Chapter 9 Graph Algorithms

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Chapter 9 Graph Algorithms

• Fundamentals of graph theory
• Algorithms for some graph problems
• Choice of data structures for graph problems

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9.1 Definition

• A graph G (V, E) consists of a set of vertices,
  V, and a set of edges, E.
• Each edge (arc) is a pair (v, w), where v,w ? V.
  An edge may have a weight (cost).
• If the pair is ordered, then the graph is
  directed - digraph.
• Vertex w is adjacent to v if and only if (v, w) ?
  E.

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9.1 Definition

• A path is a sequence of vertices w1, w2, w3, ...,
  wN, such that (wi, wi1) ? E for 1?iltN.
• The length of a path is the number of edges on
  the path, which is N-1.
• A simple path is one such that all vertices are
  distinct, except that the first and the last
  could be the same.
• A cycle in a directed graph is a path of length
  at least 1 such that w1 wN.

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9.1 Definition

• A directed graph is acyclic if it has no cycles
  (abbreviated as DAG).
• The indegree of a vertex v is the number of edges
  (u, v).
• An undirected graph is connected if there is a
  path from every vertex to every other vertex.
• A directed graph with this property is called
  strongly connected.

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9.1 Definition

• If a directed graph is not strongly connected,
  but the underlying graph (without direction to
  the arcs) is connected, then the graph is said to
  be weakly connected.
• A complete graph is a graph in which there is an
  edge between every pair of vertices.
• Examples of graph models include computer
  networks, job scheduling.
• For vertices with names, use hash table to map
  names to numbers.

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9.1 Definition
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9.1 Definition

• Matrix representation
• 2-dimensional array, A, adjacency matrix.
• For each edge (u, v), set A u v 1
  otherwise the entry is 0.
• If the edge has a weight associated with it, set
  A u v to the weight.
• Space requirement is ?(V2) alright if the
  graph is dense, i.e., E ?(V2).

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9.1 Definition

• Adjacency list representation
• For each vertex, keep a list of all adjacent
  vertices.
• For sparse graphs.
• Space requirement is ?(EV).

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9.1 Definition
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9.2 Topological Sort

• An ordering of vertices in a directed acyclic
  graph, such that if there is a path from vi to
  vj, then vj appears after vi in the ordering.

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9.2 Topological Sort

• Topological ordering is not possible if there is
  a cycle in the graph.
• A simple algorithm
• Compute the indegree of all vertices from the
  adjacency information of the graph.
• Find any vertex with no incoming edges.
• Print this vertex, and remove it, and its edges.
• Apply this strategy to the rest of the graph.
• Running time is ?(V2).

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9.2 Topological Sort

• / Figure 9.5 Simple topological sort /
• void Topsort (Graph G)
• int Counter
• Vertex V, W
• for (Counter0 CounterltNumVertex Counter)
• V FindNewVertexOfInDegreeZero ()

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9.2 Topological Sort

• if (V NotAVertex)
• Error ("Graph has a cycle")
• break
• TopNum V Counter
• for each W adjacent to V
• Indegree W --

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9.2 Topological Sort

• An improved algorithm
• Keep all the unassigned vertices of indegree 0 in
  a queue.
• While queue not empty
• Remove a vertex in the queue.
• Decrement the indegree of all adjacent vertices.
• If the indegree of an adjacent vertex becomes 0,
  enqueue the vertex.
• Running time is ?(EV).

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9.2 Topological Sort
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9.2 Topological Sort

• / Figure 9.7 Pseudocode for Topological sort /
• void Topsort (Graph G)
• Queue Q
• int Counter 0
• Vertex V, W
• Q CreateQueue (NumVertex)
• MakeEmpty (Q)
• for each vertex V
• if (Indegree V 0)

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9.2 Topological Sort

• Enqueue (V, Q)
• while (!IsEmpty (Q))
• V Dequeue (Q)
• TopNum V Counter / Next No. /
• for each W adjacent to V
• if (--Indegree W 0)
• Enqueue (W, Q)

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9.2 Topological Sort

• if (Counter ! NumVertex)
• Error ("Graph has a cycle")
• DisposeQueue (Q) / Free the memory /

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9.3 Shortest Path Algorithms

• Single-Source Shortest Path Problem
• Given an input a weighted graph, G (V, E), and
  a distinguished vertex, s, find the shortest
  weighted path from s to every other vertex in G.
• Assume there is no edge with a negative cost
  (which could create negative cost cycles).

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9.3 Shortest Path Algorithms

• Unweighted Shortest Paths for Single-Source
  Graphs
• A simple algorithm
• 1. Mark the starting vertex, s
• 2. Find and mark all unmarked vertices adjacent
  to s (vertices adjacent to s)
• 3. Find and mark all unmarked vertices adjacent
  to the marked vertices.
• 4. Repeat Step 3. until no more vertices can be
  marked.

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9.3 Shortest Path Algorithms

• The strategy is known as breadth-first search.
• For each vertex, it is required to keep track of
• whether the vertex has been marked (known)
• its distance from s (dv)
• previous vertex of the path from s (pv)
• Running time is ?(V2).

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9.3 Shortest Path Algorithms
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9.3 Shortest Path Algorithms

• / Fig. 9.16 Unweighted shortest path algorithm
  /
• void Unweighted (Table T)
• / Assume T is initialized /
• int CurrDist
• Vertex V, W
• for (CurrDist0 CurrDistltNumVertex
  CurrDist)
• for each vertex V

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9.3 Shortest Path Algorithms

• if (!T V.Known T V.Dist
• CurrDist)
• T V.Known True
• for each W adjacent to V
• if (T W.Dist Infinity)
• T W.Dist CurrDist 1
• T W.Path V

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9.3 Shortest Path Algorithms

• Improvement
• Use a queue to keep all vertices whose shortest
  distance to s is known.
• Initialize dv to ? for all vertices, except s
• Set ds to 0 and enqueue s.
• While the queue is not empty
• remove the first vertex in the queue.
• enqueue all vertices which are adjacent to the
  vertex, and have a distance of infinity from s.

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9.3 Shortest Path Algorithms

• known not used
• Running time is ?(EV).

/ Fig. 9.18 Unweighted shortest path algorithm
/ void Unweighted (Table T) / Assume T is
initialized / Queue Q Vertex V, W
Q CreateQueue (NumVertex)
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9.3 Shortest Path Algorithms

• MakeEmpty (Q)
• / Enqueue the start vertex S /
• Enqueue (S, Q)
• while (!IsEmpty (Q))
• V Dequeue Q)
• T V.Known True / Not really needed /

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9.3 Shortest Path Algorithms

• for each W adjacent to V
• if (T W.Dist Infinity)
• T W.Dist T V.Dist 1
• T W.Path V
• Enqueue (W, Q)
• DisposeQueue (Q) / Free the memory /

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9.3 Shortest Path Algorithms

• Result for Fig. 9.20, with weight1 start v3

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9.3 Shortest Path Algorithms
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9.3 Shortest Path Algorithms

• Dijkstras Algorithm (for single-source weighted
  graphs)
• A greedy algorithm, solving a problem by stages
  by doing what appears to be the best thing at
  each stage.
• Select a vertex v, which has the smallest dv
  among all the unknown vertices, and declare that
  the shortest path from s to v is known.

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9.3 Shortest Path Algorithms

• For each adjacent vertex, w, update dw dv
  cv,w if this new value for dw is an improvement.

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9.3 Shortest Path Algorithms
Application of Dijkstras algorithm for Fig. 9.20
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9.3 Shortest Path Algorithms
Details on Fig. 9.28
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9.3 Shortest Path Algorithms

• / Fig. 9.29 Declarations for Dijkstras
  Algorithm
• typedef int Vertex
• struct TableEntry
• List Header / Adjacency list /
• int Known
• DistType Dist
• Vertex Path
• defin NotAVertex (-1)
• tyepdef struct TableEntry Table NumVertex

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9.3 Shortest Path Algorithms

• / Fig. 9.30 Table initialization routine /
• void InitTable (Vertex Start, Graph G, Table T)
• int i
• ReadGraph (G, T)
• for (i0 iltNumVertex i)
• T i.Known False T i.Dist Infinity
• T i.Path NotAVertex
• T Start.Dist 0

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9.3 Shortest Path Algorithms

• / Fig. 9.31 Routine to print actual shortest
  path /
• / Assume the path exists /
• void PrintPath (Vertex V, Table T)
• if (T V.Path ! NotAVertex)
• PrintPath (T V.Path, T)
• printf (" to")
• printf ("v", V) / v is pseudocode /

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9.3 Shortest Path Algorithms

• / Fig. 9.32 Pseudocode - Dijkstras Algorithm /
• void Dijkstra (Table T)
• Vertex V, W
• for ()
• V smallest unknown distance vertex
• if (V NotAVertex)
• break
• T V.Known True
• for each W adjacent to V

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9.3 Shortest Path Algorithms

• if (!T W.Known)
• if (T V.Dist Cvw lt T W.Dist)
• / update W /
• Decrease (T W.Dist to T V.Dist
  
• Cvw)
• T W.Path V

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9.3 Shortest Path Algorithms

• Total of the running time to find the minimum dv
  is ?(V2).
• Running time for updating dw is constant per
  update, for a total of ?(E).
• Total running time is ?(EV2).
• For a sparse graph, E ?(V), the algorithm
  is slow. Its best to keep the distances in a
  priority queue.

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9.4 Network Flow Problems

• Given
• A directed graph G (V,E) with edge capacities
  cv,w and two vertices being assigned as the
  source, s and sink, t
• Through any edge, (v,w), at most cv,w units of
  flow may pass.
• Note that for any vertices other than the source
  and the sink, the amount of flow coming in must
  be the same as that coming out.

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What is the maximum flow from in to out?
The problem is to find the maximum amount of flow
that can pass from s to t.
Out-flow
In-flow
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9.4 Network Flow Problems
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9.4.1 Maximum-Flow Algorithm
Initially
Total flow 0
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm
Attempt 1 Try (S-gtb-gtd-gtT) which allows 2 units
of flow
Total flow 2
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm
Attempt 1 Then (S-gta-gtc-gtT) which allows 2 units
of flow
Total flow 4
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm
Attempt 1 Third (S-gta-gtd-gtT) which allows 1 unit
of flow
Finished!
Total flow 5
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm
Attempt 2 Try (S-gtb-gtd-gtT) which allows 3 units
of flow
Finished!
Total flow 3
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm

• Its hard to know which path to choose first in
  order to attain the maximum flow.
• Solution
• Use of augmenting paths!

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9.4.1 Maximum-Flow Algorithm
Attempt 3 Try (S-gtb-gtd-gtT) which allows 3 units
of flow
S
0
2
3
1
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c
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T
Total flow 3
Flow graph
Residual graph
Original graph
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9.4.1 Maximum-Flow Algorithm
Attempt 3 Then (S-gtb-gtd-gta-gtc-gtT) with 2 units
of flow
S
0
0
2
3
1
a
b
3
2
1
0
2
1
d
c
2
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0
0
T
Finished!
Total flow 5
Flow graph
Residual graph
Original graph
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9.5 Minimum Spanning Tree

• Given
• An undirected graph G (V,E) with edge weight
  cv,w
• The problem is to find the minimum tree which
  spans all the vertices in V
• Note a minimum spanning tree should have the
  number of edges equal V-1 (Why?)

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9.5 Minimum Spanning Tree
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9.5.1 Prims Algorithm

• Identical to Dijkstras algorithm except that the
  graph is undirected (bidirectional) and

Decrease (T W.Dist to T V.Dist Cvw)
Refer to Pg 39
T W.Dist min( T W.Dist, Cvw)
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Finished!