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GRAPH ALGORITHMS BASICS. Graph : G = {V, E} V ={v 1 , v 2 , …, v n }, E = {e 1 , e 2 , … e m }, e ij =(v i , v j , l ij ) A set of n nodes/vertices, and m edges/arcs as pairs of nodes: Problem sizes: n, m Where l ij , if present, is a label or the weight of an edge.

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Graph algorithms basics
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

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Graph algorithms basics1
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

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Graph algorithms basics

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

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Graph algorithms basics

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

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Topological sort problem 1
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).

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Topological sort problem 11
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.

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Topological sort problem 12
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)

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

Indegree(v2)=1

Indegree(v4)=1

Indegree(v5)=1

No node to choose before all nodes are ordered: Fail

v2

v5

v4

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

  • 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!

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Graph algorithms basics

TOPOLOGICAL SORTProblem 1

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).

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Graph algorithms basics

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.

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Graph algorithms basics

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.

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Graph algorithms basics

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

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Graph algorithms basics

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

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Graph algorithms basics

SINGLE SOURCE SHORTEST PATH

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

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Graph algorithms basics

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

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Graph algorithms basics

DJIKSTRA’S ALG: COMPLEXITY

  • 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.

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Graph algorithms basics

DJIKSTRA’S ALG: COMPLEXITY

  • 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.

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Graph algorithms basics

DJIKSTRA’S ALG: PROOF SKETCH

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

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Graph algorithms basics

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

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Graph algorithms basics

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.

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Graph algorithms basics

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.

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Graph algorithms basics

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?

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Graph algorithms basics

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).

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Graph algorithms basics

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

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Problem4 all pairs shortest path
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.


Graph algorithms basics

D1(4,5) = min{D0(4,5) , D0(4,1) + D0(1,5)}

= min{inf, 2+3}

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Source: Cormen et al.


Problem 5 max flow min cut optimization
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
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


Graph algorithms basics

Greedy Alg, initialization

Input G

Gflow

Gresidual

Source: Weiss

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Graph algorithms basics

G =>

<= optimum output

Gflow

Gresidual

Greedy Alg, iteration 1

Terminates

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Max flow min cut greedy strategy1
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|)


Graph algorithms basics

Returned Result, Flow = 3

Optimum Result, Flow = 5

1

Capacities Unutilized

=>

1

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Graph algorithms basics

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

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

Iteration 2

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Graph algorithms basics

Iteration 2

Iteration 3

Terminates with optimum flow=5

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Max flow backtracking strategy ford fulkerson s algorithm
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


Graph algorithms basics

Ford-Fulkerson, using backflow residual, Iteration 1

Iteration 2

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

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Graph algorithms basics

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!

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Max flow backtracking with djikstra edmonds karp algorithm
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


Graph algorithms basics

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


Graph algorithms basics

Prim’s Algorithm for MST:

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


Graph algorithms basics

Prim’s Algorithm for MST

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


Graph algorithms basics

Proof sketch:

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


Graph algorithms basics

Problem 6: MST

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


Graph algorithms basics

Kruskal’s Algorithm for MST

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


Graph algorithms basics

Kruskal’s Algorithm for MST

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


Graph algorithms basics

Proof sketch:

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
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
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?

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Recursive depth first traversal of graphs1
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


Graph algorithms basics

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

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Graph algorithms basics

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


Graph algorithms basics

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


Graph algorithms basics

Problem 7: Finding Strongly Connected Component (SCC)

in a Directed Graph

Source: Weiss, Fig 9.74

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Graph algorithms basics

SCC-detection Algorithm

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


Graph algorithms basics

SCC-detection Algorithm:

First Pass DFS

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Graph algorithms basics

SCC-detection Algorithm:

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