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Midterm

< 20 120-29 030-39 440-49 550-59 960-69 770 3

Matchings

matching M = subgraph with vertices of degree 1

Matchings

= unmatched vertex (vertex not in M)


Matchings


= unmatched vertex (vertex not in M)

perfect matching = matching with no unmatched vertices (perfect world – everybody has a buddy)

Perfect matching

Maximal / maximum

maximal = no edge can be added to M to make a bigger matching

maximum = there is no matching with more edges

maximal, NOT maximum maximum maximal

Augmenting path

path connecting two unmatched vertices with alternating non-matching and matching edges

If there exists augmenting path then the matching is NOT maximum.

Augmenting path


Augmenting path


Lucky ?

Augmenting path

Lucky ? NO

Theorem:

If matching M is not maximum then there exists an augmenting path.

Augmenting pathTheorem:

If the matching M is not maximum then there exists an augmenting path.

M = matching, not maximumB = bigger matching

Proof


MB

Proof


MB

Proof


MB

Proof


MBvertices of degree 0,1,2 paths and cycles

no odd cycles

must have a path with more edges from Bthan from M augmenting path

How to find maximum matching?

Mrepeat find augmenting path P M MPuntil no augmenting path

Augmenting path in bipartite graphs

augmenting path has odd length endpoints on different sides

call the one on the left start

non matching edges matching edges









Theorem: augmenting path exists there is a path from unmatched vertex on the left to an unmatched vertex on the right





Theorem: augmenting path exists there is a path from the blue vertex on the left to the blue vertex on the right


Finding augmenting path(or checking none exists)

one BFS (or DFS) computation

TIME = O(E)

Finding max-matching O( VE )

Mrepeat find augmenting path P M MP

until no augmenting path

Maximum-weight matchings

1010

10

55

5

5

55

Maximum-weight matchings

1010

10

55

5

5

55

weight = 30

maximum-weight matching doesn’t have to be maximum cardinality !!!

Augmenting path


+ - + +-

we > wf

e PMC f PM

must gain by the swap:

Augmenting greedily

Assume M = matching of the maximum-weight among matchings with k edges P = augmenting path with maximum gainThen MP = matching of the maximum weight among matchings with k+1 edges

THEOREM:

Augmenting greedilyAssume M = matching of the maximum-weight among matchings with k edges P = augmenting path with maximum gainThen MP = matching of the maximum weight among matchings with k+1 edges

Proof: B = max-weight with k+1 edgesMB



gain = 0



gain = 0



gain = 0



gain = wt(B) – wt(M)

How to find maximum-weight matching?

Mrepeat find augmenting path P with maximum gain M MPuntil no augmenting path

only need O(V) iterations

How to find augmenting path withthe maximum gain?

+

-

SHORTEST PATH PROBLEM


+

-

NEGATIVE CYCLE?


+

-

NEGATIVE CYCLE?


+

-

impossible – would contradict maximality (among matchings of cardinality k)


no negative cycles

can use Bellman-Fordto find the shortest path(and hence get augmenting pathwith the maximum gain)

Putting it all together



O(VE) time per iteration (Bellman-Ford)O(V2 E) time total

Putting it all together



O(Vlog V + E) time per iteration (Dijkstra)O(V2log V + VE) time total

Back to midterm

Linear ordering of vertices of a directed acyclic graph such thatno edges point backward.

2log n = n

n1/log n = (2log n )1/log n = 2 (log n)/log n = 2n2/log n = (2log n )2/log n = 2 2(log n)/log n = 4

1

n1 + 1/log n = n . n1/log n = 2nn

n2

binomial(n,2)=n(n-1)/2

Mergesort reduces an instance ofsize n to 2 instances of size n/2and the reduction takes time (n).

Assume that all degrees 2. Then

2 |E| = deg(v) 2 = 2 nvV vV

Hence there is a vertex of degree 1.

There cannot be a vertex of degree 0,since G is connected and |V|2.

Hence there is a vertex of degree = 1.

max { E[ i ], O[ j ] + 1}

max { O[ i ], E[ j ] + 1}

max { B[i-1], B[i-2] + ai }

for i from 1 to n do for j from 1 to n do A[i,j] (A[i,j] + (i+j)) mod 2

M1 largest all-ones squareM2 largest all-zeros squarereturn max (M1, M2)

for i from 1 to n do for j from 1 to n do M[i,j] 1 if i>1 and j>1 then if (A[i,j]=A[i-1,j-1]) and (A[i,j] A[i-1,j]) and (A[i,j] A[i,j-1])

M[i,j] 1+min(M[i-1,j],M[i,j-1],M[i-1,j-1])

return max M[i,j]i,j {1,...,n}

Extract-Min(H) min H[1] H[1] H[n] n n-1 Heapify(H,1) return min

H = heapn = size of the heap

input: weighted, undirected graph G (given by adjacency list)

output: minimum-cost spanning tree of G (cost of a tree is the sum of the weights of its edges)

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