Chapter 13 Graphs. The graph ADT, while relatively unstructured, provides a platform for solving some of the most sophisticated problems in computer science. Graph Abstract Data Types. Topological Sort. Minimum Path. Maximum Flow. Spanning Tree. Depth-First Search. 4. 3. 6. 6. 5. - PowerPoint PPT Presentation
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CS CS 240 Chapter Chapter 13 - Graphs - Graphs Page 1 Chapter 13 Chapter 13 Graphs Graphs The graph ADT, while relatively unstructured, provides The graph ADT, while relatively unstructured, provides a platform for solving some of the most sophisticated a platform for solving some of the most sophisticated problems in computer science. problems in computer science. Graph Abstract Data Types Graph Abstract Data Types Spanning Tree Spanning Tree Topological Sort Topological Sort Depth-First Search Depth-First Search Minimum Path Minimum Path Maximum Flow Maximum Flow
The graph ADT, while relatively unstructured, The graph ADT, while relatively unstructured, provides a platform for solving some of the provides a platform for solving some of the most sophisticated problems in computer most sophisticated problems in computer science.science. Graph Abstract Data TypesGraph Abstract Data Types
Topological Sort Algorithm Place all indegree-zero vertices in a list. While the list is non-empty:
– Output an element v in the indegree-zero list.Output an element v in the indegree-zero list.– For each vertex w with (v,w) in the edge set E:For each vertex w with (v,w) in the edge set E:
Decrement the indegree of w by one.Decrement the indegree of w by one. Place w in the indegree-zero list if its new indegree is 0.Place w in the indegree-zero list if its new indegree is 0.
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Indegree of Indegree of A:A:Indegree of Indegree of B:B:Indegree of Indegree of C:C:Indegree of Indegree of D:D:Indegree of Indegree of E:E:Indegree of Indegree of F:F:Indegree of Indegree of G:G:Indegree of Indegree of H:H:Indegree of I:Indegree of I:
Shortest-Path Algorithms (Continued)Case 2: Weighted Graphs With No Negative WeightsUse Dijkstra’s Algorithm, i.e., starting at the specified vertex, finalize
the unfinalized vertex whose current cost is minimal, and update each vertex adjacent to the finalized vertex with its (possibly revised) cost and predecessor, until all vertices are finalized. Algorithm’s time complexity: O(E+V2) = O(V2).
Note that this algorithm would not work for graphs with negative weights, since a vertex cannot be finalized when there might be some negative weight in the graph which would reduce a particular path’s cost.
Shortest-Path Algorithms (Continued)Case 3: Weighted Graphs With Negative WeightsUse a variation of Dijkstra’s Algorithm without using the concept of
vertex finalization, i.e., starting at the specified vertex, update each vertex adjacent to the current vertex with its (possibly revised) cost and predecessor, placing each revised vertex in a queue. Continue until the queue is empty. Algorithm’s time complexity: O(EV).
Note that this algorithm would not work for graphs with negative-cost cycles. For example, if edge IH in the above example had cost -5 instead of cost -3, then the algorithm would loop indefinitely.
Maximum-Flow AlgorithmAssume that the directed graph G = (V, E) has edge capacities
assigned to each edge, and that two vertices s and t have been designated the source and sink nodes, respectively.
We wish to maximize the “flow” from s to t by determining how much of each edge’s capacity can be used so that at each vertex, the total incoming flow equals the total outgoing flow.
This problem relates to such practical applications as Internet routing and automobile traffic control.
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In the graph at left, for instance, the total of the incoming capacities for node C is 40 while the total of the outgoing capacities for node C is 35. Obviously, the maximum flow for this graph will have to “waste” some of the incoming capacity at node C. Conversely, node B’s outgoing capacity exceeds its incoming capacity by 5, so some of its outgoing capacity will have to be wasted.
Minimum Spanning TreesAn alternative to Kruskal’s Algorithm (which develops a minimum spanning tree via
forests) is Prim’s Algorithm, which is just a variation of Dijkstra’s Algorithm. Starting with a minimum-cost edge, it builds the tree by adding minimum-cost edges
as long as they don’t create cycles. Like Kruskal’s, Prim’s Algorithm is O(ElogV)
Depth-First SearchA convenient means to traverse a graph is to use a depth-first
search, which recursively visits and marks the vertices until all of them have been traversed.
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Original GraphOriginal Graph
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Depth-First SearchDepth-First Search
(Solid lines are part of (Solid lines are part of depth-first spanning tree; depth-first spanning tree; dashed lines are visits to dashed lines are visits to
Articulation PointsA vertex v in a graph G is called an articulation point if its removal from
G would cause the graph to be disconnected.Such vertices would be considered “critical” in applications like
networks, where articulation points are the only means of communication between different portions of the network.
A depth-first search can be used to find all of a graph’s articulation points.
The only articulation points are the root (if it has more than one child) and any other node v with a child whose “low” number is at least as large as v’s “visit” number. (In this example: nodes B, C, E, and G.)
Euler CircuitsAn Euler circuit of a graph G is a cycle that visits every edge exactly once.Such a cycle could be useful in applications like networks, where it could provide an efficient
means of testing whether each network link is up.A depth-first search can be used to find an Euler circuit of a graph, if one exists. (Note: An
Euler circuit exists only if the graph is connected and every vertex has even degree.)
Original GraphOriginal Graph
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Splicing the first two cycles yields cycle A(BCDFB)GHA.
Splicing this cycle with the third cycle yields ABCD(FEJF)BGHA.
Splicing this cycle with the fourth cycle yields ABCD(FHIF)EJFBGHA
Note that this traversal takes O(E+V) time.
After Removing First After Removing First DFS Cycle: BCDFBDFS Cycle: BCDFB
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After Removing Second After Removing Second DFS Cycle: ABGHADFS Cycle: ABGHA
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After Removing Third After Removing Third DFS Cycle: FEJFDFS Cycle: FEJF
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After Removing After Removing Fourth DFS Cycle: Fourth DFS Cycle:
A problem is said to be an NP problem if it can be solved with a nondeterministic, polynomial-time algorithm. In essence, at a critical point in the algorithm, a decision must be made, and it is assumed that a magical “choice” function always chooses correctly.
Problem: Is there a way to pack at least T dollars worth of jewels, without exceeding the weight capacity of the knapsack?
(It’s not as easy as it sounds; three lightweight $1000 jewels might be preferable to one heavy $2500 jewel, for instance.)
