1. Topological Sort (topological order)
Let G = (V, E) be a directed graph
with |V| = n.
A Topological Sort is a sequence v1, v2,.... vn, such that
{ v1, v2,.... vn } = V, and
for every i and j with 1  i < j  n, (vj , vi)  E. 7 1 6 5 3 2 4 7 5
There are more than one such order in this graph!! 2 4 3 1 6
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2. If the graph is not acyclic, then there is no topological sort for the graph.
Not an acyclic Graph 2 4 7 9 1 3 5 8
No way to arrange vertices in this cycle without pointing backwards 6
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3. Algorithm for finding Topological Sort
Principle: vertex with in-degree 0 should be listed first. 2 1 7 2 7
Algorithm: 1
Input G=(V,E)
Repeat the following steps until no more vertex left:
Find a vertex v with in-degree 0;
If can’t find such v and V is not empty
then stop the algorithm; this graph is not acyclic
2. Remove v from V and update E
3. Process v
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4. Topological Sort (topological order) of an acyclic graph7 1 6 5 3 2 4 7 5 2 4 3 1 6
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5. Fail to find a topological sort (topological order)1 2 8 2 4 7 9 1 3 5 8
the graph is not acyclic 6
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6. Topological Sort (topological order) of an acyclic graph2 8 5 4 3 7 9 6 1 2 4 7 9 1 3 5 8 6
This graph is acyclic
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7. 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);
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8. Programming languages Operating systems Distributed processing
Edsger Wybe Dijkstra
Algorithm design
Programming languages
Operating systems
Distributed processing
Formal specification and verification
1972 Turing Award
Dijkstra Algorithm:
To find the shortest path between two vertices in a weighted graph.
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9. Common Algorithm Techniques
Back Tracking....
Greedy Algorithms: Algorithms that make decisions based on the current information; and once a decision is made, the decision will not be revised
Divide and Conquer....
Dynamic Programming.....
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
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10. Greedy Algorithms  Dynamic Programming
Problem: Minimized the number of coins for 66¢
{ 25¢, 12¢, 5¢, 1¢ }
{ 25¢, 10¢, 5¢, 1¢ }
25¢ 2
50¢
10¢ 1 5¢ 1¢ 5
66¢
25¢ 2
50¢
12¢ 1 5¢ 0¢ 1¢ 4 4¢ 7
66¢
25¢ 2
50¢
12¢ 0¢ 5¢ 3
15¢ 1¢ 1 6
66¢
16¢
16¢ 6¢ 4¢ 1¢ 4¢ 0¢ 0¢
Greedy method
Greedy method
Dynamic method
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11. Finding the shortest path between two vertices
Starting Vertex
Starting Vertex
Unweighted Graphs 
Weighted Graphs 2 5 1 2 2 3 3 5 1 1 1 5 3 3 7 7 7 2 2 1 9 10 12 8 8 2
Both are using the
greedy algorithm. w w
Final Vertex
Final Vertex
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12. Construct a shortest path Map
Starting Vertex
Starting Vertex
0,-1 1 2
1,0
1,0 3 4
2,1
2,1 5
2,2 x
w-1,y n
w,x
Final Vertex
Final Vertex
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13. Shortest paths of unweighted graphs
// <vertex, <distance, previous vertex in the path>>
typedef map<int,pair<double, int> > PathMap;
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;
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14. Construct a shortest path Map for weighted graph
Starting Vertex
Starting Vertex 
0,-1 2 5   1 1 2
2,0
3,1
5,0 1 5   3 4 7
3,1
7,1 1 9  5
10,2
8,4
w-1,y
: decided n
w,x
Final Vertex
Final Vertex
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15. 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;
.....
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16. 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;
Shortest paths (II)
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