive infinity • At each step • We add to the cloud the vertex π’ outside the cloud with the smallest distance label, π· π£ • We update the labels of the vertices adjacent to π’, in a process called edge relaxation EDGE RELAXATION • Consider an edge π = π’, π§ such that • • • • D[u] = 50 π’ is the vertex most recently added to the cloud s u e D[z] = 75 z D[y] = 20 y π§ is not in the cloud The relaxation of edge π updates distance π· π§ as follows: D[u] = 50 π· π§ ← min π· π§ , π· π’ + π. π€πππβπ‘ u s D[y] = 20 y D[z] = 60 e z EXAMPLE A 8 B 2 8 7 Pull in one of the vertices with red labels 2. The relaxation of edges updates the labels of LARGER font size 0 4 2 3 ο₯ 2 C 1. 1 9 E D ο₯ F A 8 B 2 7 5 E C 3 4 5 B 8 7 5 E 2 4 2 9 1 C 3 11 F 5 7 B 2 2 1 D 9 8 A 3 5 F 8 3 D 4 2 0 2 8 A 8 0 0 4 2 7 5 E C 3 2 1 9 D 8 F 5 3 EXAMPLE 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 C 3 5 E 2 1 9 D 8 F 5 3 EXERCISE DIJKSTRA’S ALGORITHM • Show how Dijkstra’s algorithm works on the following graph, assuming you start with SFO, i.e., π =SFO. • Show how the labels are updated in each iteration (a separate figure for each iteration). SFO PVD ORD LGA HNL LAX DFW MIA DIJKSTRA’S ALGORITHM • A locator-based priority queue stores the vertices outside the cloud • Key: distance • Element: vertex • We store with each vertex: • • distance π· π£ label locator in priority queue Algorithm Dijkstras_sssp πΊ, π Input: A simple undirected weighted graph πΊ with nonnegative edge weights and a source vertex π Output: A label π· π£ for each vertex π£ of πΊ, such that π· π’ is the length of the shorted path from π to π£ 1. π· π ← 0; π· π£ ← ∞ for each vertex π£ ≠ π 2. Let priority queue π contain all the vertices of πΊ using π· π£ as the key 3. while ¬π. empty do //π π iterations 4. //pull a new vertex π’ in the cloud 5. π’ ← π. removeMin //π log π 6. for each edge π ∈ π’. incidentEdges( ) do //π πππ π’ iterations 7. //relax edge π 8. π§ ← π. opposite π’ 9. if π· π’ + π. weight < π·[π§] then 10. π· π§ ← π· π’ + π. weight 11. π. updateKey π§ //π log π ANALYSIS • Graph operations • Method incidentEdges is called once for each vertex • Label operations • We set/get the distance and locator labels of vertex π§ π deg π§ times • Setting/getting a label takes π 1 time • Priority queue operations • Each vertex is inserted once into and removed once from the priority queue, where each insertion or removal takes π log π time • The key of a vertex in the priority queue is modified at most deg π€ times, where each key change takes π log π time • Dijkstra’s algorithm runs in π π + π log π time provided the graph is represented by the adjacency list structure • Recall that Σπ£ deg π£ = 2π • The running time can also be expressed as π π log π since the graph is connected • The running time can be expressed as a function of π, π π2 log π 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 • Parents are all initialized to null • In the edge relaxation step, we update the parent label as well WHY DIJKSTRA’S ALGORITHM WORKS • • Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. Proof by contradiction • • • • Suppose it didn’t find all shortest distances. Let πΉ be the first wrong vertex the algorithm processed. When the previous node, π·, on the true shortest path was considered, its distance was correct. But the edge π·, πΉ was relaxed at that time! Thus, so long as π· πΉ > π· π· , πΉ’s distance cannot be wrong. That is, there is no wrong vertex. s 8 0 4 2 B 2 7 7 C 3 5 E 2 9 1 D 8 F 5 3 WHY IT DOESN’T WORK FOR NEGATIVE-WEIGHT EDGES A 8 • Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance. • If a node with a negative incident edge B 8 7 C 4 2 3 ο₯ 2 were to be added late to the cloud, it could mess up distances for vertices already in the cloud. 2 0 -3 9 E D ο₯ F A 8 5 0 4 2 B 2 8 7 5 E C 3 2 9 -3 11 F D 5 4 C’s true distance is 1, but it is already in the cloud with π· πΆ = 2! 0 BELLMAN-FORD ALGORITHM • Works even with negative-weight edges • Must assume directed edges (for otherwise we would have negative-weight cycles) • Iteration π finds all shortest paths that use π edges. • Running time: π ππ • Can be extended to detect a negativeweight cycle if it exists • How? Algorithm BellmanFord πΊ, π 1. Initialize π· π ← 0 and π· π£ ← ∞ for all vertices π£ ≠ π 2. for π ← 1 … π − 1 do 3. for each π ∈ πΊ. edges do 4. //relax edge π 5. π’ ← π. source ; π§ ← π. target 6. if π· π’ + π. weight < π· π§ then 7. π· π§ ← π· π’ + π. weight • Nodes are labeled with their π·[π£] values BELLMAN-FORD EXAMPLE 0 8 First round 8 4 0 -2 ο₯ 7 -2 1 ο₯ 3 -2 ο₯ 8 7 9 ο₯ 5 0 -2 ο₯ 7 0 -2 1 4 -2 1 -2 3 5 ο₯ -2 5 4 9 Third round 8 4 1 -2 3 ο₯ Second round 8 4 -1 5 7 9 3 9 5 -2 1 1 -2 -1 9 4 5 EXERCISE BELLMAN-FORD’S ALGORITHM • Show how Bellman-Ford’s algorithm works on the following graph, assuming you start with the top node • Show how the labels are updated in each iteration (a separate figure for each iteration). 0 8 4 -2 ο₯ 7 ο₯ 3 -5 ο₯ 1 ο₯ ο₯ 5 9 ALL-PAIRS SHORTEST PATHS • Find the distance between every pair of vertices in a weighted directed graph πΊ • We can make π calls to Dijkstra’s algorithm (if no negative edges), which takes π ππ log π time. • Likewise, π calls to Bellman-Ford would take π π2 π time. • We can achieve π π3 time using dynamic programming (similar to the Floyd-Warshall algorithm). i Uses only vertices numbered π, … , π (compute weight of this edge) Uses only vertices numbered π, … , π j k Uses only vertices numbered π, … , π Algorithm AllPairsShortestPath πΊ Input: Graph πΊ with vertices labeled 1, … , π Output: Distances π· π, π of shortest path lengths between each pair of vertices 1. for each vertex pair π, π do 2. if π = π then 3. π·0 π, π ← 0 4. else if π = π, π is an edge in πΊ then 5. π·0 π, π ← π. weight 6. else 7. π·0 π, π ← ∞ 8. for π ← 1 … π do 9. for π ← 1 … π do 10. for π ← 1 … π do 11. π·π π, π ← min π·π−1 π, π , π·π−1 π, π + π·π−1 π, π 12. return π·π
