Extended notes by Oscar Tengwall from lecture given by Gunnar Kreitz.

Transcription

1 Part I Lecture 5: Graphs Extended notes by Oscar Tengwall from lecture given by Gunnar Kreitz. 1 Graph Data Structures Graphs in mathematics are usually represented by two sets V and E (vertices and edges) and denoted by G =(V,E). In computers these two sets are in turn represented by either an adjacency matrix or adjacency list (or both). Apart from bare vertices and edges additional information such as weights, directions, distances and capacities are often stored. a b c Figure 1: A directed graph 1.1 Adjacency Matrix The adjacency matrix consists of a matrix AM where values at index am ij represents an edge between the vertex i and j. For directed graphs the edge source is the first index and edge target the last index. For undirected graphs the use of the upper or lower triangular suffices for the entire graph (there are only 1 2 ( V 2 V ) possible edges). a b c a b c Figure 2: Adjacency matrix for graph in Fig. 1 a b c Figure 3: A simple undirected graph 1

2 a b c a b c Figure 4: Adjacency matrix for the simple undirected graph in Fig Adjacency Lists An adjacency list consists of a vector of linked lists (or arrays/vectors) AL. Each vertex has its own linked list AL i of edges that originate from that vertex. a [a] [b] b [a] [c] c Figure 5: Adjacency list for graph in Fig Comparison Graph types It terms of memory usage adjacency lists are better for sparse 1 graphs and adjacency matrices are better for dense graphs. Implementation difficulty Adjacency matrices are somewhat simpler to implement than adjacency lists. Constraints Adjacency lists can handle additional edge and vertex informtation as well as graphs with parallel edges. Adjacency matrices on the other hand can only handle additional edge and vertex information with added effort. They also have problems representing multiple parallel edges. Multiple parallel edge support can be obtained by adding more information at each field in the matrix. One way to store this added information is through adding a third dimension to the adjacency matrix thereby creating a three dimensional array AM ijk. In the new dimension k information about the k:th edge between i and j is stored. 1.4 Further Alternatives There are other ways of representing graphs in a computer, but adjacency matrices and lists are usually used to some extent. In addition to adjacency matrices and lists one can use a variety of support structures depending one what types of computations that will be made on the graph. Below are some examples of such support structures. 1 A graph is sparse when E V 2. 2

3 1.4.1 Index map A common problem with graphs is that the vertices may have string names instead of integer identifiers. Here an index map can be used to convert the high level description to a lower one. The index map itself is implemented by a vector or array for mapping vertex id vertex name and a map (binary search tree) or hash table is used for mapping vertex name vertex id. Complexity of O(1)(expected) or O(log V ). New vertices are inserted by first checking that they do not already exist and then inserting them into the array or vector and in turn inserting the index used in the array or vector into the map or hash table. Why go through the trouble of using an index map? We can use the same implementation of a graph algorithm repeatedly. Integer operations are faster than string operations Edge hashing The complexity of determining whether there exists an edge between two vertices in an adjacency matrix is O(1). In an adjacency list it is O( maxdeg ) where maxdeg is the maximum degree of all vertices in the graph. One way to improve the lookup time when using an adjacency list, while not storing an entire adjacency matrix, is to use a hash table or map to allow fast lookup in O(log( E ) Duplicate edge filtering Descriptions of graphs might have multiple parallel edges. There are a number of ways to filter out parallel edges when one does not want them. A simple way is to first read in the indata to an adjacency matrix (since it can only store one edge in each direction) and then convert the adjacency matrix into an adjacency list. 1.5 Common Problems Some of the problems in dealing with graphs are self-loops (an edge from a vertex to itself), multiple parallel edges and when the graph is disconnected (consists of two or more disconnected parts). 1.6 Example: Road Construction Input: Named cities Existing roads (between pairs of cities) Possible roads with construction costs Output: Cheapest way to connect all cities to form a road grid. 3

4 Model: Cities Vertices Roads Edges Construction costs Edge weights We would much rather work with V = {0,..., n 1}. Therefore we need a mapping from city name city number. Here we can use the index map described in How do we now solve the graph problem? Solve by creating a minimum spanning tree (MST) where existing roads are edges with weight=0 and new roads have weight=construction cost. The MST problem can then be solved by use of either Prim s or Kruskal s algorithm Prim s Algorithm Prim s algorithm first chooses a vertex s, marks it as visited, and then continuously chooses the cheapest edge from visited to non-visited vertices. Prim s algorithm uses arrays; p[] for storing pointers to the parent of each vertex, d[] for the cost of visiting a vertex and v[] for keeping track of visited vertices. Below s denotes the start vertex. Pseudo Code for Prim s Algorithm for u V = {0,..., n 1} p[u] 1 d[u] v[u] d[s] 0 do n times find u so that v[u] = and d[u] minimal (if d[u] =, G not connected) v[u] for all (u, w) Adj u if ( v[q]) and d[w] >wt(u, w) d[w] wt(u, w) p[w] u 1. Linear search at and use an adjacency matrix in θ(n 2 ). 2. Priority queue for d[u] and adjacency list requires O(log V ) / vertex requires O(log V ) / edge Total O( E log V ) 4

5 3. Fibonacci heap (an extension of binomial heaps) O(1) for updating O(log V ) extract minimum Total O( E + V log V ) (with amortized analysis) This is obviously a greedy algorithm but does it guarantee correct solutions? We wish to prove that the chosen edges always can be extended to form an optimal tree. Basic case: At the beginning no edges have been chosen i.e. we can obviously extend it to form an optimal tree. Induction case: We have chosen e 1,..., e i 1 and now choose e i =(u, w). According to the assumption that there exists an optimal tree T where e 1,..., e i 1 are members then if e i T then we are done. Otherwise e i T contains a cycle. Let f be the edge in T that has one vertex visited in the path from visited vertices to v. Statement: w(f) w(e i ) [else Prim does not choose e i ]. T =(T e i )\f is a tree and w(t ) w(t ) (Actually equal since T is optimal) so T is optimal and contains e 1,..., e i. 1.7 Breadth First Search (BFS) Pseudo code for BFS for all u do col[u] DONE open.push(s) while open.empty() do u open.front() col[u] DONE for each child w u do if col[u] DONE open.push(w) open.pop() As seen above the algorithm for BFS is rather simple and primarily uses a queue. Below is an example queue implementation showing how easy a queue is to implement. Please note that there already exists even better queue implementations in C++ and Java. Therefore the code provided is only meant as an example. For more functionality you can use a pair<int, int> for storing e.g. <vertex, distance>. 5

6 1.7.1 Simple C++ Queue int q[n]; int head; //first element in queue int tail; //first free space in queue int front(void) { return q[head]; } void pop(void) { ++head; } push(int u) { q[tail++] = u; } bool empty(void) { head == tail; } void clear(void) { head = tail = 0; } 1.8 Topological sort of Directed Acyclic Graph (DAG) Games (3) Money (2) Party (4) Study (5) Work (6) Partner (7) Children (8) Die (9) Sleep (1) Figure 6: A DAG where each the topological sorting of it is marked with parenthesises. Pseudo code for Topological sort of DAG Assign vertices topnr[u] {0,..., n 1} such that there path from u to w in G topnr[u] < topnr[w] for u =0to n 1 vis[u] topnr[u] = nr n 1 for u =0to n 1 if vis[u] nr df s(u, nr, vis, topnr) Pseudo code for Depth First Search (DFS) vis[u] for each w adj[u] if not vis[w] nr df s(w, nr, vis, topnr) else if topnr[w] == CYCLE! topnr[u] nr return nr 1 6

7 1.9 Strongly Connected Components (SCC) An SCC C consists of a maximum subset of vertices such that if u, w are in C then there exists a path from u w, w u. Pseudo code for SCC Sort graph in topological order (ignoring cycles). G T G with all edges flipped. for each u V G T (in topological order) if vis[u] nr df s(u, nr, vis, topnr) Not that each call at corresponds to an SCC made up of all vertices visited during the DFS. The algorithm has a complexity of O( V + E ) Lemma Let C, C be different SCC in G =(V,E). Assume that (u, w) E T where u C, w C. Then the first vertex in C will be before C in the topological ordering Shortest Paths Dijkstra We want to find dist(s, x), x V. The algorithm itself is very similar to Prim s algorithm. Thus it also uses the same arrays; p[] for storing pointers to the parent of each vertex, d[] for the cost of visiting a vertex and v[] for keeping track of visited vertices. Below s denotes the start vertex. The complexity is the same as Prim s algorithm O( E log V ). If a fibonacci heap is used it is O( E + V log V ). Pseudo Code Dijkstra for u V = {0,..., n 1} p[u] 1 d[u] v[u] d[s] 0 do n times find u so that v[u] = and d[u] minimal (if d[u] =, G not connected) v[u] for all (u, w) E if v[q] relax(u, w) Pseudo Code relax(u, w) 7

8 if (d[w] >d[u]+wt(u, w)) d[w] d[u]+wt(u, w) p[w] u Why does it work? Induction: When v[u] is then d[u] =dist(s, u). 8