CS 206 Introduction to Computer Science II 11 / 05 / 2008 Instructor: Michael Eckmann

Michael Eckmann - Skidmore College - CS Fall 2008 Today’s Topics Questions? Heapsort Graphs –traversals

Heaps There is a sorting algorithm based on heaps called HeapSort. The algorithm is simply: –start with n unsorted data items –create a maxHeap (of size n) out of these items --- store it as an array –set i = n -1 swap the root (index 0) and last node (index i)‏ reheapify (downward reheapification) the tree that starts at the root (index 0) and goes to i-1 (do not include the nodes at i and higher in the new heap)‏ –recursively do this until the size of the tree we are heapifying is one (i=0)‏ –The array is now sorted from low to high. Draw on the board.

Graphs Graphs consist of a set of vertices and a set of edges. An edge connects two vertices. Edges can be directed or undirected. Directed graphs' edges are all directed. Undirected graphs' edges are all undirected. Directed graphs are sometimes called digraphs. Two vertices are adjacent if an edge connects them. The degree of a vertex is the number of edges starting at the vertex. Two vertices v1 and v2 are on a path if there are a list of vertices starting at v1 and ending at v2 where each consecutive pair of vertices is adjacent.

Graphs The length of a path is the number of edges in the path. A simple path is one whose edges are all unique. A cycle is a simple path, starting and ending at the same vertex. A vertex is reachable from another vertex if there is a path between them. A graph is connected if all pairs of vertices in the graph have a path between them. Example on the board of a connected and an unconnected graph. A complete graph (aka fully connected graph) is a connected graph where all pairs of vertices in the graph are adjacent. Example on the board.

Graphs Connectivity of a digraph –A digraph is strongly connected if there is a path from any vertex to any other vertex (following the directions of the edges)‏ –A digraph is weakly connected if in its underlying undirected graph, there is a path from any vertex to any other vertex Degree of a vertex in a digraph –In-degree of a vertex is the number of edges entering the vertex –Out-degree of a vertex is the number of edges leaving the vertex

Graphs Edges often have a weight associated with them. An edge's weight is some numeric value. Many things in the real world can be naturally modelled with graphs. Example on the board of using graphs to represent real world problems –e.g. A weighted graph that connect cities (vertices) and the weights of the edges might be length of time to travel between the two cities. –A graph whose vertices represent buildings on campus and edges are sidewalks connecting the buildings.

Graphs Graphs can be represented in programs as many things. Two common ways to represent them are: –adjacency matrix --- each row number r represents a vertex and the value at column c is true if there's an edge from vertex r to vertex c, false otherwise –edge lists store a linked list for each vertex, v i items in the list are those vertices v j for which there's an edge from v i to v j

Graphs Graph traversal –Breadth first search (BFS)‏ Pick a vertex at which to start Visit all of the adjacent vertices to the start vertex Then for each of the adjacent vertices, visit their adjacent vertices And so on until there are no more adjacent vertices Do not visit a vertex more than once –Only vertices that are reachable from the start vertex will be visited --- example on the board. –The order that vertices in a BFS are visited are in increasing order of length of path from starting vertex. –Those that have the same path length from the start vertex can be visited in any order. –Example of BFS on the board.

Graphs Implementation of breadth first search –Need a flag for each vertex to mark it as unvisited, waiting, or visited – so we don't visit vertices more than once. –Keep a queue which will hold the vertices to be visited –Keep a list of vertices as they are visited –BFS algorithm: Mark all vertices as unvisited Initially enqueue a vertex into the queue, mark it as waiting While the queue is not empty –Dequeue a vertex from the queue –Put it in the visited list, mark it as visited –Enqueue all the adjacent vertices that are marked as unvisited to the vertex just dequeued. –Mark the vertices just enqueued as waiting

Graphs Let's see how far we get with implementation of a graph and breadth first search.

Graphs Graph traversal –Depth first search (DFS)‏ Pick a vertex at which to start Visit one of it's adjacent vertices then visit one of that one's adjacent vertices, and so on until there is no unvisited adjacent vertex of the one we're working on. Then backtrack one level and visit another adjacent vertex from that one and repeat. Do this until we're at the start vertex and there's no more unvisited adjacent vertices Do not visit a vertex more than once –Only vertices that are reachable from the start vertex will be visited –Those vertices that are adjacent to a vertex can be visited in any order. –Example of DFS on the board.

Graphs Shortest path algorithms –problem is to find the shortest path from one given vertex to each of the other vertices. –output is a list of paths from given vertex to all other vertices –what real world examples might ever want to find the shortest path?

Graphs Shortest path algorithms –problem is to find the shortest path from one given vertex to each of the other vertices. –output is a list of paths from given vertex to all other vertices –the shortest path could be in terms of path length (number of edges between vertices e.g. a direct flight has path length 1, flights with connecting flights have path length > 1 –the shortest path could be in terms of minimum weight for weighted graphs (example on the board.)‏ e.g. finding the lowest cost flights Dijkstra's algorithm solves this problem

Graphs the shortest path could be in terms of path length (number of edges between vertices) e.g. a direct flight has path length 1, flights with connecting flights have path length > 1 –Initialize all lengths to infinity –Can process the graph in a BFS starting at the given vertex –When visit a node, replace it's length with the current length. –The BFS uses a queue as we saw last time. –Let's implement this.

Graphs the shortest path could be in terms of minimum weight for weighted graphs e.g. finding the lowest cost flights Dijkstra's algorithm solves this problem –It attempts to minimize the weight at each step. Dijkstra's algorithm is a greedy algorithm. That is, its strategy is to locally minimize the weight, hoping that's the best way to get the minimum weight of the whole graph. –Sometimes the local minimum weight is not the correct choice for the overall problem. In that case, the algorithm will still work, but the initial guess was wrong. –Dijkstra's algorithm works in a similar way to BFS but instead of a queue, use a “minimum” priority queue. That is, a priority queue that returns an item whose priority is least among the items in the priority queue. –Let's see an example on the board and come up with pseudocode for this algorithm.