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GRAPH ALGORITHMSBASICS

Graph: G = {V, E}

V ={v1, v2, …, vn}, E = {e1, e2, … em}, eij=(vi, vj, lij)

A set of n nodes/vertices, and m edges/arcs as pairs of nodes: Problem sizes: n, m

Where lij, if present, is a label or the weight of an edge.

V ={v1, v2, v3, v4, v5}

E = {e1=(v1, v2), (v3, v1), (v3, v4), (v5, v2), e5=(v4, v5)}

degree(v1)=2, degree(v2)=2, … :number of arcs

For directed graph: indegree, and outdegree

v1

v3

v2

v5

v4

(C) Debasis Mitra

GRAPH ALGORITHMSBASICS

Adjacency List Representation:

v1: (v2,10), (v3, 20)

v2: (v1,10), (v5, 25)

v3: (v1,20), (v4, 30)

v4: (v3,30), (v5, 50)

v5: (v2,25), (v4, 50)

Weighted graph G:

V ={v1, v2, v3, v4, v5}

E = {(v1, v2, 10), (v3, v1, 20), (v3, v4 , 30), (v5, v2 , 25), (v4, v5 , 50)}

v1

10

20

Matrix representation:

v3

Matrix representation

for directed Graph?

v2

30

25

Matrix representation

for unweighted Graph?

v5

v4

50

(C) Debasis Mitra

Directed graph: edges are ordered pairs of nodes.

Weighted graph: each edge (directed/undirected) has a weight.

Path between a pair of nodes vi, vk: sequence of edges with vi, vk at the two ends.

Simple path: covers no node in it twice.

Loop: a path with the same start and end node.

Path length: number of edges in it.

Path weight: total wt of all edges in it.

Connected graph: there exists a path between every pair of nodes, no node is disconnected.

Complete graph: edge between every pair of nodes [NUMBER OF EDGES?].

Acyclic graph: a graph with no cycles.

Etc.

Graphs are one of the most used models of real-life problems for computer-solutions.

GRAPH ALGORITHMSBASICS

(C) Debasis Mitra

Algorithm 1:

For each node v in V do

-Steps- // (|V|)

Algorithm 2:

For each node v in V do

For each edge in E do

-Steps- // (|V|*|E|)

Algorithm 3:

For each node v in V do

For each edge e adjacent to v do

-Steps- // ( |E| ) with adjacency list,

// but (|V|* |V| ) for matrix representation

Algorithm 4:

For each node v in V do

-steps-

For each edge e of v do

-Steps- // ( |V|+|E| ) or ( max{|V|, |E| })

GRAPH ALGORITHMSBASICS

v1

10

20

v3

v2

30

25

v5

v4

50

(C) Debasis Mitra

TOPOLOGICAL SORTProblem 1

Input: directed acyclic graph

Output: sequentially order the nodes without violating any arc ordering.

Note: you may have to check for cycles - depending on the problem definition (input: directed graph).

An important data: indegree of each node - number of arcs coming in (#courses pre-requisite to "this" course).

(C) Debasis Mitra

TOPOLOGICAL SORTProblem 1

Input: directed acyclic graph

Output: sequentially order the nodes without violating any arc ordering.

A first-pass Algorithm:

Assign indegree values to each node

In each iteration: Eliminate one of the vertices with indegree=0 and its associated (outgoing) arcs (thus, reducing indegrees of the adjacent nodes) - after assigning an ordering-number (as output of topological-sort ordering) to these nodes,

keep doing the last step for each of the n number-of-nodes.

If at any stage before finishing the loop, there does not exist any node with indegree=0, then a cycle exists! [WHY?]

A naïve way of finding a vertex with indegree0 is to scan the nodes that are still left in the graph.

As the loop runs for ntimes this leads to (n2) algorithm.

(C) Debasis Mitra

TOPOLOGICAL SORTProblem 1

Input: A directed graph

Output: A sorting on the node without violating directions, or Failure

Algorithm naïve-topological-sort

1 For each node v ϵ V calculate indegree(v); // (N)

2 counter = 0;

3 While V≠ empty do // (n)

4 find a node v ϵ V such that indegree(v)= =0;

// have to scan all nodes here: (n)

// total: (n) x (while loop’s n) = O(n2)

5 if there is no such v then return(Failure)

else

6 ordering-id(v) = ++counter;

7 V = V – v;

8 For each edge (v, w) ϵ E do //nodes adjacent to v

//adjacent nodes: (|E|), total for all nodes O(n or |E|)

9 decrement indegree(w);

10 E = E – (v, w);

end if;

end while;

End algorithm.

Complexity: dominating term O(n2)

(C) Debasis Mitra

v1

Indegree(v1)=0

Indegree(v2)=1

Indegree(v3)=1

Indegree(v4)=2

Indegree(v5)=1

Ordering-id(v1) =1

Eliminate node v1, and edges (v1,v2), (v1,v3) from G

Update indegrees of nodes v2 and v3

v3

v2

v5

v4

(C) Debasis Mitra

Indegree(v2)=0

Indegree(v3)=0

Indegree(v4)=2

Indegree(v5)=1

Ordering-id(v1) =1, Ordering-id(v2)=2

Eliminate node v2, and edges (v2,v5) from G

Update indegrees of nodes v5

v3

v2

v5

v4

(C) Debasis Mitra

Indegree(v3)=0

Indegree(v4)=2

Indegree(v5)=0

Ordering-id(v1) =1, Ordering-id(v2)=2, O-id(v3)=3

Eliminate node v3, and edges (v3,v4) from G

Update indegrees of nodes v4

v3

v5

v4

(C) Debasis Mitra

Indegree(v4)=1

Indegree(v5)=0

Ordering-id(v1) =1, Ordering-id(v2)=2, O-id(v3)=3, (v5)=4

Eliminate node v5, and edges (v5,v4) from G

Update indegrees of nodes v4

v5

v4

(C) Debasis Mitra

Indegree(v4)=0

Ordering-id(v1) =1, Ordering-id(v2)=2, O-id(v3)=3, (v5)=4, (v4)=5

Eliminate node v4 from G, and V is empty now

Resulting sort: v1, v2, v3, v5, v4

v4

(C) Debasis Mitra

v1

However, now,

Indegree(v1)=0

Indegree(v2)=2

Indegree(v3)=1

Indegree(v4)=2

Indegree(v5)=1

Ordering-id(v1) =1

Eliminate node v1, and edges (v1,v2), (v1,v3) from G

Update indegrees of nodes v2 and v3

v3

v2

v5

v4

(C) Debasis Mitra

Indegree(v2)=1

Indegree(v3)=0

Indegree(v4)=2

Indegree(v5)=1

Ordering-id(v1) =1, Ordering-id(v3)=2

Eliminate node v3, and edges (v3,v4) from G

Update indegrees of nodes v4

v3

v2

v5

v4

(C) Debasis Mitra

Indegree(v2)=1

Indegree(v4)=1

Indegree(v5)=1

No node to choose before all nodes are ordered: Fail

v2

v5

v4

(C) Debasis Mitra

- A smarter way: notice indegrees become zero for only a subset of nodes in a pass,
- but all nodes are being checked in the above naïve algorithm.
- Store those nodes (who have become “roots” of the truncated graph) in a box
- and pass the box to the next iteration.
- An implementation of this idea could be done by using a queue:
- Push those nodes whose indegrees have become 0’s (when you update those values),
- at the back of a queue;
- Algorithm terminates on empty queue,
- but having empty Q before covering all nodes:
- we have got a cycle!

(C) Debasis Mitra

Algorithm Q-based-topological-sort

1 For each vertex v in G do // initialization

2 calculate indegree(v); // (|E|)with adj. list

3 if indegree(v) = =0 then push(Q, v);

end for;

4 counter = 0;

5 While Q is not empty do

// indegree becomes 0 once & only once for a node

// so, each node goes to Q only once: (N)

6 v = pop(Q);

7 ordering-id(v) = ++counter;

8 V = V – v;

9 For each edge (v, w) ϵ E do //nodes adjacent to v

// two loops together (max{N, |E|})

// each edge scanned once and only once

10 E = E – (v, w);

11 decrement indegree(w);

12 if now indegree(w) = =0 then push(Q, w);

end for;

end while;

13 if counter != N then return(Failure);

// not all nodes are numbered yet, but Q is empty

// cycle detected;

End algorithm.

Complexity: the body of the inner for-loop

is executed at most once per edge,

even considering the outer while loop,

if adjacency list is used.

The maximum queue length is n.

Complexity is (|E| + n).

(C) Debasis Mitra

SOURCE TO ALL NODES SHORTEST PATH-LENGTHProblem 2

Path length = number of edges on a path.

Compute shortest path-length from a given source node to all nodes on the graph.

Strategy: starting with the source as the "current-nodes,"

in each of the iteration expand children (adjacent nodes) of "current-nodes."

Assign the iteration# as the shortest path length to each expanded node, if the value is not already assigned.

Also, assign to each child, its parent node-id in this expansion tree (to retract the shortest path if necessary).

Input: Undirected graph, and source node

Output: Shortest path-length to each node from source

This is a breadth-first traversal, nodes are expanded as increasing distances from the source: 0, then 1, then 2, etc.

(C) Debasis Mitra

SOURCE TO ALL NODES SHORTEST PATH-LENGTHProblem 3

Once again, a better idea is to have the children being pushed at the back of a queue.

Input: Undirected graph, and source node

Output: Shortest path-length to each node from source

Algorithm q-based-shortest-path

1 d(s) = 0; // source

2 enqueue only s in Q; // (1), no loop, constant-time

3 while (Q is not empty) do

// each node goes to Q only once: (N)

4 v = dequeueQ;

5 for each vertex w adjacent to v do // (|E|)

6 if d(w) is yet unassigned then

7 d(w) = d(v) + 1;

8 last_on_path(w)= v;

9 enqueuew in Q;

end if;

end for;

end while;

End algorithm.

Complexity: (|E| + n), by a similar analysis as that of the previous queue-based algorithm.

(C) Debasis Mitra

SOURCE TO ALL NODES SHORTEST PATH-LENGTHProblem 3

Source: v1

: v1, length=0

Queue: v1

v3

v2, length=1

v3, length=1

v2

v5, length=1

Queue: v2,v3 , v5

Q: v3 ,v5

v1, no

v1, no

v5, already assigned

v1, no

v5

v1, no

v4

v4, length=2

Q: v4

v3, no

Q: empty

(C) Debasis Mitra

SOURCE TO ALL NODES SHORTEST PATH-WEIGHT

Input: Positive-weighted graph, a “source” node s

Output: Shortest distance to all nodes from the source s

v1

10

20

70

v3

v2

Breadth-first search does not work any more!

Say, source is v1

Shortest-path-length(v5) = 35, not 70

However, the concept works:

expanding distance front from s

25

30

v5

v4

50

(C) Debasis Mitra

Input: Weighted graph Dsw , a “source” node s [can not handle negative cycle]

Output: Shortest distance to all nodes from the source s

Algorithm Dijkstra

1 s.distance = 0; // shortest distance from s

2 mark s as “finished”; //shortest distance to s from s found

3 For each node w do

4 if (s, w) ϵ E then w.distance= Dsw else w.distance = inf;

// O(n)

5 While there exists any node not marked as finished do

// (n), each node is picked once & only once

6 v = node from unfinished list with the smallest v.distance;

// loop on unfinished nodes O(n): total O(n2)

7 mark v as “finished”; // WHY?

8 For each edge (v, w) ϵ E do //adjacent to v:(|E|)

9 if (w is not “finished” &(v.distance + Dvw < w.distance) ) then

10 w.distance = v.distance + Dvw;

11 w.previous = v;

// v is parent of w on the shortest path

end if;

end for;

end while;

End algorithm

(C) Debasis Mitra

DJIKSTRA’S ALG: SINGLE SOURCE SHORTEST PATH

v1 (0)

v1 (0)

v1 (0)

10

20

10

20

10

20

70

70

v2 ( )

v3 ( )

70

v3 (20)

v3 (20)

v2 (10)

v2 (10)

25

25

25

30

30

30

v5 ( )

v4 ( )

50

v5 (35)

v5 (70)

v4 ( )

v4 ( )

50

50

v1 (0)

v1 (0)

10

20

10

20

70

v3 (20)

v2 (10)

70

v3 (30)

v2 (10)

25

25

30

30

v5 (35)

v4 (50)

50

v5 (35)

v4 (50)

50

(C) Debasis Mitra

- Lines 2-4, initialization: O(n);
- Line 5: while loop runs n-1 times: [(n)],
- Line 6: find-minimum-distance-node v runs another (n) within it,
- thus the complexity is (N2);
- Can you reduce this complexity by a Queue?
- Line 8: for-loop within while, for adjacency list graph data structure,
- runs for |E| times including the outside while loop.
- Grand total: (|E| + n2) = (n2), as |E| is always n2.

(C) Debasis Mitra

- Djikstra complexity: (|E| + n2) = (n2), as |E| is always n2.
- Using a heap data-structurefind-minimum-distance-node (line 6) may be log_n,
- inside while loop (line 5): total O(n log_n)
- but as the distance values get changed the heap needs to be kept reorganized,
- thus, increasing the for-loop\'s complexity to (|E|log N).
- Grand total: (|E|log_n + n log_n) = (|E|log_n), as |E| is always n in a connected graph.

(C) Debasis Mitra

Induction base: Shortest distance for the source s itself must have been found

Induction Hypothesis: Suppose, it works up to the k-th iteration.

This means, for k nodes the corresponding shortest paths from s have been already

identified (say, set F, and V-F is U, the set of unfinished nodes).

Induction Step: Let, node p has shortest distance in U, in the k+1 –th iteration.

(1) Each node in F has its chance to update p.

(2) Each other node in U has a distance >= that of p, so, cannot improve p any more

(assuming edge weights are positive or non-negative)

(3) Hence, p is ‘finished” and so, moved from U to F

(4) Each node in F, except p, has its chance to update nodes in U.

(5) Each other node in U has a distance >= that of p, so, cannot improve

distances of nodes in U optimally (assuming edge weights are positive or non-negative)

(6) Hence, p is the only candidate to improve distances of nodes in U

and so, updates only nodes in U in the k+1 -th iteration

(C) Debasis Mitra

DJIKSTRA’S ALG:WITH NEGATIVE EDGE

- Cost may reduce by adding a new arc
- Hence, the ‘finished’ marker does not work
- The proof of above Djikstra depends on positive edge weights
- Assumption: a path’s cost is non-decreasing as new arc is added to it
- This assumption is not true for negative-weighted edges

(C) Debasis Mitra

DJIKSTRA’S ALG:WITH NEGATIVE EDGE

Algorithm Dijkstra-negative

Input: Weighted (undirected) graph G, and a node s in V

Output: For each node v in V, shortest distance w.distance from s

1 Initialize a queue Q with the s;

2 While Q not empty do

3 v = pop(Q);

4 For each w such that (v, w) ϵ E do

5 If (v.distance + Dvw < w.distance) then

6 w.distance = v.distance +Dvw;

7 w.parent = v;

8 If w is not already in Q then push(Q, w);

// if it is already there in Qdo not duplicate

end if

end for

end while

End Algorithm.

(C) Debasis Mitra

DJIKSTRA’S ALG:WITH NEGATIVE EDGE

- Negative cost on an edge or a path is all right, but negative cycle is a problem
- A path may repeat itself on a loop, thus reducing negative cost on it
- That will go on infinitely
- No limit to minimizing the cost between a source to any node on a negative cycle
- Shortest-path is not defined when there is a negative cycle.

(C) Debasis Mitra

DJIKSTRA’S ALG:WITH NEGATIVE EDGE

- Cost reduces on looping around negative cycles
- Q-Djikstra does not prevent that: it may not terminate
- But how many looping should we allow?
- Can we predict that we are on an infinite loop here?
- How many times a node may appear in the queue?
- What is the “diameter” of a graph: longest path length?

(C) Debasis Mitra

DJIKSTRA’S ALG:WITH NEGATIVE CYCLE

- “Diameter” of a graph = n
- No node should get a chance to update others more than n+1 times!
- Because, there cannot be a simple (loop free) path of length more than n
- Line 8 of Q-Djikstra is guaranteed to happen once for any node,
- or, any node will be pushed to Q at least once
- But not more than n times, unless it is on a negative loop
- So, Q-Dijkstra should be terminated if a node v gets n+1 times in to the queue
- If that happens: the graph has a negative-cost cycle with the node v on it
- Otherwise, Q-Djikstra may go into an infinite loop
- (over the cycle with negative path-weight)
- Complexity of Q-Djikstra: O(n*|E|), because each node may be updated by
- each edge at most one time (if there is no negative cycle).

(C) Debasis Mitra

Djikstra’s Algorithm is Greedy Algorithm:

pick up the best node from U in each cycle

Find all pairs shortest distance (Problem 4):

For Each node v as source

Run Djikstra(v, G)

Complexity:

O(n.n2), or O(n.n.|E|) for negative edges in G

A dynamic programming algorithm (Floyd’s alg) does better

(C) Debasis Mitra

Problem4: All Pairs Shortest Path

A variation of Djikstra’s algorithm. Called Floyd-Warshal’s algorithm. Good for dense graph.

Algorithm Floyd

Copy the distance matrix in d[1..n][1..n];

for k=1 through n do //consider each vertex as updating candidate

for i=1 through n do

for j=1 through n do

if (d[i][k] + d[k][j] < d[i][j]) then

d[i][j] = d[i][k] + d[k][j];

path[i][j] = k; // last updated via k

End algorithm.

O(n3), for 3 loops.

Problem 5: Max-flow / Min-cut (Optimization)

Input: Weighted directed graph (weight = maximum permitted flow on an edge),

a source node s without any incoming arc,

a sink node n without any outgoing arc.

Objective function: schedule a flow graph s.t. no node holds any amount (influx=outflux), except for s and n, and no arc violets its allowed amount

Output: maximize the objective function (maximize the amount flowing from s to n)

<= output max-Gflow

G =>

Max-flow / Min-cut Greedy Strategy

Input: Weighted directed graph (weight = maximum permitted flow on an edge), a source node s without any incoming arc, a sink node n without any outgoing arc.

Output: maximize the objective function (maximize the amount flowing from s to n)

Greedy Strategy:

Initialize a flow graph Gf (each edge 0) Gr, and a residual graph as G

In each iteration, find a “max path” p from s to n in Gr

Max path is a path allowing maximum flow from s->n, subject to the constraints

Add p to Gf, and subtract p from Gr, any edge having 0 is removed from Gr

Continue iterations until there exists no such path in Gr

Return Gf

Max-flow / Min-cut Greedy Strategy

Complexity:

How many paths may exist in a graph? |E| !

How many times any edge may be updated until it gets to 0? f.|E| for max-flow f

Each iteration, looks for max-path: |E| log |V|

Total: O(f. |E|2 log |V|)

Correct Alg, with oracle, not-greedy, iteration 1

Start with not-max-path s->b->d->t (2 each edge)

Iteration 2

(C) Debasis Mitra

Max-flow Backtracking StrategyFord-Fulkerson’s Algorithm

Greedy Strategy is sub-optimal

A Backtracking optimal algorithm:

Initialize a flow graph Gf (each edge 0) Gr, and a residual graph as G

In each iteration, find a path p from s to n in Gr

Add p to Gf, and Gr is updated with reverse flow opposite to p, and leaving residual amounts in forward direction => allows backtracking if necessary

Continue iterations until either all edges from s are incoming, or all edges to n are outgoing

Return Gf

Guaranteed to terminate for rational numbers input

Complexity: If f is the max flow in returned Gf,

Then, number of times any edge may get updated is f , even though each arc never gets to 0

O(f. |E|)

Random choice for a path p may run into bad input

Ford-Fulkerson, using backflow residual, Iteration 1

Iteration 2

Terminates, no outflow left from source, or inflow to sink

(C) Debasis Mitra

Problem input for Ford-Fulkerson:

May keep backtracking if starts with random path, say, s->a->b->t

Complexity: If f is the max flow in returned Gf,

O(f. |E|), or O(100000*|E|)

Why bother?

Use greedy!

(C) Debasis Mitra

Max-flow Backtracking with DjikstraEdmonds-Karp Algorithm

Another Backtracking optimal algorithm (may avoid problem of Ford-Fulkerson):

Initialize a flow graph Gf (each edge 0) Gr, and a residual graph as G

In each iteration, find a “max path” p from s to n in Gr using Djikstra

Add p to Gf, and Gr is updated with reverse flow opposite to p, and leaving residual amounts in forward direction => allows backtracking if necessary

Continue iterations until either all edges from s are incoming, or all edges to n are outgoing

Return Gf

Complexity:O(cmax|E|2 log |V|), first term is max capacity of any edge [actually logCmax in complexity!]

Min-cut = Minimum total capacity from one partition (of G)

S ϵ source to another T ϵ sink

Which edges are to cut from G to isolate source from sink?

Min-cut = Max-flow

Problem 6: Minimum Spanning Tree (Optimization)

Prim’s Algorithm

- Input: Weighted un-directed graph
- Objective function: Total weight of a spanning tree over G
- Output: minimize the objective function (minimum-wt spanning tree - MST)
- Prim’s Algorithm: Djikstra’s strategy
- Separate two sets of nodes F (in MST now) and U (not in MST yet), starting with any arbitrary node s ϵ F
- In each iteration find a ‘closest’ node v ϵ U from any node pϵF; move v from U to F; add Epv to MST: {For all xϵU, For all yϵF, |Evy| ≤ |Exy| }
- Continue until U is empty
- Return MST

- How many nodes are on a spanning tree?
- How many arcs?
- How many spanning trees exist on a graph?
- Complexity of Prim’s Alg:?

(C) Debasis Mitra

Complexity

- How many nodes are on a spanning tree?
- How many arcs?
- How many spanning trees exist on a graph?
- Complexity of Prim’s Alg:?

Exactly same as Djikstra: O(|E| log n)

(C) Debasis Mitra

Prim’s Alg starting with a in F

Closest to a is b with 1 on Eab

a

a

a

3

2

3

2

3

2

7

c

b

7

9

c

b

7

9

c

b

9

1

10

1

10

8

1

10

8

8

d

e

d

e

d

e

Closest to a or b is e with Ebe=1

Closest to a, b or e is c with Eac=3

Closest to a, b, e or c is d with Ebd=8

a

a

a

3

2

3

3

2

2

7

7

7

c

b

c

b

c

b

9

9

9

1

10

1

1

10

10

8

8

8

d

e

d

d

e

e

Done! All nodes in F.

Eab+ Ebc + Ebd+ Eac= 14, is MST weight

(C) Debasis Mitra

Prims’s Algorithm

Inductionbase: ???

Inductionhypothesis: on k-th iteration (1<=k<n), say, MST is correct for the

subgraph of G with k nodes in F

Induction Step: Epv is a shortest arc between all arcs from F to U

After adding Epv to MST, it is still correct for the new subgraph with k+1 nodes

(C) Debasis Mitra

Kruskal’s Algorithm

- Input: Weighted un-directed graph
- Objective function: Total weight of a spanning tree over G
- Output: minimize the objective function (minimum-wt spanning tree - MST)
- Kruskal’s Algorithm: Greedy strategy
- 1)Sort all arcs ϵE in low to high weights; 2)Initialize an empty MST
- In each iteration 3)find the shortest arc eϵE; 4)add it to MST unless it causes a cycle (MST may be a forest until iterations end); 5)E = E\e
- Continue iterations until E is empty // will run for |E| times

- Complexity of Kruskal’sAlg?
- log|E| sorting
- (Cycle checking in MST needs special datastructure) x |E| iterations
- O(|E| log |E|), which is O(|E| log n)

(C) Debasis Mitra

Kruskal’s Alg must start with Ebc in MST

a

a

a

3

2

3

2

3

2

7

c

b

7

9

c

b

7

9

c

b

9

1

10

1

10

8

1

10

8

8

d

e

d

e

d

e

Add Eac

Eae creates cycle, fails to be added

Add Ebd

a

a

a

3

2

3

3

2

2

7

7

7

c

b

c

b

c

b

9

9

9

1

10

1

1

10

10

8

8

8

d

e

d

d

e

e

Done! All n-1 arcs in MST.

Eab+ Ebc + Ebd+ Eac= 14, is MST weight

(C) Debasis Mitra

Kruskal’s Alg must start with Ebc in MST

a

a

a

2

3

2

3

2

3

7

c

b

7

9

c

b

7

9

c

b

9

1

10

1

10

8

1

10

8

8

d

e

d

e

d

e

Add Eac

Eae creates cycle, fails to be added

Add Ebd

a

a

a

2

3

2

3

2

3

7

7

7

c

b

c

b

c

b

9

9

9

1

10

1

1

10

10

8

8

8

d

e

d

d

e

e

Done! All n-1 arcs in MST.

Eab+ Ebc + Ebd+ Eac= 14, is MST weight

(C) Debasis Mitra

Kruskal’s Algorithm

Inductiondoes not work: growing forest is not necessarily MST until finished

Contradiction: Suppose, returned T is not MST, but a different ST P is:

{||T|| > ||P||}, double bar indicates weight

Transform T to P,

by adding arc Exy to T -> form a cycle ->

remove Elm from the cycle already in T to create a new ST ->

chain through this process -> P

|| Elm || must be <= || Exy ||, as Elm was picked up for T before Exy,

from pre-sorted list

Elm is in T but not Exy,

Exy is in P but not Elm

So, ||T|| <= ||P||. Contradiction!

(C) Debasis Mitra

Depth First Traversal of Graphs

- Creates a Spanning Forest on a finite graph
- For connected graph, it will be a spanning tree
- For dynamically developing graph (e.g. search tree),
- where finiteness is not guranteed, DFS may not terminate
- Space requirement is limited by the depth of the tree
- What is space complexity on BFS traversal?
- Time: depends on branching factor b –
- have to come back to the same node b number of times
- Stack is the typical data-structure for DFS
- Many graph algorithms need DFS traversal

(C) Debasis Mitra

Recursive Depth First Traversal of Graphs

Input: graph (V, E)

Output: a DFS spanning forest

Algorithm dfs(v)

1. mark node v as visited;

2. operate on v (e.g., print);

3. for each node w adjacent to node v do

4. if w is not marked as visited then

5. dfs(w); // an iterative version of this

//will maintain its own stack

end for;

End algorithm.

Starter algorithm

on a given graph G call dfs(any/given node in G);

End starter.

Problem with above algorithm?

(C) Debasis Mitra

Recursive Depth First Traversal of Graphs

Problem with above algorithm:

Will terminate prematurely if the graph is disconnected.

repeat until all nodes are marked

pick up next unmarked node s;

dfs(s);

end repeat;

This will generate the required DFS-spanning forest

(C) Debasis Mitra

Problem 7: Finding Articulation Points in a Graph

A

B

C

D

G

E

F

Input: Undirected, unweighted graph G

ArticulationPoint: a node, removal of which from G will disconnect the graph

Output: Articulation point(s) of G, if there exists any

(C) Debasis Mitra

Finding Articulation Points in a Graph

By ordered DFS traversal

- Strategy:
- Draw a dfs-spanning tree T for the given graph starting from any of its nodes.
- Number them according to the pre-order of their traversal (num(v)).
- Def. of “low”: For each vertex v, find the lowest vertex-num where one can go to, from v by the
- directed edges on T & possibly by a single left-out edge of G (back-edge on T):
- Do post-order DFS-traversal on T now, & assign low of each node v:
- {low(v) =min{num(v), for all children w, low(w), num(w) for each node w with an arc from v in G}
- If for any vertex v there is a childw such that num(v) low(w), then vis an articulation point,
- unless v is the root in the T (this condition is always true for the root!).
- If v is the root of the T, and has two or more children in this T,
- then v is an articulation point.

(C) Debasis Mitra

Finding Articulation Points in a Graph

By ordered DFS traversal

A

B

C

D

G

E

F

B 1/1

A 2/1

C 3/1

G 7/7

D 4/1

F 6/4

E 5/4

Complexity: O(|E| + n)

(C) Debasis Mitra

Problem 7: Finding Strongly Connected Component (SCC)

in a Directed Graph

Source: Weiss, Fig 9.74

(C) Debasis Mitra

Input: Directed Graph G

Output: Subgraphs SCC of G, each subgraph in SCC is

a strong connected component of G

Strategy:

1. Traverse G using DFS to create spanning forest of the graph T,

numbering the nodes in the post-order traversal;

2. Create Gr by reversing the directions of arcs in G;

3. Traverse using DFS the nodes of Gr in a decreasing order of the numbers,

4. DFS spanning-forest from this second traversal creates the set of SCC’s for G.

(C) Debasis Mitra

Second Pass DFS on Gr

Start DFS, stuck at G:

G is a SCC

Start DFS again: B-A-C-F is a SCC

Start DFS again, stuck at D:

D is a SCC

Start DFS again, stuck at E:

E is a SCC

Start DFS again: H-I-J is a SCC

(C) Debasis Mitra

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