1. Constructing and Using
Flow Graphs
Constructing and Using
Copyright © Curt Hill

2. Flow Charts
In program design and documentation we often use a flow chart
You will remember flow charts from earlier courses
There are several shapes connected by arrows
Circle – start or stop
Rectangle – processing block
Diamond – a decision
Notice the flow of control structures in the next picture
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3. Flow Chart Example
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4. Notes Flow charts were developed in the 1920s to describe processes
They were very popular in the early computer years (1950s to 1970s)
In this presentation we are more interested in flow graphs
Flow graphs are based graph theory
Thus they are able to use numerous theorems from graph theory
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5. Flow Graphs AKA Control Flow Graph (CFG)
A flow graph is a simplified flow chart
Looks more like a graph theory diagram
Both flow charts and flow graphs are directed graphs
Only two constructs
A node which is a circle
An arrow which is an edge between two nodes
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6. Flow Graph Example
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7. Charts and Graphs
The rectangles of a flow chart typically represent one or more sequential flow statements
Thus we often have several rectangles one leading to another
This is fine for some things but not what we want in flow graphs that we are currently considering
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8. Interest We are not interested in sequential statements
We wish to collapse them into a block
This is also a common thing for a compiler
In our normal flow graph any adjacent set of non-flow statements is collapsed into one
Even if one is a control flow node
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9. Better Example 11 1 11 1
2,3 2 6 3
4,5 4 7 8 7 6 8
From Pressman 5 9 10
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10. Observations When we collapse these nodes the following becomes true
Thus for every arc A->B
Outdegree of A > 1 or indegree of B > 1 or both
Next we take another detour through graph theory
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11. Mathematical definitions
Nodes or Vertices
A location or point
May have a number of properties
Arcs or Edges
Connector of nodes
Directional or bidirectional
May also have other properties, eg. cost 2 1 3 6 4 5
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12. Mathematical definitions2
Indegree of a node
Number of arcs ending at node
Outdegree
Number of arcs starting at node
Directed or digraph
Arcs are one directional only
Undirected
Node indegree = node outdegree
Not of interest today 1 3 6 4 5
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13. Directed graph Flow graphs are directed
Flow must start at a node and finish at a node
We denote these with arrows
We cannot back up in a program
Each node may point at zero or more arcs
Each node may be pointed at by zero or more arcs
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14. Connectivity 2
A and B together is a disconnected graph with two components
A is strongly connected
There is a path from every node to every other node
B is weakly connected
No path leads to 7 1 3 6 4 5 A B 9 8 7
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15. Graph Theory Concepts Path Reachability Domination
A series of edges from one node to another
Reachability
A path exists from one node to another
If no path exists from the start node that part of the graph can be removed – AKA dead code
Domination
If all paths from A to C go through B then B dominates C
EG. a loop header dominates the loop
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16. Consider this code
int f1, f2, total=0, counter=1; while(!infile){ cin >> f1 >> f2 if(f1==0){ total+=f1; counter++; } else if(f2 == 0){ cout << total<<counter; counter = 0; } else total -= f2;
cout << f1 << f2;
} // end while
cout << counter;
Copyright © Curt Hill

17. Flow Graph 9 1 4 3 6 5 7 8 2 Copyright © 2001- 2017 Curt Hill
From Pressman 7 8
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18. Construction There is only one start and one end
Any return statements make an arc to the one end
Predicates include the following:
If, while, for, case
Breaks make an arc to the first statement after a loop/switch
Continue makes an arc to then end of the loop
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19. More Construction
An if may will be merged with prior sequential statements
Loops will not – there is a branch to the beginning
A for distinguishes between initialization and rest of loop
The initialization may merge with preceding assignments
It is the case of the switch which is the real predicate
Multiple cases with one piece of code become one predicate
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20. Consider this code
int f1, f2, total=0, counter=1; while(!infile){ cin >> f1 >> f2 if(f1==0){ total+=f1; counter++; } else if(f2 == 0){ cout << total<<counter; counter = 0; } else total -= f2;
cout << f1 << f2;
} // end while
cout << counter;
Copyright © Curt Hill

21. Flow Graph
int f1, f2, total=0, counter=1; while(!infile){ cin >> f1 >> f2 if(f1==0){ total+=f1; counter++; } else if(f2 == 0){ cout << total<<counter; counter = 0; } else total -= f2;
cout << f1 << f2;
} // end while
cout << counter; 9 1 2 4 3 5 6
From Pressman 7 8
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22. Observation The cin and if were merged.
The two assignments in the then and else were also merged
We may merge a predecessor with an if, but not a successor
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23. Conclusion Flow graphs are a common way to represent programs
Emphasis is on flow of control
Allows us to apply graph theory to program flow
This will be needed for measures such as Cyclomatic Complexity
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