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This text discusses the concept of representing graphs using adjacency matrices and adjacency lists. It covers the space and time complexities of these representations, as well as graph traversal algorithms like depth-first and breadth-first traversal. The text also explores the relationship between breadth-first and depth-first search algorithms and their applications in directed graphs.
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Thinking about Algorithms Abstractly Searching in a Graph
Representations of Graphs • Adjacency Matrices • Adjacency Lists
A B C D A 0 1 1 1 B 1 0 0 1 C 1 0 0 1 D 1 1 1 0 Adjacency Matrices Graphs G= (V, E) can be represented by adjacency matrices G[v1..v|V |, v1..v|V |], where the rows and columns are indexed by the nodes, and the entries G[vi, vj] represent the edges. In the case of unlabeled graphs, the entries are just boolean values.
A B C D A 10 4 1 B 15 C 9 D Adjacency Matrices In case of labeled graphs, the labels themselves may be introduced into the entries. Adjacency matrices require O(|V |2) space, and so they are space-efficient only when they are dense (that is, when the graphs have many edges). Time-wise, the adjacency matrices allow easy addition and deletion of edges.
Adjacency Lists A representation of the graph consisting of a list of nodes, with each node containing a list of its neighboring nodes. This representation takes O(|V | + |E|) space.
Graph Traversal • Depth-First Traversal • Breadth-First Traversal
Depth-First Traversal • algorithmdft(x) visit(x) FOReachysuchthat(x,y)isanedgeDO IF<ywasnotvisitedyet> THENdft(y)
Depth-First Traversal • A recursive algorithm implicitly recording a “backtracking” path from the root to the node currently under consideration
Depth-First Traversal • Depth first search is another way of traversing graphs, which is closely related to preorder traversal of a tree. • the Breath-first search tree is typically "short and bushy", the DFS tree is typically "long and stringy".
Breadth-First Traversal Visit the nodes at level i before the nodes of level i+1.
Breadth-First Traversal • visit(startnode) • queue<-startnode • WHILEqueueisnotemptyDO • x<-queue • FOReachysuchthat(x,y)isanedgeandyhasnotbeenvisitedyetDO • visit(y) • queue<-y • END • END
Breadth-First Traversal • Each vertex is clearly • marked at most once, • added to the list at most once (since that happens only when it's marked), and • removed from the list at most once. • Since the time to process a vertex is proportional to the length of its adjacency list, the total time for the whole algorithm is O(m). • A tree T constructed by the algorithm is called a breadth first search tree. • The traversal goes a levelat a time, left to right within a level (where a level is defined simply in terms of distance from the root of the tree).
Breadth-First Traversal • Every edge of G can be classified into one of three groups. • Some edges are in T themselves. • Some connect two vertices at the same level of T. • The remaining ones connect two vertices on two adjacent levels. It is not possible for an edge to skip a level. • Breadth-first search tree really is a shortest path tree starting from its root.
Relation between BFS and DFS • dfs(G) • { • list L = empty • tree T = empty • choose a starting vertex x • search(x) • while(L nonempty) • remove edge (v,w) from end of L • if w not yet visited • { • add (v,w) to T • search(w) • } • } • bfs(G) • { • list L = empty tree • T = empty • choose a starting vertex x • search(x) • while(L nonempty) • remove edge (v,w) from start of L • if w not yet visited • { • add (v,w) to T • search(w) • } • } • search(vertex v) { • visit(v); • for each edge (v,w) • add edge (v,w) to end of L • }
Relation between BFS and DFS • Both of these search algorithms now keep a list of edges to explore; the only difference between the two is • while both algorithms adds items to the end of L, • BFS removes them from the beginning, which results in maintaining the list as a queue • DFS removes them from the end, maintaining the list as a stack.
BFS and DFS in directed graphs • The same search routines work essentially unmodified for directed graphs. • The only difference is that when exploring a vertex v, we only want to look at edges (v,w) going out of v; we ignore the other edges coming into v. • For BFS in directed graphs each edge of the graph either • connects two vertices at the same level • goes down exactly one level • goes up any number of levels. • For DFS, each edge either connects an ancestor to • a descendant • a descendant to an ancestor • one node to a node in a previously visited subtree.
