10/13/2015IT 328, review graph algorithms1 Topological Sort ( topological order ) Let G = (V, E) be...
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03/22/22 IT 328, review graph algorithms 1 Sort (topological order) Let G = (V, E) be a directed graph with |V| = n. 1.{ v 1 , v 2 ,.... v n } = V, and 2.for every i and j with 1 i < j n, (v j , v i ) E. 1 7 6 5 0 3 4 2 A Topological Sort is a sequence v 1 , v 2 ,.... v n , such that 1 7 6 5 0 3 4 2 There are more than one such order in this graph!!
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Transcript of 10/13/2015IT 328, review graph algorithms1 Topological Sort ( topological order ) Let G = (V, E) be...
04/19/23 IT 328, review graph algorithms 1
Topological Sort (topological order)
Let G = (V, E) be a directed graph with |V| = n.
1. { v1, v2,.... vn } = V, and2. for every i and j with 1 i < j n, (vj , vi) E.
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A Topological Sort is a sequence v1, v2,.... vn, such that
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2There are more than one such order in this graph!!
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If the graph is not acyclic, then there is no topological sort for the graph.
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Not an acyclic Graph
No way to arrange vertices in this cycle without pointing backwards
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Algorithm for finding Topological Sort
Principle: vertex with in-degree 0 should be listed first.
Algorithm:
Repeat the following steps until no more vertex left:
1. Find a vertex v with in-degree 0;
Input G=(V,E)
2. Remove v from V and update E3. Add v to the end of the sort
If can’t find such v and V is not empty then stop the algorithm; this graph is not acyclic
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Topological Sort (topological order) of an acyclic graph
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the graph is not acyclic
Fail to find a topological sort (topological order)
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This graph is acyclic
Topological Sort (topological order) of an acyclic graph
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Topological Sort in C++// return an array where the first entry is the size of g// or 0 if g is not acyclic;int * graph::topsort(const Graph & g) { int *topo; Graph tempG = g;
topo = new int[g.size()+1];topo[0] = g.size();for (int i=1; i<=topo[0]; i++) {
int v = findVertexwithZeroIn(tempG);if (v == -1) { // g is not acyclic
delete [] topo;topo = new int[1];topo[0] = 0; return topo;
}topo[i] = v;remove_v(tempG, v);
}return topo;
}
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Edsger Wybe Dijkstra 1930-2002
Dijkstra Algorithm:
To find the shortest path between two vertices in a weighted graph.
• Algorithm design • Programming languages • Operating systems • Distributed processing • Formal specification and verification
1972 Turing Award
04/19/23 IT 328, review graph algorithms 9
1. Back Tracking....2. Greedy Algorithms: Algorithms that make decisions
based on the current information; and once a decision is made, the decision will not be revised
3. Divide and Conquer....4. Dynamic Programming.....
Common Algorithm Techniques
Problem: Minimized the number of coins for change.
In real life, we can use a greedy algorithm to obtain the minimum number of coins. But in general, we need dynamic programming to do the job
04/19/23 IT 328, review graph algorithms 10
Greedy Algorithms Dynamic Programming
Problem: Minimized the number of coins for 66¢
{ 25¢, 10¢, 5¢, 1¢ } { 25¢, 12¢, 5¢, 1¢ }
25¢ 2 50¢
10¢ 1 10¢
5¢ 1 5¢
1¢ 1 1¢
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12¢ 1 12¢
5¢ 0 0¢
1¢ 4 4¢
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25¢ 2 50¢
12¢ 0 0¢
5¢ 3 15¢
1¢ 1 1¢
6 66¢
Greedy method Greedy method Dynamic method
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6¢
1¢
0¢
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0¢
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Unweighted Graphs Weighted Graphs
Starting Vertex
wFinal Vertex
Starting Vertex
w Final Vertex
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Both are using thegreedy algorithm.
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Finding the shortest path between two vertices
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Shortest paths of unweighted graphs
PathMap graph::UnweightedShortestPath(Graph & g, int s) {
PathMap SPMap; // create a bookkeeper; for (Graph::iterator itr = g.begin(); itr != g.end(); itr++) {
SPMap[itr->first] = make_pair(inft,-1); // -1 mean no previous } SPMap[s] = make_pair(0,-1); // s is not done yet. deque<int> candidates; candidates.push_back(s); while (!candidates.empty()) {
int v = candidates[0];candidates.pop_front();AdjacencyList ADJ = g[v];for (AdjacencyList::iterator w=ADJ.begin(); w != ADJ.end(); w++) { if (SPMap[w->first].first == inft) {
SPMap[w->first].first = SPMap[v].first+1; SPMap[w->first].second=v;
candidates.push_back(w->first); } // end enqueue next v} // end ADJ interation; all nextvs's next into the queue.
} return SPMap;}
// <vertex, <distance, previous vertex in the path>>typedef map<int,pair<double, int> > PathMap;
04/19/23 IT 328, review graph algorithms 13
Construct a shortest path Map for weighted graph
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Shortest paths of weighted graphs (I)
// A Dijkstra's algorithm PathMap graph::ShortestPath(Graph & g, int s) {
PathMap SPMap;
map<int,pair<double, bool> > dist_map; // A distance map // for book keeping; for (Graph::iterator itr = g.begin(); itr != g.end(); itr++) {
dist_map[itr->first] = make_pair(inft,false);SPMap[itr->first] = make_pair(inft,-1);
} dist_map[s] = make_pair(0,false); // s is not done yet. multimap<double,int> candidates ; // next possible vertices //sorted by their distance. candidates.insert(make_pair(0,s)); // start from s;..........}
04/19/23 IT 328, review graph algorithms 15
Shortest paths (II)
// A Dijkstra's algorithm PathMap graph::ShortestPath(Graph & g, int s) { ...... while (! candidates.empty()) {
int v = candidates.begin()->second;double costs2v = candidates.begin()->first;candidates.erase(candidates.begin());if (dist_map[v].second) continue; // v is done after pair
// (weight v) is inserted;dist_map[v] = make_pair(costs2v,true);AdjacencyList ADJ = g[v]; // all not done adjacent vertices
// should be candidates for (AdjacencyList::iterator itr=ADJ.begin(); itr != ADJ.end(); itr++) {
int w = itr->first; if (dist_map[w].second) continue; // this w is done; double cost_via_v = costs2v + itr->second; if (cost_via_v < dist_map[w].first) { dist_map[w] = make_pair(cost_via_v ,false); SPMap[w] = make_pair(cost_via_v,v); } candidates.insert(make_pair(cost_via_v,w));} // end for ADJ iteration;
} // end while !candidates.empty() return SPMap; }
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Minimum Spanning Tree of an undirected graph:
The lowest cost to connect all vertices
Since a cycle is not necessary, such a connection must be a tree.
A spanning is a walk such that every vertex is included
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Prim’s algorithm: grow the minimum spanning tree from a root
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Prim’s algorithm: grow the minimum spanning tree from a different root
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Kruskal’s Algorithm: construct the minimum spanning from edges
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A Hamiltonian cycle is a cycle that contains every vertex;A graph that contains a Hamiltonian cycle is called Hamiltonian
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A non-Hamiltonian graph
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