Analysis and Design of
Algorithms
Session -13
1
Single Source Shortest Path Problem

Insights
• Introduction
• Example
• Dijkstra’s Algorithm – Pseudocode
• Dijkstra’s Algorithm – Time Complexity

Single Source Shortest Path Problem (SSSP) – An Introduction
Single Source Shortest Path Problem (SSSP)
• In a Single Source Shortest Paths Problem, we are given a Graph G = (V, E), we want to
find the shortest path from a given source vertex s ∈ V to every vertex v ∈ V.

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1
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5 0
6
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8
5 is a Source Vertex

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1500
5 0
6 250
7 inf
8 inf
Updated 4 & 6 Vertices

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1500
5 0
6 250
7 inf
8 inf
Vertex 6 selected

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 inf
8 inf
Updated Vertex 4

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 inf
8 1650
Updated Vertex 8

4 -10
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 1150
8 1650
Updated Vertex 7

4 -11
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 1150
8 1650
Vertex 7 selected

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 1150
8 1650
No changes in Vertex 4 & 8

4 -13
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 inf
4 1250
5 0
6 250
7 1150
8 1650
Vertex 4 selected

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Updated Vertex 3

4 -15
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 inf
2 inf
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Vertex 8 selected

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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 inf
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Updated Vertex 1

4 -17
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 inf
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Vertex 3 selected

4 -18
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1200
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 3250
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Updated Vertex 2

4 -19
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6
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5
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1
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1700
1200
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 3250
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Vertex 2 selected

4 -20
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4
2
6
7
8
5
3
1
250
1500
900
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1400
1000
1700
1200
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1000
Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 3250
3 2450
4 1250
5 0
6 250
7 1150
8 1650
No changes in Vertex 1

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4
2
6
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8
5
3
1
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1400
1000
1700
1200
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Single Source Shortest Path Problem (SSSP)
Boston
New York
Miami
New Orleans
Los Angeles
Chicago
Denver
San Francisco
City Dist
1 3350
2 3250
3 2450
4 1250
5 0
6 250
7 1150
8 1650
Vertex 1 selected

22
Algorithm ShortestPaths (v, cost, dist, n)
// dist[j], 1<= j <=n, is set to the length of the shortest path from vertex v to vertex j in a
// di-graph G with n vertices. dist[v] is set to zero. G is represented by its cost adjacency
// matrix cost[l :n, 1:n].
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

23
Algorithm ShortestPaths (v, cost, dist, n)
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

24
Algorithm ShortestPaths (v, cost, dist, n)
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

25
Algorithm ShortestPaths (v, cost, dist, n)
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

26
Algorithm ShortestPaths (v, cost, dist, n)
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

27
Algorithm ShortestPaths (v, cost, dist, n)
{
for i :=1 to n do { // Initialize S.
S[i]:=false; dist[i] := cost[v, i];
}
S[v]:=true; dist[v] :=0.0; // Put v in S.
for num := 2 to n -1 do { // Determine n-1 paths from v.
Choose u from among those vertices not in S such that dist[u] is minimum;
S[u]:=true; // Put u in S.
for (each w adjacent to u with S[w]= false) do // Update distances.
if (dist[w] >(dist[u]+cost[u, w,])) then
dist[w] :=dist[u]+cost[u, w];
}// end of for loop
}

28
Thank You