List of Problems

1
List of Problems

• Building roads connecting all cities with the
  minimum cost
• Determining if there are train tracks linking
  city X to city Y
• Finding a shortest tour to visit all cities
• Finding a cheapest route to fly from city X to
  city Y
• Getting out of a maze
• Finding a route to fly from city X to city Y with
  the minimum number of transits
• Checking if there is a loop in the network of
  train tracks.

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Building Roads connecting Cities
3
Testing if cities X and Y are connected
X
Z
Y
4
Finding shortest tour to visit all cities
5
Finding cheapest air ticket
X
Y
6
Getting out of a Maze
7
Finding route with minimum transits
X
Y
8
Loop Detection
9
Subgraph

• Graph G1(V1,E1) is a subgraph of graph
  G2(V2,E2) if
• V1 ? V2 and E1 ? E2
• G1 is a spanning subgraph of G2 if
• V1 V2 and E1 ? E2

10
Spanning Tree

• Graph G1(V1,E1) is a spanning tree of graph
  G2(V2,E2) if
• G1 is a spanning subgraph of G2
• G1 is a tree

11
Algorithms involving graphs

• Graph traversal BFS , DFS
• Strongly connected components
• Topological sorting
• Shortest path algorithms
• In an unweighted graph shortest length between
  two vertices
• In a weighted graph smallest cost between two
  vertices
• Dijakstra
• Minimum Spanning Trees
• Using a tree to connect all the vertices at
  lowest total cost

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General Graph Search AlgorithmLooking for a node
with a property

• Set Store equal to the some node
• while ( Store is non-empty) do
• choose a node n in Store
• if n is solution, stop
• Decisions
• else add SOME sons of n to store
• What should store be?
• How do we choose a node?
• what does add mean?
• How do pick which sons to store.
• Cycles are a problem

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DFS and BFS

• Depth First Search (DFS)
• Start at a node
• Travel to new nodes as far as possible.
• If stuck, go back and repeat the process.
• Breadth First Search (BFS)
• Start at a node
• Travel to all the nodes which are 1 step away
• Repeat the process

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Breadth First Search

• G is a undirected Graph
• Each node has a boolean visited field.
• Initial node is arbitrary
• Store Queue
• Choose dequeue and mark as visited
• Add enqueue only those sons that have not been
  visited
• Properties
• Time Number of nodes to solution
• Space Number of nodes
• Guaranteed to find shortest solution

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Breadth First Search

• Breadth_First_Search( u )
• enqueue(Q,u) and cross u
• while Q not empty
• v dequeue(Q)
• process v
• for each x adjacent to v
• if (x is not crossed) // crossedvisited
• enqueue(Q,x) and cross x

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Breadth First Search
z
w
v
u
t
y
x
t u v y z w x
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Depth First Search (DFS)

• A graph traversal technique
• A generalization of preorder traversal of trees
• Idea
• Start at a node
• Travel to new nodes as far as possible.
• If cant go to a new node from the current node,
  go back to previous node and repeat the process.

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Depth First Search (DFS)

• Start at some vertex v
• Process v
• Recursively traverse all vertices adjacent from v
• Must avoid processing each node more than once
• Must avoid cycles
• When a vertex is visited, mark it as visited

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Depth First Search Pseudo-Code

• Store Stack
• Choose pop
• Initial node any node
• Add push only new sons (unvisited ones)
• keep a boolean field visited, initialized to
  false.
• When a node is popped, mark it as visitied.
• Graph connected if all nodes visited.
• Properties
• Memory cost number of nodes
• Guarantee to find a solution, if one exists (not
  shortest solution) How could we guarantee that?
• Time number of nodes

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Depth-first spanning tree Tree edges Back edges
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• preorder( Node T )
• Print T.info
• for each node W adjacent to T
• preorder( W )
• dfs( Node V )
• V.visited True
• Print V.info
• for each node W adjacent to V
• if (W.visited ! True)
• dfs( W )

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Depth First Search (DFS)

• Depth_First_Search( u )
• push(S,u) and cross u
• while S not empty
• v pop(S)
• process v
• for each x adjacent to v
• if (x is not crossed)//visited
• push(S,x) and cross x

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a
b
e
d
c
g
f
a
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a