Chapter 9 Graph algorithms Sample Graph Problems Path

Chapter 9 Graph algorithms

Sample Graph Problems • Path problems. • Connectedness problems. • Spanning tree problems.

Graph Problem – An Example Graph coloring: Given a graph G, assign colors to vertices using as few colors as possible so that if there an edge between u and v, colors assigned to u and v must be different. How many colors do we need for this graph?

Graph Problem – An Example How many colors do we need for this graph? Three colors are enough, but two is not enough.

Graph Coloring applications Graph coloring: Given a graph G, assign colors to vertices using as few colors as possible so that if there an edge between u and v, colors assigned to u and v must be different. Applications: This problem has many different applications. • map coloring – this is the direct application. Countries are vertices, and adjacent countries (that share a boundary) are connected by an edge. In real maps, adjacent countries are assigned different colors.

Graph coloring Application 2: scheduling problem. Given is a list of courses, and for each course the list of students who want to register for it. The goal is to find a time slot for each course (using as few slots as possible). If a student is registered for courses p and q, add an edge between p and q. If the resulting class can be colored with k = the number of available slots (MFW 8 -9 , MWF 9 – 10 etc. are various slots), then we can find a schedule that will allow the students to take all the courses they want to.

Another application of graph coloring Traffic signal design: At an intersection of roads, we want to install traffic signal lights which will periodically switch between green and red. The goal is to reduce the waiting time for cars before they get green signal. This problem can be modeled as a coloring problem. Each path that crosses the intersection is a node. If two paths intersect each other, there is an edge connecting them. Each color represents a time slot at which the path gets a green light.

Application we will study • We will discuss in some detail an algorithm (known as Dijkstra’s algorithm) for finding the shortest path in a weighted graph. • This algorithm can be used to in applications like map-quest. (Map-quest actually uses a variant of this algorithm. )

Path Finding Path between 1 and 8. 2 4 3 8 8 1 10 6 2 4 4 4 5 9 5 6 6 7 Path length is 20. 7 5 3 11

Another Path Between 1 and 8 2 4 3 8 8 1 10 6 2 4 4 4 5 9 5 6 6 7 Path length is 28. 7 5 3 11

Example Of No Path 2 3 8 1 10 4 5 6 9 7 No path between 2 and 9. 11

Connected Graph • Undirected graph. • There is a path between every pair of vertices.

Example of a graph Not Connected 2 3 8 1 10 4 5 6 9 7 11

Connected Graph Example 2 3 8 1 10 4 5 6 9 7 11

Connected Components 2 3 8 1 10 4 5 6 9 7 11

Connected Component • A maximal subgraph that is connected. § Cannot add vertices and edges from original graph and retain connectedness. • A connected graph has exactly 1 component.

Communication Network 2 3 8 1 10 4 5 6 9 7 Each edge is a link that can be constructed (i. e. , a feasible link). 11

Communication Network Problems • Is the network connected? § Can we communicate between every pair of cities? • Find the components. • Want to construct smallest number of feasible links so that resulting network is connected.

Cycles And Connectedness 2 3 8 1 10 4 5 6 9 11 7 Removal of an edge that is on a cycle does not affect connectedness.

Cycles And Connectedness 2 3 8 1 10 4 5 6 9 11 7 Connected subgraph with all vertices and minimum number of edges has no cycles.

Tree • Connected graph that has no cycles. • n vertex connected graph with n-1 edges. • A connected graph in which removal of any edge makes it unconnected. • An cyclic graph in which addition of any edges introduces a cycle.

Spanning Tree • Subgraph that includes all vertices of the original graph. • Subgraph is a tree. § If original graph has n vertices, the spanning tree has n vertices and n-1 edges.

Minimum Cost Spanning Tree 2 4 3 8 8 1 6 2 10 4 4 4 5 5 9 6 2 6 5 8 3 11 7 7 • Tree cost is sum of edge weights/costs.

A Spanning Tree 2 4 3 8 8 1 10 6 2 4 4 4 5 5 9 6 2 6 5 7 Spanning tree cost = 51. 7 8 3 11

Minimum Cost Spanning Tree 2 4 3 8 8 1 10 6 2 4 4 4 5 5 9 6 2 6 5 7 Spanning tree cost = 41. 7 8 3 11

Graph Representation • Adjacency Matrix • Adjacency Lists § Linked Adjacency Lists § Array Adjacency Lists

Adjacency Matrix • 0/1 n x n matrix, where n = # of vertices • A[i, j] = 1 iff (i, j) is an edge 2 3 1 4 5 1 2 3 4 5 1 0 1 0 2 1 0 0 0 1 3 0 0 1 4 1 0 0 0 1 5 0 1 1 1 0

Adjacency Matrix Properties 2 3 1 4 5 1 2 3 4 5 1 0 1 0 2 1 0 0 0 1 3 0 0 1 4 1 0 0 0 1 5 0 1 1 1 0 • Diagonal entries are zero. • Adjacency matrix of an undirected graph is symmetric. §A(i, j) = A(j, i) for all i and j.

Adjacency Matrix (Digraph) 2 3 1 4 5 1 2 3 4 5 1 0 0 0 2 0 0 1 3 0 0 1 4 1 0 0 5 0 1 0 • Diagonal entries are zero. • Adjacency matrix of a digraph need not be symmetric.

Adjacency Matrix • n 2 bits of space • For an undirected graph, may store only lower or upper triangle (exclude diagonal). § (n-1)n/2 bits • O(n) time to find vertex degree and/or vertices adjacent to a given vertex. • O(1) time to determine if there is an edge between two given vertices.

Adjacency Lists • Adjacency list for vertex i is a linear list of vertices adjacent from vertex i. • An array of n adjacency lists. a. List[1] = (2, 4) 2 3 a. List[2] = (1, 5) a. List[3] = (5) 1 4 a. List[4] = (5, 1) 5 a. List[5] = (2, 4, 3)

Linked Adjacency Lists • Each adjacency list is a chain. 2 3 1 4 5 a. List[1] [2] [3] [4] a. List[5] 2 1 5 5 2 Array Length = n # of chain nodes = 2 e (undirected graph) # of chain nodes = e (digraph) 4 5 1 4 3

Weighted Graphs • Cost adjacency matrix. § C(i, j) = cost of edge (i, j) • Adjacency lists => each list element is a pair (adjacent vertex, edge weight)

Single-source Shortest path problem • directed, weighted graph is the input • specified source s. • want to compute the shortest path from s to all the vertices.

Example:

Dijkstra’s algorithm: • Works when there are no negative weight edges. • takes time O(e log n) where e = number of edges, n = number of vertices. • suppose n ~ 105, e ~ 106, then the number of computations ~ 2 x 107

Data structures needed: . • Adjacency list rep. of graph • a heap • some additional structures (e. g. array)

key operation: relaxation on edge For each node v, the algorithm assigns a value d[v] which gets updated and will in the end become the length of the shortest path from s to v.

implementation of relaxation

Dijkstra’s algorithm

Dijkstra’s algorithm – Example

Dijkstra’s algorithm - implementation

Complexity of Dijkstra’s algorithm • Initialization: O(n) to set up d and parent arrays. • Number of iterations is n-1 since after each iteration the shortest distance from source to one of the vertices is determined (namely, the one that gets deleted from heap). • Since the heap size is initially n, there will be n deletemin operations. (# operations = O(n log n)). • There will be at most one relaxation performed on each edge. Each relaxation involves Decrease. Key operation on heap. Takes O(log n) each. So total cost is O(e log n). • Overall cost = O(n) + O(n log n) + O(e log n) = O(e log n) since e is at least n-1 (in a graph where all vertices are reachable from the source. )

Proof of correctness of Dijkstra’s algorithm Main idea: d[j] holds the distance of a path from s to j. In general, this is not the length of the shortest path. However, for the minimum value in d represents the actual shortest path. This can be shown by using the fact that there are no negative weight edges in the graph.

Topics that could not be covered § Data structures § Skip list § Red-black tree § Applications § Disjoint set representation (Ch 9) § Huffman coding § Analysis of algorithms § Average-case analysis of Quicksort § Analysis of closed hashing § Analysis of heap building algorithm

Topics for CS 415 • Algorithm design techniques • • Analysis techniques • • • Divide and conquer Greedy algorithm Dynamic programming Graph algorithms (flow, matching etc. ) Linear programming reductions Recurrence relations Summation, integral Induction Reductions Applications • • computer vision and image processing ? Data compression, social networks, games?